Classical and Quantum Effects in Plasmon

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Classical and Quantum Effects in Plasmonic Metals
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Brown, Ana Maii
(2016)
Classical and Quantum Effects in Plasmonic Metals.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z9QV3JHT.
Abstract
The field of plasmonics exploits the unique optical properties of metallic nanostructures to concentrate and manipulate light at subwavelength length scales. Metallic nanostructures get their unique properties from their ability to support surface plasmons– coherent wave-like oscillations of the free electrons at the interface between a conductive and dielectric medium. Recent advancements in the ability to fabricate metallic nanostructures with subwavelength length scales have created new possibilities in technology and research in a broad range of applications.
In the first part of this thesis, we present two investigations of the relationship between the charge state and optical state of plasmonic metal nanoparticles. Using experimental bias-dependent extinction measurements, we derive a potential- dependent dielectric function for Au nanoparticles that accounts for changes in the physical properties due to an applied bias that contribute to the optical extinction. We also present theory and experiment for the reverse effect– the manipulation of the carrier density of Au nanoparticles via controlled optical excitation. This plasmoelectric effect takes advantage of the strong resonant properties of plasmonic materials and the relationship between charge state and optical properties to eluci- date a new avenue for conversion of optical power to electrical potential.
The second topic of this thesis is the non-radiative decay of plasmons to a hot-carrier distribution, and the distribution’s subsequent relaxation. We present first-principles calculations that capture all of the significant microscopic mechanisms underlying surface plasmon decay and predict the initial excited carrier distributions so generated. We also preform ab initio calculations of the electron-temperature dependent heat capacities and electron-phonon coupling coefficients of plasmonic metals. We extend these first-principle methods to calculate the electron-temperature dependent dielectric response of hot electrons in plasmonic metals, including direct interband and phonon-assisted intraband transitions. Finally, we combine these first-principles calculations of carrier dynamics and optical response to produce a complete theoretical description of ultrafast pump-probe measurements, free of any fitting parameters that are typical in previous analyses.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Plasmonics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Thesis Committee:
Vahala, Kerry J. (chair)
Minnich, Austin J.
Faraon, Andrei
Atwater, Harry Albert
Defense Date:
20 April 2016
Funders:
Funding Agency
Grant Number
NSF
UNSPECIFIED
Link Energy Foundation
UNSPECIFIED
DOE ‘Light-Material Interactions in Energy Conversion’ Energy Frontier Research Center
DE- SC0001293
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CaltechTHESIS:04242016-093536420
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DOI:
10.7907/Z9QV3JHT
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Article adapted for Ch. 1
DOI
Article adapted for Ch. 2
DOI
Article adapted for Ch. 3
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Brown, Ana Maii
0000-0003-3008-2310
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Classical and Quantum Effects in Plasmonic Metals

Thesis by

Ana Maii Brown

In Partial Fulfillment of the Requirements for the
degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2016
Defended April 20, 2016

Ana Maii Brown
ORCID: 0000-0003-3008-2310

iii

ACKNOWLEDGEMENTS
First and foremost, and with much gratitude, I thank my adviser, Professor Harry
Atwater, for his kind guidance and never-tiring enthusiasm and support. I certainly
would not have made it through earning a PhD without his encouragement and
unwavering confidence in me. I would also like to thank all of the Atwater Group
members and staff, and especially Matt Sheldon and Ravishankar Sundararaman for
their patient mentorship, and Jennifer Blankenship, Tiffany Kimoto, Daniel TurnerEvans, and Carissa Eisler for their genuine friendship. I am thankful for the blessing
of having David Ayala in my life. David has made me a better person, and makes
life, with all its mundane and epic moments, more vivid and exquisite. Finally,
it is with deep gratitude that I thank my parents, Norm and Adelina, and my sister, Robin, for their unconditional love and steadfast support. Adelina’s inherent
strength and humble wisdom, Norm’s unending curiosity and love, and Robin’s
ingenuity and devotion to family have shaped me, and yet still leave me in awe.
I am grateful for the generous financial support for my studies and for the work
in this thesis, mainly provided by the National Science Foundation and the Link
Energy Foundation.
Ana Maii Brown
April 2016

ABSTRACT
The field of plasmonics exploits the unique optical properties of metallic nanostructures to concentrate and manipulate light at subwavelength length scales. Metallic nanostructures get their unique properties from their ability to support surface
plasmons– coherent wave-like oscillations of the free electrons at the interface between a conductive and dielectric medium. Recent advancements in the ability
to fabricate metallic nanostructures with subwavelength length scales have created
new possibilities in technology and research in a broad range of applications.
In the first part of this thesis, we present two investigations of the relationship
between the charge state and optical state of plasmonic metal nanoparticles. Using experimental bias-dependent extinction measurements, we derive a potentialdependent dielectric function for Au nanoparticles that accounts for changes in the
physical properties due to an applied bias that contribute to the optical extinction.
We also present theory and experiment for the reverse effect– the manipulation of
the carrier density of Au nanoparticles via controlled optical excitation. This plasmoelectric effect takes advantage of the strong resonant properties of plasmonic
materials and the relationship between charge state and optical properties to elucidate a new avenue for conversion of optical power to electrical potential.
The second topic of this thesis is the non-radiative decay of plasmons to a hot-carrier
distribution, and the distribution’s subsequent relaxation. We present first-principles
calculations that capture all of the significant microscopic mechanisms underlying
surface plasmon decay and predict the initial excited carrier distributions so generated. We also preform ab initio calculations of the electron-temperature dependent heat capacities and electron-phonon coupling coefficients of plasmonic metals. We extend these first-principle methods to calculate the electron-temperature
dependent dielectric response of hot electrons in plasmonic metals, including direct interband and phonon-assisted intraband transitions. Finally, we combine these
first-principles calculations of carrier dynamics and optical response to produce a
complete theoretical description of ultrafast pump-probe measurements, free of any
fitting parameters that are typical in previous analyses.

PUBLISHED CONTENT AND CONTRIBUTIONS
Portions of this thesis have been drawn from the following publications:
1. Sheldon, M. T., Van de Groep, J., Brown, A. M., Polman, A. & Atwater,
H. A. Plasmoelectric potentials in metal nanostructures. Science 346, 828–
831 (2014).
doi:10.1126/science.1258405.
2. Brown, A. M., Sheldon, M. T. & Atwater, H. A. Electrochemical Tuning
of the Dielectric Function of Au Nanoparticles. ACS Photonics 2, 459–464
(2015).
doi:10.1021/ph500358q.
3. Brown, A. M., Sundararaman, R., Narang, P., Goddard III, W. A. & Atwater,
H. A. Non-Radiative Plasmon Decay and Hot Carrier Dynamics: Effects of
Phonons, Surfaces and Geometry. ACS Nano 10, 957–966 (2015).
doi:10.1021/acsnano.5b06199.
4. Brown, A. M., Sundararaman, R., Narang, P., Goddard III, W. A. & Atwater,
H. A. Ab initio phonon coupling and optical response of hot electrons in
plasmonic metals. Physical Review B (2016). Submitted.
5. Brown, A. M., Sundararaman, R., Narang, P., Schwartzberg, A. & Atwater,
H. A. Ultrafast carrier experimental and ab initio dynamics in plasmonic
nanoparticles (2016). In preparation.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Properties and Applications of Plasmonic Metallic Nanoparticles . . 1
1.2 The Drude and Lorentz-Drude Models . . . . . . . . . . . . . . . . 3
1.3 Mie Theory: Scattering of Light by Nanospheres . . . . . . . . . . . 4
1.4 Beyond the Lorentz-Drude Todel and Mie Theory . . . . . . . . . . 5
1.5 Scope of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter II: The Plasmoelectric Effect . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Charge Dependent Dielectric Function . . . . . . . . . . . . . . . . 8
2.2 Thermodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Temperature Calculations . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Calculation of Steady-State Nanoparticle Surface Potential and Charge 17
2.5 Charge Accumulation Shell Model . . . . . . . . . . . . . . . . . . 18
2.6 Nanoparticle Emissivity . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 FDTD Simulation Methods . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Plasmoelectric Simulations . . . . . . . . . . . . . . . . . . . . . . 21
2.9 Kelvin Probe Force Microscopy Experiments . . . . . . . . . . . . . 23
2.10 Optical Measurements of Nanoparticles Under Monochromatic Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Chapter III: Electrical Tuning of the Dielectric Function and Optical Properties of Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Previous Work and Overview . . . . . . . . . . . . . . . . . . . . . 33
3.2 Electrochemical Cell Fabrication . . . . . . . . . . . . . . . . . . . 34
3.3 Optical Measurements of Nanoparticles Under Applied Bias . . . . . 36
3.4 Modeling Optoelectronic Effects with FDTD Simulations . . . . . . 40
3.5 Comparison of Simulation and Experimental Results . . . . . . . . . 43
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter IV: Complete ab initio Description of Non-Radiative Plasmon Decay 49
4.1 Motivation and Previous Work . . . . . . . . . . . . . . . . . . . . . 49
4.2 Experimental Decay Rate . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Plasmon Decay via Direct Transitions . . . . . . . . . . . . . . . . . 53
4.5 Plasmon Decay via Phonon-assisted Transitions . . . . . . . . . . . 54

vii
4.6 Phonon Modes and Matrix Elements . . . . . . . . . . . . . . . . . 56
4.7 Surface-assisted Transitions . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Estimate of Resistive Losses . . . . . . . . . . . . . . . . . . . . . . 58
4.9 Results for Common Plasmonic Metals . . . . . . . . . . . . . . . . 60
4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Chapter V: Ab initio Description of Hot Electron Relaxation in Plasmonic
Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Motivation and Previous Work . . . . . . . . . . . . . . . . . . . . . 66
5.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Electronic Density of States and Heat Capacity . . . . . . . . . . . . 69
5.4 Phononic Density of States and Heat Capacity . . . . . . . . . . . . 72
5.5 Electron-phonon Matrix Element and Coupling . . . . . . . . . . . . 74
5.6 Temperature Dependent Dielectric function . . . . . . . . . . . . . . 79
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter VI: Experimental and ab initio Ultrafast Carrier Relaxation in Plasmonic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Ab initio theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Chapter VII: Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . 106
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix A: Derivations of Expressions for Direct and Phonon-Assisted
Plasmon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.1 Direct Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Phonon-Assisted Transitions . . . . . . . . . . . . . . . . . . . . . . 123
A.3 Final Expressions for Direct and Phonon-assisted Plasmon Decay
After Accounting For Reverse Transitions . . . . . . . . . . . . . . . 129
Appendix B: Tabulated Electronic Heat Capacity and Electron-Phonon Coupling Factor as a Function of Electron Temperature . . . . . . . . . . . . 131
Appendix C: Dielectric Function Temperature Dependence Prefactor Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

viii

LIST OF ILLUSTRATIONS

Number
Page
1.1 Sketch of surface plasmon . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Absorption cross section spectrum of a Ag nanoparticle and the plasmoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Plasmoelectricly Induced Electron Density . . . . . . . . . . . . . . 10
2.3 Plasmoelectricly Induced Temperature Increase . . . . . . . . . . . . 15
2.4 Au nano particle heat diffusion and temperature . . . . . . . . . . . . 17
2.5 Electron Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Plasmoelectric Potential and Absorption . . . . . . . . . . . . . . . . 20
2.7 Nanoparticle Emissivity . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Modeled plasmoelectric response for Ag nanoparticles . . . . . . . . 22
2.9 KPFM Surface Potential Map . . . . . . . . . . . . . . . . . . . . . 24
2.10 KPFM Control Measurement . . . . . . . . . . . . . . . . . . . . . 25
2.11 Plasmoelectric effect on dense Au nanoparticles on ITO/glass . . . . 26
2.12 Schematic of white vs. monochromatic illumination experimental
setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.13 Plasmoelectric effect for 60 nm Au colloids suspended in water . . . 30
3.1 Electrochemical Cell Geometery . . . . . . . . . . . . . . . . . . . . 35
3.2 Extinction spectra and changes in spectra of Au colloids in electrochemical cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Experimental and simulated peak parameters as a function of applied
bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Raw Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 FDTD simulation geometry . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Physical parameters as a function of applied bias . . . . . . . . . . . 44
3.7 Comparison of experiment and simulation over full wavelength range 47
4.1 Schematic of Plasmon Decay . . . . . . . . . . . . . . . . . . . . . 50
4.2 Comparison of linewidths . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Comparison of contribution to Im(ε) . . . . . . . . . . . . . . . . . 63
4.4 Carrier Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Fundamental parameters for describing hot electron relaxation . . . . 68
5.2 Electronic density of states . . . . . . . . . . . . . . . . . . . . . . . 70

ix
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10

Electronic heat capacity as a function of electron temperature . . . . 71
Phonon density of states . . . . . . . . . . . . . . . . . . . . . . . . 73
Lattice heat capacity as a function of lattice temperature . . . . . . . 74
energy resolved electron-phonon coupling strength . . . . . . . . . . 77
Electron-phonon coupling factor G as a function of electron temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Ab initio dielectric function at room temperature . . . . . . . . . . . 82
Complex dielectric function for Te = 400K . . . . . . . . . . . . . . 83
Change in dielectric function for Te =300 K to 400 K . . . . . . . . . 83
Complex dielectric function for Te = 1000K . . . . . . . . . . . . . . 84
Change in dielectric function for Te =300 K to 1000 K . . . . . . . . 84
Complex dielectric function for Te = 5000K . . . . . . . . . . . . . . 85
Change in dielectric function for Te =300 K to 5000 K . . . . . . . . 85
Critical interband transitions for dielectric function temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Ultrafast transient absorption setup . . . . . . . . . . . . . . . . . . 89
Differential extinction map as a function of probe wavelength and
delay time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Transient differential extinction spectral and kinetic traces . . . . . . 92
Extinction of Au colloids used in transient absorption measurements . 92
Electron distribution relaxation evolution . . . . . . . . . . . . . . . 94
Kinetic signal amplitude dependence on pump power . . . . . . . . . 101
Transient signal temporal behavior dependence on pump power . . . 101
Nonthermalized and thermalized electron distribution dependence
on pump power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Transient extinction signal for a variety of probe wavelengths . . . . 104
Transient extinction signal for a variety of probe wavelengths . . . . 104

LIST OF TABLES
Number
Page
3.1 Simulation parameter limits and resolution . . . . . . . . . . . . . . 42
4.1 ab initio parameters for metals . . . . . . . . . . . . . . . . . . . . . 60
5.1 Parameters for dielectric function temperature dependence . . . . . . 81
B.1 Tabulated electronic heat capacity and electron-phonon coupling factor as a function of electron temperature . . . . . . . . . . . . . . . . 131

Chapter 1

INTRODUCTION
1.1

Properties and Applications of Plasmonic Metallic Nanoparticles

The field of plasmonics exploits the unique optical properties of metallic nanostructures to route and manipulate light at subwavelength length scales. Recent advancements in the ability to fabricate metallic nanostructures with subwavelength
length scales have created new possibilities in technology and research focused on
the manipulation of light. Plasmonic metallic nanostructures have been used most
successfully to achieve extreme concentration of light, as in plasmonic antennas,
lenses and resonators.[1]
Traditionally, light concentration has been achieved with dielectric lenses and resonators but these devices have limitations that are far surpassed by plasmonic-based
metallic nanostructures. The smallest spot size that can be produced with dielectric
lenses and resonators is approximately λ/2. Therefore the electromagnetic mode
volume in which an optical signal can be concentrated is approximately equal to
(λ/2) 3 ; metallic nanostructures can be used to concentrate light to volumes much
smaller than these traditional limits.
Metallic nanostructures get their unique properties from their ability to support surface plasmons– coherent wave-like oscillations of the free electrons at the interface
between a conductive and a dielectric medium (Figure 1.1). Surface plasmons are
produced when a time-varying electric field constituting incident light produces a
force on the free electrons inside the metal which drives them into a collective oscillation in resonance with the incident electric field. Surface plasmons are transverse
magnetic in character, and the generation of surface charge requires an electric field
normal to the surface. This leads to the field component normal to surface being enhanced near the surface and decaying exponentially with distance away from
it. This field component is evanescent and prevents power from propagating away
from the surface. The decay length of the field in the dielectric interfacing with
the metal is roughly half of the wavelength of the incident light, whereas the decay
length inside the metal is determined by the skin depth.
Surface plasmons come in two varieties: one form, termed a local surface plasmon,
is a resonant standing wave on a conductive nanostructure with dimensions on or-

Figure 1.1: Schematic of surface plasmon showing electric fields and oscillating
charges. Figure adapted from Reference 2.
der of the exciting optical wavelength; the other form, termed a surface plasmon
polariton, is a non-resonant charge-oscillation wave which propagates down the
surface of a conductor. Surface plasmon polaritons occur in metal structures with
one or more dimensions greater than or equal to the wavelength of incident light;
the charge oscillation can propagate down the conducting surface as a surface plasmon polariton mode. Local surface plasmons occur in metallic nanostructures with
dimensions significantly shorter than the dimensions of the incident light, where
the whole particle is subject to virtually a uniform incident electric field at any one
point in time. This results in dipolar charge separation and resonant frequencies,
much like the resonances of a standing wave, which produce a very strong charge
displacement and associated field concentration.
Plasmons provide a pathway to manipulate electromagnetic radiation at nanometer
length scales [1–3] and at femtosecond time scales.[4] Research is currently being
conducted to create new passive and active plasmonic devices to generate, guide,
modulate, and detect light. Excitation, propagation, and localization of plasmons
can be tailored by nanoscale control of size, shape, and architecture.[1, 5] Metal
nanostructures exhibiting subwavelength optical confinement [6, 7] have enabled
nanoscale photonic waveguides[8, 9], modulators[10], surface plasmons amplified
by stimulated emission of radiation (SPASERs)[11, 12], light trapping structures for
photovoltaics[13, 14], field enhancement for Raman spectroscopy, and biological
labeling techniques[1, 15]. Coupling of laser light to plasmonic structures can result in excitation of extremely high energy densities and efficient localized heating,

because the resonant absorption cross section for plasmonic nanostructures is much
larger than their physical cross sections.[16] Recently, there has been considerable
interest in plasmonic energy relaxation and conversion mechanisms. Plasmons have
relatively short (< 10 fs) lifetimes and can decay into single particle excitations, notably hot electron-hole pairs.[17] Plasmonic resonators have been recently reported
to emit optically excited hot electrons across a rectifying metal-semiconductor interface.[18, 19]
Despite more than a decade of intensive scientific exploration, new plasmonic phenomena continue to be discovered, including quantum interference of plasmons,
observation of quantum coupling of plasmons to single particle excitations, and
quantum confinement of plasmons in single-nanometer scale plasmonic particles.
Also, plasmonic structures find widening applications in integrated nanophotonics, biosensing, photovoltaic devices, single photon transistors, and single molecule
spectroscopy.
1.2

The Drude and Lorentz-Drude Models

In 1900 Paul Drude proposed a model of electrical conduction to explain the transport and optical properties of electrons in metals.[20, 21] His model approximated
a conductor as a collection of free electrons and stationary positively charged ions.
In this model, equation of motion for an electron subject to an external electric field
~ is
d~r
d 2~r
~ r ,t)
(1.1)
m 2 = −mγ − e E(~
dt
dt
where e is electron charge, m is electron mass, r is the electron displacement from
equilibrium, and γ is the harmonic damping constant. By considering the conductivity of this system and the relation = 1 + 4πiσ/ω, we arrive at the Drude
dielectric function:
ω2p
4πne2 /m
(ω) = 1 −
=1− 2
( ω2 + iγω)
(ω + iγω)

(1.2)

where n is the electron density and the plasma frequency is defined as ω p ≡ 4πne2 /m.
The plasma frequency is the resonant frequency of the electrons in the metal. This
complex dielectric function accounts for free-electron behavior in metals, and has
been used to approximate the dielectric function of metals in countless scientific
investigations. However, this model completely neglects the treatment of interband
transitions, which become important when incident light has an energy higher than
the interband transition threshold.

In 1905, Hendrick Antoon Lorentz extended the Drude model to account for electrons which are bound by a damped harmonic oscillator to an atom or ion in the
solid.[22, 23] The equation of motion for an electron in this Lorentz-Drude model
in the presence of an external electric field E is given by the Drude-Lorentz equation:
d 2~r
d~r
~ r ,t)
m 2 = −mω02~r − mγ − e E(~
(1.3)
dt
dt
where ω0 is the natural frequency of the oscillator, and the other parameters are the
same as those in the Drude model. The dielectric function for this system is given
by
ω2p
4πne2 /m
(ω) = 1 + 2
=1+ 2
(1.4)
(ω0 − ω2 − iγω)
(ω0 − ω2 − iγω)
The Lorentz-Drude model has been used in investigations to approximate the interband contributions to the dielectric function. An extension of the Lorentz-Drude
model to more than one oscillator has been used to better approximate the dielectric
functions of Ag and Au.[24]
1.3

Mie Theory: Scattering of Light by Nanospheres

Mie theory, which describes the absorption and scattering profiles of a subwavelength sphere illuminated by a plane wave, is named after Gustav Mie, who was the
first to electrodynamically describe localized surface plasmon polaritons in metallic nanoparticles in 1908.[25] Using Maxwell’s equations, Bohren and Huffman[26]
derive the classical analytical solution for the scattering and extinction cross section
of the sphere:
2π X
Csca = 2
(2n + 1)(|an | 2 + |bn | 2 )
(1.5)
k n=1

Cext =

2π X
(2n + 1)Re(an + bn )
k 2 n=1

(1.6)

with the scattering coefficients
an =

µm2 jn (mx)[xhn (x)]0 − µ1 jn (x)[mx jn (mx)]0
µm2 jn (mx)[xhn(1) (x)]0 − µ1 hn(1) (x)[mx jn (mx)]0

bn =

µ1 jn (mx)[x jn (x)]0 − µ jn (x)[mx jn (mx)]0

(1.7)

(1.8)
µ1 jn (mx)[xhn(1) (x)]0 − µhn(1) (x)[mx jn (mx)]0
where jn and hn are the spherical Bessel functions, x is the size parameter x =
ka, m is the relative refractive index m = k 1 /k, k and k 1 are the wave-vectors in
the particle and surrounding medium, and µ and µ1 are the permeabilities of the
particles and the surrounding medium, respectively.

1.4

Beyond the Lorentz-Drude Todel and Mie Theory

Mie theory was first experimentally verified in the 1970’s, and as advances in technology have made fabrication and characterization of nanoparticles a common research area in recent decades, Mie theory along with the Lorentz-Drude model has
been commonly used to describe the optical properties of metallic nanoparticles in
a variety of systems. However there are many shortcomings of the Lorentz-Drude
model and Mie theory which make them unable to predict or describe nonidealities
in real systems, experiments with more complex systems, or microscopic processes
related to localized surface plasmons. For example, surfaces and interfaces can
change electronic structure and damping, and these effects become more important
as the size of nanoparticles decreases.[27, 28] Static or dynamic charge transfer
such as those in a system with an applied bias or difference in work function between the plasmonic resonator the the surrounding medium results in a change in
the dielectric function and thus to the optical properties.[29, 30] Excitation by incident light leads to an increase in temperature of the electron and phonon bath,
resulting in changes in the dielectric function as well as in the heat capacity and
electron-phonon coupling.[17, 31, 32] When we examine plasmonic behavior at
very short time scales, quantum effects become important.[33–35] Without corrections for these and other effects, Mie theory with the Lorentz-Drude dielectric function will not accurately predict experimental results. [27, 28] The projects included
in this thesis go beyond Mie theory and the Lorentz-Drude Model to achieve more
accurate descriptions of plasmonic phenomenon in real systems.
1.5

Scope of This Thesis

Though a variety of projects are presented in this thesis, each study was aimed
at furthering the understanding of physical phenomena fundamentally related to
plasmons. The chapters are organized as follows:
The Plasmoelectric Effect
Chapter 2 presents the theoretical framework and experimental evidence for a
novel avenue for the conversion of optical power to an electric potential. This
plasmoelectric phenomenon of an optically induced electrochemical potential
is described in an all-metal geometry and is based on the plasmon resonance
in metal nanostructures. Under certain conditions, we predict that when illuminated with monochromatic light off-resonance, plasmonic systems will
absorb or expel electrons from an accessible ground to come into resonance

with the incident light and reach a thermodynamically favorable state. We
provide experimental evidence for the plasmoelectric effect in arrays of gold
nanoparticles on an indium tin oxide substrate, where we observed plasmoelectric surface potentials as large as 100 millivolts under monochromatic illumination of 100 milliwatts per square centimeter. A spectroscopic analysis
of Au nanoparticles in solution showed further evidence for the plasmoelectric effect. Plasmoelectric devices may enable the development of all-metal
optoelectronic devices that can convert light into electrical energy.
Electrical Tuning of the Dielectric Function and Optical Properties of
Nanoparticles
In Chapter 3 we discuss the tunable dielectric response of plasmonic Au
nanoparticles under electrochemical bias. We show that the changes in the
optical properties of the Au nanoparticles as a function of applied bias can be
interpreted in terms of changes in the surface charge density, surface damping, and the near-surface volume fraction of the nanoparticles that experience
a modified dielectric function, as well as changes in the index of refraction
of the surrounding electrolyte medium. Using experimental bias-dependent
extinction measurements, we derive a potential-dependent dielectric function
for Au nanoparticles that accounts for changes in the physical properties contributing to optical extinction.
Complete ab initio Description of Non-Radiative Plasmon Decay
Chapter 4 describes first-principles calculations that capture all of the significant microscopic mechanisms underlying surface plasmon decay and predict the initial excited carrier distributions so generated. We present the first
ab initio predictions of phonon-assisted optical excitations in metals, which
are critical to bridging the frequency range between resistive losses at low
frequencies and direct interband transitions at high frequencies. In the commonly used plasmonic materials, gold, silver, copper, and aluminum, we find
that resistive losses compete with phonon-assisted carrier generation below
the interband threshold, but hot carrier generation via direct transitions dominates above threshold.
Ab initio Description of Hot Electron Relaxation in Plasmonic Metals
In Chapter 5 we present ab initio predictions of the electron-temperaturedependent electronic heat capacities and electron-phonon coupling coefficients of plasmonic metals. We find substantial differences from free-electron

and semi-empirical estimates, especially in noble metals above transient electron temperatures of 2000 K, because of the previously neglected strong dependence of electron-phonon matrix elements on electron energy. We also
present first-principles calculations of the electron temperature dependent dielectric response of hot electrons in plasmonic metals, including direct interband and phonon-assisted intraband transitions, facilitating complete ab initio predictions of the time-resolved optical probe signatures in ultrafast laser
experiments.
Experimental and ab initio Ultrafast Carrier Relaxation in Plasmonic
Nanoparticles
Chapter 6 presents ultrafast transient absorption measurements and first-principles
calculations of the relaxation of hot electrons excited by plasmon decay. We
use results from Chapters 5 and 6 to achieve an improved nonlinearized Boltzmann method that fully accounts for the nonthermal character of the excited
electron distribution and electronic-structure effects. The model uses our ab
initio results for the density of states, electron-phonon matrix elements, dielectric function, and electron distribution produced via plasmon decay to
replace approximations of these parameters used by previous investigations.
Importantly, our model is free of any fitting parameters, which have been relied upon heavily by other studies to achieve agreement between experiment
and theory.

Chapter 2

THE PLASMOELECTRIC EFFECT
It is well known that the plasmon resonance frequency, ω p , of metallic plasmonic
materials is dependent on the electron density. Recent work has demonstrated control over ω p of metal nanostructures when an external electrostatic field alters the
carrier density in the metal [29]. Increasing the carrier density in a noble metal
nanoparticle results in a blueshift of the resonance, whereas decreasing it results in
a redshift, as illustrated for a 20-nm-diameter Ag nanoparticle in vacuum in Figure
2.1. But until recently, the reverse effect, the generation of an electrostatic potential
due to an optically driven change in carrier density in a plasmonic nanostructure,
had not been observed. Thermodynamically, however, such a effect, coined the
plasmoelectric effect, is expected to occur. This chapter first discusses the thermodynamic model which predicts the plasmoelectric effect, and then presents direct
experimental evidence of plasmoelectric potentials in the range 10 to 100 mV on
colloidal assemblies, in qualitative agreement with a thermodynamic model. These
results may provide a new route to convert optical energy into electrical power.
2.1

Charge Dependent Dielectric Function

Our theoretical framework for the plasmoelectric effect requires a model for the
dielectric function dependence on electron density, which then determines the material’s optical properties dependence on electron density. In this work, we choose
to work with two common plasmonic metals, silver and gold. For the dielectric
function of Au we use a Brendel-Bormann Gaussian oscillator model, and use the
method outlined by Rakic and coworkers [24]. For the dielectric function of Ag
we apply a 6th-order multiple oscillator Lorenz-Drude model, fit to data from the
Palik Handbook [36]. These dielectric functions accurately reproduce the observed
extinction spectra of spherical gold or silver nanoparticles when input into the exact analytic solutions to Maxwell’s equations provided by Mie theory [17, 26]. To
introduce the explicit dependence on electron density, n, we assume that all terms
in the dielectric function that depend on the bulk plasma frequency, ω p , depend on
electron density, n, according to a simple Drude model relation,

Figure 2.1: (a) Schematic: Spontaneous charge transfer to or from the nanoparticle
is thermodynamically favored when the consequent spectral shift increases the absorption, raising the temperature. Irradiation on the blue side of the resonance leads
to a negative charge on the particle; irradiation on the red side leads to a positive
charge. (b) Calculated absorption cross section for a 20-nm-diameter Ag nanoparticle in vacuum with bulk carrier density n Ag and carrier densities that are reduced
or increased by 1%.

ne2
ωp =
o m∗e

! 1/2
(2.1)

where e is the electron charge, o is the permittivity of free space, and m∗e is the electron effective mass [30]. This strategy is consistent with other work that examined
carrier density-dependent plasmon shifts, for example in doped semiconductors,
electrochemical cells, or at metal surfaces during ultrafast pump-probe measurements [10, 37–39].
Figure 2.2 shows the calculated relative change of the absorption cross section of a
20-nm-diameter Ag nanoparticle in vacuum, calculated with Mie theory, as a func-

Figure 2.2: Relative change of the absorption cross section, Cabs (n, λ), of a 20nm-diameter Ag nanoparticle in vacuum, as a function of wavelength and electron
density, and normalized by the absorption cross section for a neutral Ag silver particle with electron density n Ag .
tion of wavelength and electron density, and normalized by the absorption cross
section for a neutral Ag silver particle with electron density n Ag [26]. Note that
to the blue of the plasmon resonance increasing electron density increases the absorption cross section, and to the red of the plasmon resonance increasing electron
density decreases the absorption cross section.
2.2

Thermodynamic Model

Free Energy Minimization
To model the experimentally observed plasmoelectric effect, we consider a metal
nanostructure placed on a grounded conducting substrate and illuminated with monochromatic radiation at a wavelength just below the plasmon resonance, λ p . Random
charge fluctuations between particle and substrate will cause the plasmon resonance spectrum to vary by minute amounts (Figure 2.2). If an electron is randomly
added to the particle, the resonance will shift toward the blue, leading to an increase in light absorption of the particle, which in turn leads to a small increase
in the nanoparticle temperature. The changes in number of electrons N and temperature T (N ) change the free energy F of the particle, and an equilibrium charge

11
density is achieved when the free energy is at a minimum:
∂F (N,T )
∂F
∂F
dT
=0
∂N
∂N T
∂T (N ) N dN

(2.2)

Here, we assume that both the intensity and wavelength of the illumination are
∂F
)T ,
constant. Using, by definition, the electrochemical potential µ(N,T ) ≡ ( ∂N
where µ, and the entropy of the particle S(N,T ) ≡ −( ∂F
∂T ) N , we find that the free
energy minimum corresponds to a configuration with a number of electrons, N,
such that
µ(N,T ) = S(N,T )

dT (N )
dN

(2.3)

Equation (2.3) shows that under illumination, the plasmonic particle adopts an electrochemical potential that is proportional to dT/dN. This quantity, which is determined only by the plasmon resonance spectrum and the heat flow from the particle
to the substrate, provides the unique thermodynamic driving force in the system. It
favors charge transfer to or from the particle that increases absorption, and thereby
temperature, in order to lower the free energy. The factor dT/dN is largest on the
steepest parts of the resonance spectrum; it is positive for irradiation on the blue
side of the resonance, leading to a positive chemical potential for the electrons and
hence a negative voltage. The reverse is observed for irradiation on the red side of
the resonance.
Equation (2.2) can be used to quantitatively estimate the equilibrium plasmoelectric potential by writing F (N,T ) as the sum of the free energies of electrons and
phonons, using the well-known free energy functions of an electron and phonon
gas [40, 41]. The electronic term is composed of a contribution due to the chemical
potential of the electrons that is directly given by the Fermi function, and an electrostatic contribution due to charging of the metal nanosphere; the phonon term is
given by the Debye model. Taking the derivatives with respect to N and T, we find
analytical expressions for µ(N,T ) and S(N,T ) (see section 2.2) that are then input
into Eq. (2.3).
Our thermodynamic model posits that there is a well-defined, constant temperature, T, of a plasmonic nanoparticle during steady-state illumination at a single
wavelength, λ, and intensity, Iλ . This constant T results from the requirement that
the optical power absorbed and thermal power conducted away or radiated by the

12
nanoparticle must be equal in steady state. This temperature is a function of the
absorption cross section, Cabs , of the nanostructure, which also depends on the
electron density, n, due to the strong dependence on n in the complex dielectric
function of the metal. Then, T (n, λ) is the unique thermodynamic state function
that distinguishes an illuminated plasmonic absorber from one in the dark.
We assert that it is reasonable to define such a temperature for a resonantly heated
particle, because electronic relaxation processes are fast (∼ 10-100’s fs), after which
the electronic system and lattice are in equilibrium. We note that similar arguments
are central to, for example, the detailed balance calculation of the limiting efficiency
of a photovoltaic cell [42]. This well-defined temperature describes a local thermal
equilibrium in a thermodynamic system enclosing the particle, and motivates a thermodynamic equilibrium argument based on free energy minimization.
Calculation for the Configuration of Minimum Total Free Energy
We determine the total free energy, Ftot , of the nanoparticle by considering the
separate contributions from the electrons, Fe , and the phonons, Fp :
Ftot (N,T (N )) = Fe (N,T (N )) + Fp (T (N ))

(2.4)

Note that for the optical power densities and time scales we consider, because of
the fast electronic relaxation rate and electron-phonon coupling rate in a metal, the
electron temperature and phonon temperature are equivalent [17]:
Telectron ≈ Tphonon = T (N )

(2.5)

Fe is defined in terms of the chemical potential of the electrons, µe , and the electrostatic potential on the particle, φ, as
Z N
Z N−No
Fe (N,T (N )) =
µe (N ,T (N ))dN +
φ(N 0 − No )dN 0
(2.6)

where No is the number unbound electrons on the neutral particle, and µe is the
Fermi function with a small (< 0.1%) temperature correction [40]
1 πk bT (N, λ)
µe (N,V,T (N, λ)) = F (N,V ) *1 −
3 2 F (N,V )

(2.7)

13
with
~ 3π 2 N
F (N,V ) =
2m

! 2/3
(2.8)

and φ is the electrostatic potential for a conducting sphere:
φ(N − No ) =

e2 (N − No )
4πR 0 m

(2.9)

where R is the sphere radius, e is the electron charge, and 0 and m are the permittivity of free space and the relative permittivity of the surrounding medium,
respectively.
Fp is defined in terms of the speed of sound in the particle, vs , via the Debye temperature, θ, with [41, 43]
Fp (T (N )) = 3k bT (N ) A0 ln
− k bT (N ) A0
(2.10)
T (N )
and
~vs 6π 2 A0
θ=
kb

! 1/3
(2.11)

where k b is Boltzmann’s constant, and A0 the number of atoms in the particle
which here is equal to No . For Au, θ ∼ 170K [40]. Expression (2.11) is the
high-temperature limit of the quantum Debye model, valid for T > θ.
The total electrochemical potential µ and the entropy S can now be derived by
∂F
applying µ ≡ ( ∂N
)T and S ≡ −( ∂F
∂T ) N to expression (2.4) for the free energy. We
find
! 2/3
mk b 2T (N ) 2 πV
e2 (N − N0 )
µ(N,T (N )) = F (N,V ) −
(2.12)
3N
4πR 0 m
6~2
and
S(N,T (N )) =

Z N

1 π 2 k b 2T (N 0 ) 0
+ 4k b A0
dN − 3k b A0 ln
3 2 F (N 0 )
T (N )

(2.13)

The steady-state charge configuration (value of N) that corresponds to the minimum
free energy can then be obtained by using eqs. (2.12) and (2.13) to solve Eq. (2.3).
This leads to

! 2/3
mk b 2T (N ) 2 πV
e2 (N − N0 )
0 = F (N ) − F (N0 ) −
3N
4πR 0 m
6~2
Z N
1 π k b T (N ) 0 dT (N )
dN
dN
0 3 2 F (N )
dT (N )
dT (N )
+ 3k b A0 ln
− 4k b A0
(2.14)
T (N )
dN
dN
This equation can be solved to find the steady-state value for N (for a given λ and
Iλ ) using as input the function dT (N )/dN, which is determined by the plasmon
resonance spectrum and the heat conducted or radiated away from the particle.
Theoretical Efficiency Limit
This thermodynamics model shows that an excited plasmonic resonator behaves
as a heat engine that can convert absorbed off-resonant optical power into a static
electrochemical potential. In principle, optical-to-electrical energy conversion by
this mechanism could be optimized to perform with an efficiency at the Carnot
limit, as with any generalized heat engine [44].
2.3

Temperature Calculations

Ag Nanoparticle in Vacuum
To calculate the plasmoelectric potential for a Ag nanoparticle in vacuum we calculate the temperature of the nanoparticle, T, given the absorption cross section
Cabs (n, λ) from section 2.1 and incident power density. Under steady state illumination, the power absorbed by the particle must equal the power emitted:
Pin = Pout

(2.15)

Pin constitutes the absorbed monochromatic optical radiation plus the absorbed
thermal radiation from the ambient background (at Tamb = 293K ).
Pin = Cabs (n, λ)Iλ + σ ATamb

(2.16)

Here, σ is the Stefan Boltzmann constant, A is the surface area of the nanoparticle,
and is the emissivity. The nanoparticle emissivity = 0.01 was experimentally determined using FTIR absorption spectroscopy on gold colloids, assuming
reciprocity of the absorption and emission in the 2-5 µm wavelength range of the

Figure 2.3: Calculated temperature for a 20-nm-diameter Ag particle in vacuum as
a function of illumination wavelength (Iλ = 1 mW/cm2 ).
measurement (see section 2.6). This value is close to that of bulk gold and silver
in the infrared, as reported elsewhere [45] and that predicted by Mie theory. For a
Ag nanoparticle in vacuum the only loss channel for the power is thermal radiation,
and thus
Pout = σ AT 4

(2.17)

Solving Equation (2.15) for the temperature of the particle then gives

T (n, λ) = *

Cabs (n, λ)Iλ + σ ATamb

σ A

1/4

(2.18)

Figure 2.3 shows the calculated temperature for a 20-nm-diameter Ag nanoparticle
in vacuum as a function of illumination wavelength for an incident power density
of 1 mW/cm2 ; it peaks at 400 K. The calculated temperature and its dependence on
N, dT (N )/dN (for a given λ and Iλ ) are then used as input in solving Eqn. (2.14)
for the case of a Ag nanoparticle in vacuum (see section 2.4).
Au Nanoparticle on ITO
To calculate the temperature of a Au nanoparticle on ITO/glass we first calculate the
absorption cross section spectrum Cabs (n, λ) using finite-difference-time-domain
(FDTD) full wave electromagnetic simulations. In our experiments, described later

16
in this chapter, the exact dielectric surrounding of the nanoparticle is very complex;
there are large variations in inter-particle spacing as well as clusters with different
configurations, both inducing near-field coupling and thereby red-shifted resonance
wavelengths. Furthermore, Kelvin Probe Force Microscopy (KPFM) experiments
were performed in ambient conditions, such that adsorption of water to the particleITO interface is likely. Since the exact influence of such conditions on the plasmoelectric potential is beyond the scope of this work, we simplified the geometry
to a single 60-nm-diameter Au nanoparticle on an ITO substrate. We assumed a
uniform background index of n = 1.4 to take into account the red-shift due to the
dielectric surrounding described above, such that the modeled scattering spectrum
represented the measured scattering spectrum. We then calculated the steady-state
power balance to obtain the nanoparticle temperature. Heat flow into the substrate
was modeled using a 1D heat transfer model, as sketched in Figure 2.4, with the
particle on a glass substrate with thickness d = 1 mm covered by a thin layer of
ITO. We used the thermal heat conductivity for glass, σT = 1.05 Wm-1 K -1 [46].
The heat of the particle can be transferred from the particle (at Tpart ) through a
cylinder of glass with cross sectional area equal to the contact area Acontact between
the glass and the particle, to the bottom of the substrate (at ambient temperature,
Tamb = 293 K). The heat flow is given by [46]
Pdi f f = σT

(Tpart − Tamb )
Acontact

(2.19)

Figure 2.4 shows the calculated temperature of the Au nanoparticle as a function
of incident wavelength for the three different intensities used in Figure 2.11b. In
these calculations the parameter Acontact was tuned such that the calculations of the
surface potential (see section 2.4) best match the experimentally observed surface
potential for the largest power densities. The best fit for the contact area was 2.5
times the geometrical particle cross section, a reasonable value given the simple
heat flow model. The same contact area was then used for the lower power densities to calculate the temperature profiles in Figure 2.4 and the surface potentials
in Figure 2.4c. We found the same power-dependent trend for slightly more complicated 2D and 3D heat diffusion models, with the assumed contact area between
the particle and ITO dominating the magnitude of the temperature. The maximum
temperature calculated in Figure 2.4 amounts to 308 K, 15 degrees above ambient
temperature, which is a reasonable temperature to sustain for an Au nanoparticle on
a substrate. Note that the increase of temperature due to the plasmoelectric effect

Figure 2.4: (Left) Sketch of 1D heat diffusion model for a Au nanoparticle on
ITO/glass. (Right) Au nanoparticle temperature as a function of illumination wavelength, corresponding to the experiments in Figure 2.11b.
compared with a neutral particle is only of order of up to tens of mili-Kelvin, with
some variation depending on geometry and illumination intensity. The calculated
temperature and its dependence on N, dT (N )/dN (for a given λ and Iλ ) were used
as input in solving Equation (2.14) for the case of Au nanoparticles on ITO/glass
(see section 2.4).
2.4

Calculation of Steady-State Nanoparticle Surface Potential and Charge

Using the temperature and values for dT (N )/dN described in section 2.3 we solved
Equation (2.14) to determine the steady-state surface potential and corresponding
charge on the nanoparticle as a function of irradiation wavelength. Figure 2.5 shows
the calculated charge density change (N − No ) for Au nanoparticles on ITO/glass,
for the highest experimental power density of Figure 2.4b (Iλ =1000 mW/cm2 ). The
calculations use 6.67 × 106 Au atoms for a 20-nm-diameter Au sphere. As can be
seen, the excess electron number (N − No ) is modulated around a value of −104.69
electrons, observed at the peak of the resonance (where there is no plasmoelectric
effect, dN/dT = 0). This corresponds to the number of electrons for which the total
electrostatic charging energy equals the Fermi energy for Au, EF = 5.02 eV. In the
dark, the offset between the Fermi level of the particle and the substrate induces
a compensating electrostatic charge on the particle, which is the usual condition
for electrochemical equilibration between two dissimilar conductors. Note that the
presence of the ITO substrate can be easily accounted for by subtracting the Fermi

Figure 2.5: Calculated excess electron number N − N0 (right-hand scale) for a 60nm-diameter Au particle on ITO/glass as a function of illumination wavelength (Iλ
= 1000 mW/cm2 ). The average number of electrons generating the plasmoelectric
effect is shown by the right- hand scale.
energy of the ITO substrate, such that the offset observed in Figure 2.5 is in fact
the difference in Fermi energy between the particle and the ITO. Correcting for this
“dark” charge (right-hand scale in Figure 2.5), we see that in the time-averaged situation less than one electron on average is added or removed from the nanoparticle to
generate the observed plasmoelectric potential. The data in Figure 2.5 can be converted to surface potential, as shown in Figure 2.11c. Additionally, we note that the
thermoelectric effect (omitted in our analysis) induces only a minor contribution to
the potential state of the particle, even for large temperature gradients between the
substrate and particle, because of the small Seebeck coefficient of metals, ∼ µV/∆K
[47].
2.5

Charge Accumulation Shell Model

In the calculations described above we assumed that any additional charge carriers
that are transferred to the particle are uniformly distributed throughout the nanoparticle. However, electrostatic models require that any surplus charges reside on the
surface of a metal object, since no static electric fields can exist inside the metal
[48]. On the other hand, the surface plasmon resonance is a dynamic phenomenon,
with electromagnetic fields that penetrate into the metal as defined by the electromagnetic skin-depth, which is ∼ 1-5 nm for Ag and Au at optical frequencies, and
an electron mean free path that is larger than the Fermi screening length. Addition-

19
ally, the plasmon resonance frequency is predominately determined by the electron
density of the portion of the particle within the optical skin depth of the metal, as
reported experimentally [27] and verified by us using FDTD simulations. Therefore, electrostatic arguments suggest that any additional charges will reside near
the surface of the nanoparticle, and electrodynamic arguments suggest that only the
electron density near the surface needs to be increased in order to blue-shift the
plasmon resonance. If excess charge resides only near the surface, the plasmoelectric effect thus requires a smaller number of electrons to obtain the same frequency
shift, and thereby dT (N )/dN in Equation (2.3) is larger.
To study the influence of non-uniform distributions of charge density in the particle, we consider a simplified shell model, in which we assume that all the additional
charge carriers transferred due to the plasmoelectric effect reside in the outer shell
with thickness δ (see inset of Figure 2.6). To implement this shell model in Equation
(2.14), one has to express the Fermi energy, temperature, and N and V in terms of
the number of electrons in the shell volume rather than in the whole particle. Equation (2.14) can then be solved for the number of electrons in the shell, assuming a
certain shell thickness.
Figure 2.6 shows the plasmoelectric potential (a) and the relative increase in absorption (b) for a 20-nm-diameter Ag particle in vacuum, as a function of illumination
wavelength, with Iλ = 10 mW/cm2 , and for different shell thicknesses. Figures 2.6
(a) and (b) clearly show the effect of a thinner shell: the transferred electrons induce a relatively larger increase in electron density and thereby a larger increase
in absorption. Note that for shell thicknesses larger than 1 nm, the magnitude of
the potential rapidly converges to that obtained with uniform charge distribution
(δ = 10 nm, green curves). Uniform charge distribution is assumed for calculations
throughout this chapter. Further arguments for assuming a uniform distribution of
charge include the fact that the area under the absorption spectrum of a dipolar plasmon mode should only scale with the total number of valence electrons in the metal,
irrespective of their location inside the particle.
2.6

Nanoparticle Emissivity

We used FTIR absorption spectroscopy to estimate the emissivity of the gold colloids in this study as ε ≈ 0.01 (the same order of magnitude as bulk gold), by
assuming reciprocity of the absorption and emission in the frequency band of this
measurement, between 2-5 µm. This suggests that the IR absorptivity and emis-

Figure 2.6: Calculated induced potential (a) and relative increase in absorption compared with a neutral particle (b) for a 20-nm-diameter Ag nanoparticle in vacuum
as a function of illumination wavelength for shell thicknesses of 1.0, 2.5, 5.0 and 10
nm (black, red, blue and green trace respectively). The illumination power density
is 10 mW/cm2 .
sivity of the gold colloids is comparable with bulk gold in the IR, as has been
reported elsewhere [45]. Figure 2.7 examines the effect of emissivity on the predicted increase in extinction due to the plasmoelectric effect under monochromatic
illumination versus white light illumination (see section 2.10). We find that the difference between the extinction increase for ε = 1 (which represents a perfect black
body) and ε = 0.01 (similar to bulk gold) is about one order of magnitude. In the
simulations shown in this chapter, we use our experimentally determined emissivity
of 0.01.
2.7

FDTD Simulation Methods

We performed full wave modeling using Lumerical finite-difference time-domain
(FDTD) [49] simulations to determine the scattering and absorption spectra for Au
particles on ITO. Optical constants for Au were, as described in section 2.1, taken
from Johnson and Christy [50], and we used the Brendel and Bohrmann model [17]
for calculation of the dielectric function. For ITO, optical constants were taken
from spectroscopic ellipsometry measurements of sputtered ITO films.
We modeled a 60-nm-diameter Au sphere on top of an ITO slab and a background
index of n = 1.4 (see section 2.3). The simulations employed Perfectly Matching Layer (PML) boundary conditions on all boundaries. We used a Total Field
Scattered Field (TFSF) source to launch and extinguish broadband plane waves (λ
=460-650 nm). The scattering and transmission due to the sphere were recorded

Figure 2.7: Simulated extinction increase compared with white light illumination when Au 60-nm-diameter particles in water are illuminated with 20 mW/cm2
monochromatic illumination for emissivity values, ε, ranging from 1 (perfect black
body) to 0.01 (∼bulk gold).
and used to calculate the absorption and extinction spectra.
2.8

Plasmoelectric Simulations

Before applying the model to an experimental geometry (e.g. see Figure 2.11),
we first calculated the plasmoelectric potential (i.e., the electrochemical potential
gained by the particle from the electron transfer induced by optical absorption) for
a spherical 20-nm-diameter Ag nanoparticle in vacuum under monochromatic illumination. For these particles, analytical Mie theory can be used to calculate the
absorption cross section spectrum, Cabs (λ, n), for a given electron density in the
nanoparticle by taking into account the dependence of the bulk plasma frequency,
ω p , in the complex dielectric function of the metal on carrier density, ω p ∝ n1/2
(as described in section 2.1). To calculate the nanoparticle temperature, we used a
steady-state heat-flow model in which heat is dissipated from the nanoparticle by radiation (see section 2.3). Figure 2.8a shows the calculated plasmoelectric potential
of the Ag nanoparticle as a function of illumination wavelength at an incident flux
of 1 mW/cm2 , under which the particle obtains a maximum temperature of ∼ 400K
(see section 2.3). The model predicts a clear negative surface potential below the
plasmon resonance and a positive one, up to 150 mV, above it. The asymmetry in
the plasmoelectric potential below and above the resonance wavelength is due to

Figure 2.8: A 20-nm-diameter Ag particle in vacuum is illuminated with monochromatic light (Iλ = 1 mW/cm2 ). (a) Plasmoelectric potential and (b) relative absorption increase as a function of incident wavelength.
the intrinsic nonresonant interband absorption in the metal. Figure 2.8b shows the
corresponding relative absorption increase for the Ag nanoparticle, which ranges
up to a factor of 2.5 × 10−5 .
The model for the simple geometry in Figure 2.8 describes the key factors in the
plasmoelectric effect: an increase in carrier density under illumination at wavelengths shorter than the resonance peak, inducing a negative plasmoelectric potential and enhanced absorption relative to the neutral particle. Similarly, radiation at
longer wavelengths induces a lower carrier density, a positive plasmoelectric potential, and enhanced absorption. These results demonstrate that an excited plasmonic
resonator behaves as a heat engine that can convert absorbed off-resonant optical
power into a static electrochemical potential.
Next, we used the model to calculate the wavelength-dependent and power-dependent
plasmoelectric potential for the experimental geometry in Figure 2.11a, a spherical
60-nm-diameter Au particle on an ITO/glass substrate. We calculated the factor
dT/dN in Equation (2.3) using FDTD simulations of the absorption spectra for
an Au particle on an ITO/glass substrate to take into account radiative damping
from the substrate not captured by simple Mie theory (see section 2.7)and a onedimensional model for heat conduction into the substrate (see section 2.3). Because of variations in interparticle coupling and clustering (see scanning electron
microscopy (SEM) image in Figure 2.11a) and possible adsorption of water onto
the particle-ITO interface, the ensemble dielectric environment is complex. The
broadened, redshifted scattering spectrum in Figure 2.11a is evidence for some particle aggregation. In the FDTD simulations, we modeled these effects by assuming

23
a background index of n = 1.4 for the medium above the ITO substrate, such that
the absorption spectrum matched the experimentally observed spectrum. As can be
seen in Figure 2.11c for the high-power data, the modeled trends correspond well
with the experimental trends: the modeled minimum potential occurs at 530 nm
(experimental: 500 nm), the modeled zero potential occurs at 545 nm (experimental: 560 nm), and a large positive potential is observed for wavelengths above the
resonance, both in model and experiment. The extent of the measured potential to
wavelengths up to 640 nm in Figure 2.11b is in agreement with the long-wavelength
tail in the spectrum of Figure 2.11a. At 555 nm, the modeled particle temperature
peaks at 308 K, 15 K above ambient, in good agreement with other experimental
observations [45].
2.9

Kelvin Probe Force Microscopy Experiments

To experimentally probe the plasmoelectric potential, we measured the surface electrostatic potential of films of 60-nm-diameter Au colloid nanoparticles deposited on
ITO under optical excitation while varying the wavelength of monochromatic illumination as shown in Figure 2.11. Kelvin probe force microscopy (KPFM) was
used to characterize the surface potential. Briefly, a conductive atomic force microscope (AFM) tip employed in non-contact mode determined the static potential
difference between the tip and sample surface, indicating the work function difference as well as any trapped charges or other induced potentials at that location.
This technique allowed us to measure plasmoelectric potentials induced on the Au
particle surfaces as a function of illumination wavelength.
Colloid Desposition
60-nm-diameter Au colloids (BBI International, EM.GC60 Batch #15269, OD1.2)
were used for KPFM measurements. Samples for these measurements were prepared on indium tin oxide (ITO) coated glass substrates (SPI brand, 30-60 Ω, 06430,
ITO layer thickness: 100 nm) that were first ultrasonicated overnight in a solution
containing an equal volume of acetone, methanol, toluene, and isopropyl alcohol
and then dried with nitrogen gas. Substrates were placed individually in glass scintillation vials (ITO-side up) with 600 µL Au colloid diluted by 1.2 mL of deionized
(DI) water; for control samples, DI water was used instead of the Au colloid solution. Then 60 µL 0.1 M HCl was added, and the vials were immediately centrifuged
at 2000 rpm (∼670 g-force) for 40 minutes. After centrifuging, the solution had become clear and the ITO film had a noticeable red color due to Au nanoparticles

Figure 2.9: KPFM surface potential map of three 60-nm-diameter Au nanoparticles
shows the work function offset of Au vs. ITO while the sample was in the dark.
deposited on the surface. Subsequently, the substrates were rinsed thoroughly with
DI water and heated to 290 C by placing a heat gun 5 cm above the sample surface for 20 minutes. The scattering spectra of the deposited samples (as in Figure
2.11a) were obtained using a Zeiss Axio Observer inverted microscope equipped
with a 20× dark-field objective, with illumination from a halogen lamp. Figure
2.11a shows the measured scattering spectrum for a sample, showing a clear plasmon resonance around λ = 550 nm.
KPFM Measurement Methods
To probe the local static potential difference between the tip and sample surface at
room temperature, we used an Asylum Research MFP-3D AFM in scanning Kelvin
probe microscopy mode with a n+ silicon conductive tip (Nanosensor ATEC-EFM20) to measure the surface potential of the ITO-coated glass substrate samples, with
the ITO film connected to ground [51]. Figure 2.9 shows that the technique gives
mV resolution of the work function difference between Au nanoparticles and ITO,
detecting the greater surface electron density of Au compared with ITO when the
sample was dark.
Focused optical excitation (∼ 900 µm2 spot size) adjacent to the AFM tip was
provided by a supercontinuum pulsed laser (40 MHz, Fianium SC400-4) that was
frequency-selected by an acousto-optic tunable filter (Fianium AOTF) with ∼15 nm
FWHM bandwidth. During illumination the surface potential in a region of bare

Figure 2.10: KPFM-measured surface potential from a control experiment of bare
ITO/glass under 1W/cm2 scanned monochromatic illumination, plotted on the same
scale as Figure 2.11b.
ITO substrate adjacent to the nanoparticle array was compared with the dark signal.
The sample was measured at the limit of highest available particle coverage, and
it was anticipated that the measured region of ITO/glass adjacent to the Au particles was equipotential with the particles. Direct illumination of the AFM tip was
avoided, and there was no dependence of the magnitude of signal based on the
distance from the optical spot, within ∼10’s of µm. A control measurement of
ITO/glass free of Au colloids shown in Fig 2.10 shows no detectable wavelengthdependent signal. The absolute surface potential determined by the KPFM technique is a convolution of the work function difference between the sample and the
particular tip used, and factors relating to the tip-sample geometry that define capacitance. Therefore, to emphasize wavelength-dependent changes of potential, the
reported data (Figure 2.11b) are plotted relative to the surface potential in the same
spot measured in the dark before optical excitation.
Measured Plasmoelectric Potentials
The illumination wavelength was scanned through the plasmon resonance spectrum,
from 480 to 650 nm, while we probed the potential of the illuminated Au nanoparticle array. A clear optically-induced surface potential was observed, which varied
with illumination wavelength (Figure 2.11b). We observed negative induced po-

Figure 2.11: Plasmoelectric effect on dense Au nanoparticle arrays on ITO/glass.
(a) Dark-field scattering spectrum of 60-nm-diameter Au nanoparticles on
ITO/glass. The inset shows a SEM image of the nanoparticle array. (b) KPFM
measurements of the surface potential as a function of illumination wavelength (15nm bandwidth) for three different illumination intensities. The surface potential of
a flat region of ITO/glass adjacent to the nanoparticle array was monitored during
scanned monochromatic illumination (see schematic geometry). A control measurement of an ITO/glass substrate without nanoparticles (1 W/cm2 ) is also plotted
(gray). (c) Modeled plasmoelectric potential for 60-nm-diameter Au nanoparticles
on ITO/glass for the three illumination intensities in (b).

27
tentials during excitation to the blue side of the neutral-particle plasmon resonance
wavelength near 550 nm, and positive potentials during excitation on the red side
of the resonance, with the measured potential changing sign near the peak of the
plasmon resonance. The magnitude of the signal increases with increasing optical
power density, except at the neutral particle resonance. These data provide evidence
for induced plasmoelectric potentials and are consistent with the modeled plasmoelectric response. Notably, these trends are not consistent with the thermoelectric
effect. The thermoelectric potential would be maximized at the plasmon resonance,
due to the maximal induced heat absorption at that frequency. By contrast, we observe no induced potentials at the plasmon resonance frequency. A thermoelectric
potential would also not be expected to change sign based on the wavelength of
illumination. Further, even a temperature increase in the particles several hundred
Kelvin greater than predicted by our model would induce thermoelectric potentials
of only a few hundred µV based on the Seebeck coefficient of bulk Au [47]. This
small voltage is below the limit of our measurement sensitivity. However, some offset in the data may be due to the thermoelectric response of the substrate. Control
samples of ITO with no Au particles show a weak constant positive potential, ∼ 1
mV, during wavelength scans at high optical intensity. Direct illumination of the
AFM tip was avoided during measurements.
For comparison, the modeled plasmoelectric response from an individual 60 nm Au
particle in this geometry, as described in section 2.8, is displayed (Figure 2.11c).
The deviation of the measured surface potential trend as compared to the modeled
curves may result from several factors. The KPFM measurement is a non-contact
technique, with the conductive AFM tip nominally 300 nm from the sample during
our experiments. Therefore the modeled curves (Figure 2.11c) assume the electrostatic potential from the particle decays inversely with distance, exhibiting the
potential profile of a charged sphere at the separation distance of the KPFM tip.
However, details of the local geometry will strongly determine the magnitude of
the measured signal. Further, if the nanoparticles reach a higher or lower temperature than our model anticipates, the plasmoelectric voltage will be proportionally
affected. The spectral broadening at longer wavelengths observed in the optical
signal (Figure 2.11a), which may be due to particle aggregation during deposition, will also reduce the plasmoelectric response, similar to the effect of particle size polydispersity in Figure 2.13. This contribution may also account for the
greater symmetry observed in the magnitude of the negative and positive potentials compared with the theoretical curves, as spectral broadening towards longer

28
wavelengths would more significantly decrease positive plasmoelectric potentials.
Circuit non-idealities, such as contact resistance between the Au and ITO, may also
contribute to a reduction in the plasmoelectric potential. In our studies it was necessary to anneal samples for more robust signal, presumably to remove residual
surface ligands from the Au colloid, reducing the contact resistance.
2.10

Optical Measurements of Nanoparticles Under Monochromatic Illumination

The plasmoelectic model predicts that when illuminated to either side of the plasmon resonance with monochromatic radiation, the charge density of a plasmonic
resonator in contact with a electron reservoir will change such that the absorption
cross section will shift towards the illumination wavelength. It follows that the spectra of Au colloids measured with scanned monochromatic light will show a broader
resonance peak than the resonance peak in a spectra measured with white light. We
performed optical extinction spectroscopy for 60-nm-diameter Au colloids in water to characterize this plasmoelectric response. The changes in the surface charge
density of the particles can result from induced polarization across the metal-water
interface.
Experimental Setup
To measure the extinction of 60-nm-diameter Au colloids (BBI International, EM.GC60
Batch #15269, OD1.2) in water under broadband (white) and monochromatic illumination, we used lock-in amplifiers (LIA) in combination with two photodiode
power meters and a monochromator. The measurements setup is shown in Figure
2.12. For broadband illumination, the cuvette with Au colloids was positioned in
front of the monochromator (sample position 1). For the monochromatic illumination conditions, the cuvette was placed in between the two photodiodes. The
extinction can easily be obtained by comparing the signal on both power meters for
each illumination wavelength and normalizing by spectra of a cuvette with DI water
placed in the same two positions as the sample.
Measured Spectral Dependence on Illumination Properties
We compare extinction measurements under white light and monochromatic excitation conditions. As noted above, scanned monochromatic illumination leads to
increased optical extinction when a plasmoelectric potential is manifest as compared with the extinction of neutral particles. In contrast, if a particle is excited

Figure 2.12: Schematic of white vs. monochromatic illumination experimental
setup
with broadband (‘white light’) illumination, so that the excitation power density
per unit spectral interval is equivalent both at higher and lower frequencies relative to the plasmon absorption maximum, then there is no thermodynamic driving
force for the plasmoelectric effect, so no spectral shift is expected and the particle
will remain neutral. Figure 2.13a shows the measured optical extinction spectra
of Au colloids during both white light (black curve) and scanned monochromatic
excitation (red curve), both with intensities of 20 mW/cm2 . A small, but clearly
distinguishable difference in extinction is observed on both sides of the spectrum.
Figure 2.13b shows the relative extinction change, derived for the data in Figure
2.13a; data for illumination at 0.2 and 2.0 mW/cm2 are also shown. A systematic
increase of the spectrally-integrated extinction is observed during monochromatic
illumination when compared with spectra obtained during white light illumination.
The observed wavelength-dependence is in good agreement with the modeled plasmoelectric response trends shown in Figure 2.13c (see section 2.10). Specifically,
the minimum increase of extinction is correlated with the extinction maximum at
the plasmon resonance near 530 nm. Moreover, the asymmetric absorption increase
when comparing each side of the extinction maximum, which is characteristic of
the non-plasmonic interband absorption in Au, as described above, is clearly reproduced in the measurement, providing further evidence of induced plasmoelectric
potentials during scanned monochromatic illumination.

Figure 2.13: Plasmoelectric effect for 60 nm Au colloids suspended in water: (a)
Measured extinction spectra for broadband (black curve) and monochromatic (red
curve) illumination (both 20 mW/cm2 ). (b) Relative increase in extinction for
monochromatic illumination derived from the data in (a). Data for 0.2 and 2.0
mW/cm2 are also shown. (c) Modeled increase in extinction for same fluxes as in
(b).

31
Modeled Optical Plasmoelectric Response
To model the increase in extinction during monochromatic illumination in a water environment, we calculate the thermal energy of the system in diffusion-limited
conditions (i.e. particles are in thermal equilibrium with the water) that correspond
with the experimental procedure. Assuming that the water in the cuvette is a thermally isolated system, all the power absorbed by the nanoparticle suspension will
heat the water (thermal radiation is negligible), and the system will increase its temperature. Using the laser power (∼ 1 mW max), scan speed (1 nm/s), the extinction
(measured, see Figure 2.13a), the fraction of extinction that goes into absorption
(known from Mie theory), the heat capacity of water, and the volume of water in
the cuvette, we calculate that the temperature increase during monochromatic illumination is ∼1 mK per second at the highest illumination intensity in our study. We
then calculate the thermal energy of a volume of water heated per particle under
illumination. Given the particle concentration of the suspension (2.6 × 1010 mL−1 ),
this corresponds to a maximum ∼ 1.6 × 10−13 J of thermal energy provided per
particle. To solve the governing relation Equation (2.7), the change of this thermal energy per change of surface charge density on the particle (due to changes of
absorption) is equated to the change of electrochemical potential. Full wave simulations (FDTD method) were used to calculate the extinction based on the charge
density that satisfied Equation (2.7) at each incident wavelength and power, and is
depicted in Figure 2.13c.
Our calculations also account for the influence of the colloid polydispersity on the
plasmoelectric response, assuming an ensemble of particles with a size distribution that is typical for high-quality Au colloids used in this study (8% coefficient of
variation). Relative to a monodisperse colloid or individual particles, polydispersity
increases the absorption at all incident wavelengths. For excitation at the ensemble
absorption maximum, some fraction of particles larger or smaller than the average size can change surface charge density to enhance absorption. We note that
the sub-linear increase of the experimentally measured changes in extinction with
incident power (Figure 2.13b) as compared to the modeled linear increase with incident power (Figure 2.13c) suggests an intensity-dependent dissipation mechanism
not accounted for by our simple thermal analysis, despite otherwise good agreement with the modeled trend and relative magnitude of the wavelength-dependent
response.

32
2.11

Conclusions

To aid interpretation of our findings, we comment briefly on other mechanisms for
generating photopotentials in metals. A thermoelectric effect is several orders of
magnitude weaker (∼ µV/K) than the observed potentials as discussed above [47].
Hot carrier-induced effects would require rectifying contacts, which are not present
in our geometry. Moreover, both of these effects would not result in a bisignated
signal [18]. “Plasmon drag” or similar direct photon-to-electron momentum transfer mechanisms on Au colloids would not produce a bisignated signal [52]. Future
work in this area will benefit from further insight into the microscopic mechanisms
that contribute to the observed effect.
The observed plasmoelectric phenomenon takes advantage of the remarkable spectral tailorability of plasmonic nanostructures and can be extended to a variety of
material systems, absorber geometries, and radiation environments. Plasmoelectric
devices may enable the development of entirely new types of all-metal optoelectronic devices that can convert light into electrical energy by replacing the usual
function of doped semiconductors with metal nanostructures that are optically excited off-resonance.

33
Chapter 3

ELECTRICAL TUNING OF THE DIELECTRIC FUNCTION
AND OPTICAL PROPERTIES OF NANOPARTICLES
3.1

Previous Work and Overview

The plasmonic response of metal nanoparticles has generated great scientific interest, motivating both fundamental investigations and exploration of a variety of applications, such as photovoltaic cells [13], photocatalytic fuel cells [30, 53] surface
enhanced Raman Spectroscopy (SERS), [54–57] cell labeling, and molecular sensing [27, 28, 56, 58–60]. These complex environments modify the plasmonic behavior in several ways due to the material dynamic response under electrochemical potential changes, by the equilibration of the particle with the surrounding Fermi-level
offset, and the effects of chemical reactivity, solvent polarizability, and interface
damping [29, 61–67]. In order to achieve precise manipulation of the properties of
metallic resonators in tunable plasmonic systems, we are fundamentally interested
in how changes of charge density and the properties of the surrounding environment, such as refractive index and electronic surface states, affect the plasmonic
absorption. The Drude model can be used as a first approximation to predict how a
change in charge density produced by an applied bias tunes the plasmonic resonance
[27, 30, 60, 68]. However, it is well known that plasmon resonances are dependent
on their environment, [13, 26, 54, 56, 57, 64, 69] and a simple Drude model omits a
description of property changes other than a uniform change in charge density. The
Drude model does not take into account other effects that happen in parallel with
changes in charge density in a real system, such as changes in electronic surface
states due to chemical interactions with the surrounding medium and the resultant
changes in damping and index of refraction (e.g. adsorbates, oxidation, electric
double layer) [27, 28, 56, 59, 60, 64, 67, 70]. Furthermore, using a Drude model
to represent a uniform change in charge density in a plasmonic structure does not
account for the excess charge residing at the conductor surface. Thus, in order to
predict optoelectronic behavior of plasmonic systems, we desire a model that will
account not only for changes in charge density, but also for changes in damping,
index of refraction, and penetration depth of surface charge and damping. This will
allow us to assess the relative spectral contributions of each effect, and how they
act in concert in an electrically tuned plasmonic system. Better quantification of

34
the various contributions that shape plasmonic resonances is crucial for current research in tunable plasmonics, and more broadly, for any system in which plasmonic
elements will be used an a non-neutral state, e.g. as catalysts or electrical contacts.
Previous experiments have shown that externally biased Au nanoparticles (AuNPs)
are not adequately described by the Drude model [17, 26, 27, 60, 64, 68, 71]. A
potential-dependent modified dielectric function and T-matrix-based fitting routine
has been proposed to analyze interface damping and uniform charging effects for Au
nanorods in an electrochemical cell [68]. This work provided a systematic model
for bias-dependent extinction, but did not explicitly address index changes in the
surrounding media or non-uniform distribution of charge at the particle surface [68,
72]. Separately, the effects of various surface layers on the dielectric response of
Au nanoparticles in an electrochemical cell were investigated, in comparison with
modeling that relied on analytical Mie Theory, though analysis of broadening effects and the role of the refractive index of the substrate were not considered [24,
27, 72]. These researchers found evidence of increased damping at a positive applied bias, which they attributed to a lossy layer at the particle surface. Previous
research has not analyzed the effects of changes in the surrounding dielectric environment, changes in damping, and concentration of excess charge at the particle
surface all occurring in unison. In this Chapter, we present mechanisms for biasdependent optical extinction of Au nanoparticles, using full wave electromagnetic
simulations in conjunction with experimental optical spectra to characterize the response of arrays of colloidal Au nanoparticles immobilized on indium tin oxide
(ITO) substrates in an electrochemical cell. Through this combined simulation and
experimental approach, we account for the influence of refractive index due to the
substrate, index changes in the near-surface environment of the nanoparticle, and
a variable-thickness shell of modified damping and charge density at the surface
of the gold nanoparticle (AuNP). We compare the simulations with experimental
results to quantitatively analyze the contributions from these effects.
3.2

Electrochemical Cell Fabrication

ITO coated glass substrates (SPI brand, 30-60 Ω, 06430) were used as the top
and bottom electrodes in the electrochemical cell (Figure 3.1). The substrates
were cleaned by ultrasonicating overnight in a solution containing equal volume
of acetone, methanol, toluene, and isopropyl alcohol and then dried with nitrogen gas. Parafilm wax was used to cover one half of the bottom electrode ITO
slide to prevent Au colloid deposition in the subsequent step, thereby reserving

Figure 3.1: Geometry of electrochemical cell with 60-nm-diameter AuNPs on the
bottom electrode. The top electrode is grounded, and potentials are applied to the
bottom electrode. An optical beam passes through the cell for spectral measurements
half of the bottom electrode for use as a control in the normalization of optical
spectra; this allowed us to carefully control for a possible optical response from
the ITO substrate as a function of applied bias. The bottom electrode was placed
in a glass scintillation vial (ITO-side up) with 300 µL of 60-nm-diameter Au colloids in water (BBI International, EM.GC60 Batch #16516 OD1.2) and 1.5 mL
of deionized (DI) water. 60 µL 0.1 M HCl was added to the vial, and the vial
was immediately centrifuged at 2000 rpm (∼670 g-force) for 40 minutes. After
centrifuging, the Au colloid solution had become clear and the bottom ITO electrode has a noticeable red color due to Au nanoparticles deposited on the surface.
The Parafilm wax was removed from the bottom electrode and the electrode was
rinsed thoroughly with DI water then soaked in toluene for one hour to remove
residue from the Parafilm wax. Both electrodes were plasma etched with a direct plasma at 110 W and 300 mTorr O2 for 20 minutes to remove ligands and
any organic mater from the AuNPs and ITO surface. Substrates were then vacuum annealed at 350 C for 20 minutes. The two electrodes were mounted faceto-face with two pieces of Teflon tape used as a spacer between the slides at the
slide edges. Under inert atmosphere, Diethylmethyl(2- methoxyethyl)ammonium
bis(trifluoromethylsulfonyl)imide (DEME) electrolyte (Sigma Aldrich, 727679) was
used to fill the space between the slides and the cell edges were sealed with 5minute-setting epoxy to prevent water moisture and oxygen from entering the cell.

36
3.3

Optical Measurements of Nanoparticles Under Applied Bias

Experimental Setup and Procedures
In our experiment, the spectra of an ensemble of 60-nm-diameter Au colloids in
an electrochemical cell were recorded as a function of applied bias. The applied
bias was swept from 0V to 2.25V and then to -2.25V and back to 0V in 0.25V
steps. The top electrode was grounded and potentials were applied to the bottom
electrode using a DC voltage source (Tektronix PS282).
The extinction spectra of an array of AuNPs in the electrochemical cell were obtained using a spectral response and lock-in amplifier technique. The beam from a
supercontinuum pulsed laser (20 MHz, Fianium SC400-2) was chopped at 100 Hz
and directed into a monochromator (Oriel 777000) that was optically in-series with
two Si photodiodes (the first photodiode was used as a reference). The spectral resolution of the monochromator was approximately 1.5 nm. The photodiode signals
were passed through transimpedance amplifiers (DL Instruments 564) and were detected by lock-in amplifiers (SR830 DSP). Spectra with the optical beam passing
through the portions of the bottom electrode with and without Au colloids were
collected at each applied bias step. The spectra taken in the portion of the bottom
electrode without Au colloids were used as normalization spectra. Approximately
20 minutes elapsed between each voltage step and the corresponding spectral measurement.
Optical Measurement Results
Figure 3.2(a) shows the extinction spectra for AuNPs in the electrochemical cell at
selected values of applied bias, which were smoothed with a Savitsky-Golay filter.
A red-shift of the peak is observed at positive applied bias and a blue-shift and increase in peak height is observed under negative applied bias. The shift of the peak
is more evident in the extinction change spectra ( Figure 3.2b ) where a red-shift
is manifest as a positive change in extinction to the red of the peak and a negative
change in extinction to the blue of the peak, and vice versa for a blue-shift. Figure
3.3 shows the change of the spectral properties as a function of applied bias. Over
the range of applied potentials, the wavelength of peak extinction shifted by 3nm
and the full width half maximum of the peak varied by 4nm ( Figure 3.3 a & c ). The
total peak extinction change was 7% of the initial 0V peak extinction ( Figure 3.3b
). The hysteresis in the experimental results indicated some nonreversible process
and that this electrochemical system did not exhibit ideal reversible charging. Additionally, we found that at negative applied potentials, the peak height increased.

Figure 3.2: (a) Extinction versus wavelength at selected applied potentials as indicated by line color. (b) Extinction change versus wavelength at the same applied
potentials in (a).
This is in contrast to what we would expect based on the Drude model and Mie theory [17, 24, 26], which predict a peak height change that decreases monotonically
as a function of increasing electron density. This suggests that, at the very least,
there was significant damping in the experimental system at positive potentials that
is not captured by a Drude and Mie theory model.
Raw Spectra and Noise Level
Below we provide spectra of the particles in the electrochemical cell with no smoothing (Figure 3.4). These spectra have been normalized by the spectra taken in the
part of the electrochemical cell with no particles to account for contributions from
the substrate or other scatterers in the system. Part (a) shows the spectra over the
same wavelength range as that in Figure 3.2. Part (b) shows the spectra over the
entire measured wavelength range. As can been seen in the figures, there is very
little noise. The signal to noise ratio was roughly 400:1.

Figure 3.3: Experimental results are displayed in blue; simulation results are displayed in red. (a) Peak position (b) peak extinction, and (c) full with half max
change relative to the first 0V applied bias point versus applied bias. For a-c, the
markers for the applied bias points for the middle portion of the cycle, from 2V
to -2.25V in -0.25V steps are hollow. (d) Experimental extinction spectra at 0V,
+2.25V and -2.25V and their best-fit simulated spectra.

Figure 3.4: Normalized raw spectra at the extremes of the applied bias cycle (2.25V
and -2.25V) as well as the first 0V point ( 0V Start) and the 0V point in the middle
of the applied bias cycle (0V Middle).

Figure 3.5: Full wave FDTD simulation, x-component of the electric field in a ynormal cross-section. The 60-nm-diameter particle is modeled by a neutral core
defined by the BB model and a shell with a modified dielectric that allows for
variations in damping and charge density at the surface. The substrate is defined
by measured n and k data and the electrolyte is modeled as a uniform dielectric
above the substrate. The varied simulation parameters are shell charge density,
shell damping, shell thickness, and the refractive index of the electrolyte. (a) Uses
a linear color scale to show the dominant dipole field, and (b) uses a log color scale
to show fields extending into the particle and ITO substrate.

40
3.4

Modeling Optoelectronic Effects with FDTD Simulations

Three-dimensional full wave electromagnetic simulations were performed using finite difference time domain methods to model and analyze the experimental system.
The simulation geometry consisted of a 60-nm-diameter Au nanosphere on an ITO
substrate and in a surrounding medium with uniform index of refraction ( Figure
3.5 ). The ITO substrate was defined by the real and imaginary part of the complex
index of refraction, measured with ellipsometry. The AuNP was simulated as a core
defined by the Brendel and Bormann (BB) [56, 57, 72, 73] dielectric function with
the additional feature of a variable-thickness shell with a modified dielectric function. The BB model uses a superposition of an infinite number of oscillators (termed
a BB oscillator) to replace the single Lorentz oscillator used in the Lorentz-Drude
model [24, 60, 72, 74]. The BB dielectric function is defined as:
ω2p

BB (ω) =1 −

ω(ω − iΓo ) +
ωp =

(3.1a)
χ j (ω)

j=1

no e2
o me

(3.1b)

where Γo is the damping constant of bulk gold, χ j (ω) is a BB oscillator, ω p is the
plasma frequency in the Drude model, no is the electron density of bulk gold, e is
the elementary charge, and me is the effective electron mass. Six BB oscillators
were used in our parameterization. We obtain an analytic function that satisfies
the Kramers-Kronig reciprocity relations for real and imaginary components of the
dielectric function. This model has been shown to accurately model the optical
properties of gold in the wavelength range relevant to this study [24, 28, 75–78].
Voltage-dependent Dielectric Function
A modified dielectric function at the surface of the particle was used to model
changes in the electronic states and population near the surface:
ω p,v (s) 2

shell (ω, s) =1 −

ω (ω − i (Γo + ΓV (s))) +
ω p,V (s) =

n(s)e2
o me

(3.2a)
χ j (ω)

j=1

(3.2b)

41
Here, ω p,V (s) and n(s) are state (s) dependent variables, where the state describes
the applied bias (V) applied to the cell and the microscopic state of the system, and
these state-dependent variables have replaced the respective constant values used
in the BB model. An additional, state-dependent damping term ΓV (s) was added
to the bulk damping to allow for changes for damping at the particle surface as a
function of the state of the system (largely the applied bias).
Parameter Sweeps and FDTD Simulations
The independent parameters that were varied in our analysis are: the index of surrounding electrolyte, and ΓV and n(s) in the shell dielectric function, as well as
the thickness of this shell. Additionally, to account for polydispersity in the size
of particles (8% coefficient of variation as reported by the supplier), we performed
electromagnetic simulations for particles with 28, 30, and 32 nm radii and took a
weighted average of the resulting spectra to produce simulated extinction spectra
for polydisperse samples for each parameter set (surrounding-index, ΓV , n(s), and
shell-thickness). We performed calculations where each of these parameters was
varied independently. For each unique parameter set, a full-wave electromagnetic
simulation was performed with Lumerical FDTD Solutions software to simulate
the extinction spectra corresponding to that parameter set. This approach allowed
us to calculate a set of predicted extinction spectra from the effect of changes in
index, interface damping changes, and surface charging occurring in parallel. We
began with broad parameter sweeps where the values for parameters found in current literature describing damping, surface effects, charging, or electric double layer
were well within the sweep limits [24, 27, 28, 71, 75]. The limits for the parameter
sweeps are shown in Table 3.1.
Next, we preformed an iterative process for finding the values of parameters that
produced simulated extinction spectra with smallest root mean square error (RMSE)
in comparison to the Savitsky-Golay smoothed experimental extinction spectra. After each parameter sweep, new parameter sweeps with finer resolution of the parameter space around the parameter values that had produced the lowest error were
preformed. This iterative processes was preformed until an acceptable resolution
(shown in the Table 3.1) was reached for all parameters and at this point, the parameter set with the smallest RMSE overall was selected as the champion set. Using
this iterative process, we determined the parameter values that most closely reproduced the experimental spectra at each applied bias and achieved RMSE errors below 1%. This process was carried out for each applied bias and thus resulted in one

42
champion set of parameters for each applied bias step. The champion parameters
values for these simulations then represented the identifiable physical properties
and changes in the system as determined by our model. These parameters are those
shown in Figure 3.6.
Physical Interpretation of Simulation Parameters
We believe the model described above represents the most comprehensive analysis
to date of the microscopic mechanisms that can contribute to bias-dependent optical extinction for metal nanoparticles. We now discuss the physical basis for the
observed variation of nanoparticle parameters. Because gold oxide has a larger dielectric constant than the electrolyte solution, oxidation of the AuNP surface will
cause changes of the index of refraction at the surface of the metal nanoparticle
relative to an unoxidized Au particle [28, 56, 57, 73, 79]. Further, the refractive
index of the surrounding medium could also change due to alignment of dipoles in
the electrolyte, as in an electric double layer [60, 64, 67, 70, 71, 74]. Damping may
also be modified by an applied bias because of variations of population or depopulation of electronic surface states at the AuNP surface [28, 75–78]. For example,
an applied bias could facilitate enhanced chemical reactivity and adsorption at the
particle surface and electrons may become trapped in empty adsorbate states, causing an increase in damping[28, 79]. Alternatively, an increase in electron density
due to a negative applied bias could result in electron spill-out from the surface and
repulsion of solvent molecules from the AuNP surface [53, 80, 81]. This could conceivably give rise to a decreased damping due to a decrease in chemical reactivity
and trap state occupation at the particle surface. It is also reasonable to assume that
the applied bias will alter the total number of electrons in the AuNP and thereby
alter the plasma frequency in accordance with the Drude model [29, 65]. In the
electrostatic limit, mobile charges will rearrange to minimize the electric fields in
the bulk of a conducting particle, so we assume that excess electrons or holes reside
near the particle surface. Finally, the modified electronic states and electron density
Table 3.1: Simulation parameter limits and resolution
Parameter
Surrounding Index
Shell Charge (n/n Au %)
Shell Thickness (nm)
Damping (rad/sec)

Lower Limit
1.0
0.95

Upper Limit
1.7
1.05
3e14

Finest Resolution
0.001
0.005
0.25
2.5e11

43
at the surface may have a variable penetration depth that depends on the applied
bias because of the aforementioned electron spill-out, adsorbate states, and the optical versus the static skin depth of excess charge residing near the surface. We
account for changes in the penetration depth of altered electronic states by varying
the thickness of the modified dielectric shell in our simulations.
3.5

Comparison of Simulation and Experimental Results

By varying the charging, damping, shell thickness, and electrolyte index independently of one another in our parametric calculations and by computing their net
influence on the extinction spectra, we were able to closely model the experimental
changes in the extinction as a function of applied bias with full wave electromagnetic simulations. Figure 3.3 shows a comparison of the spectral properties of the
simulated and experimental spectra. We see that the best-fit simulations closely
track the peak position, height and changes in width. Furthermore, in Figure 3.3d,
we show the experimental and best-fit simulated spectra at the extrema of the applied bias range (2.25V and -2.25V) as well as at the first 0V applied bias point,
with very good agreement between the curves. We note that the absolute FWHM is
sensitive to peak broadening due to particle size polydisparity and other ensemble
effects[29, 71] that are not well captured by our FDTD simulations, though changes
of FWHM (Figure 3.3c) are primarily due to changes in damping, and therefore are
expected to be tracked well by our analysis method. From Figure 3.6a we note that
the maximum RMSE of the fits of the simulated and experimental spectra was less
than 0.8% of the experimental 0V peak extinction.
The results from performing simulations to obtain the best-fit simulation for each
experimental spectra provides a parameter set for the model that is appropriate at
each applied bias step. These results are shown in Figures 3.6b-e. The height
of the colored band in the plots represents the uncertainty due to the finite step
size used in the parameter sweeps. The color of the band goes from red to blue
in time to make the time direction and hysteresis of the traces more clear. Our
analysis indicates that at the initial 0V data point, before any external voltage has
been applied to the electrochemical cell, there is an elevated electron density of
1% at the surface of the particle compared to neutral bulk Au (Figure 3.6b). This
may be due to Fermi level offsets between the ITO, electrolyte, and AuNPs or due
to interactions with the electrolyte at the surface of the particles. Note that we
do not claim a uniform increase of electron density throughout the particle, but
only a 1% increase in the surface shell region of the metal particle. As the applied

Figure 3.6: Simulation parameters that produced the spectra best-fit to the experimental spectra. (a) Root mean square error of the simulated spectra fit to the experimental spectra, normalized as a percentage of the initial 0V peak extinction.
(b) Charge (electron) density and (c) total damping in the modified-dielectric particle shell versus applied bias. (d) Thickness of the modified-dielectric particle shell
versus applied bias. (e) Refractive index of the surrounding medium above the ITO
substrate versus applied bias. For parts b-e the height of the colored band represents
the uncertainty due to the finite step size used in the parameter sweeps. For (c), the
step size for Γ is smaller than the thickness of the black line. The color of the band
goes from red to blue in time.

45
bias becomes more positive, the electron density decreases, and then increases as
the applied bias becomes more negative, as expected; however, there is significant
hysteresis which is not consistent with reversible, ideal charging. The overall shell
charge density range for the applied potentials used was within 3% of the neutral
charge density. The hysteresis upon charging/discharging is strongest in the first
half of the applied bias cycle (from 0 V to 2.25V and back to 0V). This is a common
feature in cyclic voltammetry measurements and may be due to initial chemical
reactions or “settling” of the system during this initial application of an electric
field.
The results for the damping parameter exhibit an overall trend of increased damping
at positive biases and decreased damping at negative biases ( Figure 3.6c ). The
damping coefficient used in the BB model to best describe neutral, bulk Au metal
is Γo = 7.596 × 1013 rad/sec [67]. At the first 0V data point, we found the damping
constant was 1 × 1013 rad/sec less than this value, indicating that the damping of
plasmons on the AuNPs in this system is less than in bulk gold. Recalling that this
data point corresponds to a 1% elevation of electron density in the particle shell,
we speculate that the decreased damping could be due to electron spill-out and a
resulting interlayer where the solvent is repelled and electron trapping in surface or
adsorbate states is decreased, resulting in an electron configuration that experiences
less damping than in the bulk. At +2.25V, we found ΓV = 3 × 1013 rad/sec. The
increase in damping at positive applied voltages is consistent with prior published
work [27, 28, 56, 58], and may be due to excitation of sp-electrons in adsorbate
states at the surface of the particle. Again, at negative voltages, as in the first 0V
case, we found that the damping was lower than for neutral, bulk Au metal, (at
-2.25V, ΓV = −6.6 × 1013 rad/sec) and we similarly attribute this to decreased
reactivity and electron spill-out at the particle surface.
The thickness of the modified-dielectric particle shell of the best-fit simulations is
shown as a function of applied bias in Figure 3.6d. We found that the shell thickness
varies between 1-3nm. This thickness is greater than the electrostatic skin depth of
bulk metal (i.e. the Fermi screening length is ∼0.3 nm for Au) and less than the
optical skin depth of the nanoparticle (∼10 nm), as determined from full wave simulations. Our simulations suggest that the shell thickness increases during the first
rise in applied bias, and then continues to increase more gradually at negative applied biases. The initial increase in shell thickness may be due to reactivity with
the electrolyte as the system “settles” under the initial application of voltage. The

46
electronic structure at the surface changes significantly during this period, enough
for the effects on the dielectric constant to penetrate a few nanometers into the surface of the particle. It is also reasonable to suggest that the shell thickness increases
at negative applied biases because the associated increase in electron density at
the surface might result in the zone with a modified dielectric function protruding
deeper into the particle. We believe that the thickness of the metallic shell with a
modified dielectric function corresponds to the region of the particle whose electronic structure is modified due to adsorbates, surface states and an excess or deficit
of electrons at the surface.
Finally, Figure 3.6e shows the results for the index of the surrounding medium
as a function of applied bias. We found less than a 1% change in the refractive
index of the surrounding medium. There is an overall trend indicating a slight
increase in index as the applied bias goes from positive to negative. This small
change could be due to a difference in the index of refraction of the cation and
anion of the DEME electrolyte and alignment of these dipoles in an electric double
layer [70, 71]. Significantly, we did not see an increase in the index at positive
voltages as would be expected in the case of oxidation of the AuNP surface in to
Au x O x . We note that the formation of Au halide surface compounds, induced by
chemical reaction with impurities in the DEME electrolyte (<1% by supplier assay)
for example, is similarly inconstant with our observations, as Au halides are also
expected to exhibit a larger refractive index.
Comparison of Simulation and Experiment Over Full Wavelength Range
Figure 3.7 shows the best-fit simulation and experimental spectra for the same voltage points shown in Figure 3.3d (the 0V curves are for the first 0V applied bias point
at the start of the voltage cycle). It is clear from part (a) of Figure 3.7 that the simulations do not reproduce the broadness of the experimental spectra. We attribute
the amplified broadening of the experimental spectra to non-idealities present in the
experimental system that were not captured in the simulations. These non-idealities
include roughness of the ITO substrate, particle-particle interactions, and heterogeneity of the particles beyond the small changes in radius that are accounted for
by the simulations. Part (b) of Figure 3.7 shows the absolute full width at half max
for the experimental spectra and the best-fit simulated spectra. The changes in this
parameter, relative to the first 0V applied bias point are shown in Figure 3.3c. There
we show that the general trends of the changes in experimental peak broadness as
a function of voltage are reproduced relatively well by the best-fit simulation peak

Figure 3.7: Experimental extinction spectra at 0V, +2.25V and -2.25V and their
best fit simulated spectra. (b) Absolute full width half max for experimental (blue)
and simulated (red) spectra. The markers for the applied bias points for the middle
portion of the cycle, from 2V to -2.25V in -0.25V steps are hollow.
broadness changes. However, due to the non-idealities discussed above, the full
width at half max is offset between the experimental and simulated spectra by approximately 6nm throughout the applied bias cycle.
3.6

Conclusions

In conclusion, we have shown that by using full wave electromagnetic simulations
to model a AuNP with a variable-thickness conducting shell with a modified dielectric function, a variable electrolyte index, and an explicitly modeled ITO substrate,
we are able to find good fits to experimental extinction spectra and track changes
in the spectra as a function of applied bias. The best-fit simulation spectra correspond to a set of simulation parameters as a function of applied bias, and provide
insight into the physical phenomena occurring in the experimental system under
bias. Using this analysis, we have modeled the changes in the surface charge density, surface damping, and penetration depth of the resultant modified dielectric
function, as well as changes in the index of refraction of the surrounding electrolyte
medium. Our approach allowed us to vary these parameters independently, but also

48
to understand the result of the effects acting in parallel. We find that the changes
in surface damping and charge density play the largest role in modifying the optical response of AuNPs under applied bias, with a smaller dependence on changes
induced in the surrounding electrolyte. Based on our analysis we can relate the applied bias to changes in charge density in the AuNPs; the most non-neutral charge
state of the shell, a decrease of electron density by 2% compared to bulk, occurred
at +2.25V applied bias and corresponds to roughly 12,000 holes in the particle shell
(calculated assuming a 2% change of charge density in a 3nm-thick shell of a 60nm-diameter Au nanosphere). This analysis is useful as a guide to understanding
optical properties of plasmonic nanostructures in non-neutral states or for which
the surrounding electrochemical environment is dynamically modified, e.g. due to
changes of solvent, ion concentration or photochemical changes. Thus the results
presented here allow us to assess the relative importance of interface damping, surface charging, and index changes on optical extinction of plasmonic nanostructures
spanning a large range of conditions relevant for chemical and biological applications.

49
Chapter 4

COMPLETE AB INITIO DESCRIPTION OF NON-RADIATIVE
PLASMON DECAY
4.1

Motivation and Previous Work

Illumination of a metallic structure produces strong optical near fields that initiate
a cascade of processes with multiple outcomes, including the excitation of surface
plasmons, their radiative decay to photons, and their non-radiative decay in the material.[82]. Non-radiative plasmon decay includes the generation of electron-hole
pairs. The energies of these electrons and holes depend on the material and the plasmon energy, and are considered ‘hot’ when they are significantly larger than those
of thermal excitations at ambient temperatures. Decay of surface plasmons to hot
carriers is a new direction that has recently attracted considerable interest in many
applications of plasmonics. The decay of plasmons that determines generated carrier energy distributions and the subsequent scattering and transport of these carriers
are both essential to the design of plasmonic hot carrier devices. Typically, scattering events thermalize the carriers and bring their energies closer to the Fermi level of
the metal. However, plasmonic hot carrier applications require carriers far from the
Fermi level to more efficiently drive both solid state and chemical processes. Hot
carrier ejection into semiconductor and molecular systems has been clearly demonstrated in several recent device applications ranging from energy conversion and
photocatalysis to photodetection. In particular, demonstrations of photochemistry
driven by both hot electrons[83, 84] and hot holes[85] raise interesting questions
regarding the timescales of plasmonic hot carrier generation and transport.[86, 87]
Yet a theoretical understanding of plasmon decay and the underlying microscopic
mechanisms has previously been incomplete.
Losses in metals can proceed either through classical resistive dissipation or singleparticle excitations. For plasmons, these single-particle excitations constitute Landau damping that results in generating highly energetic carriers. Direct optical
excitation of carriers in most metals is allowed only above an interband threshold energy due to crystal momentum conservation. Below this threshold, which
typically corresponds to optical frequencies, phonons provide the necessary momentum to circumvent this selection rule. Additionally, in metals, confinement of

Figure 4.1: (a) Schematic for excitation and decay of surface plasmons. Surface
plasmons excited, for example, through coupling to a grating or prism subsequently
decay via direct and phonon-assisted transitions to generate hot electrons and holes.
(b) Illustrations of direct, surface-assisted and phonon-assisted transitions on the
band structure of gold. Surface-assisted transitions constitute the small but nonzero probability of non-vertical transitions due to the momentum distribution of the
plasmon. The intermediate virtual state (empty circle) requires a sum over states
(filled circles) in perturbation theory. When the intermediate state is a real state
on the band structure (goes ‘on shell’), it corresponds to a sequential process of
electron-phonon scattering followed by a direct transition (or vice versa).
fields to the surface breaks translational invariance that can also provide the momentum necessary to excite intraband transitions.[88] These ‘surface-assisted’ and
phonon-assisted transitions are important contributors to losses in metals at infrared
frequencies, and hence are important to understand from both a fundamental and
technological perspective.[89–92]
First principles calculations provide an opportunity to quantitatively analyze individually each microscopic mechanism underlying plasmon decay (Figure 4.1(a))
and gauge their relative contributions in different materials and at different frequencies. These calculations can examine the processes at various time scales, separating effects due to the initial distribution of hot carriers and its subsequent transport.
Such a detailed understanding, which is extremely challenging to extract from experiment, elucidates opportunities to enhance plasmonic hot carrier devices as well
as their fundamental limits.
Previously, Sundararaman et al. studied in detail direct interband transitions in plasmonic metals [33] and showed that the plasmon-generated hot carrier distribution
is extremely sensitive to details of the electronic band structure. Specifically, they
found that in noble metals the positions of the d-bands relative to the Fermi level
result in much hotter holes than electrons, and subsequent studies confirmed these

51
results.[93] Sundararaman et al. also showed that the decay of surface plasmon
polaritons is representative of decays in plasmonic nanostructures and that geometry effects on the generation of carriers are significant only at dimensions below
∼10 nm.
In this Chapter, we report first-principles calculations that describe all of the significant microscopic mechanisms underlying surface plasmon decay and hot carrier
generation. The work presented here completes the theoretical picture of surface
plasmon decay by adding ab initio calculations of phonon-assisted transitions and
resistive losses, allowing us to quantitatively predict the initial excited carrier distributions resulting from decay. Within a quasiparticle picture we include electronelectron and electron-lattice interactions in the quasiparticle energies as a part of
the underlying electronic structure calculation. We also calculate electron-electron
and electron-phonon scattering contributions to the quasiparticle linewidth, which
determines carrier lifetimes and transport. We calculate all of these processes that
have significant contributions and dominate in the relevant energy ranges and length
scales, but ignore higher order processes such as decays involving multiple electronhole pairs or multiple phonons, as these do not dominate in any regime. Previous
first principles calculations of phonon-assisted transitions treat indirect-gap semiconductors below their optical gap.[94, 95] We report the first ab initio calculations
of phonon-assisted optical excitations in metals, which are critical to bridging the
frequency range between resistive losses at low frequencies and direct interband
transitions at high frequencies. In this extension of calculations to metals, we show
that it is necessary to treat carefully the energy-conserving ‘on-shell’ intermediate
states, that correspond to sequential processes (Figure 4.1(b)). We predict the relative contributions of these processes and direct transitions and compare the absolute
decay rates to those estimated from experimentally-measured complex dielectric
functions for frequencies ranging from infrared to ultraviolet.
4.2

Experimental Decay Rate

In order to compare various contributions to surface plasmon decay with experiment
on equal footing, we calculate contributions to the imaginary part of the dielectric
tensor Im¯ (ω) and relate the complex dielectric function to the plasmon decay rate.
The decay rate per unit volume is the energy loss per unit volume (see for example
equation 4.26 in Ref.96) divided by the photon energy. Specifically, the decay
1 ~∗
~ r ) at a point in the material where
E (~r ) · Im¯ (ω) · E(~
rate per unit volume is 2π~
~ r ). For a surface plasmon polariton with wave-vector ~k and
the electric field is E(~

52
angular frequency ω on the surface of a semi-infinite metal slab extending over z <
0, substituting the electric field profile of a single quantum [97, 98] and integrating
over space yields the total decay rate
Γ=

λ~ ∗ · Im¯ (ω) · λ.
2L(ω)|γ(z < 0)|

(4.1)

Here, L(ω) is the quantization length for the plasmon determined by normalizing
the energy density of the mode, |γ(z < 0)| is the inverse decay length of the plas~ ≡ k̂ − ẑk/γ(z < 0) is the polarization vector. All of
mon into the metal, and λ
these quantities are fully determined by the experimental dielectric function and
described in detail in Refs. 98 and 33.
We calculate the total ‘experimental’ decay rate of plasmons as a function of frequency by using expression 4.1 directly with the complex dielectric functions measured by ellipsometry.[36]
4.3

Electronic Structure

We require an approximation to quasiparticle energies and optical matrix elements
to describe the decay of surface plasmons to quasiparticle excitations (equations
4.2, 4.3). We use the relativistic DFT+U approach that Sundararaman et al. previously established[33] to best reproduce experimental photoemission spectra in
contrast to semilocal density-functional or even quasiparticle self-consistent GW
methods.[99] Strong screening in metals renders electron-hole interactions and excitonic effects negligible, so that we can work at the independent quasiparticle level
rather than with the more expensive Bethe-Salpeter equation[100] that accounts for
those effects.
Following Ref. 33, we perform density-functional calculations in the open-source
code JDFTx[101] with full-relativistic (spinorial) norm-conserving pseudopotentials at a plane-wave cutoff of 30 Eh (Hartrees). We use the PBEsol[102] exchangecorrelation approximation and a rotationally-invariant localized DFT+U correction[103] for the d-electrons in noble metals (U = 1.63 eV, 2.45 eV and 2.04 eV
respectively for copper, silver, and gold). Ref. 33 provides more details regarding
the selection of the electronic structure method and shows that this method produces
accurate electronic band structures in agreement with angle-resolved photoemission
(ARPES) measurements within 0.1 eV.
We perform the self-consistent ground state calculations using a 12 × 12 × 12 uniform k-point mesh centered at the Γ point along with a Fermi-Dirac smearing of

53
0.01 Eh to resolve the Fermi surface. The optical matrix elements correspond to
~ + me [~r , V̂N L ], which accounts for the
the momentum operator p~ˆ ≡ mi~e [~r , Ĥ] = ~i ∇
i~
fact that the nonlocal DFT+U and pseudopotential contributions (V̂N L ) to the DFT
Hamiltonian ( Ĥ) do not commute with the position operator, ~r . These nonlocal
corrections are usually insignificant for s and p-like electrons, but are critical for
describing optical transitions involving the d-electrons in the noble metals.[104]
Finally, we interpolate the electronic energies and matrix elements to arbitrary kpoints in the Brillouin zone using a basis of maximally-localized Wannier functions.[105, 106] Specifically, we use an sp3 basis with 4 Wannier bands for aluminum and a relativistic d 5t 2 basis with 14 Wannier bands for the noble metals
(where t is an orbital centered on the tetrahedral void sites of the face-centered cubic lattice). These Wannier functions exactly reproduce the orbital energies and
matrix elements within the maximum surface plasmon energy of the Fermi level
for all metals. We then evaluate equation 4.2 by Monte Carlo sampling 6.4 × 106 q~
values in the Brillouin zone for the noble metals (9.6 × 107 for aluminum), and
histogram contributions by plasmon and carrier energies to get the direct-transition
results in Figures 4.2 and 4.4. Note that aluminum requires more q~ samples to get
similar statistics since it contributes fewer pairs of bands per q~.
4.4

Plasmon Decay via Direct Transitions

Within the random phase approximation, direct interband transitions contribute[33]
d~q X
4π 2 e2
q~ 2
~ · h~
0 )δ(ε q
0 −ε q
−~ω)
λ ·Im¯direct (ω)· λ = 2 2
n0n ,
me ω BZ (2π) 3 n 0 n
(4.2)
where ε q~n and f q~n are the energies and occupations of electronic quasiparticles
q~
with wave-vectors q~ (in the Brillouin zone BZ) and band index n, and h~
pin 0 n are
momentum matrix elements. (See Appendix Section A.1 for a derivation of equation 4.2). Note that the factor ( f q~n − f q~n 0 ) rather than f q~n (1 − f q~n 0 ), as usually
found in Fermi’s Golden rule, accounts for the difference between the forward and
reverse processes (see Appendix Section A.3). This is appropriate for the steady
state change of plasmon number due to interactions with the electrons rather than
the decay rate of a single plasmon mode. However, the two expressions are identical for ~ω >> k BT, which is true for all relevant plasmon energies. To account for
finite carrier lifetimes, the energy-conserving δ-function is replaced by a Lorentzian
with half-width ImΣq~n + ImΣq~n 0 , where ImΣq~n is the total carrier linewidth due to
electron-electron and electron-phonon scattering as calculated in Reference 33.

54
Substitution of equation 4.2 into expression 4.1 results in exactly the same plasmon
decay rate as previously derived by Sundararaman et al. using Fermi’s Golden rule
within a fully quantum many-body formalism of the electrons and plasmons.[33]
We calculate the energies and matrix elements with the relativistic DFT+U method
described above (Section 4.3) which produces band structures in excellent agreement with photoemission spectra. Since we use a spinorial electronic structure
method to fully treat relativistic effects, and the band indices include spin degrees
of freedom.
4.5

Plasmon Decay via Phonon-assisted Transitions

The contribution due to phonon-assisted transitions from second-order perturbation
theory, as derived in the Appendix Section A.2, is [94, 95]
2 e2 Z
d~q0 d~q X
4π
( f q~n − f q~ 0 n 0 )
λ~ · Im¯phonon (ω) · λ~ = 2 2
me ω BZ (2π) 6 n 0 nα±
1 1
× nq~ 0−~q,α + ∓
δ(ε q~ 0 n 0 − ε q~n − ~ω ∓ ~ωq~ 0−~q,α )
2 2
q~ 0 −~q,α

× λ~ ·

gq~ 0 n 0,~qn h~
pi n 1 n

q~ 0

q~ 0 −~q,α

h~
pin 0 n1 gq~ 0 n ,~qn

*.
+/ , (4.3)
~ω
iη
~ω
iη
q~ n1
q~ n
q~ n
q~ −~q,α
n1 , q~ n1

where ~ω~kα is the energy of a phonon with wave-vector ~k and polarization index
α; n~kα is the corresponding Bose occupation factor and gq~kα0 n 0,~qn is the corresponding
electron-phonon matrix element with electronic states labeled by wave-vectors q~, q~0
and band indices n, n0 (with ~k = q~0 − q~ for crystal momentum conservation). The
sum over ± accounts for phonon absorption as well as emission. Since the ab
initio matrix elements couple all pairs of wave-vectors in the Brillouin zone, they
implicitly account for wrap-around (Umklapp) processes.
We calculate the phonon energies and electron-phonon matrix elements consistently
using the same relativistic DFT+U approximation as for the electronic states (Section 4.3). We use a Wannier representation to efficiently interpolate the phonon
energies and matrix elements to calculate the Brillouin zone integrals in expression 4.3 accurately (see Section 4.6 for details on the phonon states and matrix
elements).
Extrapolation to Eliminate Sequential Processes
The imaginary part of the energy denominator, η, in the second line of equation 4.3
corresponds to the linewidth of the intermediate electronic state (with band index

55
n1 ). The value of η does not affect the phonon-assisted absorption at photon energies less than the optical gap of materials previously considered [94, 95] and is
usually treated as a numerical regularization parameter. However, above the optical
gap (the interband threshold for metals), the real part of the denominator crosses
zero when the intermediate state conserves energy (i. e. is ‘on shell’), making the
resulting singular contributions inversely proportional to η. These singular contributions correspond to sequential processes: electron-phonon scattering followed by
a direct interband transition or vice versa (Figure 4.1(b)). For a metal, including
contributions from these sequential processes would lead to a multiple counting of
the direct transition. Scattering events preceding the optical transition are a part
of the equilibrium Fermi distribution, while scattering events following the optical
transition correspond to the subsequent inelastic relaxation of the generated carriers. We avoid multiple counting by taking advantage of the η independence of the
non-singular part and the η −1 variation of the singular part and extrapolating from
calculations done using two values of η, as described below.
By taking the limit η → 0 in equation 4.3 and noting that |1/(x + iη)| 2 → πδ(x)/η,
we can show that
e−ph

λ~ ∗ · Im¯phonon (~q0 n0, q~ n) · λ~ =

ImΣq~n 0

~ ∗ · Im¯direct (~q n0, q~ n) · λ~

e−ph

~ ∗ · Im¯direct (~q0 n0, q~0 n) · λ~
+ λ

ImΣq~n

+ O(η 0 ) + O(η 1 ) + · · · , (4.4)

where Im¯ (~q0 n0, q~ n) denotes the contribution to Im¯ due to a specific pair of initial
e−ph
and final electronic states. Here, ImΣq~n is the electron line width due to electronphonon scattering, given by

e−ph
ImΣq~n = π

1 1
Ωd~q0 X
0 −~
,α
2 2
BZ (2π) n 0 α±

q~ 0 −~q,α

× δ(ε q~ 0 n 0 − ε q~n ∓ ~ωq~ 0−~q,α ) gq~ 0 n 0,~qn , (4.5)
where the electronic states, phonon modes, and electron-phonon matrix elements
are computed exactly as for the phonon-assisted decay (see Ref. 34).
The above expansion in η (equation 4.4) clearly illustrates that the singular contributions correspond to sequential processes. The first term on the right hand side

56
corresponds to a direct transition followed by electron-phonon scattering while the
second term corresponds to the reverse. If we substitute the intermediate state
linewidth ImΣq~n for η as the formalism prescribes,[94, 95] and for simplicity foe−ph
cus only on electron-phonon scattering contributions ImΣq~n (which are dominant
for low energy carriers), then we see that the η-singular part reduces to simply twice
the direct contribution (expression 4.2). For a metal, this contribution should not
be counted as a separate decay rate since scattering events preceding and following a transition are part of the initial Fermi distribution and the subsequent carrier
transport, respectively.
We eliminate the singular contributions using an extrapolation scheme designed to
exploit the fact that the η dependence is different for on-shell and off-shell processes. To retain the non-singular O(η 0 ) terms while canceling the O(η −1 ) singular
terms discussed above, we modify equation 4.3 as
(4.3)corrected = 2 (4.3)| 2η − (4.3)| η .

(4.6)

We find that η = 0.1 eV, which was previously used for semiconductors,[94] is
sufficiently large to keep the singular terms resolvable for effective subtraction and
sufficiently small to have negligible impact on the physical non-singular contributions. We note that this extrapolation only has an effect and is necessary above the
optical gap of the material. Previous ab initio studies of phonon-assisted processes
did not deal with this issue since they focused on predictions for semiconductors
above the indirect gap and below the direct (optical) gap.
4.6

Phonon Modes and Matrix Elements

We calculate the ab initio force matrix for phonons and electron-phonon matrix elements from direct perturbations of atoms in a 4 × 4 × 4 supercell using exactly the
same electronic DFT parameters as above in JDFTx.[101] All four metals considered here have a single atom basis and hence exactly three acoustic phonon modes.
We cast these phonon energies and matrix elements into a Wannier basis to enable
interpolation for a dense sampling of the Brillouin zone integrals. (See Ref. 107
for a detailed introduction; we implement an analogous method in JDFTx, with
additional support for spinorial relativistic calculations.)
We use the aforementioned Wannier basis to cover the energy range close to the
Fermi level, and add random-initialized maximally-localized Wannier orbitals orthogonal to the first set to extend the energy range of included unoccupied states.
We use a total of 24 Wannier bands for aluminum and 46 spinorial Wannier bands

57
for the noble metals that exactly reproduce the orbital energies, and optical and
phonon matrix elements up to at least 50 eV above the Fermi level. We find this
energy range of unoccupied states sufficient to fully converge the sum over states in
the second order perturbation theory expression 4.3 at all plasmon energies considered.
Finally, we evaluate the double integral over the Brillouin zone in expression 4.3
by Monte Carlo sampling with 2 × 107 {~q, q~0 } pairs for the noble metals (3 × 108
pairs for aluminum to get similar statistics). We use standard temperature, T =
298 K, to calculate the Fermi occupations for electrons and Bose occupations for
phonons. Note that such low electronic temperatures (compared to the Fermi energy
∼ 105 K) necessitate extremely dense Brillouin zone sampling, which is, in turn,
made practical by the Wannier interpolation.[107] Histogramming by plasmon and
carrier energies, we collect the phonon-assisted contributions to Figures 4.2 and
4.4 (after incorporating the extrapolation discussed above to eliminate sequential
process contributions).
4.7

Surface-assisted Transitions

In metals, the strong confinement of fields at the surface introduces an additional
mechanism for intraband transitions. The exponential decay of the fields in the
metal with inverse decay length |γ(z < 0)| introduces a Lorentzian distribution in
the momentum of the plasmon normal to the surface with width ∼ |γ(z < 0)|. (This
can also be interpreted in terms of the uncertainty principle.) This momentum distribution allows diagonal intraband transitions on the band structure (Figure 4.1(b)),
which contributes a ‘surface-assisted’ loss[88, 108]:
ω2p

2k 2
(4.7)
Im sur f ace (ω) = 3 × |γ(z < 0)|vF 2
k + |γ(z < 0)| 2
Here, ω p = 4πne2 /me is the the bulk plasma frequency of the metal and vF =
√3
(~/me ) 3π 2 n its Fermi velocity, where n is the bulk carrier density of the metal.
In nano-confined geometries, such as spherical nanoparticles, the probability of
intraband transitions due to crystal momentum non-conservation can be greatly enhanced, as shown by several numerical studies using free-electron jellium models.[109–111] We can show from a full quantum-mechanical treatment of the states
of a spherical nanoparticle that geometry-assisted transitions effectively contribute
ω2p

6π 2 vF
Im sphere (ω) = 3 ×

(4.8)

58
where R is the radius of the spherical nanoparticle. This is similar to expression 4.7
except for dimensionless prefactors and the particle radius setting the length scale
instead of the skin depth. See Ref. 108 for detailed derivations of these contributions.
The direct, surface/geometry-assisted, and phonon-assisted transitions considered
above are the lowest-order processes for the decay of a plasmon to single-particle
excitations, which correspond to the Landau damping of the plasmon on the Fermi
sea.[112–114] Higher-order processes including multiple electron-hole pairs or multiple phonons are suppressed by phase-space factors at low energies and become
important only at higher energies that are not usually accessed by surface plasmons.[4]
4.8

Estimate of Resistive Losses

Apart from Landau damping, an additional source of plasmon loss is the intrinsic
lifetime of the electronic states comprising the collective oscillation and results
in heating rather than production of a few energetic carriers. This corresponds
to a resistive loss in the material. Single electron-hole pair generation dominates
the plasmon decay at high frequencies, while resistive loss in the metal dominates
at frequencies close to 0 eV. Here, we estimate these losses from the frequencydependent resistivity calculated ab initio within a linearized Boltzmann equation
with a relaxation time approximation.
The Boltzmann equation for electron occupations f q~n (t) in a uniform time-dependent
~ is[115]
electric field E(t)
∂ f q~n
∂ f (t)
∂ f q~n (t)
~ · q~n
+ e E(t)
∂t
∂t coll
∂ p~

(4.9)

We then substitute f q~n (t) = f q~n + δ f q~n (t), where the first term is the equilibrium
Fermi distribution and the second contains perturbations to linear order due to the
applied electric field, and collect contributions at first order in E(t).
To first order, the collision integral on the right-hand side of equation 4.9 can be
, where τq~−1
is the difference between rates of scattering out
written as −δ f q~n τq~−1
of and into the electronic state q~ n. Within the relaxation time approximation, we
assume that τq~−1
is approximately constant for carriers near the Fermi level, and ren
place it by an average τ −1 (inverse of momentum relaxation time). This is an excellent approximation for metals where electron-phonon scattering dominates carrier
relaxation near the Fermi level,[115] which is the case for most elemental metals

59
(except those with partially occupied d-shells) including aluminum and the noble
metals.
Switching equation 4.9 to the frequency domain, linearizing, invoking the relaxation time approximation, and rearranging, we get
δ f q~n (ω) =

−e f q~0 n
τ −1 − iω

~vq~n · E(ω),

(4.10)

∂ε

where ~vq~n ≡ ∂~qq~ n is the group velocity of electronic state q~ n and f q~0 n is the energy
derivative of the Fermi-Dirac distribution. We then calculate the current density
~j = Pq~n e f q~n~vq~n , and obtain the conductivity tensor by factoring out ~vq~n . Averaging
over directions, the isotropic conductivity is then
vq~2n
d~q X
σ(ω) =
e τ
(− f q~n )
(4.11)
1 − iωτ
BZ (2π)
{z
≡σ0

where σ0 is the zero-frequency (DC) conductivity.
Finally we calculate the momentum relaxation time τ using Fermi golden rule
for electron-phonon scattering. In the average, we weight the scattering rates by
v2

(− f q~0 n ) q3~ n since that determines the relative contributions to the conductivity above.
It is then straightforward to show that τ −1 = Γsum /w sum , where
2π
Γsum =

vq~n − ~vq~n · ~vq~ 0 n 0
Ωd~q d~q0 X
(−
q~ n
BZ (2π)
n 0 nα±
!#
× nq~ 0−~q,α + ∓
− f q~ 0 n 0
q~ 0 −~q,α

× δ(ε q~ 0 n 0 − ε q~n ∓ ~ωq~ 0−~q,α ) gq~ 0 n 0,~qn

(4.12)

with all the ab initio electron and phonon properties defined exactly as before, and
where the denominator for normalizing the weights is
vq~2n
d~q X
(− f q~n )
(4.13)
w sum =
BZ (2π)
Note that with this definition, we can simplify the DC conductivity, σ0 = e2 τw sum =
2 /Γ
e2 w sum
sum .
Given the frequency-dependent conductivity of the metal, we can calculate the resistive losses Im = Im [4πiσ(ω)/ω], which, upon simplification gives
Im resistive (ω) =

4πσ0
ω(1 + ω2 τ 2 )

(4.14)

60
Table 4.1: ab initio momentum relaxation times and resistivities of plasmonic metals at T = 298 K, compared to experimental resistivities from Ref. 116.
Metal
Al
Cu
Ag
Au

τ [fs]
12.0
35.6
36.4
26.3

ρ0 = σ0−1 [Ωm]
2.46 × 10−8
1.58 × 10−8
1.58 × 10−8
2.23 × 10−8

Expt ρ0 [Ωm]
2.71 × 10−8
1.71 × 10−8
1.62 × 10−8
2.26 × 10−8

We calculate w sum and Γsum using Monte Carlo sampling of the Brillouin zone
integrals with 1.6 × 106 q~ values for the single integral and 5 × 107 {~q, q~0 } pairs
for the double integral, which converges τ and σ0 within 1%. Table 4.1 lists the
momentum-relaxation time and resistivity we predict for the common plasmonic
metals. The excellent agreement with experimental resistivities demonstrates the
quantitative accuracy of the ab initio electron-phonon coupling (better than 10% in
all cases).
4.9

Results for Common Plasmonic Metals

Fig. 4.2 compares the plasmon linewidth and decay rates estimated directly from the
experimentally-measured complex dielectric functions with theoretical predictions
for cumulative contributions from direct, surface-assisted, phonon-assisted transitions and resistive losses. For all the common plasmonic metals, aluminum and the
noble metals, we find that direct transitions dominate above the interband threshold
(∼ 1.6 − 1.8 eV for aluminum, gold, and copper and ∼ 3.5 eV for silver). All other
contributions add to less than 10% above threshold, and hence the cumulative results overlay the direct transition lines. In silver, the maximum plasmon frequency
coincides with the interband threshold and hence there is no accessible frequency
range for which direct transitions dominate. In aluminum, direct transitions are in
fact possible at all frequencies due to a band crossing near the Fermi level,[33] but
an additional channel for direct transitions with much higher density of states opens
up at the effective threshold of ∼ 1.6 eV.
Below the threshold, direct transitions are forbidden (or for aluminum, are weak)
and the contributions due to the other processes become important. Surface-assisted
processes contribute only a small fraction (at most 5%) of the experimental linewidth
over the entire frequency range below threshold. Phonon-assisted transitions and
resistive losses compete significantly and dominate this frequency range. The relative importance of phonon-assisted transitions increases slightly with frequency,

a) Al

10-2

10-6

Linewidth [eV]

100

10-2

Direct
+ Surface
+ Phonon
+ Resistive
Experimental

10-4

100

b) Ag

Decay rate [fs-1]

100

10-4
10-6

Plasmon Energy [eV]

c) Au

d) Cu

100

10-2

Decay rate [fs-1]

Linewidth [eV]

10-2

10-4

10-6

Plasmon Energy [eV]

Figure 4.2: Comparison of calculated and experimental linewidths (left axis) and
decay rates (right axis) in (a) Al (b) Ag (c) Au and (d) Cu. The theoretical curves
indicate cumulative contributions from direct transitions alone (‘Direct’), including surface-assisted transitions (‘+Surface’), including phonon-assisted transitions
(‘+Phonon’), and including resistive losses (‘+Resistive’).
with resistive and surface-assisted losses dominating at very low frequencies (close
to 0 eV in these plots), an approximately even split between the three processes
at ∼ 1 eV, and a greater contribution from phonon-assisted transitions just below
threshold.
The total theoretical prediction including all these contributions agrees very well
with experiment over the entire range of frequencies.[36] Above threshold for the
noble metals, the theoretical predictions overestimate experiment by ∼ 10 − 20%,
which is the typical accuracy of optical matrix elements involving d electrons in
density-functional theory.[104] Below threshold, the total theory result underestimates the experimental value but it is typically within a factor of two from it. This
is, in part, because material non-idealities could contribute additional losses and our
theoretical calculations estimate an ideal lower bound. In fact, the largest discrepancy is for silver because these ideal losses are the smallest and the non-idealities
become more important relatively. Also note that there is a significant spread in tab-

62
ulated experimental dielectric functions for the noble metals,[36] with discrepancies
a factor of two or higher in the imaginary parts at infrared frequencies. (We used
the measurements that covered the greatest frequency range.) Therefore, more careful experimental measurements in that frequency range with higher quality samples
would be necessary and useful for a stricter comparison.
The results in Fig. 4.2 are based on calculations at standard room temperature,
T = 298 K. We expect the direct and surface-assisted contributions to be approximately independent of temperature, the resistive contributions to decrease almost
linearly with temperature, and the phonon-assisted contributions to reduce by a factor of two upon lowering the temperature (phonon emission persists while phonon
absorption freezes out). Therefore, at very low temperatures, phonon-assisted transitions will dominate below threshold, while direct transitions continue to dominate
above threshold.
Fig. 4.3 compares the relative contributions due to all the above processes to absorption in the surface of bulk gold and spherical gold nanoparticles of various
sizes. Geometry-assisted intraband contributions are negligible for a semi-infinite
surface, and are comparable to the phonon-assisted and resistive contributions for a
40 nm diameter sphere below threshold. With decreasing particle size, the relative
contributions of the geometry-assisted transitions increase in inverse proportion and
dominate the sub-threshold absorption in gold spheres smaller than 20 nm. However, direct transitions continue to dominate above the interband threshold even for
these small spheres. Therefore, simplified treatment of localized plasmon decay
in nanoparticles using jellium models that preclude direct transitions[109–111] is
only reasonable for frequencies below the interband threshold. Those approximations are therefore reasonable for silver, where the interband threshold exceeds the
dipole resonance frequency, but not for gold, copper or aluminum.
Figure 4.4 shows the initial carrier distributions generated via direct and phononassisted transitions, which we calculate by histogramming the integrands in expressions 4.2 and 4.3 by the initial (hole) and final (electron) state energies. The carrier
distributions are plotted as a function of carrier energy (horizontal axis) and plasmon / photon energy (axis normal to the page). The color scale indicates the fraction
of carriers generated by direct or phonon-assisted transitions. Note that carrier energies may exceed the plasmon energy with low probability because of the Lorentzian
broadening due to carrier linewidths in energy conservation (equivalently, a consequence of the uncertainty principle); this causes the small contributions from direct

63
Phonon

(a) Surface

80
60
40

Direct
Geometry

100
Contribution [%]

Contribution [%]

100

Resistive

80
60
40

Geometry

Resistive
Plasmon Energy [eV]

Direct
Geometry

Resistive
Plasmon Energy [eV]

100
Contribution [%]

Contribution [%]

Phonon

(b) 40 nm
sphere

Plasmon Energy [eV]
100

Phonon

80
60
40

(d) 10 nm
sphere
Direct

Geometry

Resistive
Plasmon Energy [eV]

Figure 4.3: Comparison of resistive, geometry-assisted, phonon-assisted and direct
transition contributions to absorption in gold as a function of frequency for (a) a
semi-infinite surface, or (b) 40 nm, (c) 20 nm or (d) 10 nm diameter spheres. The
surface/geometry contributions are negligible for the semi-infinite surface, are comparable to the phonon-assisted and resistive contributions for a 40 nm sphere, and
increase in inverse proportion with decreasing sphere diameter. Direct transitions
dominate above threshold in all cases.
transitions below threshold seen for the noble metals in Figure 4.4.
Direct transitions, shown in blue, dominate at high energies and exhibit the strong
material dependence previously discussed in detail in Ref. 33. For copper and gold,
direct transitions occur from the d-bands to unoccupied states above the Fermi level,
which results in holes that are much more energetic than electrons. Aluminum
exhibits a relatively flat distribution of electrons and holes, while silver exhibits a
bimodal distribution of hot electrons as well as holes from direct transitions in a
very narrow frequency range close to the maximum plasmon frequency.
Phonon-assisted transitions, shown in red, exhibit a flat distribution of electrons and
holes extending from zero to the plasmon energy for all the metals. In aluminum,

Figure 4.4: Probability density energy distributions of hot carriers, P(ω, ε), generated by the decay of surface plasmons due to phonon-assisted and direct transitions, as a function of plasmon frequency (ω) and carrier energy (ε), in (a) Al
(b) Ag (c) Au and (d) Cu. The color-scale indicates the relative contributions of
phonon-assisted (red) and direct (blue) transitions. For each frequency, P(ω, ε) is
normalized such that it equals 1 for a flat distribution, similar to the red plateaus
below threshold where phonon-assisted transitions dominate.
direct transitions are also possible below the threshold at 1.6 eV and contribute
∼ 25% of the generated carriers. Geometry-assisted intraband transitions (in the
surface or sphere cases) have a similar phase space to phonon-assisted transitions
and also generate flat distributions of electron and hole energies. Resistive losses
compete with phonon-assisted transitions but dissipate thermally and do not generate energetic hot carriers. Due to these losses, below threshold ∼ 30 − 50% of the
absorbed energy is dissipated without hot carrier generation. Therefore, plasmonic
hot-carrier applications could benefit from the higher efficiency above threshold,
where direct transitions dominate by far and result in high-energy carriers. Additionally, we predict aluminum to be an excellent candidate for general hot carrier
applications since it allows direct transitions at all energies and has the smallest
fraction of resistive loss (despite its absolute resistivity being higher than other met-

65
als).
4.10

Conclusions

We preformed the first ab initio calculations of phonon-assisted optical excitations
in metals, allowing us to link the energy range between resistive losses at low energies (microwave-infrared) and direct interband transitions at high energies (visibleultraviolet). Along with surface-assisted transitions due to field confinement in metals,[88] this completes the theoretical picture of surface plasmon decay, accounting
for all relevant mechanisms.
We find good agreement with experimental measurements for the total decay rate,
but we additionally predict the relative contributions of all these processes and the
initial generation of hot carriers in plasmonic metals. We find that direct transitions dominate above threshold and generate hot carriers, while below threshold,
hot carrier generation by phonon-assisted transitions is diminished by competition
from resistive losses. We also find that surface-assisted transitions are enhanced in
nano-confined geometries and become dominant below threshold for particle sizes
∼ 10 nm, but that direct transitions remain dominant above the interband threshold
even for small particles.
We suggest that aluminum is quite promising as a general-purpose plasmonic hot
carrier generator since it generates hot carriers efficiently for the widest frequency
range, and generates high-energy electrons and holes with equal probability. Compared to the noble metals, aluminum also exhibits the best transport properties for
high energy holes. In the future, a detailed analysis of the transport of energetic
carriers in real metal nanostructures, based on the initial carrier distributions and
scattering rates predicted here, could enable directed design of optimal hot carrier
devices.

66
Chapter 5

AB INITIO DESCRIPTION OF HOT ELECTRON RELAXATION
IN PLASMONIC METALS
5.1

Motivation and Previous Work

In the previous Chapter, we discussed the non-radiative decay of plasmons into hot
electron-hole pairs. After this decay has taken place and a nonthermal distribution
of electrons and holes has thus been established, a series of decay processes take
place in which the electrons thermalize among themselves and with the lattice until
a new equilibrium state is reached. Understanding the energy transfer mechanisms
during these equilibration processes is critical for a wide array of applications.
In the system we consider here, an incident ultrashort laser pulse excites a plasmon on the surface of a metal. This plasmon may decay into a hot (non-thermal)
electron-hole pair, as discussed in Chapter 4. The electron bath is then warmed
by the energy of the hot electron via electron-electron scattering. The electron
dynamics during this thermalization process are determined by the electronic density of states and the related electronic heat capacity. The electron bath then cools
by transferring energy to the lattice via electron-phonon scattering. The rate of
this electron-phonon equilibration process is determined by the electron-phonon
scattering matrix element and the density of states. Finally, the lattice cools via
phonon-phonon scattering with the surrounding environment. See Figure 5.1 for a
schematic representation of these processes.
Describing the evolution of the excited non-equilibrium electron distribution has
been the subject of intense research for two decades.[117–121] The dynamics of
the nonthermalized and thermalized electrons and phonons are most efficiently observed with pulsed laser measurement techniques in transient absorption experiments, where resolution of tens of femtoseconds can be achieved.[122–128] The
majority of investigations of hot carrier relaxation have employed various approximate models for analysis of experimental data, typically based on free-electron
models and empirical electron-phonon interaction parameters, to calculate the energy absorption, electron-electron thermalization, and electron-phonon relaxation.
[32, 129–133] A complete ab initio description of the time evolution and optical response of this non-equilibrium electron gas from femtosecond to picosecond time

67
scales had not yet been achieved, and few analyses have avoided empirical treatment of electron-phonon interactions.[31]
The initial internal electron thermalization via electron-electron scattering is qualitatively described within Landau Fermi liquid theory.[134–137] The subsequent
relaxation of the high temperature electron gas with the lattice is widely described
by the two-temperature model (TTM),[31, 32, 126–128] given by coupled differential equations for the electron and lattice temperatures, Te and Tl , respectively,
dTe
= 5 · (κ e 5 Te ) − G(Te ) × (Te − Tl ) + S(t)
dt
dTl
= 5 · (κ p 5 Tp ) + G(Te ) × (Te − Tl ).
Cl (Tl )
dt

Ce (Te )

(5.1)

Here, Ce (Te ) and Cl (Tl ) are the electronic and lattice heat capacities, G(Te ) is the
electron-phonon coupling factor, S(t) is the source term which describes energy
deposition by a laser pulse, and κ e and κ p are the thermal conductivities of the
electrons and phonons. In nanostructures, temperatures rapidly become homogeneous in space and the contributions of the thermal conductivities drop out. A vast
majority of studies treat the material parameters Ce (Te ), Cl (Tl ) and G(Te ) as phenomenological temperature-independent constants or as linear approximations.
A key challenge in the quantitative application of the two temperature model is the
determination of these temperature-dependent material parameters. With pulsed
lasers, it is possible to absorb sufficient energy in plasmonic nanostructures to melt
the metal once the electrons and lattice have equilibrated.[138] The highest electron
temperature, Temax , accessible in repeatable measurements is therefore limited only
by the equilibrated lattice temperature being less than the melting temperature Tm
R T ma x
RT
of the metal,[116] which yields the condition T e dTe Ce (Te ) = T m dTl Cl (Tl ).
Starting at room temperature T0 = 300 K and using our ab initio calculations of the
electron and lattice heat capacities, Ce (Te ) and Cl (Tl ) (see Sections 5.3, 5.4), we find
Temax ≈ 5700, 8300, 7500, and 6700 K, respectively, for aluminum, silver, gold, and
copper. These maximum electron temperatures are much higher than the melting
temperatures of the metals because the lattice heat capacity is on the order of two
times larger than the electronic heat capacity. For gold and copper in particular,
these maximum electron temperatures are sufficient to change the occupations of
the d-band electrons ∼ 2 eV below the Fermi level. Consequently, it is important
to derive the temperature dependence of these material parameters from electronic
structure calculations rather than free-electron like models.[31]

Figure 5.1: Evolution of the non-equilibrium ‘hot’ electrons with time with different regimes dominated by distinct material properties. The electronic density of
states (DOS) determines the electronic heat capacity Ce (Te ) and the temperature
to which the electrons equilibrate. The electron-phonon coupling G(Te ) determines
the dynamics of energy transfer from the electrons to the phonons. The phonon density of states determines the lattice heat capacity Cl and the temperature to which
the lattice equilibrates. All of these properties are particularly sensitive to the electron temperature Te , and are essential for a quantitative description of the ultrafast
response of plasmonic metals under laser excitation.
To accurately predict the transient optical response of metal nanostructures, we
account for the electron temperature dependence of the electronic heat capacity,
electron-phonon coupling factor, and dielectric function. These properties, in turn,
require accurate electron and phonon band structures as well as electron-phonon
and optical matrix elements. In Chapter 4, we showed that ab initio calculations
can quantitatively predict optical response, carrier generation, and electron transport in plasmonic metals in comparison with experiment, with no empirical parameters.[34] In this Chapter, we calculate Ce (Te ), G(Te ) and the temperature and
frequency-dependent dielectric function, (ω,Te ) from first principles. This is the
first ab initio, temperature dependent dielectric function presented for metals, which
is a great advancement over the Drude model dielectric function with Matthiessen’s
rule for temperature dependence, which has been heavily relied upon by previous investigations.[31, 121, 139] Our calculations implicitly include electronicstructure effects in the density of states and electron-phonon interaction matrix
elements, and implicitly account for processes such as Umklapp scattering. We
show substantial differences between our fully ab initio predictions and those from

69
simplified models due to the energy dependence of the electron-phonon matrix elements, especially at high electron temperatures where the d-bands contribute.
5.2

Computational Methods

We perform ab initio calculations of the electronic states, phonons, electron-phonon
and optical matrix elements, as well as several derived quantities based on these
properties, for four plasmonic metals: aluminum, copper, silver, and gold. We use
the PBEsol+U density functional theory methods described in Section 4.3.
We calculate phonon energies and electron-phonon matrix elements using perturbations on a 4 × 4 × 4 supercell. In ab initio calculations, these matrix elements implicitly include Umklapp-like processes. We then convert the electron and phonon
Hamiltonians to a maximally-localized Wannier function basis,[106] with 123 kpoints in the Brillouin zone for electrons. Specifically, we employ 24 Wannier centers for aluminum and 46 spinorial centers for the noble metals, which reproduces
the DFT band structure exactly to at least 50 eV above the Fermi level.
Using this Wannier representation, we interpolate the electron, phonon, and electronphonon interaction Hamiltonians to arbitrary wave-vectors and perform dense Monte
Carlo sampling to accurately evaluate the Brillouin zone integrals for each derived
property below. This dense Brillouin zone sampling is necessary because of the
large disparity in the energy scales of electrons and phonons, and directly calculating DFT phonon properties on dense k-point grids is computationally expensive and
impractical. See Chapter 4 for further details on the ab initio calculation methods
and benchmarks of the accuracy of the electron-phonon coupling (e.g. resistivity
within 5% for all four metals).
5.3

Electronic Density of States and Heat Capacity

The electronic density of states (DOS) per unit volume is given by:
d ~k X
δ(ε − ε~kn ),
g(ε) =
BZ (2π)

(5.2)

where ε~kn are energies of quasiparticles with band index n and wave-vector ~k in
the Brillouin zone BZ. The density of states directly determines the electronic heat
capacity and is an important factor in the electron-phonon coupling and dielectric
response of hot electrons. Above, the band index n implicitly counts spinorial orbitals in our relativistic calculations, and hence we omit the explicit spin degeneracy
factor.

DOS [1029 eV-1 m-3] DOS [1029 eV-1 m-3]

2.5

a) Al

0.4
0.3

1.5

0.2
0.1

b) Ag

ab initio
Lin et al. 2008
free electron

0.5

c) Au

d) Cu

1.5

0.5

-10

-5

ε-εF [eV]

-10

-5

ε-εF [eV]

Figure 5.2: Comparison of electronic density of states of for (a) Al, (b) Ag, (c) Au
and (d) Cu from our relativistic PBEsol+U calculations (ab initio), previous semilocal PBE DFT calculations[31] (less accurate band structure), and a free electron
model.
Figure 5.2 compares the density of states predicted by our relativistic PBEsol+U
method with a previous non-relativistic semi-local estimate[31] using the√PBE func
3/2
2 2
tional,[140] as well as a free electron model ε~k = ~2mke for which g(ε) = 2π2 2m
The free electron model is a reasonable approximation for aluminum and the PBE
and PBEsol+U density-functional calculations also agree reasonably well in this
case. The regular 313 k-point grid used in Reference 31 for Brillouin zone sampling introduces the sharp artifacts in the density of states, compared to the much
denser Monte Carlo sampling in our calculations with 640,000 k-points for Au, Ag,
and Cu, and 1,280,000 k-points for Al.
For the noble metals, the free electron model and the density functional methods
agree reasonably near the Fermi level, but differ significantly for energies ∼2 eV
below the Fermi level where d-bands contribute. The free electron models ignore
the d-bands entirely, whereas the semi-local PBE calculations predict d-bands that
are narrower and closer to the Fermi level than the PBEsol+U predictions. The
U correction[103] that we employ accounts for self-interaction errors in semi-local
DFT and positions the d bands in agreement with angle-resolved photoemission

Ce [105 J/m3K]

a) Al

b) Ag

ab-initio
Lin et al. 2008
Sommerfeld

c) Au

d) Cu

20
15

Te [10 K]

Figure 5.3: Comparison of the electronic heat capacity as a function of electron
temperature, Ce (Te ), for (a) Al, (b) Ag, (c) Au and (d) Cu, corresponding to the
three electronic density-of-states predictions shown in Figure 5.2. The free electron
Sommerfeld model underestimates Ce for noble metals at high Te because it neglects d-band contributions, whereas previous DFT calculations[31] overestimate it
because their d-bands are too close to the Fermi level.
spectroscopy measurements (to within ∼ 0.1 eV).[33] Additionally, the density of
states in the non-relativistic PBE calculations strongly peaks at the top of the dbands (closest to the Fermi level), whereas the density of states in our relativistic
calculations is comparatively balanced between the top and middle of the d-bands
due to strong spin-orbit splitting, particularly for gold. Below, we find that these
inaccuracies in the density of states due to electronic structure methods previously
employed for studying hot electrons propagate to the predicted electronic heat capacity and electron-phonon coupling.
The electronic heat capacity, defined as the derivative of the electronic energy per
unit volume with respect to the electronic temperature (Te ), can be related to the
density of states as
Z ∞
∂ f (ε,Te )
(5.3)
Ce (Te ) =
dεg(ε)ε
∂Te
−∞
where f (,Te ) is the Fermi distribution function. The term ∂ f /∂Te is sharply

72
peaked at the Fermi energy ε F with a width of approximately k BTe , and therefore the heat capacity depends only on electronic states within a few k BTe of the
Fermi level. For the free electron model, Taylor expanding g(ε) around ε F and analytically integrating expression 5.3 yields the linear Sommerfeld model Ce (Te ) =
π 2 ne k 2B
~2 k F
2 3
2ε F Te , which is valid for Te
TF (∼ 10 K). Above, ne = 3π k F , ε F = 2me
and k F are the number density, Fermi energy, and Fermi wave-vector of the free
electron model, respectively.
At temperatures Te
TF , the electronic heat capacities are much smaller than the
lattice heat capacities,[119, 126, 136] which makes it possible for laser pulses to
increase Te by 103 − 104 Kelvin, while Tl remains relatively constant.[127, 141,
142] Figure 5.3 compares Ce (Te ) from the free-electron Sommerfeld model with
predictions of expression 5.3 using the density of states from PBE and PBEsol+U
calculations. The free-electron Sommerfeld model is accurate at low temperatures
(up to ∼ 2000 K) for all four metals.
With increasing Te , ∂ f /∂Te in expression 5.3 is non-zero increasingly further away
from the Fermi energy, so that deviations from the free electron density of states
eventually become important. For aluminum, the density of states remains freeelectron-like over a wide energy range and the Sommerfeld model remains valid
even at high temperatures. For the noble metals, the increase in density of states
due to d-bands causes a dramatic increase in Ce (Te ) once Te is high enough that
∂ f /∂Te becomes non-zero in the d-band energy range. Copper and gold have shallower d-bands and deviate at lower temperatures compared to silver. Additionally,
the d-bands are too close to the Fermi level in the semi-local PBE calculations of
Reference 31 which results in an overestimation of Ce (Te ) compared to our predictions based on the more accurate relativistic PBEsol+U method.
See Appendix Chapter B for a table of the electronic heat capacity as a function of
electron temperature predicted by expression 5.3
5.4

Phononic Density of States and Heat Capacity

Similarly, the phonon density of states per unit volume is given by
d~q X
δ(ε − ~ωq~ α ),
D(ε) =
BZ (2π)

(5.4)

DOS [1029 eV-1 m-3] DOS [1029 eV-1 m-3]

300
ab initio
Debye

240

300

a) Al

180

120

c) Au

240

180

120

0.02

0.04

d) Cu

240

180

b) Ag

240

0.06

ε [eV]

0.02

0.04

0.06

ε [eV]

Figure 5.4: Comparison of ab initio phonon density of states and the Debye model
for (a) Al, (b) Ag, (c) Au, and (d) Cu.
where ~ωq~ α is the energy of a phonon with polarization index α and wave-vector
q~. The phonon density of states directly determines the lattice heat capacity,
Z ∞
∂n(ε,Tl )
(5.5)
Cl (Tl ) =
dεD(ε)ε
∂Tl
where n(ε,Tl ) is the Bose occupation factor.
Within the Debye model, the phonon energies are approximated by an isotropic
linear dispersion relation ωq~ α = vα q up to a maximum Debye wave-vector qD
chosen to conserve the number of phonon modes per unit volume. This model yields
ε2 P
the analytical phonon density of states, D(ε) = (2π
2)
α θ(~qD vα − ε)/(~vα ) ,
where vα = {v L , vT , vT } are the speeds of sound for the one longitudinal and two
degenerate transverse phonon modes of the face-centered cubic metals considered
here.[116]
Figure 5.4 compares the ab initio phonon density of states with the Debye model
predictions, and shows that the Debye model is a good approximation for the density of states only up to 0.01 eV. However, Figure 5.5 shows that the corresponding
predictions for the lattice heat capacities are very similar, rapidly approaching the
equipartition theorem prediction of Cl = 3k B /Ω at high temperatures, which is in-

Cl [105 J/m3K]

a) Al

20
ab initio
Debye

b) Ag

c) Au

d) Cu

0.5

1.5

0.5

Tl [10 K]

1.5

Tl [10 K]

Figure 5.5: Comparison of ab initio and Debye model predictions of the lattice heat
capacity as a function of lattice temperature, Cl (Tl ), for (a) Al, (b) Ag, (c) Au, and
(d) Cu. Despite large differences in the density of states (Figure 5.4), the predicted
lattice heat capacities of the two models agree within 10%.
sensitive to details in the phonon density of states. In fact, the largest deviations of
the Debye model heat capacity are below 100 K and deviate less than 10% from
the ab initio predictions for all four metals. We therefore find that a simple Debye
model of the phonons is adequate for predicting the lattice heat capacity, in contrast
to the remaining quantities we consider below which are highly sensitive to details
of the phonons and their coupling to the electrons.
5.5

Electron-phonon Matrix Element and Coupling

In Section 5.3 we showed that the electronic heat capacity, which determines the
initial temperature that the hot electrons equilibrate to, is sensitive to electronic
structure especially in noble metals at high Te where d-bands contribute. Now we
analyze the electron-phonon coupling which determines the subsequent thermalization of the hot electrons with the lattice. We show that details in the ab initio electron-phonon matrix elements also play a significant role, in addition to the
electronic band structure, and compare previous semi-empirical estimates of the
temperature dependent phonon coupling to our direct ab initio calculations.

75
The rate of energy transfer per unit volume from electrons at temperature Te to the
lattice at temperature Tl via electron-phonon scattering is given by Fermi’s golden
rule as
dE
≡ G(Te )(Te − Tl )
dt
2π
Ωd ~kd ~k 0 X
δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )
~ BZ (2π) 6 nn 0 α
~0 ~

(5.6)

STe ,Tl (ε~kn , ε~k 0 n 0 , ~ω~k 0−~k,α )
× ~ω~k 0−~k,α g~k0 −0k,α
k n ,kn

(5.7)
with
STe ,Tl (ε, ε0, ~ω ph ) ≡ f (ε,Te )n(~ω ph ,Tl )(1 − f (ε0,Te ))
− (1 − f (ε,Te ))(1 + n(~ω ph ,Tl )) f (ε0,Te ). (5.8)
Here, Ω is the unit cell volume, ~ωq~ α is the energy of a phonon with wave-vector
~0 ~
q~ = ~k 0 − ~k and polarization index α, and g k − k,α is the electron-phonon matrix
~k 0 n 0 ,~kn

element coupling this phonon to electronic states indexed by ~kn and ~k 0 n0.
Above, S is the difference between the product of occupation factors for the forward
and reverse directions of the electron-phonon scattering process ~kn + q~ α → ~k 0 n0,
with f (ε,Te ) and n(~ω,Tl ) being the Fermi and Bose distribution function for the
electrons and phonons, respectively. Using the fact that STe ,Tl = 0 for an energyconserving process, ε + ~ω ph = ε0, by detailed balance, we can write the electronphonon coupling coefficient as
2π
G(Te ) =

Ωd ~kd ~k 0 X
δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )
BZ (2π)
nn 0 α
~0 ~

× ~ω~k 0−~k,α g~k0 −0k,α
( f (ε~kn ,Te ) − f (ε~k 0 n 0 ,Te ))
k n ,kn

n(~ω~k 0−~k,α ,Te ) − n(~ω~k 0−~k,α ,Tl )
Te − Tl

. (5.9)

This general form for ab initio electronic and phononic states is analogous to previous single-band / free electron theories of the electron-phonon coupling coefficient;
see for example the derivation by Allen et al.[143]
The direct ab initio evaluation of G(Te ) using expression 5.9 requires a six-dimensional
integral over electron-phonon matrix elements from DFT with very fine k-point

76
grids that can resolve both electronic and phononic energy scales. This is impractical without the recently-developed Wannier interpolation and Monte Carlo sampling methods for these matrix elements,[34, 107] and therefore our results are the
first fully ab initio predictions of G(Te ). See Appendix Chapter B for a table of the
electron-phonon coupling factor as a function of electron temperature predicted by
expression 5.9.
Previous theoretical estimates of G(Te ) are semi-empirical, combining DFT electronic structure with empirical models for the phonon coupling. For example, Wang
et al.[144] assume that the electron-phonon matrix elements averaged over scattering angles are independent of energy and that the phonon energies are smaller than
k BTe , and then approximate the electron-phonon coupling coefficient as
Z ∞
−∂ f (ε,Te )
πk B
dεg 2 (ε)
λh(~ω) i
(5.10)
G(Te ) ≈
~g(ε F )
∂ε
−∞
where λ is the electron-phonon mass enhancement parameter and h(~ω) 2 i is the
second moment of the phonon spectrum.[31, 117, 145] Lin et al.[31] treat λh(~ω) 2 i
as an empirical parameter calibrated to experimental values of G at low electron
temperatures obtained from thermoreflectance measurements, and extrapolate it to
higher electron temperatures using expression 5.10. See Refs. 144 and 31 for more
details.
For clarity, we motivate here a simpler derivation of an expression of the form of
equation 5.10 from the general form (equation 5.9). First, making the approximation ~ωq~ α
Te (which is reasonably valid for Te above room temperature) allows
us to approximate the difference between the electron occupation factors in the second line of equation 5.9 by ~ωq~ α ∂ f /∂ε (using energy conservation). Additionally,
for Te
Tl , the third line of equation 5.9 simplifies to k B /(~ω~k 0−~k,α ). With no
other approximations, we can then rearrange equation 5.9 to collect contributions
by initial electron energy,
Z ∞
−∂ f (ε,Te )
πk B
dεh(ε)g 2 (ε)
(5.11)
G(Te ) ≈
~g(ε F ) −∞
∂ε
with
2g(ε F )
h(ε) ≡ 2
g (ε)

Ωd ~kd ~k 0 X
δ(ε − ε~kn )
BZ (2π)
nn α
~0 ~

. (5.12)
× δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )~ω~k 0−~k,α g~k0 −0k,α
k n ,kn

h(ε) [meV2]

800
600
ab initio
Lin et al. 2008

200

-10

-5

ε-εF [eV]

c) Au

-6

-3

ε-εF [eV]

300

d) Cu

250
200

150
100

b) Ag

100

400

h(ε) [meV2]

150

a) Al

50
-8

-4

ε-εF [eV]

-8

-4

ε-εF [eV]

Figure 5.6: Energy-resolved electron-phonon coupling strength h(ε), defined by
expression 5.12, for (a) Al, (b) Ag, (c) Au, (d) Cu. For the noble metals, h( F )
is substantially larger than its value in the d-bands, which causes previous semiempirical estimates[31] using a constant h(ε) to overestimate the electron-phonon
coupling (G(Te )) at Te ∼ 3000 K, as shown in Figure 5.7.
Comparing this with expression 5.10, we can see that the the primary approximation
in previous semi-empirical estimates[31, 144] is the replacement of h(ε) by an
energy-independent constant λh(~ω) 2 i, used as an empirical parameter.
Figure 5.6 compares ab initio calculations of this energy-resolved electron-phonon
coupling strength, h(ε), with previous empirical estimates of λh(~ω) 2 i, and Figure 5.7 compares the resulting temperature dependence of the electron-phonon coupling, G(Te ), from the ab initio (expression 5.9) and semi-empirical methods (expression 5.10). For noble metals, G(Te ) increases sharply beyond Te ∼ 3000 K
because of the large density of states in the d-bands. However, h(ε) is smaller by
a factor of 2 − 3 in the d-bands compared to near the Fermi level. Therefore, assuming h(ε) to be an empirical constant equal to its value at the fermi level[31, 32]
results in a significant overestimate of the contributions of the d-bands to G(Te ) at
high Te , compared to the direct ab initio calculations. Additionally, the shallowness
of the d-bands in the semi-local PBE band structure used in Reference 31 lowers

G [1017 W/m3K]

Hostetler et al. 1999
Lin et al. 2008
ab-initio

0.8

a) Al

0.6

0.4

0.2
Hohlfeld et al. 2000
Hostetler et al. 1999
Lin et al. 2008
ab-initio

1.2

Groeneveld et al. 1995 b) Ag
Groeneveld et al. 1990
Lin et al. 2008
ab-initio

c) Au

Hohlfeld et al. 2000
d) Cu
Elsayed-Ali et al. 1987
Lin et al. 2008
ab-initio

0.8
0.4

Te [10 K]

Figure 5.7: Comparison of predictions of the the electron-phonon coupling strength
as a function of electron temperature, G(Te ), for (a) Al, (b) Ag, (c) Au and (d) Cu,
with experimental measurements where available.[124, 130, 131, 146, 147] The
DFT-based semi-empirical predictions of Lin et al.[31] overestimate the coupling
for noble metals at high temperatures because they assume an energy-independent
electron-phonon coupling strength (Figure 5.6) and neglect the weaker phonon coupling of d-bands compared to the conduction band. The experimental results (and
hence the semi-empirical predictions) for aluminum underestimate electron-phonon
coupling because they include the effect of competing electron-electron thermalization which happens on the same time scale.
the onset temperature of the increase in G(Te ), and results in further overestimation
compared to our ab initio predictions.
The ab initio predictions agree very well with the empirical measurements of G(Te )
available at lower temperatures for noble metals.[124, 130, 131, 146, 147] In fact,
the semi-empirical calculation based on λh(~ω) 2 i underestimates the room temperature electron-phonon coupling for these metals; the significant overestimation
of G(Te ) seen in Figure 5.7 is in despite of this partial cancellation of error. This
shows the importance of ab initio electron-phonon matrix elements in calculating
the coupling between hot electrons and the lattice.
Experimental measurements of the electron-phonon coupling in noble metals are

79
reliable because of the reasonably clear separation between a fast electron-electron
thermalization rise followed by a slower electron-phonon decay in the optical transient signal. In aluminum, these time scales significantly overlap making an unambiguous experimental determination of G difficult. Consequently, the value of G
for Al is not well agreed upon.[148] For example, in Reference 146, measurements
showed no fast transient free-electron spike and G was extracted from the lattice
temperature variation instead. However, the measured rate for the lattice temperature rise includes competing contributions from electron-electron and electronphonon thermalization; attributing the entire rate to electron-phonon coupling only
provides a lower bound for G. Indeed, Figure 5.7(a) shows that this experimental estimate[146] and its phenomenological extension to higher Te [31] significantly
underestimate our ab initio predictions by almost a factor of two. Note that density
functional theory is highly reliable for the mostly free-electron-like band structure
of aluminum, and the ab initio electron-phonon matrix elements are accurate to
within 5%.[34] We therefore conclude that electron-electron thermalization is only
about two times faster then electron-phonon thermalization in aluminum, causing
the significant discrepancy in experimental measurements. This further underscores
the importance of ab initio calculations over phenomenological models of electronphonon coupling.
5.6

Temperature Dependent Dielectric function

The final ingredient for a complete ab initio description of hot electron relaxation
dynamics is the electron-temperature-dependent dielectric function of the material.
In Chapter 4 we showed that we could predict the imaginary part of the dielectric
function Im (ω) of plasmonic metals in quantitative agreement with ellipsometric
measurements for a wide range of frequencies by accounting for the three dominant
contributions,
Im (ω) =

4πσ0
+ Im direct (ω) + Im phonon (ω).
ω(1 + ω2 τ 2 )

(5.13)

Here we focus on electron temperature dependence of these contributions; see
Chapter 4 for a detailed description of this expression.
The first term of expression 5.13 accounts for the Drude response of the metal due
to free carriers near the Fermi level, with the zero-frequency conductivity σ0 and
momentum relaxation time τ calculated using the linearized Boltzmann equation
with ab initio collision integrals. The second and third terms of expression 5.13,

80
derived in Appendix Chapter A and discussed in Chapter 4,
d~q X
4π 2 e2
q~ 2
Im direct (ω) = 2 2
( f q~n − f q~n 0 )δ(ε q~n 0 − ε q~n − ~ω) λ̂ · h~
pin 0 n
me ω BZ (2π) n 0 n
(5.14)
and
4π 2 e2
d~q0 d~q X
Im phonon (ω) = 2 2
( f q~n − f q~ 0 n 0 )
me ω BZ (2π) 6 n 0 nα±
1 1
× nq~ 0−~q,α + ∓
δ(ε q~ 0 n 0 − ε q~n − ~ω ∓ ~ωq~ 0−~q,α )
2 2
q~ 0 −~q,α

× λ̂ ·

gq~ 0 n 0,~qn h~
pin1 n

q~ 0

q~ 0 −~q,α

h~
pin 0 n1 gq~ 0 n ,~qn

*.
+/ , (5.15)
ε − ε q~n − ~ω + iη ε q~ 0 n1 − ε q~n ∓ ~ωq~ 0−~q,α + iη
n1 , q~ n1

capture the contributions due to direct interband excitations and phonon-assisted inq~
traband excitations, respectively. Here h~
pin 0 n are matrix elements of the momentum
operator, λ̂ is the electric field direction (results are isotropic for crystals with cubic symmetry), and all remaining electron and phonon properties are exactly as described previously in Chapter 4. The energy-conserving delta functions are replaced
by a Lorentzians of width equal to the sum of initial and final electron linewidths,
to account for the finite lifetime of the quasiparticles.
The dielectric function calculated using expressions 5.13-5.15 depends on the electron temperature in two ways. First, the electron occupations f q~n directly depend
on electron temperature. Second, the phase-space for electron-electron scattering
increases with electron temperature, which increases the momentum relaxation rate
(τ −1 ) in the first Drude term of expression 5.13 and the Lorentzian broadening in
the energy conserving δ-function in equations 5.14 and 5.15.
We calculate ab initio electron linewidths using Fermi golden rule calculations for
electron-electron and electron-phonon scattering at room temperature, as detailed in
Chapter 4. Because these calculations are computationally expensive and difficult
to repeat for several electron temperatures, we instead use the ab initio linewidths at
room temperature with an analytical correction for the Te dependence. The electronphonon scattering rate depends on the lattice temperature, but is approximately independent of Te because the phase space for scattering is determined primarily by
the electronic density of states and electron-phonon matrix elements, which depend strongly on the electron energies but not on the occupation factors or electron
temperature. The phase space for electron-electron scattering, on the other hand,

81
Table 5.1: Parameters to describe the change in dielectric function with electron
temperature using expression 5.17, extracted from fits to ab initio calculations. The
energies and effective masses for the parabolic band approximation for the d → s
transition in noble metals are indicated Figure 5.15(a).
Al
Ag
Physical constants:
ω p [eV/~]
15.8
8.98
−1
τ [eV/~]
0.0911 0.0175
Fits to ab initio calculations:
De [eV−1 ]
0.017
0.021
3/2
A0 [eV ]
70
ε c [eV]
0.31
ε 0 [eV]
3.36
mv∗ /mc∗
5.4

9.01
0.0240

10.8
0.0268

0.016
22
0.96
1.25
3.4

0.020
90
0.98
1.05
16.1

depends on the occupation factors and electron temperature because an electron at
an energy far from the Fermi level can scatter with electrons close to the Fermi
level. The variation of this phase-space with temperature is primarily due to the
change in occupation of states near the Fermi level, and we can therefore estimate
this effect in plasmonic metals using a free electron model.
Within a free electron model, the phase-space for electron-electron scattering grows
quadratically with energy relative to the Fermi level, resulting in scattering rates ∝
(ε − ε F ) 2 at zero electron temperatures, as is well-known.[123, 149] We can extend
these derivations to finite electron temperature to show (see Appendix Chapter C for
derivation) that the energy and temperature-dependent electron-electron scattering
rate
De
−1
[(ε − ε F ) 2 + (πk BTe ) 2 ]
(5.16)
τee
(ε,Te ) ≈
for |ε − ε F |
ε F and Te
ε F /k B . Within
the constant
√ the free electron
q model,
4ε F ε S
m e e4
4ε F
−1
of proportionality De =
2 0 2 3/2 √
4ε F +ε S + tan
ε S , where the back4π~ ( b ) ε S

εF

ground dielectric constant 0b and the Thomas-Fermi screening energy scale ε S are
typically treated as empirical parameters.[123] Here, we extract De by fitting (5.16)
to the ab initio electron-electron scattering rates at room temperature T0 .[34] The
resulting fit parameters are shown in Table 5.1. We then estimate the total scattering
rates at other temperatures by adding (De /~)(πk B ) 2 (Te2 − T02 ) to the total ab initio
results (including electron-phonon scattering) at T0 .
Finally, we use the Kramers-Kronig relations to calculate Re( (ω,Te )) from Im( (ω,Te )).
Given the dearth of published temperature-dependent dielectric functions in the lit-

(ω/ωp)2 ε(ω)

a) Al

0.5

Im(ε) ab initio
Im(ε) experiment
Re(ε) ab initio
Re(ε) experiment

-0.5
-1

c) Au

1.5

(ω/ωp)2 ε(ω)

b) Ag

1.5
0.5
-0.5

0.5

-0.5

-1

d) Cu

1.5

Frequency [eV]

-1

Frequency [eV]

Figure 5.8: Predicted complex dielectric functions for (a) Al, (b) Ag, (c) Au, (d) Cu
at room temperature (300 K) compared with ellipsometry measurements.[36] The
y-axis is scaled by ω2 /ω2p in order to represent features at different frequencies such
as the Drude pole and the interband response on the same scale.
erature, we have published detailed tables of our ab initio predictions for electron
temperatures up to 8000 K, and spanning frequencies from the infrared to the ultraviolet, in the supplementary information of Reference [35].
Figure 5.8 compares the predicted dielectric functions with ellipsometry measurements[36] for a range of frequencies spanning from near-infrared to ultraviolet.
Note that we scale the y-axis by (ω/ω p ) 2 , where the free-electron plasma frequency
ω p = 4πe2 ne /me , in order to display features at all frequencies on the same scale.
We find excellent agreement for aluminum within 10% of experiment over the entire
frequency range, including the peak around 1.6 eV due to an interband transition.
The agreement is reasonable for noble metals with a typical error within 20%, but
with a larger error ∼ 50% for certain features in the interband d → s transitions due
to inaccuracies in the d-band positions predicted by DFT (especially for silver).
Figures 5.9, 5.11 and 5.13 show the ab initio dielectric functions at electron temperatures of 400, 1000, and 5000 K compared to those at 300 K. As electron temperature increases, features in the dielectric functions become increasingly broad. Note

(ω/ωp) ε(ω)

a) Al

0.5

(ω/ωp) ε(ω)

-0.5

1.5
0.5
-0.5
-1

c) Au

2.5
1.5
0.5
-0.5

Im(ε) Te=400K
Im(ε) Te=300K
Re(ε) Te=400K
Re(ε) Te=300K

b) Ag

d) Cu

1.5
0.5
-0.5

Frequency [eV]

-1

Frequency [eV]

Figure 5.9: Ab initio predicted complex dielectric functions for (a) Al, (b) Ag, (c)
Au, (d) Cu at 400 K compared to 300 K with the same scaling as in Figure 5.8.

a) Al

(ω/ωp) ∆ε(ω)

0.0004

0.004

-0.0002

-0.004

-0.0004

-0.008

0.0002

c) Au

(ω/ωp) ∆ε(ω)

0.004

b) Ag

0.008

Im(ε) ab initio
Im(ε) d→s model
Re(ε) ab initio
Re(ε) d→s model

d) Cu

0.004

0.002

-0.002

-0.002
-0.004

-0.004

Frequency [eV]

Figure 5.10: Change in the predicted complex dielectric function for (a) Al, (b)
Ag, (c) Au, (d) Cu from room temperature (300 K) to electron temperature Te =
400 K (with the lattice remaining at room temperature) with the same scaling as
in Figure 5.8. The analytical model given by expression 5.17, with fit parameters
summarized in Table 5.1, captures the essential features of the ab initio data for
noble metals at lower temperatures, but misses the contributions of broadening due
to electron-electron scattering at higher temperatures.

(ω/ωp) ε(ω)

a) Al

0.5

(ω/ωp) ε(ω)

-0.5

1.5
0.5
-0.5
-1

c) Au

2.5
1.5
0.5
-0.5

Im(ε) Te=1000K
Im(ε) Te=300K
Re(ε) Te=1000K
Re(ε) Te=300K

b) Ag

d) Cu

1.5
0.5
-0.5

Frequency [eV]

-1

Frequency [eV]

Figure 5.11: Ab initio predicted complex dielectric functions for (a) Al, (b) Ag, (c)
Au, (d) Cu at 1000 K and 300 K with the same scaling as in Figure 5.8.

a) Al

0.04

-0.002

-0.08

c) Au

0.02

Im(ε) ab initio
Im(ε) d→s model
Re(ε) ab initio
Re(ε) d→s model

-0.04

-0.004

(ω/ωp) ∆ε(ω)

b) Ag

0.08

0.002

(ω/ωp) ∆ε(ω)

0.004

d) Cu
0.02

0.01

-0.01

-0.02

Frequency [eV]

Figure 5.12: Change in the predicted complex dielectric function for (a) Al, (b)
Ag, (c) Au, (d) Cu from room temperature (300 K) to electron temperature Te =
1000 K (with the lattice remaining at room temperature) with the same scaling as
in Figure 5.8.

(ω/ωp) ε(ω)

a) Al

0.5

(ω/ωp) ε(ω)

-0.5

1.5
0.5
-0.5
-1

c) Au

2.5
1.5
0.5
-0.5

Im(ε) Te=5000K
Im(ε) Te=300K
Re(ε) Te=5000K
Re(ε) Te=300K

b) Ag

d) Cu

1.5
0.5
-0.5

-1

Frequency [eV]

Figure 5.13: Ab initio predicted complex dielectric functions for (a) Al, (b) Ag, (c)
Au, (d) Cu at 5000 K compared to 300 K with the same scaling as in Figure 5.8.

a) Al

0.2

-0.03

-0.2

-0.06

-0.4

c) Au

Im(ε) ab initio
Im(ε) d→s model
Re(ε) ab initio
Re(ε) d→s model

d) Cu

0.2

0.1

-0.1

(ω/ωp) ∆ε(ω)

0.2

b) Ag

0.4

0.03

(ω/ωp) ∆ε(ω)

0.06

-0.2

Frequency [eV]

-0.2

Frequency [eV]

Figure 5.14: Change in the predicted complex dielectric function for (a) Al, (b)
Ag, (c) Au, (d) Cu from room temperature (300 K) to electron temperature Te =
5000 K (with the lattice remaining at room temperature) with the same scaling as
in Figure 5.8.

86
that the change in dielectric function (relative to 300 K) is not noticeable on the
scale of the dielectric functions for all but the highest temperatures; we therefore
directly plot the change of the complex dielectric function upon increasing the electron temperature Te from room temperature to 400 K, 1000 K and 5000 K while the
lattice remains at room temperature in Figures 5.10, 5.12, and 5.14, respectively.
For all four metals, the response from infrared to ultraviolet frequencies is dominated by ‘sharp’ features due to interband transitions that broaden with increasing
temperature, rather than a change in the Drude response which would be the only
contribution in a free-electron model.
The strongest temperature dependence in noble metals results from transitions between the highest occupied d-band to the Fermi level near the L point, as shown
in Figure 5.15(a). Assuming a parabolic dispersion and a constant transition matrix element, and accounting for the change in the Drude response, this temperature
dependence can be modeled as[32, 150]

ω2p
∆ (ω) = −∆ 
 ω(ω + i(τ −1 + De (πk BTe ) 2 ))

Z ∞

A0
dε(1 − f (ε,Te ))
+K
 . (5.17)
m∗v
(~ω) 2 −ε c
(~ω
(ε
))
(ε
c 
m∗c
The denominator in the second term captures the joint density of states for transitions between the bands, and the numerator counts unoccupied states near the Fermi
level, which introduces the temperature dependence. Above, K fills in the real part
of the dielectric function, given the imaginary part using the Kramers-Kronig relation.
Table 5.1 lists the parameters for the parabolic band approximation obtained from
the ab initio band structures. Figure 5.10 shows that this approximation captures
the correct shape of ∆ε(ω) for small changes in Te . However, this model underestimates the Te dependence for higher electron temperatures because it ignores the
quadratic increase in broadening of the electronic states due to increased electronelectron scattering, as Figures 5.12 and 5.14 show. Aluminum exhibits a sharp
change in the dielectric function around ~ω ≈ 1.5 eV, which results from several
transitions to/from the Fermi level near the W point as Figure 5.15(b) shows. Additionally, two of the involved bands are not parabolic, making it difficult to construct
a simple model like expression 5.17. Therefore, simplified models are adequate
for qualitative analysis of lower temperature excitation experiments in noble met-

ε - εF [eV]

-4

-2

(a) Noble metals

(b) Aluminum

Figure 5.15: Critical interband transitions determining the ‘sharp’ features in the
dielectric function change for (a) noble metals (gold shown; similar shapes for silver and copper) and (b) aluminum. A parabolic band model around the L point
(parameters in Table 5.1) approximates the critical transition in noble metals. This
is difficult in aluminum because of four such transitions in a narrow energy range
≈ 1.3 − 1.6 eV.
als,[32] but ab initio dielectric functions are necessary for a quantitative analysis of
higher temperature experiments and a wider range of materials and probe frequencies.
5.7

Conclusions

In this chapter we presented Ab initio calculations of electron-phonon coupling,
electron and lattice heat capacities, and dielectric functions and show qualitative
differences from free-electron and previous semi-empirical estimates because of the
substantial energy dependence of electron-phonon matrix elements and electronic
density of states. These changes are particularly important for gold and copper at
transient electron temperatures greater than 2000 K because of the change in occupations of the d-bands situated ∼ 2 eV below the Fermi level in these metals. Our
ab initio temperature dependent dielectric function is, to our knowledge, the first of
its kind and a great improvement over the free-electron based models and parabolic
band approximations that are commonly used. We show that while simple models
can account for some of the qualitative features of the change in dielectric function
for small changes in temperature, ab initio treatment is essential to quantitatively
account for the complete frequency and temperature dependence, including effects
such as carrier linewidth broadening and transitions between multiple non-parabolic
bands. The temperature dependence of the optical response is, in particular, important for a wide range of applications beyond understanding ultrafast transient

88
measurements.
This work has direct implications for analysis of experimental pump-probe studies
of metal nanostructures, such as those discussed in Chapter 6. The ab initio material
properties predicted in this Chapter allow a parameter-free description of the spectra obtained in transient absorption studies since we implicitly account for all the
microscopic processes in the non-equilibrium dynamics of electrons in plasmonic
metals.

89
Chapter 6

EXPERIMENTAL AND AB INITIO ULTRAFAST CARRIER
RELAXATION IN PLASMONIC NANOPARTICLES
6.1

Motivation and Background

Plasmonic hot carriers provide tremendous opportunities for combining efficient
light capture with energy conversion and catalyst activity at the nano scale.[83,
151, 152] Hot carriers can be used to directly drive chemical reactions at the metal
surface, or they can be transferred to a semiconductor for use in photovoltaics[153,
154] and photoelectrochemical systems.[155, 156] Dynamics of hot carriers are
typically studied via ultrafast pump-probe measurements of plasmonic nanostructures.[122–128, 157] Ultrafast transient absorption measurements have been used
to investigate microscopic processes on femtosecond time scales, such as plasmon
decay and hot electron relaxation through electron-electron, electron-surface, and
electron-phonon scattering.[17, 119, 123, 130, 131, 158–162]
In ultrafast transient absorption measurements on metals, a laser pulse, referred to
as the pump pulse, is used to excite a plasmon on the surface of the metal (see Figure 6.1). The plasmon can then decay via absorption by an electron and produce

Figure 6.1: Ultrafast transient absorption measurement setup. A pump pulse excites
the sample. A white light probe pulse is delayed with respect to the pump pulse and
collected by a spectrometer. By varying the pump-probe delay time, the absorption
or extinction of the sample may be monitored as a function of time.

90
a hot electron-hole pair. This process of plasmon decay typically occurs within 10
fs[17]; an ab initio description of this process was discussed in Chapter 4. The
excited high-energy nonthermal electron loses energy to other electrons through
electron-electron scattering and heats the thermal electron bath. Electron thermalization has been shown to be on the order of 500 fs in the noble metals, in contrast
to previous assumptions of instantaneous thermalization.[17, 32, 159, 163] The hot
electron bath then equilibrates with the phonon bath via electron-phonon scattering on a time scale of a few picoseconds.[17, 119, 164, 165] Ab initio descriptions
of the processes of electron-electron and electron-phonon thermalization were discussed in Chapter 5. In ultrafast absorption measurements, these processes are
monitored using a low-power laser pulse (probe pulse) which is delayed in time with
respect to the pump pulse. The probe and pump pulse duration are on a time scale
much shorter than that of electron-electron and electron-phonon thermalization. By
changing the delay between the pump and probe pulse, the optical response of the
sample can be mapped over time, and from this we can draw conclusions about the
electron and phonon dynamics which produce the observed optical changes.
Ultrafast pulsed lasers produce much larger excitations than their continuous counterparts. For example, an absorbed 1 ps pulse of 1 µJ deposits an energy density
which is six orders of magnitude larger than that of a 1 W continuous laser per picosecond, and it is reasonable to assume that this 1 ps pulse in the metal will excite
on the order of one conduction electron per atom.[147] It is this capacity for strong
excitation and femtosecond time resolution that has allowed researchers to investigate the microscopic processes of electron and phonon relaxation after plasmon
decay.
Figure 6.2 shows a representative map of the optical signal (differential extinction
cross section) as a function of pump-probe delay time and probe wavelength. Taking a slice of the map at one probe wavelength reveals the temporal behavior of
the electron relaxation (Fig. 6.3(b)). The typical temporal behavior is characterized
by an initial fast rise (100’s fs) attributed to electron-electron scattering that converts fewer high-energy excited carriers into several more lower-energy carriers,
followed by a slower decay (1-10 ps) attributed to electron-phonon scattering. By
properly choosing the probe wavelength at which to monitor the temporal response,
different mechanisms can be investigated, and studies have taken advantage of this
to investigate electron-electron scattering, electron-phonon coupling, electron-spin
relaxation, and electronic transport.[32, 123, 125, 131, 166–169]

Figure 6.2: Map of the differential extinction cross section as a function of pumpprobe delay time and probe wavelength for a pump pulse of 68 µJ/cm2 energy
density at 380 nm. At time 0 ps, the pump pulse excites the sample. As the electrons
thermalize, extinction near the absorption peak (533 nm) decreases while extinction
in the wings to either side of the absorption peak increases. After ∼ 500 fs, the
electrons began to thermalize with the lattice and the differential extinction decays.
A contour line is drawn in black at zero extinction change.
Taking a slice of the map at one time gives the spectral response, as shown in
Figure 6.3(a) for a set of times relative to the delay time with maximum signal,
t max = 700 fs. In noble metals excited by a pump pulse, the largest changes in the
electron occupation occur near the interband transition threshold, and consequently
the changes in the dielectric function and thus the differential optical signal are
largest near these energies.[35] The highest sensitivity to the electron temperature
is achieved by choosing a probe wavelength near this energy.[147]
Transient absorption measurements of noble metal nanoparticles with a plasmon
resonance will display a spectral feature around the plasmon resonance energy, in
addition to the usual feature around the interband transition threshold. The differential signal associated with the plasmon resonance is a result of the enhancement
of the nonlinear optical response by dielectric confinement around the plasmon res-

600

∆ CExt [nm2]

-600
tmax - 200fs
tmax
tmax + 500fs
tmax + 1ps
tmax + 2ps

-1200
-1800
450

600

-600
480 nm
510 nm
530 nm
550 nm
590 nm

-1200
-1800

500
550
600
Probe Wavelength [nm]

650

Delay Time [ps]

Figure 6.3: Spectral and kinetic traces from differential extinction map in Figure 6.2, for a pump pulse of 68 µJ/cm2 energy density at 380 nm. Differential
extinction (a) as a function of probe wavelength at various times relative to the
pump-probe delay time with maximum signal, t max = 700 ps; and (b) as a function
of pump-probe delay time at various probe wavelengths.

2.6

Photon Energy [eV]
2.4
2.2

0.6

Extinction

0.5

-hωp

0.4

-hωib

0.3
0.2
0.1
450

500

550
600
Wavelength [nm]

650

Figure 6.4: Extinction spectra of 60 nm diameter Au colloids in water. The positions of the plasmon resonance, ~ω p , at 2.33 eV and the interband transition threshold, ~ωib , at 2.4 eV are indicated.

93
onance.[141] For the 60-nm-diameter Au nanoparticles used in the measurements
presented in this Chapter, the plasmon resonance (533 nm) is close to the interband
transition energy (see Figure 6.4), and the spectral features due to the interband transition and plasmon resonance overlap. The overall spectral shape of the differential
optical signal seen in Figure 6.3 (a) can be understood as a thermal broadening of
the plasmon absorption resonance, which is the dominant absorption mechanism
in the visible wavelength range considered here. The thermal broadening of the
resonance results in a decrease in extinction (negative signal) near the plasmon resonance, and an increase in extinction (positive signal) in the wings to either side of
the peak.
Recent literature has focused on the contributions of thermalized and nonthermalized electrons to the optical signal in ultrafast pump-probe measurements.[32, 123,
141, 159, 160, 163] However, a detailed ab initio understanding of thermal and nonthermal carrier dynamics in conjunction with theory-directed experimental studies
has been conspicuously absent. This Chapter addresses the excitation and relaxation dynamics of hot carriers in metals across timescales ranging from 10 fs–10 ps
with a parameter-free description of the optical signature.
This Chapter is organized as follows. We start with the theoretical background and
computational methods used to calculate the ultrafast pump-probe response which
includes the optical response of the metal and the dynamics of the excited carriers,
including electron-electron and electron-phonon scattering (Section 6.2). To accurately predict the transient optical response of metal nanostructures without using
empirical parameters, we account for the electron-temperature dependence of the
electronic heat capacity, electron-phonon coupling factor, and dielectric functions,
from fully ab initio calculations using the method presented in Chapter 5. In Section 6.3, we describe the experimental details and methodology of the pump-probe
measurements performed. We then present the key results of this Chapter: the experimental and theoretical dependence of signal amplitude and temporal behavior
on pump power and probe wavelength. We show that the experimental trends are
captured by our theoretical results and we analyze these trends in terms of contributions from thermal and nonthermal electron populations to the optical signal.
6.2 Ab initio theory
Theoretically describing pump-probe measurements of hot carrier dynamics in plasmonic systems involves two major ingredients. First, the optical response of the

94
0.3

t=0
100 fs
700 fs
2 ps
4 ps

0.2

∆ f (ε)

0.1
-0.1
-0.2
-0.3
-2

-1

ε - εF [eV]

Figure 6.5: Difference of the predicted time-dependent electron distribution from
the fermi distribution at 300 K, induced by a pump pulse at 560 nm with intensity
of 110 µJ/cm2 . Starting from the carrier distribution excited by plasmon decay
at t = 0, electron-electron scattering concentrates the distribution near the Fermi
level with the peak optical signal at ∼700 fs, followed by a return to the ambienttemperature Fermi distribution and a decay of the optical signal due to electronphonon scattering.
metal (and its environment) determines the excitation of carriers by the pump as
well as the subsequent signal measured by the probe pulse. Second, the dynamics
of the excited carriers, including electron-electron and electron-phonon scattering,
determines the time dependence of the probe signal. In Chapter 5 we presented ab
initio theory and predictions for both the optical response and the dynamics within
a two temperature model, where the electrons are assumed to be in internal equilibrium albeit at a different temperature from the lattice. Here, we additionally treat the
response and relaxation of non-thermal electron distributions from first principles,
without assuming an effective electron temperature at any point.
For the optical response, we calculate the imaginary part of the dielectric function Im (ω) accounting for direct interband transitions, phonon-assisted intraband
transitions and the Drude (resistive) response, and calculate the real part using the
Kramers-Kronig relations. Specifically, we start with density-functional theory calculations of electron and phonon states as well as electron-photon and electronphonon matrix elements using the JDFTx code,[101] convert them to an ab initio
tight-binding model using Wannier functions,[106] and use Fermi Golden rule and
linearized Boltzmann equation for the transitions and Drude contributions respectively. The theory and computational details for calculating (ω) are presented in

95
detail in Chapters 4 and 5. All these expressions are directly in terms of the electron
occupation function f (ε), and we can straightforwardly incorporate an arbitrary
non-thermal electron distribution instead of Fermi functions.
We use the ab initio metal dielectric function for calculating the initial carrier distribution as well as the probed response. The initial carrier distribution following
the pump pulse is given by
f (ε,t = 0) = f 0 (ε) + U

P(ε, ~ω)
g(ε)

(6.1)

where f 0 is the Fermi distribution at ambient temperature T0 , U is the pump pulse
energy absorbed per unit volume, g(ε) is the electronic density of states (see Ch. 5
Section 5.3), and P(ε, ~ω) is the energy distribution of carriers excited by a photon of energy ~ω (see Ch. 4). We then evolve the carrier distributions and lattice
temperature in time to calculate f (ε,t) and Tl (t) as described below. From those,
we calculate the variation of the metal dielectric function (ω,t), and in turn, the
extinction cross section using Mie theory.[25, 26, 170] To minimize systematic errors between theory and experiment, we add the ab initio prediction for the change
in the dielectric function from ambient temperature (see Ch. 5 Section 5.6), to the
experimental dielectric functions from ellipsometry.[36]
We calculate the time evolution of the carrier distributions using the nonlinear
Boltzmann equation
f (ε,t) = Γe−e [ f ](ε) + Γe−ph [ f ,Tl ](ε),
dt

(6.2)

where Γe−e and Γe−ph , respectively, are the contributions due to electron-electron
and electron-phonon interactions to the collision integral. For simplicity, we assume
that the phonons remain thermal at an effective temperature Tl (t) and calculate the
time evolution of the lattice temperature using energy balance,
dE
dTl
= −Cl (Tl )
dt e−ph
dt

(6.3)

where the term on the left corresponds to the rate of energy transfer from the lattice
to the electrons due to Γe−ph , and Cl is the ab initio lattice heat capacity (see Ch. 5
Section 5.3).
The ab initio collision integrals are extremely expensive to calculate repeatedly for
directly solving expression 6.2. We therefore use simpler models for the collision
integrals parametrized using ab initio calculations. For electron-electron scattering

96
in plasmonic metals, the calculated electron lifetimes exhibit the inverse quadratic
energy dependence τ −1 (ε) ≈ (De /~)(ε −ε F ) 2 characteristic of free electron models
within Fermi liquid theory (see Ch. 5 Section 5.6). We therefore use the freeelectron collision integral for initial electron states with energies ε, ε 1 and final
electron states with energies ε 2 , ε 3 ,[32, 123, 171]

g(ε 1 )g(ε 2 )g(ε 3 )
g 3 (ε F )
× δ(ε + ε 1 − ε 2 − ε 3 ) f (ε 2 ) f (ε 3 )(1 − f (ε))(1 − f (ε 1 ))

2De
Γe−e [ f ](ε) =

dε 1 dε 2 dε 3

− f (ε) f (ε 1 )(1 − f (ε 2 ))(1 − f (ε 3 ))

(6.4)

with the constant of proportionality De extracted from ab initio calculations of electron lifetimes (see Appendix Section C).
We calculate the electron-phonon collision integral for the interaction of an arbitrary
hot electron distribution, f (ε), with a thermal phonon distribution n(ω,Tl ), given
by the Bose distribution at lattice temperature Tl . We start with the rate of energy
transfer between the electrons and lattice per unit volume, which is exactly as in
Chapter 5 expressions 5.7 and 5.8, except that we allow f (ε) to be an arbitrary
distribution instead of restricting it to a Fermi distribution:
2π
dE
dt e−ph

Ωd ~kd ~k 0 X
δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )~ω~k 0−~k,α
BZ (2π)
n 0 nα
~0 ~

(6.5)

× g~k −~k,α
S(ε~kn , ε~k 0 n 0 ,ω~k 0−~k,α )
0 0
kn,k n

2π

Ωd ~kd ~k 0 X
dε
δ(ε~kn − ε)δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )~ω~k 0−~k,α
BZ (2π)
n 0 nα

~0 ~

× g~k −~k,α
S(ε, ε + ~ω~k 0−~k,α ,ω~k 0−~k,α ),
0 0
kn,k n

(6.6)
where Ω is the unit cell volume, ~ω~k 0−~k,α is the energy of a phonon with wave-vector
~0 ~

q~ = ~k 0 − ~k and polarization index α, and g~k0 −0k,α
is the ab initio electron-phonon
k n ,kn

matrix element coupling this phonon to electronic states indexed by ~kn and ~k 0 n0,
described in Chapter 4, Section 4.6. The occupation term S(ε, ε0,ω) is defined as
S(ε, ε0,ω) ≡ f (ε)n(ω)(1 − f (ε0 )) − (1 − f (ε))(1 + n(ω)) f (ε0 ).

(6.7)

Using energy conservation and assuming that phonon energies are negligible on the
electronic energy scale (an excellent approximation for optical frequency excita-

97
tions in metals), we simplify the occupation term
S(ε, ε0,ω) = f (ε)n(ω)(1 − f (ε + ~ω))
− (1 − f (ε))(1 + n(ω)) f (ε + ~ω)
∂f
≈ f (ε)n(ω) 1 − f (ε) −
~ω
∂ε
∂f
− (1 − f (ε))(1 + n(ω)) f (ε) +
~ω
∂ε
∂f
= − f (ε)(1 − f (ε)) −
~ω[1 + n(ω) − f (ε)]
∂ε
k BTl
∂f
~ω
≈ − f (ε)(1 − f (ε)) −
∂ε
~ω
∂f
= −
k B (Tl − Teff [ f (ε)]).
∂ε

(6.8)

(6.9)

(6.10)
(6.11)
(6.12)

The effective electron temperature Teff is defined as
Teff [ f (ε)] ≡

f (ε)(1 − f (ε))
∂f ,
k B − ∂ε

(6.13)

which is exactly Te when f (ε) is a Fermi distribution.
Substituting equation 6.12 into equation 6.6 gives
2π
dE
Ωd ~kd ~k 0 X
dε
δ(ε~kn − ε)δ(ε~k 0 n 0 − ε~kn − ~ω~k 0−~k,α )
dt e−ph
(2π) 6 n 0 nα
∂f
~k 0 −~k,α 2
× ~ω~k 0−~k,α g~ ~ 0 0 −
k B (Tl − Te f f [ f (ε)])
kn,k n
∂ε
(6.14)
∂f
2πk B
dε −
H (ε)(Tl − Teff [ f (ε)]),
(6.15)
∂ε
where we define the electron-phonon coupling strength term, H (ε) as
Ωd ~kd ~k 0 X
~k 0 −~k,α 2
H (ε) ≡
δ(ε
ε)δ(ε
~ω
)~ω
~kn
~k 0 n 0
~kn
~k 0 −~k,α
~k 0 −~k,α ~ ~ 0 0
kn,k n
(2π) 6 n 0 nα
(6.16)
or, in terms of the electron-phonon coupling strength h(ε) defined in Chapter 5,
H (ε) =

g 2 (ε)
h(ε).
2g(ε F )

(6.17)

To find the electron-phonon collision integral Γe−ph ( f (ε,t),Tl ) = d fdt(ε)
we
e−ph
note that the contribution to dE/dt (expression 6.3) from electrons with energy ε

98
corresponds to energy exchange between the lattice and electrons of energy ε + ~ω,
where ~ω is negligible on the energy scale of the electrons. Therefore we can equate
the energy flow from the electrons to the lattice to an energy flow from electrons
with energy ε to electrons with energy ε + dε:
∂f
d f (ε + dε)
d f (ε)
dε −
H (ε)(Tl − Teff [ f (ε)]) = g(ε + dε)
− g(ε)
(6.18)
∂ε
dt
dt
∂f
d f (ε)
H (ε)(Tl − Teff [ f (ε)]) =
[g(ε)
].
(6.19)
∂ε
∂ε
dt
Integrating by parts over ε yields the desired collision integral
−1 ∂
∂f
d f (ε)
H (ε)(Tl − Teff [ f (ε)]) .
Γe−ph ( f (ε,t),Tl ) =
dt e−ph g(ε) ∂ε
∂ε
(6.20)
Expressions 6.4, 6.20, and 6.3 give us the time derivatives of the electron occupations and lattice temperature, allowing us to evolve expression 6.2 in time and
numerically solve for the electron distribution as a function of time. We use the
results for the electron distribution at an array of points in time to calculate the
ab initio dielectric function, as described in Chapter 5 Section 5.6, at each time.
It is then simple to obtain the differential cross section as a function of time, and
compare this to experimental results.
Combining the above first-principles calculations of carrier dynamics and optical
response produces a complete theoretical description of pump-probe measurements,
free of any fitting parameters that are typical in previous analyses.[121, 139, 150,
160] This theory accounts for detailed energy distributions of excited carriers (Figure 6.5) instead of assuming flat distributions,[141, 159, 171] and accounts for
electronic-structure effects in the density of states, electron-phonon coupling and
dielectric functions beyond the empirical free-electron or parabolic band models
previously employed.[31, 32, 121, 123, 132, 139, 141, 144, 150, 171] We use the ab
initio energy dependent electron-phonon matrix elements, as opposed to the common practice of replacing the matrix element with an empirical constant, which we
showed in Chapter 5 Section 5.5 to be a poor approximation.[31, 132, 144, 171]
All of these advancements result in a model that is more accurate than previous
models, and importantly is free of any fitting parameters, which can easily cause
a cancelation of errors and omissions in a model and obscure the correct physical
interpretation of experimental data.

99
6.3

Experimental Methods

We use a ultrafast transient absorption system with a tunable pump and white light
probe to measure the extinction of Au colloids in solution as a function of pumpprobe delay time and probe wavelength.
The laser system consists of a regeneratively amplified Ti:sapphire oscillator (Coherent Libra), which delivers 1mJ pulse energies centered at 800 nm with a 1 kHz
repetition rate. The pulse duration of the amplified pulse is approximately 50 fs.
The laser output is split by an optical wedge to produce the pump and probe beams.
The pump beam wavelength is tuned using a coherent OperA OPA. The probe beam
is focused onto a sapphire plate to generate a white-light continuum probe. The
time-resolved differential extinction spectra are collected with a commercial Helios
absorption spectrometer (Ultrafast Systems LLC). The temporal behavior is monitored by increasing the path length of the probe pulse and delaying it with respect
to the pump pulse with a linear translation stage capable of step sizes as small as 7
fs. Our sample is a solution of 60-nm-diameter Au colloids in water with a concentration of 2.6 × 1010 particles per milliliter (BBI International, EM.GC60, OD1.2)
in a quartz cuvette with a 2 mm path length.
6.4

Results

The initial excitation by the pump pulse generates an electron distribution that is far
from equilibrium, for which temperature is not well-defined. Our ab initio predictions of the carrier distribution at t = 0 in Figure 6.5 exhibits high-energy holes in
the d-bands of gold and lower energy electrons near the Fermi level. These highly
non-thermal carriers rapidly decay within 100 fs, resulting in carriers closer to the
Fermi level which thermalize in several 100 fs, reaching a peak higher-temperature
thermal distribution at ∼ 700 fs in the example shown in Figure 6.5. These thermalized carriers then lose energy to the lattice via electron-phonon scattering over
several picoseconds.
The conventional two-temperature analysis is only valid in that last phase of signal
decay (beyond 1 ps) once the electrons have thermalized; non-thermal character
of the electrons plays an important role in the initial rise of the response at earlier
times.[32] This initial response consists of two contributions mentioned above: the
short-lived highly non-thermal carriers excited initially and the longer-lived thermalizing carriers near the Fermi level.[160]
These two contributions introduce distinct temporal and spectral signatures. Due

100
to the smaller lifetimes of higher energy carriers (due to higher electron-electron
scattering rates), the non-thermal carriers exhibit faster rise and decay times than
the thermal carriers closer to the Fermi level.[32, 160] Additionally, the larger energy range of the carrier distribution produces a broader spectral contribution (as a
function of probe wavelength) from non-thermal electrons, compared to the thermal electrons which contribute primarily near the resonant d-band to Fermi level
transition.[32, 35] We can therefore detect response of non-thermal electrons at low
pump powers, where thermal contributions are much smaller due to lower temperature changes, and at probe wavelengths far from the interband resonance, where
thermal electrons contribute less. Combining ab initio predictions and experimental measurements of 60-nm colloidal gold solutions, we quantitatively identify these
signatures of thermal and non-thermal electrons, first as a function of pump power
and then as a function of probe wavelength.
Pump power dependence
Increasing the pump power generates a greater number of initial carriers, producing
a higher thermalized electron temperature, which increases the overall amplitude
of the measured signal, but also increases electron-electron thermalization times
and reduces electron-phonon thermalization times. Figure 6.6 shows that our ab
initio predictions of electron dynamics and optical response quantitatively capture
the absolute extinction cross section as a function of time for various pump pulse
energies. In the remainder of this section, we examine the cross section time dependence normalized by their peak values to more clearly observe the changes in rise
and decay time scales.
Decay of the measured signal is because of energy transfer from electrons to the
lattice via electron-phonon scattering. At higher pump pulse energies, the electrons
thermalize to a higher temperature. For Te < 2000 K, the electron heat capacity
increases linearly with temperature, whereas the electron-phonon coupling strength
does not appreciably change with electron temperature.[35, 123] Therefore, the
electron temperature, and correspondingly the measured probe signal, decays more
slowly at higher pump powers as shown in Figure 6.7(a,b). Again, we find quantitative agreement between the measurements and ab initio predictions with no empirical parameters.
Rise of the measured signal arises from electron-electron scattering which transfers
the energy from few excited non-thermal electrons to several thermalizing electrons

101

∆ CExt [nm2]

21 µJ/cm

-400

-800

-1200

110

-1600

Delay Time [ps]

∆ CExt (normalized)

Figure 6.6: Measured (circles) and computed (solid line) differential extinction
cross section for a pump pulse at 560 nm with intensity of 21, 34, 68, and
110 µJ/cm2 , and monitored with a 530 nm probe wavelength. As pump power
is increased, the signal amplitude increases because of increasing perturbation of
the electron distribution and higher peak electron temperatures.

0.8

110 µJ/cm
68 µJ/cm2
34 µJ/cm
21 µJ/cm

0.6
0.4

0.2

∆ CExt (normalized)

Delay Time [ps]

0.8
0.6
0.4
0.2

0.1 0.2 0.3 0.4
Delay Time [ps]

Figure 6.7: The (a) measured and (b) calculated data from Fig. 6.6 normalized
to unity. Parts (a) and (b) show that the signal decay slows as pump intensity is
increased. Parts (c) and (d) show the data from parts (a), (b) in the first 0.5 ps after
excitation, and demonstrate the faster rise time at higher pump intensities.

∆ CExt (normalized)

102

0.5
-0.5
-1
-0.2

110 µJ/cm2
34 µJ/cm2

0.2 0.4 0.6
Delay Time [ps]

Figure 6.8: The (a) measured and (b) calculated differential cross section for a pump
pulse at 380 nm with intensity of 34 and 110 µJ/cm2 and monitored at 560 nm probe
wavelength, normalized to unity. Contributions from the nonthermalized electron
distribution dominate at lower pump power, resulting in a fast signal rise and decay,
while contributions from the thermalized electron distribution dominate at higher
pump powers, resulting in slower signal rise and decay. The small signals at this
probe wavelength result in the higher noise to signal ratio seen in part (a).
closer to the Fermi level. Higher power pump pulses generate a greater number
of initial non-thermal carriers, requiring fewer electron-electron collisions to raise
the temperature of the background of thermal carriers. Additionally, the electronelectron collision rate increases with temperature because of increased phase space
for scattering.[123] Both these effects lead to a faster rise time at higher pump
powers, as seen in the measurements shown in Figure 6.7(c), as well as in the ab
initio predictions shown in Figure 6.7(d), which are in quantitative agreement.
Finally, we examine the variation of the ratio of thermal and non-thermal electron
contributions with pump power. Figure 6.8 shows the sub-picosecond variation of
measured response for two different pump powers, but now with a pump wavelength
of 380 nm with a higher energy photon that excites non-thermal carriers further
from the Fermi level. Additionally, the probe wavelength of 560 nm is far from
the interband resonance at ∼ 520 nm, so that the thermal electrons contribute less
to the measured response. For the lower pump power, the thermal contribution is
small at this probe wavelength making the non-thermal contribution relatively more
important, resulting in a faster rise and decay time. At higher power, the thermal
contribution response broadens, changing the ratio between the nonthermal and
thermal contributions at this probe wavelength. The thermal contribution dominates
at higher power, resulting in a slower signal rise and decay time. Once again, the

103
measurements and ab initio calculations, which include all these effects implicitly,
are in quantitative agreement.
Probe wavelength dependence
The contributions of the nonthermal and thermal electron populations to the optical
signal vary as a function of probe energy, as discussed above. Thus, the temporal behavior of the nonthermalized and thermalized electron signal contributions
can be separated by properly selecting the probe energy with which to monitor the
electron-relaxation process. At probe energies far away from the interband transition energy the contribution from thermalized electrons is reduced and the short
time response of the metal (<1 ps) is dominated by the nonthermal character of the
electron distribution. These probe energies allow for direct and sensitive observation of the internal thermalization of the electron gas. For probe energies close to the
interband transition energy, the signal contributions from the thermalized electrons
dominate, and the signal rise and decay is much slower than that at probe energies
where the nonthermal contributions dominate. At these probe energies close to the
interband transition energy, only the distribution change close to the Fermi level is
detected because of the high transition probability to those states, and although the
signal at these probe energies a priori detects the electron thermalization over the
full excited region, the signal is dominated by electron-electron collisions close to
the Fermi level. Because of Pauli exclusion effects, these are the slowest scattering
processes involved in the internal thermalization and result in the slow rise time
of the signal– a longer "local" thermalization time, as predicted by Fermi-liquid
theory.[32, 123]
Figure 6.9 shows the measured and theoretical response of the gold nanoparticles
excited with a pump pulse at 560 nm with an intensity of 110 µJ/cm2 and monitored
at a probe energy near (530 nm) and far (620 nm) from the interband transition
threshold. As expected, where the thermalized distribution contribution dominate
at 530 nm, the measured signal reaches its maximum at a later time and exhibits a
slower decay than where the nonthermal contributions dominate at 620 nm. This
behavior is captured by the theoretical results in part (b) of the Figure.
Figure 6.10 shows data with the same excitation parameters as those in Figure 6.9
for a variety of probe wavelengths. Our theoretical results are in qualitative agreement with the temporal behavior of the signal at the various probe wavelengths.
Both the measured and theoretical data show the fastest signal rise and decay when

∆ CExt (normalized)

104

0.8

520 nm
620 nm

0.6
0.4
0.2

Delay Time [ps]

∆ CExt (normalized)

Figure 6.9: The (a) measured and (b) calculated differential cross section for a
pump pulse at 560 nm with an intensity of 110 µJ/cm2 normalized to unity, for
probe wavelengths of 520 and 620 nm. Where contributions from the thermalized
distribution dominates at 530 nm, the measured signal reaches its maximum at a
later time and exhibits a slower decay than where the nonthermal distribution contributions dominate at 620 nm.

0.8

480nm
510nm
620nm

0.6
0.4
0.2
-0.2

1 2 3 4 5
Delay Time [ps]

Delay Time [ps]

Figure 6.10: The (a) measured and (b) calculated differential cross section for a
pump pulse at 560 nm with an intensity of 110 µJ/cm2 normalized to unity, for
probe wavelengths of 480, 510, and 620 nm.
the probe is furthest from the interband transition energy (620 nm probe). The
crossing points of the data traces and the relative positions, especially at long delay
times, are in good agreement for the experimental and theoretical results.
6.5

Conclusions

In this Chapter, we presented experimental and theoretical results that explored
the effects of pump power and probe wavelength on the spectral and temporal
behavior of the transient extinction of 60 nm Au colloids in solution. We com-

105
bined first-principles calculations of carrier dynamics and optical response from
Chapters 4 and 5 to achieve an improved nonlinearized Boltzmann method that
fully accounts for the nonthermal character of the exited electron distribution and
electronic-structure effects. The theoretical calculations use our ab initio results for
the excited electron distribution produced via plasmon decay, electronic density of
states, and electron-temperature dependent electron-phonon coupling and dielectric
function to replace approximations of these parameters used by previous investigations. We presented experimental and theoretical results which investigated the
contributions from thermal and non-thermal electrons to the optical signal, as function of both pump power and probe wavelength. Our ab initio predictions captured
the signature temporal and spectral features seen experimentally for these two contributions. This complete theoretical description of hot carrier dynamics will be
useful as researchers investigate new roles for hot carriers in technologies ranging
from energy conversion and photocatalysis to photodetection.

106
Chapter 7

SUMMARY AND OUTLOOK
The work presented in this thesis focused on the classical relationship between
charge density and the plasmon resonance and on a quantum description of plasmon decay and hot carrier relaxation.
The first investigation discussed in this thesis laid out a thermodynamic framework which predicts optically induced electrochemical potentials on plasmonic resonators. We showed experimental and optical measurements which showed evidence of the predicted plasmoelectric effect. Future work to find further evidence
of a plasmoelectric response could involve repeating the optical experiments detailed in Chapter 2 with Au colloids suspended in a gel electrolyte that is highly
polarizable. In our experiments, we used Au colloids suspended in water, as received from the supplier. We did not add electrolytes to the solution because this
would cause the particles to aggregate and precipitate out of solution. However, a
gel electrolyte matrix could be used to immobilize the particles as suspended single
particles and the high polarizability of the matrix would allow increased charging
of the particles and a larger plasmoelectric response. Other optical plasmoelectric
experiments could include monitoring the change in extinction of Au colloids under white light illumination (as a control) and under white light illumination plus
a single monochromatic optical pump to either side of the neutral plasmon resonance. Under these conditions, measuring a blue-shift of the extinction profile with
a pump to the blue of the resonance, and a red-shift of the extinction profile with a
pump to the red of the resonance would provide further evidence of a plasmoelectric
response.
Future applications could use a plasmoelectric system to convert optical power to
electrical power by engineering a system with two components: one that will take
on a decreased charge density and one that will take on an increased charge density,
and connecting the components to produce a voltage difference. For example, this
could be done by using two components with the same resonance and illuminating
one with light to the blue of the resonance and the other with light to the red of the
resonance, or by using one color of light to illuminate two components– one with a
resonance to the blue of the illumination wavelength and another with a resonance

107
to the red of the illumination wavelength. This would allow for a new avenue for
harnessing the power of the sun as a renewable energy source using an all-metal
system.
The second investigation presented in this thesis explored the non-ideal effects
which play a role in the electrical tuning of the optical properties of Au nanoparticles. We developed simulations that used a modified dielectric function to account
for voltage-dependent changes in the system, and used these simulations to quantify
the contributions of various effects to the changes in the optical properties. Tunable
plasmonic systems have a multitude of applications, including applications in energy conservation. For example tunable plasmonic systems consisting of plasmonic
resonators dispersed in a transparent conductive medium are currently being investigated as a means of controlling the amount of near-infrared and visible light that
is transmitted through windows.[172, 173] These technologies could decrease the
energy requirements for air-conditioning. The simulations methods discussed in
Chapter 3 could be used to analyze and optimize tunable plasmonic systems for
such applications.
In the last three projects discussed in this thesis we preformed ab initio calculations
to achieve a complete theoretical description of plasmon decay and the relaxation of
plasmonicaly excited hot carriers. We showed that the electronic structure and the
role of phonons, which have been previously ignored, are critical for an accurate
description of plasmon decay. We also explored the effect of surfaces and geometric confinement in nanoscale systems. Our fitting-parameter-free theory, which
we have shown to be in good agreement with experimental results, is a significant
advancement over previously used models that relied on empirical parameters and
fitting-parameters to achieve agreement with experimental results.[121, 139, 150,
160] Moreover, these previously used empirical and fitting parameters may obscure
the correct physical interpretation of experimental data.
Currently, researchers are investigating the use of plasmonicaly excited hot carriers as catalysts for photochemistry such as the dissociation of hydrogen gas and
reduction of carbon dioxide, as well as collection of hot carriers across a metalsemiconductor boundary to avoid the loss of absorption at energies below the bandgap of the semiconductor.[33, 156, 174, 175] These applications take advantage of
the excellent light absorption and confinement provided by plasmons, but very low
quantum efficiencies have been achieved so far and it is experimentally difficult to
separate effects of hot carriers from overall local heating, making it hard to use ex-

108
perimental results to optimize efficiency. For improved efficiencies in applications,
we must collect or use hot carriers before they thermalize. Our complete theoretical description of plasmon decay and hot carrier relaxation from first-principles
gives us a better understanding that can be used to aid advancement of hot carrier
applications in photocatalysis, energy conversion, and photodetection.
The next steps in this work would be to combine our theoretical description of
hot carrier generation and relaxation with a theoretical description of hot carrier
transport to provide a complete picture of temporal and spatial dynamics that could
be used to improve the efficiency of hot carrier applications. In this way, one could
investigate the interplay of material electronic structure and nanoscale geometry
and predict space and time resolved carrier distributions and response. This would
allow the computational design of nanostructures and interfaces for the efficient use
and collection of hot carriers with localized carrier generation.

109

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121
Appendix A

DERIVATIONS OF EXPRESSIONS FOR DIRECT AND
PHONON-ASSISTED PLASMON DECAY
A.1

Direct Transitions

Here we sketch the derivation of equation 4.2, presented in Reference 33. For
further details and discussion please see Reference 33.
The initial system we consider is a surface plasmon with wave vector ~k on the
surface of a semi-infinite slab with the surface normal in the z direction, and a
Fermi sea of qausiparticles within the slab. Fermi’s golden rule for the decay of a
plasmon to a single electron-hole pair is
Γdirect =

2π X
δ(ε q~ 0 n 0 − ε q~n − ~ω)|Mqn,q
0n0 | ,
0 0

(A.1)

q~ n~q n

where q~, q~0 are the electron and hole wave vectors, n, n0 are the electron and hole
band indices, and Mqn,q
0 n 0 is the electron-dipole transition matrix element.
Given an approximation of the quasiparticle orbitals ψq~σn (~r ) with energies q~n , the
many-body real-space annihilation operator (also called the field operator) can be
written as Ψσ (~r ,t) = q~n ψq~σn (~r )e−i q~ n t/~ cq~n , where cq~n is the fermionic annihilation operator for state q~ n. The index σ in the orbitals is used to fully treat relativistic effects such as spin-orbit coupling. The plasmon-quasiparticle interaction is
P R
approximated using the lowest order vertex as Ĥe−pl = mee σ d~r Ψσ† Â· p̂Ψσ , with
p̂ the electronic momentum operator. The vector potential operator for plasmons in
~ˆ r ,t) = P k u~~ (~r ,t) a~ˆ~ + h.c., with a~ˆ~ the annihilation operator and
this system is A(~
u~~k the normalized mode functions of wave vector k and angular frequency ω,
u~~k (~r ,t) =

2π~ *
k ~zˆ + i γ(z)z+~k·~r −ωt
k̂ −
ωSL(ω) ,
γ(z) -

(A.2)

where S is the surface area for plasmon quantization with periodic boundary conditions for discretizing the modes and L(ω) is a normalization length chosen so that
each mode has energy ~ω. The wavenumber z satisfies γ 2 (z) = (z)ω2 /c2 − k 2 ,
with (z) = (ω) for z < 0 and (z) = 1 for z > 0.

122
Then, neglecting the photon momentum and the higher-order magnetic field coupling, the electron-dipole transition matrix element is:
Mqn,q
q n, q~0 n0, ~k | Ĥe−pl |0i
0 n 0 = h~

(A.3a)

h0| ĉq~†n ĉq~ 0 n 0 a k



e X
(A.3b)
d~r Ψσ Â · p̂Ψσ  |0i
= q
 m

f q~n (1 − f q~ 0 n 0 )n pl 
= f q~n (1 − f q~ 0 n 0 )n pl


Ω X
~∇ + [~r , V̂N L ] σ
σ∗
×
ψq~n (~r ) 
d~r ψq~ 0 n 0 (~r ) u~~k (~r ) ·
me
 L z S σ L z S

(A.3c)
!2
u k (~r )
≈ δ q 0 q f q~n (1 − f q~n 0 )n pl
u k (0) L z S


XZ
e(~∇ + [~r , V̂N L ]) σ
σ∗

d~r ψqn
(~
× u~~k (0) ·
(~
q~ n
im


(A.3d)
= δ q 0 q f q~n (1 − f q~n 0 )n pl
2L z |γ(z < 0)|


XZ
e(~∇ + [~r , V̂N L ]) σ

d~r ψq~σ∗
× u~~k (0) ·
(~
(~
n0
q~ n
im


(A.3e)
q~
= δ q 0 q f q~n (1 − f q~n 0 )n pl
[~u (0) · eh p~ˆin 0 n ]
(A.3f)
2L z |γ(z < 0)| ~k
q~
with momentum matrix elementsh p~ˆin 0 n
XZ
~∇ + [~r , V̂N L ] σ
q~
h p~in 0 n ≡
d~r ψq~σ∗
(~r )
ψq~n (~r ).
n0
ime

(A.3g)

n pl is the plasmon occupation number. We quantize the quasiparticles in a box of
area S on the surface that extends a depth L z into the surface. L z >> 1/|γ(z < 0)|,
the decay length of the plasmon mode into the metal. The factor of Ω/L z S in the
third line above accounts for the fact that the orbitals are normalized on the unit cell
of volume Ω instead of on the quantization volume L z S.

123
Plugging this into equation A.1:
Γdirect =

2π X
δ(ε q~ 0 n 0 − ε q~n − ~ω)|Mqn,q
0n0 |
0 0

(A.4a)

q~ n~q n

f q~n (1 − f q~n 0 )n pl
2π X
δ(ε q~ 0 n 0 − ε q~n − ~ω)δ q 0 q
2L z |γ(z < 0)|
q~ n~q 0 n 0
2π~
k ẑ
q~
k̂ −
· eh p~ˆin 0 n
ωSL(ω)
γ(z < 0)
(A.4b)

(A.4d)
λ~ ~k ≡ k̂ −

k ẑ
γ(z < 0)

(A.4e)

where the quantization number Nq = L z S/Ω. See Section A.3 for the final expression for direct transitions after accounting for reverse transitions.
A.2

Phonon-Assisted Transitions

Fermi’s golden rule for plasmon decay through a phonon-assisted transition is calculated similarly to the direct case in the previous section. For a phonon-assisted
transition involving a phonon with energy ~ω~k 0 α , indexed by wavevector ~k 0 and
mode α,
Γindirect =

2π

k k α± 2
δ(ε q~ 0 n 0 − ε q~n − ~ω ∓ ~ω~k 0 α )|Tqn,q
0n0 | .

(A.5)

q~ n~q 0 n 0 k 0 α±

The second-order perturbation theory transition matrix elements are:
k, k 0 α+
Tqn,q
0n0

k, k α−
Tqn,q
0n0

(A.6a)
(A.6b)

124
with the electron-plasmon interaction hamiltonian:
e X
Ĥe−pl =
d~r Ψσ† Â · p̂Ψσ
me σ

(A.7a)

using the interband approximation:

2π~ P
~ ·eh p~ˆiq~ 0 ak c† 0 cq~ n +h.c.]
~ n 0 n δq
~q
~ 0 [λ ~
2L z |γ(z <0) | ωS L(ω)
~n
n n

(A.7b)

π~e 2
~ ·h p~ˆiq~ 0 ak c† 0 cq~ n +h.c.]
~ n 0 n δq
~q
~ 0 [λ ~
ω L(ω) |γ(z <0) |L z S
~n
n n

(A.7c)

~ labeling the nuclear disand the electron-phonon interaction hamiltonian with Rs
placement modes for the degree of freedom s in the unit cell located at R:
XZ
Ĥe−ph =
x̂ Rs
d~r Ψσ† ∂Rs
r )Ψσ
(A.8a)
~ Vnuc (~
Rs

NR~ Ω

writing this in terms of phonon ladder operators:
1 XXX
~0
= q
δ q~+~k 0,~q 0 gnk0,~αqn (b~† 0 + b−~k 0 α )cq~†0 n 0 cq~n
k α
N~k 0 ~k 0 α q~n q~ 0 n 0
gnk0,~αqn
~0

with the standard definition: s
X ~0
f αs
d~r ψ ∗ ~ 0 0 (~r )ψq~n (~r )

q~+ k ,n

(A.8b)

~0 ~
r ).(A.8c)
ei k · R ∂Rs
~ Vnuc (~
2m s ω~k 0 α

Now we take an aside to simplify the second-order perturbation theory transition
matrix elements in a general framework. For simplicity, use a lumped electron
state index (not labeling k-points explictly), and let the interaction hamiltonian be
H1 = i, j α1i j (a1 ci† c j + a1† c†j ci ) and similarly for H2 . The general transition matrix
elements are:
X hF | Ĥ1 |MihM | Ĥ2 |Ii
+(1 ↔ 2),
(A.9)
T =
E M − EI
{z
T12

where I, M, and F are the many-body initial, intermediate, and final states. Let
I = |0i be the normalized initial state with some distribution of electrons and two
types of bosons (call them type 1 and type 2; in our case, they are plamsons and
phonons), and set the reference energy, EI = 0. Let F = √cc cv a1 a2 |0i be the
(1− f c ) f v n1 n2

normalized final state with an electron-hole pair and two less bosons compared to
the initial state, with relative energy EF = ε c − ε v − ω1 − ω2 .
The intermediate state should be summed over all possible many-body states. The
H2 expectation value will be non-zero only if M differs from I by a single electronhole pair and a single boson of type 2 (in our case, a phonon). Likewise the H1

125
expectation value will be non-zero only if M differs from F by a single electronhole pair and a single boson of type 1 (in our case, a plasmon). Therefore M =
c† c a |0i

, indexed by two electron indices l, m with l , m, and energy E M =

√l m 2

(1− f l ) f m n2
ε l − ε m − ω2 .

T12 ≡

X hF | Ĥ1 |MihM | Ĥ2 |Ii
E M − EI

(A.10a)

cl a2 c†y cz† |0i
α1wx α2yz h0 a2† a1† cv† cc a1 cw† cx cl† cm a2 |0ih0|a2† cm
(ε
)(1
(1 − f c ) f v n1 n2
l,m,w,x,y,z

(A.10d)
Simplifying the bosonic sector:
h0|cm
cl c†y cz |0i = h0|cm
(δl y − c†y cl )cz |0i

(A.11a)

† †
= h0|(cm
cz δl y − cm
cy cl cz )|0i

(A.11b)

= δl y δ mz f m − f m f y (δ mz δl y − δ ml δ yz )

(A.11c)

= δl y δ mz f m − f m f y δ mz δl y (y , z, l , m)

(A.11d)

(A.11e)

f m (1 − f l )δ mz δl y

Plugging this back into T12 and simplifying:
T12 =

h0|cv† cc cw† cx cl† cm |0i f m (1 − f l )δ mz δl y
n1 n2
α1wx α2yz
(1 − f c ) f v l,m,w,x,y,z
(ε l − ε m − ω2 )(1 − f l ) f m
(A.12a)

h0|cv† cc cw† cx cl† cm |0i
n1 n2
α1wx α2lm
(1 − f c ) f v l,m,w,x
ε l − ε m − ω2

(A.12b)

126
h0|cv† cc cw† cx cl† cm |0i = h0|(δcw cv† cx cl† cm − cv† cw† cc cx cl† cm )|0i

(A.13a)

= h0|(δcw δl x cv† cm − δcw cv† cl† cx cm

− δl x cv† cw† cc cm + cv† cw† cc cl† cx cm )|0i
(A.13b)

= h0|(δcw δl x cv† cm − δcw cv† cl† cx cm − δl x cv† cw† cc cm
+ δlc cv† cw† cx cm − cv† cw† cl† cc cx cm )|0i
(A.13c)
usingl , m, w , x, y , z :
= δcw δl x (δ mv f v ) − δcw (δvm δl x f v f l )
− δl x (δvm δwc f v f c ) + δlc (−δvx δwm f v f m )
− (−δvx δwm δlc f v f m f c − δvm δwc δl x f v f c f l )
(A.13d)
= δv x δwm δlc (− f v f m + f v f m f c )
+ δvm δwc δl x ( f v − f v f l − f v f c + f v f c f l )
(A.13e)
= −δvx δwm δlc f v (1 − f c ) f m + δvm δwc δl x f v (1 − f l )(1 − f c )
(A.13f)
= f v (1 − f c )[δvm δwc δl x (1 − f l ) − δvx δwm δlc f m ]

(A.13g)

127
T12 =

n1 n2
f v (1 − f c )[δvm δwc δl x (1 − f l ) − δvx δwm δlc f m ]
α1wx α2lm
(1 − f c ) f v l,m,w,x
ε l − ε m − ω2

(A.14a)
[δvm δwc δl x (1 − f l ) − δv x δwm δlc f m ]
n1 n2 f v (1 − f c )
α1wx α2lm
ε l − ε m − ω2
l,m,w,x

(A.14b)

fm
(1 − f l )
+/
α1mv α2cm
n1 n2 f v (1 − f c )
α1cl α2lv
ε l − ε v − ω2 m,v,c
ε c − ε m − ω2
,l,v,c
(A.14c)

Relabeling dummy index m → l

α1cl α2lv
α2cl α1lv +
n1 n2 f v (1 − f c ) *.
(1 − f l )
fl
ε l − ε v − ω2 l,v,c ε l − ε c + ω2
l,v,c
(A.14d)

Using energy conservation:
X"
(1 − f l )
= n1 n2 f v (1 − f c )
l,v,c

α1cl α2lv
α2cl α1lv
+ fl
ε l − ε v − ω2
ε l − ε v − ω1

(A.14e)

Substituting equation A.14e back into equation A.9,

= T12 + T21
(A.15a)
X  (1 − f l ) α1cl α2l v + f l α2cl α1l v 
ε l −ε v −ω2
ε l −ε v −ω1 

n1 n2 f v (1 − f c )
α1cl α2l v  (A.15b)
α2cl α1l v
+(1
ε l −ε v −ω1
ε l −ε v −ω2 
l,v,c 
α1cl α2lv
α2cl α1lv
(A.15c)
n1 n2 f v (1 − f c )
ε l − ε v − ω2 ε l − ε v − ω1
l,v,c
X " α1cl α2lv
n1 n2 f v (1 − f c )
+ (1 ↔ 2) .
(A.15d)
l,v,c

Note that the occupation factors correspond intuitively to whether the particle is
being absorbed or emitted. To switch from absorption to emission, n → n + 1
for each boson, and f → 1 − f for each fermion. Also, in switching from absorption to emission, the signs on the corresponding energy in the denominator and
energy-conserving delta are reversed, and we take the complex conjugate of the
corresponding matrix element coefficient (α).

128
Returning to equation A.6 and using the above results to simplify:
π~e2
k k α+
Tqn,q
f (1 − f q~ 0 n 0 )n pl n~k 0 α
0n0 = δ
q~+~k ,~q
ωL(ω)|γ(z < 0)|L z S N~k 0 q~n
~0
q~ 0 ~k 0 α
q~

h p~ˆin 0l gl,~
gnk0,~αql h p~ˆiln
X 


× λ ~k ·

ε̄
(ε
~ω
ε̄
(ε
~ω)
~k 0 α
q~ n
q~l
q~ n
l  q~ l
(A.16a)
Using energy conservation,~q0 = q~ + ~k 0 :
π~e2
= δ q~+~k 0,~q 0
f (1 − f q~ 0 n 0 )n pl n~k 0 α
ωL(ω)|γ(z < 0)|N~k 0 L z S q~n
~0
q~+~k 0 ~k 0 α
q~
h p~ˆin 0l gl,~
gnk0,~αql h p~ˆiln 
X 


× λ~ ~k ·
ε̄
~ω
ε̄
~ω

~k α
q~l
q~ n
q~+~k ,n
l  q~+~k ,l
(A.16b)
Similarly,
k k 0 α−
qn,q 0 n 0

= δ q~+~k 0,~q 0

π~e2
f (1 − f q~ 0 n 0 )n pl (n~k 0 α + 1)
ωL(ω)|γ(z < 0)|N~k 0 L z S q~n
~0
q~+~k 0 ~k 0 α
q~
h p~ˆin 0l gl,~
gnk0,~αql h p~ˆiln 
X 


× λ~ ~k ·
ε̄
~ω
ε̄
~ω

~k 0 α
q~l
q~ n
q~+~k 0 ,n 0
l  q~+~k 0 ,l
(A.17)

Combining equations A.16b and A.17 gives
π~e2
1 1
k k α±
Tqn,q 0 n 0 = δ q~+~k 0,~q 0
f (1 − f q~ 0 n 0 )n pl n~k 0 α + ∓
ωL(ω)|γ(z < 0)|N~k 0 L z S q~n
2 2

q~
q~+ k k α
gnk0,~αql h p~ˆiln 
h p~ˆin 0l gl,~
X


× λ~ ~k ·
ε̄
~ω
ε̄
~ω

~k 0 α
q~l
q~ n
q~+~k 0 ,n 0
l  q~+~k 0 ,l
(A.18)
Above, ε̄ q~n ≡ ε q~n + iImΣq~n is the complex eigenvalue of the intermediate state.

129
And finally, plugging this into equation A.5 and simplifying:
2π X
k k 0 α± 2
Γindirect =
δ(ε q~ 0 n 0 − ε q~n − ~ω ∓ ~ω~k 0 α )|Tqn,q
0n0 |
0 0 0

(A.19a)

q~ n~q n k α±

2π
π~e2
~ ωL(ω)|γ(z < 0)|N~k 0 L z S
1 1
f q~n (1 − f q~+~k 0,n 0 )n pl n~k 0 α + ∓
δ(ε q~+~k 0,n 0 − ε q~n − ~ω ∓ ~ω~k 0 α )
2 2
0 0
q~ nn k α±

 2
~k 0 α ˆ q~
q~+~k 0 ~k 0 α
h p~ˆin 0l gl,~
X 

ln
qn
n ,~ql

× λ~ ~k ·
− ε q~+~k 0,n 0 ∓ ~ω~k 0 α ε̄ q~l − ε q~n − ~ω 
 ε̄
l  q~+~k 0 ,l

(A.19b)

2π 2 e2
ωL(ω)|γ(z < 0)|Ω Nq~ N~k 0
1 1
f q~n (1 − f q~+~k 0,n 0 )n pl n~k 0 α + ∓
δ(ε q~+~k 0,n 0 − ε q~n − ~ω ∓ ~ω~k 0 α )
2 2
q~~k 0 nn 0 α±

 2
~k 0 α ˆ q~
q~+~k 0 ~k 0 α
h p~ˆin 0l gl,~
X 

ln
qn
n 0 ,~ql

× λ~ ~k ·
− ε q~+~k 0,n 0 ∓ ~ω~k 0 α ε̄ q~l − ε q~n − ~ω 
 ε̄
l  q~+~k 0 ,l

(A.19c)
and repeating equation A.4d here for comparison :
2π 2 e2
1 X
~ ~ ·h p~ˆiq~ 0 | 2 .
Γdirect =
f q~n (1− f q~n 0 )n pl δ(ε q~n 0 −ε q~n −~ω)| λ
nn
ωL(ω)|γ(z < 0)|Ω Nq~
q~ nn

(A.20)
A.3

Final Expressions for Direct and Phonon-assisted Plasmon Decay After
Accounting For Reverse Transitions

To calculate the net transition rate from plasmon occupation number n pl → (n pl −1),
we need to account for the above absorption process as well as the corresponding
emission processes that go from (n pl − 1) → n pl . By detailed balance, the matrix
element is exactly the same for absorption and emission and only the occupation
factors for the fermions change: f v (1 − f c ) → (1 − f v ) f c . The difference between
the two processes has a fermion factor of f v (1 − f c ) − (1 − f v ) f c = f v − f c . Thus
the final transition rates accounting for this are:
2π 2 e2
1 X
q~
( f q~n − f q~n 0 )n pl δ(ε q~n 0 − ε q~n − ~ω)| λ~ ~k · h p~ˆin 0 n | 2
Γdirect =
ωL(ω)|γ(z < 0)|Ω Nq~
q~ nn

(A.21)

130
Γindirect =

2π 2 e2
ωL(ω)|γ(z < 0)|Ω Nq~ N~k 0
1 1
δ(ε q~+~k 0,n 0 − ε q~n − ~ω ∓ ~ω~k 0 α )
( f q~n − f q~+~k 0,n 0 )n pl n~k 0 α + ∓
2 2
q~~k 0 nn 0 α±

 2
~k 0 α ˆ q~
q~+~k 0 ~k 0 α
h p~ˆin 0l gl,~
X 

ln
qn
n 0 ,~ql

× λ~ ~k ·
− ε q~+~k 0,n 0 ∓ ~ω~k 0 α ε̄ q~l − ε q~n − ~ω 
 ε̄
l  q~+~k 0 ,l

(A.22)
These are directly related to the expressions (4.2 and 4.3) for plasmon decay via
direct and indirect transitions presented in Chapter 4.

131
Appendix B

TABULATED ELECTRONIC HEAT CAPACITY AND
ELECTRON-PHONON COUPLING FACTOR AS A FUNCTION
OF ELECTRON TEMPERATURE
Here we provide tabulated ab initio electronic heat capacity from expression 5.3
and electron-phonon coupling factor from expression 5.9 as a function of electron
temperature for aluminum, silver, gold, and copper. See Chapter 5 for discussion
of these parameters.
Table B.1: Tabulated electronic heat capacity and electron-phonon coupling factor
as a function of electron temperature

C e [105 J/m3 K]
Ag

0.279649
0.376354
0.472946
0.568945
0.664438
0.759542
0.854331
0.948845
1.043110
1.137150
1.230990
1.324650
1.418170
1.511560
1.604860
1.698070
1.791220
1.884320
1.977390
2.070440
2.163480
2.256530
2.349590
2.442670
2.535800
2.628960
2.722190
2.815480
2.908850
3.002310
3.095860

0.180129
0.239219
0.298001
0.356496
0.414751
0.472777
0.530547
0.588019
0.645156
0.701925
0.758311
0.814307
0.869918
0.925154
0.980032
1.034570
1.088790
1.142700
1.196330
1.249690
1.302800
1.355670
1.408310
1.460740
1.512980
1.565050
1.616970
1.668780
1.720530
1.772260
1.824040

0.184019
0.245105
0.306298
0.367593
0.428952
0.490335
0.551705
0.613042
0.674336
0.735593
0.796824
0.858050
0.919296
0.980596
1.041990
1.103550
1.165350
1.227490
1.290100
1.353330
1.417370
1.482440
1.548750
1.616580
1.686190
1.757850
1.831870
1.908520
1.988090
2.070850
2.157060

0.277610
0.369473
0.460837
0.552057
0.643278
0.734512
0.825722
0.916856
1.007870
1.098750
1.189480
1.280090
1.370680
1.461390
1.552470
1.644260
1.737240
1.832020
1.929310
2.029960
2.134900
2.245110
2.361640
2.485540
2.617820
2.759470
2.911410
3.074450
3.249310
3.436580
3.636740

Te [K]
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300

G [1017 W/m 3 K]
Ag

5.039960
5.208740
5.312170
5.381720
5.431590
5.468990
5.497950
5.520900
5.539410
5.554560
5.567120
5.577650
5.586580
5.594240
5.600880
5.606710
5.611880
5.616510
5.620710
5.624550
5.628080
5.631370
5.634440
5.637340
5.640080
5.642680
5.645170
5.647560
5.649860
5.652080
5.654240

0.302574
0.305198
0.307018
0.308271
0.309100
0.309607
0.309869
0.309941
0.309867
0.309681
0.309408
0.309067
0.308674
0.308240
0.307774
0.307285
0.306777
0.306256
0.305725
0.305189
0.304649
0.304107
0.303565
0.303026
0.302488
0.301955
0.301426
0.300903
0.300387
0.299878
0.299377

0.245301
0.248132
0.250093
0.251564
0.252719
0.253649
0.254410
0.255039
0.255563
0.256005
0.256379
0.256701
0.256979
0.257224
0.257441
0.257638
0.257819
0.257991
0.258160
0.258331
0.258510
0.258706
0.258925
0.259178
0.259473
0.259820
0.260230
0.260715
0.261286
0.261955
0.262734

0.985599
1.010650
1.025440
1.034980
1.041480
1.046040
1.049250
1.051500
1.053030
1.054020
1.054590
1.054830
1.054830
1.054620
1.054270
1.053840
1.053360
1.052900
1.052540
1.052340
1.052400
1.052800
1.053670
1.055100
1.057220
1.060140
1.063990
1.068870
1.074900
1.082180
1.090820

Continued on next page

132
Table B.1 – Continued from previous page
Al

C e [105 J/m3 K]
Ag

3.189520
3.283280
3.377170
3.471180
3.565320
3.659590
3.754010
3.848560
3.943260
4.038110
4.133100
4.228240
4.323530
4.418960
4.514530
4.610250
4.706100
4.802080
4.898200
4.994440
5.090800
5.187280
5.283870
5.380560
5.477360
5.574250
5.671230
5.768300
5.865440
5.962660
6.059940
6.157280
6.254680
6.352120
6.449600
6.547120
6.644670
6.742240
6.839820
6.937420
7.035020
7.132620
7.230220
7.327800
7.425360
7.522900
7.620400
7.717870
7.815300
7.912680
8.010010

1.875940
1.928050
1.980480
2.033330
2.086730
2.140830
2.195760
2.251700
2.308810
2.367270
2.427280
2.489010
2.552670
2.618460
2.686580
2.757230
2.830610
2.906900
2.986320
3.069030
3.155210
3.245040
3.338680
3.436270
3.537960
3.643870
3.754120
3.868800
3.988020
4.111840
4.240320
4.373520
4.511460
4.654180
4.801660
4.953920
5.110920
5.272640
5.439030
5.610030
5.785580
5.965600
6.150000
6.338690
6.531550
6.728490
6.929360
7.134060
7.342440
7.554370
7.769710

2.246960
2.340780
2.438720
2.540940
2.647590
2.758800
2.874650
2.995200
3.120500
3.250550
3.385330
3.524820
3.668940
3.817610
3.970740
4.128210
4.289890
4.455620
4.625270
4.798660
4.975620
5.155980
5.339570
5.526190
5.715660
5.907820
6.102460
6.299430
6.498540
6.699630
6.902520
7.107070
7.313110
7.520500
7.729100
7.938770
8.149380
8.360810
8.572940
8.785660
8.998870
9.212480
9.426380
9.640490
9.854730
10.069000
10.283300
10.497500
10.711600
10.925400
11.139000

3.850120
4.076910
4.317190
4.570880
4.837810
5.117680
5.410090
5.714540
6.030460
6.357220
6.694130
7.040450
7.395420
7.758260
8.128180
8.504390
8.886120
9.272590
9.663060
10.056800
10.453200
10.851500
11.251100
11.651500
12.052000
12.452300
12.851800
13.250100
13.646800
14.041500
14.433900
14.823700
15.210700
15.594600
15.975200
16.352300
16.725700
17.095300
17.461100
17.822800
18.180400
18.533800
18.883100
19.228000
19.568700
19.905000
20.237000
20.564700
20.888100
21.207200
21.522000

Te [K]
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
6400
6500
6600
6700
6800
6900
7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
8100
8200
8300
8400

G [1017 W/m 3 K]
Ag

5.656340
5.658390
5.660400
5.662370
5.664310
5.666230
5.668130
5.670020
5.671890
5.673770
5.675640
5.677510
5.679390
5.681280
5.683190
5.685100
5.687030
5.688980
5.690950
5.692940
5.694950
5.696990
5.699050
5.701130
5.703240
5.705370
5.707530
5.709710
5.711920
5.714150
5.716410
5.718680
5.720990
5.723310
5.725650
5.728020
5.730400
5.732800
5.735220
5.737660
5.740110
5.742570
5.745050
5.747540
5.750040
5.752550
5.755070
5.757600
5.760130
5.762670
5.765210

0.298886
0.298406
0.297939
0.297485
0.297049
0.296631
0.296234
0.295862
0.295517
0.295203
0.294924
0.294683
0.294486
0.294337
0.294240
0.294200
0.294224
0.294315
0.294480
0.294723
0.295051
0.295469
0.295982
0.296596
0.297317
0.298149
0.299098
0.300168
0.301365
0.302693
0.304156
0.305758
0.307503
0.309395
0.311436
0.313630
0.315978
0.318483
0.321146
0.323970
0.326954
0.330100
0.333408
0.336877
0.340508
0.344300
0.348251
0.352360
0.356626
0.361046
0.365619

0.263635
0.264670
0.265848
0.267182
0.268681
0.270355
0.272211
0.274257
0.276501
0.278947
0.281599
0.284461
0.287536
0.290824
0.294327
0.298042
0.301969
0.306105
0.310446
0.314989
0.319728
0.324659
0.329775
0.335069
0.340535
0.346167
0.351955
0.357893
0.363973
0.370187
0.376526
0.382984
0.389551
0.396221
0.402986
0.409837
0.416768
0.423771
0.430840
0.437967
0.445146
0.452370
0.459634
0.466932
0.474258
0.481607
0.488974
0.496353
0.503741
0.511134
0.518526

1.100880
1.112450
1.125580
1.140320
1.156690
1.174720
1.194410
1.215740
1.238700
1.263240
1.289320
1.316880
1.345860
1.376190
1.407790
1.440580
1.474470
1.509370
1.545200
1.581870
1.619280
1.657360
1.696020
1.735170
1.774740
1.814650
1.854830
1.895210
1.935720
1.976310
2.016910
2.057480
2.097960
2.138310
2.178490
2.218450
2.258170
2.297610
2.336740
2.375540
2.413980
2.452040
2.489700
2.526960
2.563790
2.600180
2.636120
2.671600
2.706620
2.741180
2.775250

Continued on next page

133
Table B.1 – Continued from previous page
Al

C e [105 J/m3 K]
Ag

8.107280
8.204490
8.301620
8.398690
8.495670
8.592570
8.689380
8.786090
8.882700
8.979210
9.075610
9.171900
9.268060
9.364110
9.460020
9.555800
9.651450
9.746950
9.842310
9.937520
10.032600
10.127500
10.222200
10.316800
10.411200
10.505400
10.599500
10.693300
10.787000
10.880500
10.973800
11.067000
11.159900
11.252600
11.345100
11.437400
11.529500
11.621400
11.713100
11.804600
11.895800
11.986900
12.077700
12.168200
12.258600
12.348700
12.438600
12.528200
12.617600
12.706800
12.795700

7.988310
8.210020
8.434690
8.662170
8.892310
9.124950
9.359940
9.597130
9.836360
10.077500
10.320400
10.564800
10.810700
11.058000
11.306400
11.555800
11.806200
12.057300
12.309100
12.561500
12.814200
13.067300
13.320600
13.573900
13.827300
14.080600
14.333700
14.586500
14.839000
15.091000
15.342600
15.593500
15.843800
16.093400
16.342300
16.590300
16.837400
17.083600
17.328800
17.573100
17.816200
18.058300
18.299300
18.539100
18.777700
19.015100
19.251200
19.486100
19.719700
19.952000
20.183000

11.352300
11.565300
11.777800
11.989900
12.201600
12.412700
12.623300
12.833400
13.042800
13.251700
13.459900
13.667400
13.874300
14.080400
14.285900
14.490600
14.694600
14.897800
15.100200
15.301900
15.502800
15.702800
15.902100
16.100600
16.298200
16.495000
16.691000
16.886200
17.080500
17.273900
17.466500
17.658300
17.849200
18.039200
18.228400
18.416700
18.604100
18.790700
18.976400
19.161200
19.345200
19.528300
19.710500
19.891900
20.072400
20.252000
20.430800
20.608700
20.785800
20.961900
21.137300

21.832600
22.139000
22.441200
22.739200
23.033200
23.323100
23.609100
23.891200
24.169300
24.443700
24.714300
24.981300
25.244600
25.504300
25.760600
26.013400
26.262800
26.508900
26.751800
26.991500
27.228100
27.461600
27.692200
27.919800
28.144500
28.366400
28.585600
28.802100
29.015900
29.227200
29.435900
29.642200
29.846100
30.047600
30.246800
30.443700
30.638500
30.831100
31.021500
31.210000
31.396400
31.580800
31.763300
31.943900
32.122700
32.299700
32.475000
32.648500
32.820300
32.990500
33.159200

Te [K]
8500
8600
8700
8800
8900
9000
9100
9200
9300
9400
9500
9600
9700
9800
9900
10000
10100
10200
10300
10400
10500
10600
10700
10800
10900
11000
11100
11200
11300
11400
11500
11600
11700
11800
11900
12000
12100
12200
12300
12400
12500
12600
12700
12800
12900
13000
13100
13200
13300
13400
13500

G [1017 W/m 3 K]
Ag

5.767760
5.770300
5.772850
5.775400
5.777940
5.780490
5.783030
5.785560
5.788100
5.790620
5.793140
5.795650
5.798150
5.800640
5.803120
5.805590
5.808040
5.810490
5.812920
5.815330
5.817740
5.820120
5.822490
5.824840
5.827180
5.829490
5.831790
5.834070
5.836330
5.838570
5.840780
5.842980
5.845150
5.847310
5.849440
5.851540
5.853630
5.855690
5.857720
5.859740
5.861720
5.863680
5.865620
5.867530
5.869410
5.871270
5.873100
5.874910
5.876680
5.878430
5.880160

0.370341
0.375211
0.380225
0.385380
0.390673
0.396100
0.401658
0.407343
0.413151
0.419078
0.425120
0.431273
0.437534
0.443897
0.450358
0.456914
0.463560
0.470292
0.477105
0.483996
0.490961
0.497995
0.505095
0.512256
0.519475
0.526748
0.534071
0.541441
0.548854
0.556306
0.563795
0.571317
0.578869
0.586448
0.594051
0.601675
0.609318
0.616975
0.624647
0.632328
0.640018
0.647714
0.655413
0.663114
0.670814
0.678512
0.686206
0.693894
0.701574
0.709244
0.716904

0.525914
0.533295
0.540665
0.548021
0.555361
0.562680
0.569977
0.577248
0.584493
0.591708
0.598893
0.606044
0.613160
0.620240
0.627282
0.634286
0.641249
0.648171
0.655051
0.661887
0.668680
0.675428
0.682131
0.688788
0.695399
0.701962
0.708479
0.714949
0.721370
0.727744
0.734069
0.740347
0.746576
0.752756
0.758888
0.764972
0.771008
0.776995
0.782935
0.788826
0.794669
0.800465
0.806213
0.811914
0.817568
0.823174
0.828735
0.834249
0.839717
0.845139
0.850515

2.808850
2.841980
2.874620
2.906780
2.938460
2.969670
3.000400
3.030660
3.060450
3.089780
3.118640
3.147060
3.175020
3.202530
3.229610
3.256260
3.282480
3.308280
3.333670
3.358650
3.383230
3.407420
3.431220
3.454640
3.477690
3.500380
3.522710
3.544690
3.566320
3.587620
3.608580
3.629220
3.649550
3.669560
3.689280
3.708690
3.727810
3.746650
3.765210
3.783490
3.801510
3.819270
3.836770
3.854030
3.871030
3.887800
3.904340
3.920650
3.936730
3.952600
3.968250

Continued on next page

134
Table B.1 – Continued from previous page
Al

C e [105 J/m3 K]
Ag

12.884400
12.972900
13.061100
13.149000
13.236800
13.324200
13.411400
13.498400
13.585100
13.671500
13.757700
13.843700
13.929300
14.014800
14.099900
14.184800
14.269400
14.353800
14.437900
14.521800
14.605400
14.688700
14.771800
14.854600
14.937100
15.019400
15.101400
15.183100
15.264600
15.345800
15.426700
15.507400
15.587800
15.667900
15.747800
15.827400
15.906700
15.985800
16.064600
16.143200
16.221400
16.299500
16.377200
16.454700
16.531900
16.608900
16.685600
16.762100
16.838200
16.914200
16.989800

20.412600
20.640900
20.867900
21.093500
21.317700
21.540500
21.762000
21.982000
22.200700
22.418000
22.633800
22.848300
23.061400
23.273100
23.483400
23.692300
23.899800
24.106000
24.310800
24.514200
24.716300
24.917000
25.116300
25.314400
25.511100
25.706500
25.900500
26.093300
26.284800
26.475000
26.663900
26.851600
27.038000
27.223200
27.407200
27.589900
27.771500
27.951800
28.131000
28.309000
28.485800
28.661500
28.836000
29.009400
29.181700
29.352900
29.523000
29.692000
29.859900
30.026800
30.192600

21.311700
21.485300
21.658100
21.830000
22.001000
22.171200
22.340600
22.509100
22.676800
22.843600
23.009600
23.174800
23.339200
23.502700
23.665400
23.827300
23.988400
24.148700
24.308200
24.466900
24.624800
24.781900
24.938300
25.093800
25.248600
25.402600
25.555900
25.708400
25.860100
26.011100
26.161300
26.310900
26.459600
26.607700
26.755000
26.901600
27.047500
27.192700
27.337200
27.481000
27.624100
27.766500
27.908200
28.049300
28.189700
28.329400
28.468500
28.606900
28.744700
28.881900
29.018400

33.326200
33.491700
33.655700
33.818300
33.979400
34.139200
34.297600
34.454600
34.610400
34.764800
34.918100
35.070100
35.220900
35.370600
35.519100
35.666600
35.812900
35.958200
36.102500
36.245700
36.388000
36.529300
36.669600
36.809100
36.947600
37.085200
37.222000
37.357900
37.493000
37.627400
37.760900
37.893600
38.025600
38.156900
38.287400
38.417200
38.546300
38.674800
38.802600
38.929700
39.056200
39.182100
39.307400
39.432100
39.556200
39.679700
39.802700
39.925100
40.047000
40.168300
40.289200

Te [K]
13600
13700
13800
13900
14000
14100
14200
14300
14400
14500
14600
14700
14800
14900
15000
15100
15200
15300
15400
15500
15600
15700
15800
15900
16000
16100
16200
16300
16400
16500
16600
16700
16800
16900
17000
17100
17200
17300
17400
17500
17600
17700
17800
17900
18000
18100
18200
18300
18400
18500
18600

G [1017 W/m 3 K]
Ag

5.881850
5.883520
5.885160
5.886770
5.888350
5.889900
5.891420
5.892920
5.894380
5.895820
5.897230
5.898600
5.899950
5.901270
5.902550
5.903810
5.905040
5.906230
5.907400
5.908530
5.909640
5.910710
5.911750
5.912760
5.913740
5.914690
5.915610
5.916500
5.917350
5.918180
5.918970
5.919730
5.920470
5.921160
5.921830
5.922470
5.923070
5.923650
5.924190
5.924700
5.925180
5.925620
5.926040
5.926420
5.926770
5.927090
5.927380
5.927640
5.927860
5.928060
5.928220

0.724551
0.732184
0.739802
0.747404
0.754988
0.762553
0.770097
0.777621
0.785123
0.792602
0.800058
0.807488
0.814893
0.822272
0.829624
0.836949
0.844245
0.851513
0.858752
0.865960
0.873139
0.880287
0.887403
0.894489
0.901542
0.908563
0.915552
0.922508
0.929432
0.936322
0.943178
0.950001
0.956791
0.963546
0.970267
0.976955
0.983608
0.990226
0.996810
1.003360
1.009880
1.016360
1.022800
1.029210
1.035590
1.041930
1.048240
1.054510
1.060750
1.066960
1.073130

0.855847
0.861133
0.866375
0.871573
0.876727
0.881837
0.886904
0.891928
0.896909
0.901848
0.906746
0.911601
0.916416
0.921189
0.925922
0.930615
0.935269
0.939882
0.944457
0.948993
0.953490
0.957950
0.962371
0.966756
0.971103
0.975414
0.979689
0.983927
0.988130
0.992298
0.996430
1.000530
1.004590
1.008620
1.012620
1.016580
1.020510
1.024410
1.028270
1.032110
1.035910
1.039680
1.043410
1.047120
1.050800
1.054450
1.058060
1.061650
1.065210
1.068740
1.072240

3.983690
3.998930
4.013970
4.028810
4.043460
4.057920
4.072200
4.086300
4.100220
4.113980
4.127560
4.140980
4.154240
4.167340
4.180290
4.193080
4.205730
4.218230
4.230590
4.242810
4.254890
4.266840
4.278660
4.290340
4.301910
4.313340
4.324660
4.335860
4.346940
4.357900
4.368760
4.379500
4.390130
4.400660
4.411080
4.421390
4.431610
4.441730
4.451750
4.461670
4.471500
4.481230
4.490870
4.500430
4.509890
4.519270
4.528560
4.537760
4.546880
4.555920
4.564880

Continued on next page

135
Table B.1 – Continued from previous page
Al

C e [105 J/m3 K]
Ag

17.065200
17.140400
17.215300
17.289900
17.364300
17.438400
17.512300
17.585900
17.659200
17.732300
17.805200
17.877800
17.950200
18.022300

30.357400
30.521200
30.683900
30.845700
31.006400
31.166200
31.325000
31.482800
31.639700
31.795700
31.950700
32.104700
32.257900
32.410200

29.154300
29.289600
29.424200
29.558200
29.691700
29.824500
29.956800
30.088400
30.219500
30.350000
30.479900
30.609300
30.738100
30.866300

40.409500
40.529300
40.648700
40.767500
40.885900
41.003800
41.121300
41.238400
41.354900
41.471100
41.586900
41.702200
41.817100
41.931600

Te [K]
18700
18800
18900
19000
19100
19200
19300
19400
19500
19600
19700
19800
19900
20000

G [1017 W/m 3 K]
Ag

5.928350
5.928450
5.928510
5.928550
5.928550
5.928530
5.928470
5.928380
5.928260
5.928100
5.927920
5.927700
5.927460
5.927180

1.079260
1.085360
1.091430
1.097460
1.103460
1.109430
1.115360
1.121250
1.127120
1.132950
1.138740
1.144510
1.150240
1.155930

1.075710
1.079150
1.082570
1.085960
1.089320
1.092650
1.095960
1.099240
1.102490
1.105720
1.108920
1.112100
1.115250
1.118380

4.573760
4.582560
4.591280
4.599930
4.608500
4.616990
4.625410
4.633760
4.642040
4.650250
4.658390
4.666460
4.674460
4.682390

136
Appendix C

DIELECTRIC FUNCTION TEMPERATURE DEPENDENCE
PREFACTOR DERIVATION
The energy dependence of the electron-electron scattering rate is given by τ1ee =
De (E − EF ) 2 , with prefactor De given in Reference [123], equation 10 (converted
to atomic units) as:
 √
(4π) (E − EF )
4EF 
2 EF E s
(C.1)
+ arctan
×
τee (E) 64π 3 ε 2 Es3/2 EF  4EF + Es
Es 

 √
2 EF E s
4EF 
.
De =
(C.2)
× 
+ arctan
3/2 √
Es 
4πε b Es
EF  4EF + Es
To derive an expression for the prefactor, which we will call A, for the temperature
dependent form τ1ee = ATe2 , we begin with equation 8 from Reference [123]
s  Ẽmax
 √
Ẽ
Ẽ 
d f (E = EF ,T )

|e−e =
dE
dE
arctan
 Ẽ + Es
dt
Es 
Es
32π 3 ε 2b Es EF
 Ẽmin
× {[1 − f (E1 )] f (E2 ) f (E3 ) + f (E1 )[1 − f (E2 )][1 − f (E3 )]}
(4π) 2

Z Z

(C.3)
with Ẽmax = inf

√

E1 +

√
2 √
√ 2
√ 2
√ 2 √
and Ẽmin = sup
EF + E2
E1 − E3 ;
EF − E2 .

Take Ẽmin = 0. Ẽmax is slowly varying in E1 and E2 , therefore, approximate Ẽmax
as a constant equal to its value at E1 = E2 = EF : Ẽmax ≈ 4EF

s  Ẽmax  √
s  4EF
 √
Ẽ
Ẽ 
Ẽ
Ẽ 


arctan
arctan
 Ẽ + Es
 Ẽ + Es
Es 
Es 
Es
Es
 Ẽmin 
0
4EF
2 EF
+ √ arctan
4EF + Es
Es
Es

(C.4a)

(C.4b)

The temperature dependence of expression C.3 lies in the occupation factors term:
F (E1 , E2 ) = (1 − f (E1 )) f (E2 ) f (E3 ) + f (E1 )(1 − f (E2 ))(1 − f (E3 ))

(C.5)

137
We can simplify this expression using
f (E) f (E1 )(1 − f (E2 ))(1 − f (E3 )) = (1 − f (E))(1 − f (E1 )) f (E2 ) f (E3 ) (C.6)
to write f (E3 ) in terms of the other occupation factors as
f (E3 ) =

f (E) f (E1 )(1 − f (E2 ))
f (E) f (E1 ) + f (E2 ) − f (E) f (E2 ) − f (E1 ) f (E2 )

(C.7)

and replacing E by EF and using f (E = EF ) = 1/2:
2 f (E1 )(1 − f (E2 ))
(E
(E
(E
(E

f (E3 ) = 1

(C.8)

Substituting this into the occupation factors term:
2 f (E1 )(1 − f (E2 ))
F (E1 , E2 ) = (1 − f (E1 )) f (E2 ) 1
2 f (E1 ) + 2 f (E2 ) − f (E1 ) f (E2 )

(C.9)

This is only nonzero around EF in both the E1 and E2 direction, so we evaluate the
term at E1 = E2 = EF :
2 f (EF )(1 − f (EF ))
F (EF , EF ) = (1 − f (EF )) f (EF ) 1
2 f (EF ) + 2 f (EF ) − f (EF ) f (EF )
111
11
222
2 2 12 12 + 12 12 − 12 12
111
222

1+1−1
F (EF , EF ) = 1/8

(C.10a)
(C.10b)
(C.10c)
(C.10d)

138
Substituting this into expression C.3
Z Z
d f (E = EF ,Te )
(4π) 2
|e−e ≈
dE1 dE2
dt
32π 3 ε 2b Es EF
 √
2 EF
4EF 
 F (E1 , E2 )
× 
+ √ arctan
Es 
Es
 4EF + Es
(C.11a)
 √
(4π) 2
2 EF
4EF 
+ √ arctan
Es 
Es
32π 3 ε 2b Es EF  4EF + Es
Z Z
dE1 dE2 F (E1 , E2 )
(C.11b)
 √
(4π) 2
 2 EF + 1 arctan
Es
32π 3 ε 2b Es EF  4EF + Es

4EF  π 2 2
 T
Es  2 e
(C.11c)
√  √
(4π) 2
E s  2 EF
4EF  2 2
 π Te
+ √ arctan
√ 
Es 
Es
64π 3 ε 2b Es EF Es  4EF + Es
(C.11d)
 √
(4π) 2
2 EF E s
4EF  2 2
π T
+ arctan
Es 
64π 3 ε 2b Es3/2 EF  4EF + Es
(C.11e)
= π 2 DeTe2

(C.11f)

= ATe2 .

(C.11g)

Thus, A = π 2 De (in atomic units); the prefactors for the energy and temperature
dependence of the electron-electron scattering rate are related by a factor of π 2 .