High Temperature Electron-Phonon and Mag

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High Temperature Electron-Phonon and Magnon-Phonon Interactions
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Yang, Fred Chae-Reem
(2019)
High Temperature Electron-Phonon and Magnon-Phonon Interactions.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/1KP5-CJ98.
Abstract
Computational materials discovery and design has emerged in order to meet the surge in demand for new materials for applications ranging from clean alternative energy to human welfare. This acceleration of materials discovery is exhilarating, but the applications of new advanced materials can be limited by their thermodynamic stability. Accurate calculations of the Gibbs free energy, a measure of thermodynamic stability, require a deep understanding of atomic vibrations, a main source of entropy in materials. This deep understanding of atomic vibrations requires us to treat phonons (quantized lattice vibrations) beyond the harmonic model by considering their interactions with various excitations. In this thesis, I present the effects of high temperature interactions of phonons with electrons and magnetic excitations on the thermodynamics of FeTi, vanadium, and Pd
Fe.
A combination of
ab initio
calculations, inelastic neutron scattering (INS), and nuclear resonant inelastic x-ray scattering (NRIXS) showed an anomalous thermal softening of the M
phonon mode in B2-ordered FeTi and a thermal stiffening of the longitudinal acoustic N phonon mode in body-centered-cubic vanadium. Computational investigations involving electronic band unfolding were performed to identify the nesting features on Fermi surfaces crucial to high temperature electron-phonon interactions in FeTi and vanadium. These investigations showed that the Fermi surface of FeTi undergoes a novel thermally driven electronic topological transition (ETT), in which new features of the Fermi surface arise at elevated temperatures. This ETT was also observed in vanadium, but the effects were overtaken by the thermal smearing of the Fermi surface that decreased the rate of electron-phonon scattering.
Iron phonon partial densities of states of Pd
Fe were measured with NRIXS from room temperature through the Curie transition at 500 K. The experimental results were compared to
ab initio
spin-polarized calculations that modeled the finite-temperature thermodynamic properties of Pd
Fe with magnetic special quasirandom structures (SQSs) of magnetic moments. The scattering measurements and first-principles calculations showed that the iron partial vibrational entropy is close to what is predicted by the quasiharmonic approximation owing to a cancellation of effects: phonon-phonon and magnon-phonon interactions approximately cancel a ferromagnetic optical phonon stiffening.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Materials thermodynamics; electron-phonon interaction; electronic structure; fermi surface; band unfolding; electronic topological transition; magnon-phonon interaction
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Fultz, Brent T.
Thesis Committee:
Minnich, Austin J. (chair)
Bernardi, Marco
Schwab, Keith C.
Fultz, Brent T.
Defense Date:
25 March 2019
Funders:
Funding Agency
Grant Number
Department of Energy (DOE)
DE-NA-0002006
Record Number:
CaltechTHESIS:03272019-174351662
Persistent URL:
DOI:
10.7907/1KP5-CJ98
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DOI
Article adapted for Chapter 2.
DOI
Article adapted for Chapter 4.
ORCID:
Author
ORCID
Yang, Fred Chae-Reem
0000-0002-5615-5170
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
11433
Collection:
CaltechTHESIS
Deposited By:
Chae-Reem Yang
Deposited On:
01 May 2019 23:51
Last Modified:
04 Oct 2019 00:25
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High Temperature Electron-Phonon and Magnon-Phonon
Interactions

Thesis by

Fred Chae-Reem Yang

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

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2019
Defended March 25, 2019

Fred Chae-Reem Yang
ORCID: 0000-0002-5615-5170

iii

To my loving parents.

ACKNOWLEDGEMENTS
My time at Caltech has been a difficult yet satisfying period of professional and
personal growth. I would not have been able to make it past the obstacles and
emerge a better person without the positive support I had received from many
people over the course of my graduate studies.
First, I would like to express my gratitude to my advisor, Professor Brent Fultz, for
giving me the opportunity to work with him over the past few years. He had allowed
me to explore topics and projects that I found interesting while grounding me in
reality when needed, sometimes seeing the big picture when I would be missing the
forest for the trees. I appreciated his guidance and accessibility as he helped me
develop into an independent scientist.
I would like to thank the members of my thesis committee, Professors Austin
Minnich, Keith Schwab, and Marco Bernardi, for their guidance and support. I look
forward to thoughtful discussions yet to come.
I would like to thank the former and current members of the Fultz group. I have
had the great pleasure of collaborating with Jorge Muñoz. His mentorship helped
me navigate the start of my graduate studies, and I appreciated his many research
insights and his career advice. I owe my heartfelt thanks to Matthew Lucas for his
help and insight with the FeTi project and for his expertise that allowed the Pd3 Fe
beamtime experiment to come into fruition. Olle Hellman allowed me to make the
jump into computational materials science, helping me tackle research problems I
would not have been able to explore otherwise. I appreciated his patience as he
answered my many questions. I have learned a lot about neutron scattering and
running experiments from Hillary Smith, and it was always an immense pleasure to
work with her. I want to give her my heartfelt thanks for always providing assistance
with my experiments without any sense of hesitation.
I have enjoyed going to beamtime experiments with Claire Saunders, Nick Weadock,
Jane Herriman, and my officemates Dennis Kim and Yang Shen. I have enjoyed
our conversations and learning from you all. I want to extend my thanks to group
members Channing, Jiao, Tian, Lisa, Sally, Hongjin, Heng, Max, Bryce, Peter,
Camille, Stefan, and Cullen and beamtime collaborators Chen, Andrew, Tabitha,
Sarah, and Xiao. Everyone I met in the group helped motivate me to always better
myself.

This research would not have been possible without the generous support from the
Capital-DOE Alliance Center (CDAC) and the dedication of Dr. Russell Hemley
and Dr. Steve Gramsch. I would also like to thank the staff scientists I worked with,
including Yuming Xiao and Paul Chow at Argonne National Laboratory; Doug
Abernathy and Matt Stone at Oak Ridge National Laboratory; and Nick Butch,
Terry Udovic, and Juscelino Leao at the NIST Center for Neutron Research.
I want to thank many people outside of the group. This would include everyone I
had met in my class and the APhMS programs (special thanks to Stephen for our
collaborations on our problem sets and for his research insights), the Steuben house,
people I had met from our graduate student orientation, the Global Throopers
softball team, and several Caltech staff who have always gone the extra mile to
provide assistance and make sure I was welcome at Caltech.
I want to thank a few of my off-campus friends who have kept in touch with me
after Berkeley. Dara helped me stay sane throughout the past few years and provided
crucial career advice. Jeremy also kept me sane, provided alternate perspectives that
I needed, and helped me explore LA and its culture when I would have otherwise
been stuck in Pasadena.
Finally, I want to give special thanks to my parents. I would not have made it this
far without their unconditional love. My mother Anna has been my rock, giving
me encouragement and emotional support all throughout these years. I want to give
special thanks to my father Dr. Woo Sun Yang for being my biggest inspiration for
going into science, and for always taking the time out of his busy schedule to fuel
my interest in research. I hope you will be glad that there may soon be another Ph.D.
in the family.

ABSTRACT
Computational materials discovery and design has emerged in order to meet the
surge in demand for new materials for applications ranging from clean alternative
energy to human welfare. This acceleration of materials discovery is exhilarating,
but the applications of new advanced materials can be limited by their thermodynamic stability. Accurate calculations of the Gibbs free energy, a measure of
thermodynamic stability, require a deep understanding of atomic vibrations, a main
source of entropy in materials. This deep understanding of atomic vibrations requires us to treat phonons (quantized lattice vibrations) beyond the harmonic model
by considering their interactions with various excitations. In this thesis, I present
the effects of high temperature interactions of phonons with electrons and magnetic
excitations on the thermodynamics of FeTi, vanadium, and Pd3 Fe.
A combination of ab initio calculations, inelastic neutron scattering (INS), and
nuclear resonant inelastic x-ray scattering (NRIXS) showed an anomalous thermal
softening of the M5− phonon mode in B2-ordered FeTi and a thermal stiffening
of the longitudinal acoustic N phonon mode in body-centered-cubic vanadium.
Computational investigations involving electronic band unfolding were performed to
identify the nesting features on Fermi surfaces crucial to high temperature electronphonon interactions in FeTi and vanadium. These investigations showed that the
Fermi surface of FeTi undergoes a novel thermally driven electronic topological
transition (ETT), in which new features of the Fermi surface arise at elevated
temperatures. This ETT was also observed in vanadium, but the effects were
overtaken by the thermal smearing of the Fermi surface that decreased the rate of
electron-phonon scattering.
Iron phonon partial densities of states of Pd3 Fe were measured with NRIXS from
room temperature through the Curie transition at 500 K. The experimental results
were compared to ab initio spin-polarized calculations that modeled the finitetemperature thermodynamic properties of Pd3 Fe with magnetic special quasirandom
structures (SQSs) of magnetic moments. The scattering measurements and firstprinciples calculations showed that the iron partial vibrational entropy is close to
what is predicted by the quasiharmonic approximation owing to a cancellation of
effects: phonon-phonon and magnon-phonon interactions approximately cancel a
ferromagnetic optical phonon stiffening.

vii

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] F. C. Yang, J. A. Muñoz, O. Hellman, L. Mauger, M. S. Lucas, S. J. Tracy, M. B.
Stone, D. L. Abernathy, Y. Xiao, and B. Fultz. “Thermally Driven Electronic
Topological Transition in FeTi”. Physical Review Letters 117, 076402 (2016).
DOI: 10.1103/PhysRevLett.117.076402.
F.C.Y analyzed the data, performed the computational analysis, and conducted
the writing of the manuscript.
[2] H. L. Smith, C. W. Li, A. Hoff, G. R. Garrett, D. S. Kim, F. C. Yang, M.
S. Lucas, T. Swan-Wood, J. Y. Y. Lin, M. B. Stone, D. L. Abernathy, M.
D. Demetriou, and B. Fultz. “Separating the configurational and vibrational
entropy contributions in metallic glasses”. Nature Physics 13, 900 (2017).
DOI: 10.1038/nphys4142.
F.C.Y participated in the experiment and commented on the manuscript.
[3] F. C. Yang, O. Hellman, M. S. Lucas, H. L. Smith, C. N. Saunders, Y.
Xiao, P. Chow, and B. Fultz. “Temperature dependence of phonons in Pd3 Fe
through the Curie temperature”. Physical Review B 98, 024301 (2018). DOI:
10.1103/PhysRevB.98.024301.
F.C.Y conceptualized the project, prepared the sample and conducted the experiment, reduced and analyzed the data, performed all calculations, and conducted
the writing of the manuscript.
[4] H. L. Smith, Y. Shen, D. S. Kim, F. C. Yang, C. P. Adams, C. W. Li, D. L.
Abernathy, M. B. Stone, and B. Fultz. “Temperature dependence of phonons
in FeGe2 ”. Physical Review Materials 2, 103602 (2018). DOI: 10.1103/PhysRevMaterials.2.103602.
F.C.Y performed one of the measurements, reduced the data, and commented
on the manuscript.
[5] F. C. Yang, O. Hellman, and B. Fultz. “Thermal Evolution of Electron-Phonon
Interactions in Vanadium”. In Preparation.
F.C.Y conceptualized the project, performed all calculations, and conducted
the writing of the manuscript.

viii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Harmonic Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . 6
1.4 Magnon-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . 9
Chapter II: Thermally Driven Electronic Topological Transition in FeTi . . . 12
2.1 Main Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Supporting Information . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter III: Thermal Evolution of Electron-Phonon Interactions in Vanadium 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Chapter IV: Temperature Dependence of Phonons in Pd3 Fe Through the Curie
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Supporting Information . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter V: Concluding Remarks and Future Directions . . . . . . . . . . . . 75
5.1 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . 75
5.2 Magnon-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . 77
Appendix A: Harmonic Model . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.1 Einstein and Debye Models . . . . . . . . . . . . . . . . . . . . . . 80
A.2 Born von Kármán Model . . . . . . . . . . . . . . . . . . . . . . . 80
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

LIST OF ILLUSTRATIONS

Number
Page
1.1 (Left) The harmonic Sh , quasiharmonic Sqh , and measured vibrational
entropy Svib of α-iron compared with the total entropy from the
SGTE database [6]. Vibrational entropies calculated and measured
by Mauger et al. [7]. (Right) The harmonic, quasiharmonic, and
measured vibrational entropy of vanadium [10] compared with the
total entropy from the SGTE database [6] and laser-flash calorimetry
measurements by Takahashi et al. [9]. . . . . . . . . . . . . . . . . 4
1.2 The effects of temperature (from bottom to top) on phonons (left)
and electrons (right), and the effects of the adiabatic electron-phonon
interaction (EPI) when phonons alter the electronic band structure in
the presence of electron excitations. Also shown are illustrations of
the quasiharmonic approximation and phonon-phonon interactions
(PPI). Increasing width of light shading indicates increasing thermal
energy spreads. From Ref. [4]. . . . . . . . . . . . . . . . . . . . . . 5
1.3 (Left) Phonon densities of states (DOS) of vanadium from 10 to
1273 K, as measured by Delaire et al. [10]. (Right) Electronic DOS
(eDOS) of vanadium at 0 and 1000 K. The 0 K eDOS was calculated
with a static density functional theory (DFT) calculation. The 1000 K
eDOS was calculated from a convolution of the 0 K eDOS with a
Lorentzian with a full-width-at-half-maximum (FWHM) of 380 meV
[10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Trends in the temperature dependence of phonon energies ω on adiabatic electron-phonon interactions for different electronic densities
of states N(E), as summarized by Delaire, et al. [26]. The dotted
line represents quasiharmonic (QH) behavior. . . . . . . . . . . . . . 9

1.5

2.1

2.2

2.3

2.4

(Left) The nonharmonic vibrational entropy ∆Snh from phonon DOS
spectra measured by Mauger et al. compared to the magnetization of
bcc Fe [29] and the magnetic vibrational entropy Smag , obtained by
subtracting Svib and Sel [32] from the SGTE total entropy [6]. From
Ref. [7]. (Right) Average Fe phonon energies of cementite from
NRIXS measurements (points). The dashed lines are quasiharmonic
energies from experiment (“QH γT Model”) and computation (“QH
DFT”). From Ref. [31]. . . . . . . . . . . . . . . . . . . . . . . . .
Calculated FeTi phonon dispersions at temperatures from 300 to
1500 K. Also shown are phonon DOS curves for the motions of all
atoms (total) and iron atoms (Fe partial). . . . . . . . . . . . . . . .
Temperature dependence of the M5− phonon energy calculated from
TDEP (squares). The colors of the squares are identical to those
shown in Fig. 2.1. The green and blue markers are mean phonon
energies obtained from Lorentizan fits to the Fe NRIXS DOS and
the INS DOS, respectively. The dashed line is the thermal softening
of the M5− phonon from quasiharmonicity alone. The inset shows
the agreement in the slopes of the experimental and computational
phonon energies without quasiharmonic contributions. For convenience in showing the slopes, the NRIXS data were offset by −0.76
meV and the INS by −0.35 meV. . . . . . . . . . . . . . . . . . . . .
Experimental FeTi phonon DOS curves. The neutron-weighted DOS
curves were obtained from INS measurements and the Fe partial DOS
curves from NRIXS measurements. Error bars are from counting
statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a)-(c) Approximated finite-temperature Fermi surfaces in the Brillouin zone. The dark blue surfaces are the unshifted 0 K Fermi
surfaces, and the lighter blue surfaces are at energies shifted from the
Fermi energy by factors of 1.8 k BT. (d)-(f) Finite-temperature band
structures from supercell AIMD calculations, produced by BandUP.
The Fermi surface and band structure at 523 K resemble those at 0 K
without any broadening. . . . . . . . . . . . . . . . . . . . . . . . .

xi
2.5

2.6

2.7
2.8

2.9
2.10
2.11
2.12

3.1

3.2

Histogram of spanning vectors that couple the new states at one of the
R symmetry points with the rest of the Fermi surface along the [ξξ0]
and 12 ξ0 directions at 1035 K, displayed together with the changes
in energies of the TA and LA branches from 300 K to 1035 K along
the same directions. . . . . . . . . . . . . . . . . . . . . . . . . . .
A view of the displacement pattern of the M5− phonon, in which Fe
(orange) and Ti (blue) atoms move along the 11̄0 directions. The
dashed lines are the 1nn Fe–Ti interactions, and the solid lines are
the 2nn Fe–Fe interactions. . . . . . . . . . . . . . . . . . . . . . . .
Neutron scattering function S (Q, E) spectrum of FeTi at 300 K. . . .
Phonon DOS curves for FeTi at elevated temperatures. The neutronweighted DOS curves were obtained from INS measurements and
the Fe partial DOS curves from NRIXS measurements. The two data
sets were combined to obtain neutron-weight-corrected DOS curves
and Ti partial DOS curves. Error bars from counting statistics. . . . .
FeTi phonon dispersions calculated in the quasiharmonic model. . . .
Calculated 0 K electron-phonon linewidths displayed over the 0 K
FeTi phonon dispersion. . . . . . . . . . . . . . . . . . . . . . . . .
Electronic DOS for FeTi from 0 to 1035 K. Obtained from AIMD
and static DFT calculations. . . . . . . . . . . . . . . . . . . . . . .
FeTi phonon dispersions calculated after individually exchanging
the (a) Fe–Fe 2nn longitudinal and (b) Fe–Ti 1nn transverse force
constants at 300 K with those at 1500 K. . . . . . . . . . . . . . . . .
(a) The phonon DOS curves of vanadium calculated with the s-TDEP
method at temperatures from 0 (dark purple) to 1650 K (orange). (b)
Average phonon energies of vanadium calculated with the s-TDEP
method (identical colors to those shown in (a)), shown together with
average phonon energies from inelastic neutron scattering measurements (Refs. [10, 84, 98]). The dashed red curve corresponds to
quasiharmonic (QH) behavior as calculated from first-principles. . . .
The 300 K spectral function calculated with s-TDEP along the highsymmetry directions, plotted together with measurements from thermal diffuse x-ray scattering (crosses) [116] and inelastic x-ray scattering (dots) [117]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

18
20

21
23
24
24

xii
3.3

3.4
3.5

3.6

3.7

3.8

4.1

4.2

Phonon dispersion curves of vanadium calculated with the s-TDEP
method at temperatures from 0 (dark purple) to 1650 K (orange).
Vector coordinates are written in simple cubic lattice coordinates. . . 35
Unfolded electronic bands at 1100 K, compared with 0 K electronic
bands in dark red. The Fermi level is represented as a spread in energies. 36
The {100} cross sections of the Fermi surface of vanadium at (a) 0 K
and (b) 1100 K (cross section indices are expressed in simple cubic
coordinates). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
(Top) 0 K electron-phonon coupling strengths of phonon modes calculated with Quantum ESPRESSO displayed over the 0 K vanadium
phonon dispersion calculated with s-TDEP. (Bottom) Autocorrelation
of the Fermi surface at 0 and 1100 K. . . . . . . . . . . . . . . . . . 39
{100} cross sections of the Fermi surface nesting strengths of vanadium for the transverse phonon mode at q = [0.24, 0, 0] at (a) 0 K
and (b) 1100 K, the longitudinal N phonon mode at (c) 0 K and (d)
1100 K, and the H phonon mode at (e) 0 K and (f) 1100 K. Sample
[0.24, 0, 0] and [0.5, 0.5, 0] spanning vectors are shown in (a) and (c). 42
The electrical resistivities of bcc transition metals relative to their
300 K values. The plotted data are from measurements on highpurity samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
(a) Schematic of supercells with Fe atoms (dark blue) stochastically
displaced from their ideal positions (light blue) in the 0 K ferromagnetic calculations, where the magnetic moments (red arrows) are
aligned in the same direction. (b) Supercells with randomly oriented
magnetic moments and stochastically displaced Fe atoms in the 800 K
paramagnetic calculations. Each set of randomly oriented magnetic
moments is a magnetic special quasirandom structure (SQS). Pd
atoms are not shown for this illustration. . . . . . . . . . . . . . . . . 52
The 57 Fe nuclear forward scattering spectra from L12 -ordered Pd3 Fe
at several temperatures. The fits (black curves) overlay experimental
data (points). The spectra are displayed using a log scale, and offset
for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

xiii
4.3

4.4

4.5

4.6
4.7

4.8

4.9

The magnetization curve of Pd3 Fe obtained from an empirical fit of a
magnetic shape function [172] to hyperfine magnetic fields obtained
from the NFS spectra in this study (green) and Mössbauer data from
a study by Longworth [154] (orange). The shaded region indicates
the temperature range where Pd3 Fe exhibits ferromagnetic order. . . .
The normalized 57 Fe pDOS extracted from NRIXS measurements at
various temperatures. The spectra from measurements above 298 K
are offset and compared with the 298 K pDOS (black curve). Error
bars are from counting statistics. . . . . . . . . . . . . . . . . . . . .
Average energies of the Fe pDOS from NRIXS measurements (blue
points) plotted with the average Fe phonon energies from the Grüneisen
parameter model (green line) and the QH DFT model (red line). . . .
Total, Pd partial, and Fe partial phonon DOS curves of Pd3 Fe calculated with the s-TDEP method from 0 to 800 K. . . . . . . . . . . . .
(a) NRIXS 57 Fe pDOS curves compared at 298 and 786 K. (b) s-TDEP
Fe pDOS curves compared at 300 and 800 K. Phonon difference
spectra are shown for both NRIXS and s-TDEP. . . . . . . . . . . . .
(a) The Fe partial vibrational entropy from the NRIXS measurements
compared with the entropy from the Grüneisen parameter model (QH
γT ) and the QH DFT model. (b) The s-TDEP Fe partial vibrational
entropy calculated for Pd3 Fe with changing magnetic order (blue),
ferromagnetic order (green), and the absence of phonon-phonon interactions (orange). The red line is the entropy from the QH DFT
model. The insets in (a) and (b) show the nonharmonic contributions
to the vibrational entropy. . . . . . . . . . . . . . . . . . . . . . . .
Calculated phonon dispersions for the ferromagnetic and paramagnetic states at 800 K. The dispersions displayed do not include effects
from phonon-phonon interactions. Displacement patterns are shown
for two high-energy optical phonon modes that soften with decreasing magnetization. The orange and green spheres represent Fe and
Pd atoms, respectively. The Fe partial phonon DOS curves of Pd3 Fe
calculated with the s-TDEP method for the ferromagnetic and paramagnetic states at 800 K are shown in the lower left. . . . . . . . . .

57
58

xiv
4.10

4.11

4.12

4.13

4.14

4.15

4.16
4.17

Pd3 Fe spectral functions (logarithmic intensity scale) calculated with
s-TDEP along the high-symmetry directions at 0, 300, and 800 K.
Measurements of the 80 K phonon dispersion by inelastic neutron
scattering [176] are shown on top of the 0 K spectral function. . . . .
Pd3 Fe phonon lineshapes at the X high symmetry point at 800 K. The
orange and green peaks are the optical modes that shift with changing
magnetic order. The black dashed peak is the lineshape of the optical
mode after the Pd−Pd 1NN cubic force constant is set to zero. . . . .
X-ray diffraction patterns of Pd57
3 Fe (“Ordered-II”) collected on a
Cu Kα laboratory diffractometer. Measurements were performed
on the sample after the heat treatment (“Pre-NRIXS”) and after the
NRIXS measurements (“Post-NRIXS”). The labeled peaks include
both fundamental and superlattice peaks. The intensity is displayed
in a logarithmic scale. . . . . . . . . . . . . . . . . . . . . . . . . .
(Top) Room temperature Mössbauer spectrum of the annealed Pd57
3 Fe
foil sample before the NRIXS experiment (Ordered-II). (Bottom)
Pd57
3 Fe Mössbauer spectra for the “Ordered-I” state, “Ordered-II”
state, and after the NRIXS experiment. . . . . . . . . . . . . . . . .
Two-dimensional synchrotron diffraction patterns of Pd3 Fe recorded
on a CCD detector plate at (a) 298 K and (b) 786 K. (c) Onedimensional synchrotron x-ray diffraction patterns of Pd57
3 Fe from
298 to 786 K. The black dashed lines are the locations of the 298 K
diffraction peaks. The dips in intensity for 627 and 786 K are where
extraneous diffraction peaks from the aluminum foil were masked. . .
(Left) Raw NRIXS scattering spectra showing 57 Fe vibrational excitations in Pd57
3 Fe as a function of scattering energy. Spectra are
collected over a range of temperatures. (Right) The elastic line of
the raw scattering spectrum of 57 Fe at room temperature, used as the
instrument resolution function for the NRIXS measurements. . . . . .
The multi-phonon components of the NRIXS spectra of Pd57
3 Fe at
(left) 298 K and (right) 786 K. . . . . . . . . . . . . . . . . . . . . .
Lamb-Mössbauer factors obtained from NRIXS (blue squares) and
NFS (green triangles). . . . . . . . . . . . . . . . . . . . . . . . . .

72
72
73

xv
The room temperature 57 Fe pDOS of the Pd57
3 Fe sample measured in
this study (purple) compared with prior measurements of the room
temperature 57 Fe pDOS of ordered Pd57
3 Fe (black) and the disordered
fcc alloy Pd0.75 57 Fe0.25 (red). . . . . . . . . . . . . . . . . . . . . . 74
5.1 The frequency shifts and linewidth of the Eg 187 cm−1 Raman line
versus temperature in FeF3 , measured by Shepherd [197]. . . . . . . 78

4.18

xvi

LIST OF TABLES
Number
Page
3.1 Electron-phonon interaction parameters and superconducting transition temperatures of nonmagnetic bcc transition metals. . . . . . . . 44
4.1 Hyperfine fields H and change in the hyperfine field ∆H in the Pd57
3 Fe
sample at different stages of this study, shown together with measurements performed by Longworth [154]. For the Pd57
3 Fe sample, the
hyperfine field for “Ordered-II” is treated as a reference for ∆H. For
the measurements performed by Longworth, the hyperfine field for
the ordered sample is treated as a reference for ∆H. . . . . . . . . . . 68
4.2 The Fe partial vibrational heat capacity calculated from the integration of the 57 Fe phonon DOS. . . . . . . . . . . . . . . . . . . . . . 72
4.3 The vibrational kinetic energy calculated from the integration of the
57 Fe phonon DOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 The mean force constant calculated from the integration of the 57 Fe
phonon DOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Chapter 1

INTRODUCTION
“A theory is the more impressive the greater the simplicity of its
premises, the more different kinds of things it relates, and the more
extended its area of applicability. Therefore the deep impression that
classical thermodynamics made upon me. It is the only physical theory
of universal content which I am convinced will never be overthrown,
within the framework of applicability of its basic concepts.”
— Albert Einstein
1.1

Overview

The traditional development of materials for industrial applications typically takes
decades, from initial discovery to market. The demand for new advanced materials
that address pertinent challenges including clean energy, national security, and
human welfare has only grown stronger over time. In response, recent research
efforts and initiatives including the Materials Genome Initiative have aimed to
accelerate the process of discovering, manufacturing, and mass producing new
materials [1].
High performance computing (HPC) has become a major part of this push towards
accelerated materials discovery. Computational materials discovery would allow
us to predict candidate materials given desired macroscopic properties, minimizing
the time and cost of experimental discovery. Recent developments to computational materials discovery have been made through high-throughput computing, in
which large volumes of ab initio calculations are combined with data mining of
large databases using supercomputer architectures [2]. With the explosion of new
materials discoveries from these new methods, we are living in exciting times.
Now we must, however, make sure to stay grounded in reality. These predictive
tools must be able to predict the existence of existing materials, and hypothetical
materials must be able to be developed under realistic conditions. In order to carry
out reality checks on developments in this new scientific field, we turn to the older
but fundamental branch of physics, thermodynamics. In particular, we turn to the
Gibbs free energy, which allows us to assess the thermodynamic stability of phases

of materials:
G = U + PV − T S.

(1.1)

The stable phase is the one that minimizes the free energy given conditions including temperature, pressure, mechanical stress, and magnetism [3]. Calculating and
understanding the free energy of a phase and of competing phases provides essential information on material stability under different operational temperatures, an
important consideration concerning the processing and performance of the material
in technological service.
The entropic contribution to the free energy of solids, −T S, changes rapidly with
temperature. The entropy S enumerates the way heat is stored in a material within
the many degrees of freedom of the material. For ordered crystalline solids, the
vibrations of atoms make up the largest entropic contribution. With increasing
temperature, we see increased populations of phonons, quanta of lattice vibrations,
and greater vibrational excursions of atoms. The vibrational entropy Svib increases
with temperature because the material explores a larger volume in the phase space
of position and momentum [4]. Understanding the vibrational entropy and the
lattice dynamics of a material is crucial for accurately calculating the temperaturedependent behavior of its free energy.
1.2

Harmonic Lattice Dynamics

We can begin our discussion of lattice vibrations by building off of the harmonic
model, in which lattice vibrations are calculated from a Taylor expansion of the
potential energy of the crystal up to the second order of atomic displacement. In
this model, normal modes do not transfer energy to other normal modes, such that
phonons with energies εi may be considered as independent harmonic oscillators
that persist forever. This treatment allows us to write the partition function of a
harmonic solid with N atoms and 3N independent oscillators as the product of
individual oscillator partition functions:
ZN =

3N
e−βεi /2
Zi =
1 − e−βεi

(1.2)

where β ≡ (k BT)−1 . From this partition function, we can calculate the vibrational
entropy by differentiating the phonon free energy Fvib = −kBT ln Z N :
3N
∂Fvib
βεi
−βεi
Svib = −
− ln 1 − e
= kB
+ βε
∂T V
e i −1

(1.3)

From the vibrational entropy, we can calculate thermodynamic quantities such as
the heat capacity from lattice vibrations:
3N
e βεi
∂S
= kB
CV (T) = T
(βεi )2 βε
∂T V
(e i − 1)2

(1.4)

The mean occupation of these bosonic oscillators can be described with the Planck
distribution
nT (εi ) = βε
(1.5)
e i −1
with which we can rewrite Eq. 1.3 as
Svib = k B

[(nT (εi ) + 1) ln(nT (εi ) + 1) − nT (εi ) ln(nT (εi ))] .

(1.6)

We often work with a phonon density of states (DOS) g(ε), a distribution of phonon
modes with respect to energy. Measurements or calculations of phonon frequencies
allow us to calculate the vibrational entropy and thermodynamic quantities:
(1.7)
Svib = 3N k B g(ε)[(nT (ε) + 1) ln(nT (ε) + 1) − nT (ε) ln(nT (ε))]dε.
In this harmonic approximation of the vibrational entropy, the only temperature
dependence is from the Planck distribution nT (ε). Phonon frequencies and the DOS
can be calculated following the Einstein, Debye, or Born von Kármán model [5], as
discussed in Appendix A.
The harmonic model often explains physical phenomena at low temperatures. Fig.
1.1 shows that the harmonic vibrational entropy Sh of α-iron deviates from both the
total entropy [6] and vibrational entropy measured with inelastic x-ray scattering
[7] at high temperatures. We can rectify some shortcomings of the harmonic model
with the quasiharmonic (QH) model, which introduces effects of thermal expansion.
Phonons are still treated as independent harmonic oscillators, but the frequencies
are scaled by the change in volume with increasing temperature:
∆V
∆ω
= −γ

(1.8)

Entropy (kB/atom)

SGTE
Takahashi, et al.
Svib (INS)
Svib (QH)
Svib (Harmonic)
200 400 600 800 1000 1200 1400 1600
Temperature (K)

Figure 1.1: (Left) The harmonic Sh , quasiharmonic Sqh , and measured vibrational
entropy Svib of α-iron compared with the total entropy from the SGTE database [6].
Vibrational entropies calculated and measured by Mauger et al. [7]. (Right) The
harmonic, quasiharmonic, and measured vibrational entropy of vanadium [10] compared with the total entropy from the SGTE database [6] and laser-flash calorimetry
measurements by Takahashi et al. [9].
where the proportionality constant γ is the Grüneisen parameter [8]. Using the
expression for the vibrational entropy (Eq. 1.6), its temperature dependence is
introduced from both the Planck distribution nT (ε) and how the phonon DOS gT (ε)
is scaled with Eq. 1.8. Fig. 1.1 shows that the QH model improves on the harmonic
model, but still falls short of the measured vibrational entropy. Fig. 1.1 also shows
us how the vibrational entropy from the QH model deviates from the total entropy
of vanadium [6, 9] and fails to model the vibrational entropy of vanadium measured
with inelastic neutron scattering [10]. We note for Fig. 1.1 that the difference
between the total and vibrational entropies of α-Fe is the sum of the electronic and
magnetic entropies, and the difference between the total and vibrational entropies
of vanadium is the electronic entropy.
We see that even the QH approximation is too simple for modeling the behavior
of phonons in real crystals. The treatment of phonons as independent harmonic
oscillators fails to explain macroscopic phenomena like thermal expansion or finite
thermal conductivities. More realistic models of phonon behavior must take into
account the previously neglected higher order terms in atomic displacements of the
potential energy. With these additional terms, we can no longer treat phonons as
independent oscillators. We need to take into account phonon interactions to make
accurate predictions of the free energy of a material. The nonharmonic contributions
to the vibrational entropy may appear to be minor numerical corrections. These

Figure 1.2: The effects of temperature (from bottom to top) on phonons (left)
and electrons (right), and the effects of the adiabatic electron-phonon interaction
(EPI) when phonons alter the electronic band structure in the presence of electron excitations. Also shown are illustrations of the quasiharmonic approximation
and phonon-phonon interactions (PPI). Increasing width of light shading indicates
increasing thermal energy spreads. From Ref. [4].
seemingly minor numerical corrections, however, can correct for when the QH
model incorrectly predicts phase instabilities [7, 11–14].
The anharmonic phonon-phonon interaction (PPI) underlies phenomena like the
thermal expansion or finite thermal conductivities. Electron-phonon interactions
(EPIs) are responsible for Cooper pairing in conventional superconductors and play
a role in the temperature dependence of the electrical resistivity in metals and the
carrier mobility in semiconductors. Phonons can also interact with magnetic excitations, impacting the thermodynamics of magnetic systems, such as the phase
transitions of the Ni2 MnGa magnetic shape memory system [12]. From experimental measurements of phonons in materials at high temperatures, we observe that
these phonon interactions shift and broaden their energies (Fig. 1.2).

We are interested in expanding our understanding of these phonon interactions,
particularly at high temperatures. While we study the effects of phonon-phonon
interactions in this thesis, the effect of anharmonicity on thermodynamic properties is not the focal point. The remainder of this introductory chapter provides
overviews of the electron-phonon and magnon-phonon interactions, which follows
the organization of this thesis.
1.3

Electron-Phonon Interactions

When a phonon is excited, atoms are moved from their equilibrium positions. The
effective electrostatic potential acting on an electron is changed by this atomic
displacement. As a result, this electron is likely to be scattered. This scattering
process can alter the electronic screening of the atom if the phonon wavelength is
comparable to the electron screening length [15].
This interaction at low temperatures has been mathematically modeled through
perturbation theory. As detailed by Ziman [15] and Grimvall [16], an important
component of perturbation theory is the electron-phonon matrix element:
κα
~ Õ eqν
(1.9)
gmn
(k, q) =
√ hmk + q|∂qκαV |nki,
2ωqν κα mκ
where mκ is the nuclear mass of atom κ, eqν is the phonon displacement eigenvector,
and ∂qκαV is the variation of the electronic potential with respect to displacement of
atom κ in the Cartesian direction α. This matrix element describes the interaction
of the phonon mode with wavevector q and polarization ν with the electronic states
|nki and |mk + qi.
ν (k, q) is included in electron-phonon scattering rates. The
The matrix element gmn
scattering rate of phonons by electron-phonon interactions derived from either the
phonon self-energy [16] or Fermi’s golden rule [17, 18] is
2π Õ
dk ν
g (k, q) ( fnk − fmk+q )
τqν
~ mn
ΩBZ mn
(1.10)
× δ(εm (k + q) − εn (k) − ~ωqν ),

where fnk is the Fermi-Dirac distribution for electrons and εn (k) is the electronic
energy.
One takeaway from Eq. 1.10 is that the crystal momentum is conserved in this
interaction. A phonon with wavevector q interacts with two electronic states with

wavevectors k and k0 if their wavevectors follow the condition
q = k0 − k − g,

(1.11)

where g is an arbitrary reciprocal lattice vector. A nonzero g describes electronphonon Umklapp processes, which is relevant only if the electronic structure is
treated with the reduced zone scheme [15].
Another takeaway from Eq. 1.10 is that the energy is conserved in the electronphonon interaction:
ε(k0) = ε(k) ± ~ωqν,
(1.12)
where we account for both phonon absorption and creation.
The difference ( fnk − fmk+q ) in Eq. 1.10 describes how an electron is scattered
from an occupied state into an unoccupied state. If we also take into consideration
how phonon energies are small in scale in comparison to electron energies, we see
that electron-phonon interactions involve states near the Fermi surface, a map of
k-points in the Brillouin zone (reciprocal space primitive cell) where the electronic
energy equals the Fermi energy: ε ≡ E − EF = 0.
An electron-phonon interaction involving a phonon mode qν is particularly strong
if there is a high density of electronic state pairs |nki and |mk + qi on the Fermi
surface. In other words, an electron-phonon interaction involving a phonon mode qν
is strong if there are nesting features, or parallel sheets separated by one wavevector
q, on the Fermi surface. This causes an electronic screening that reduces the energy
of the phonon mode qν, causing a Kohn anomaly [19]. These strong interactions
can contribute to a high superconducting transition temperature Tc [20, 21] and a
high average electron-phonon coupling constant λ:
λ=

N(EF )

ν (k, q)| 2
|gmn

m ω2

(1.13)

where N(EF ) is the electronic density of states at the Fermi energy, hh· · · iiFS is the
average over all possible combinations of electronic states on the Fermi surface, and
ω2 is the second moment of the Éliashberg coupling function [16].
Our understanding of electron-phonon interactions up to this point in the thesis has
been based on concepts from low temperatures. We cannot say for certain how
well the concepts behind low temperature electron-phonon interactions translate to
high temperatures, where the Fermi surface is altered by thermal excitations. The

0.08
0.06
0.04
0.02
0.00

10 K
300 K
773 K
1023 K
1273 K
10

20
30
Energy (meV)

eDOS (states/eV)

DOS (1/meV)

0K
1000 K
(Delaire)
0.5
0.0
Energy E - EF (eV)

0.5

Figure 1.3: (Left) Phonon densities of states (DOS) of vanadium from 10 to 1273 K,
as measured by Delaire et al. [10]. (Right) Electronic DOS (eDOS) of vanadium at
0 and 1000 K. The 0 K eDOS was calculated with a static density functional theory
(DFT) calculation. The 1000 K eDOS was calculated from a convolution of the 0 K
eDOS with a Lorentzian with a full-width-at-half-maximum (FWHM) of 380 meV
[10].
thermodynamic importance of the adiabatic EPI at high temperatures had been a
controversial topic [22–24].
Studies by Delaire et al. showed that the adiabatic EPI has significant effects on the
high temperature thermodynamics of the A15 V3 Si and V3 Ge compounds and bcc
vanadium. First-principles calculations for these materials showed that a thermal
broadening smears the electronic DOS, reducing the number of electronic states
at the Fermi level. A thermal phonon stiffening (increase in energy) opposed the
expected softening (decrease in energy) from thermal expansion due to the decrease
in the number of electrons available for screening, as illustrated in Fig. 1.3. The
suppressed vibrational entropy and heat capacity were attributed to this interplay
between electrons and phonons, which was observed at temperatures above 1000 K
[10, 25].
The opposite effect was observed in a study by Delaire et al. on FeSi. The electronic
DOS at the Fermi level was observed to increase with temperature as the narrow
band-gap was filled in the semiconductor-to-metal transition. The anomalous thermal phonon softening was due to the increased electronic screening efficiency [26].
Fig. 1.4 summarizes the observed temperature-dependent interplay of electrons and
phonons.
The aim of Chapters 2 and 3 of this thesis is to expand our knowledge of the high
temperature adiabatic electron-phonon interaction. With scattering measurements
and state-of-the-art computational methods, we can study the temperature-dependent
lattice dynamics and the electronic structure in greater detail than what is provided

Figure 1.4: Trends in the temperature dependence of phonon energies ω on adiabatic
electron-phonon interactions for different electronic densities of states N(E), as
summarized by Delaire, et al. [26]. The dotted line represents quasiharmonic (QH)
behavior.
by the phonon DOS and electronic DOS. We study the B2-ordered compound FeTi
in Chapter 2 and revisit bcc vanadium in Chapter 3.
1.4

Magnon-Phonon Interactions

As the temperature increases through a magnetic transition temperature, the vibrations of atoms away from their equilibrium positions can modify the interactions of
magnetic moments. The resulting changes in magnetic interactions between atoms
can in turn alter their vibrations against each other. Magnon-phonon interactions,
the exchange of energy between quanta of magnetic excitations and phonons, can
be modeled with a coupling Hamiltonian Hmp [27]:
Hmp =
φk,q bk−q bk aqν − a−qν ,
(1.14)
k,q
where aqν
and a−qν are phonon creation and annihilation operators, b†k−q and bk are
ν is an interaction coefficient.
magnon creation and annihilation operators, and φk,q

Figure 1.5: (Left) The nonharmonic vibrational entropy ∆Snh from phonon DOS
spectra measured by Mauger et al. compared to the magnetization of bcc Fe [29]
and the magnetic vibrational entropy Smag , obtained by subtracting Svib and Sel [32]
from the SGTE total entropy [6]. From Ref. [7]. (Right) Average Fe phonon
energies of cementite from NRIXS measurements (points). The dashed lines are
quasiharmonic energies from experiment (“QH γT Model”) and computation (“QH
DFT”). From Ref. [31].
Magnon-phonon interactions can have an impact on bulk properties of materials,
such as the negligible thermal expansion in Invar materials [28].
The investigation of magnon-phonon interactions at elevated temperatures in this
thesis was motivated by recent studies of vibrational thermodynamics impacted by
thermal magnetic disorder. Mauger et al. have observed through nuclear resonant
inelastic x-ray scattering (NRIXS) measurements an anomalous thermal softening
of phonons in bcc α-iron stronger than predicted by the QH model [7]. The strong
deviation from the QH model tracked the rapid decrease in the magnetization of
α-Fe [29] and the rise in magnetic entropy, as seen in the left panel of Fig. 1.5.
The second-nearest-neighbor (2NN) longitudinal force constants associated with an
abnormal 2NN exchange interaction [30] were observed to soften by 40 to 60% from
30 K to the Curie temperature. The sharp increase in the vibrational entropy from
the magnon-phonon interaction was reported to help extend the stability of the bcc
phase of iron well past its Curie temperature.
The study by Mauger et al. of ferromagnetic cementite Fe3 C showed more subtle
impacts of the Curie transition on lattice dynamics. The average Fe phonon energies
measured with NRIXS were observed to be nearly constant to the Curie temperature,
as seen in the right panel of Fig. 1.5. At higher temperatures where cementite is
paramagnetic, the phonon energies began to soften. A magnon-phonon interaction
was observed to stiffen phonons below the Curie temperature and possibly drive
changes in the elastic constants [31].

11
Temperature-dependent magnon-phonon interactions have also been observed in
computational studies. Körmann et al. analyzed the impact of magnetic short-range
order on the lattice vibrations above the Curie temperature of α-Fe by combining
the NRIXS data provided by Mauger et al. with a computational framework [33].
A more recent example is the study by Stockem et al. that replicated the abnormal
temperature dependence of the thermal conductivity above the magnetic transition
temperature by dynamic coupling of spin fluctuations and lattice vibrations [34].
The aim of Chapter 4 of this thesis is to not only extend our knowledge of
temperature-dependent magnon-phonon interactions, but to further develop the computational methods for predicting the lattice dynamics of materials with increasing
thermal magnetic disorder. With experiments and computations, we analyze the
lattice dynamics of Pd3 Fe across its Curie temperature at ambient pressure.

12
Chapter 2

THERMALLY DRIVEN ELECTRONIC TOPOLOGICAL
TRANSITION IN FETI
2.1

Main Text

An electronic topological transition (ETT), first identified by Lifshitz [35], occurs
when changes to a metal cause new features to appear in the topology of the Fermi
surface [36]. Structural, mechanical, and electronic properties are usually altered by
an ETT, which can be induced by alloying [37–39] or pressure [40–42]. Recently a
novel temperature-induced ETT has been reported to alter magnetoresistivity [43].
In this present work, we show through first-principles calculations and ancillary
experiments how a thermally-driven ETT drives anomalous changes in phonon
dynamics.
FeTi is a thermodynamically stable [44–46] nonmagnetic [47] intermetallic compound with a bcc-based B2 structure and a melting point of approximately 1600 K.
FeTi is of interest for its hydrogen absorption capabilities [48–50] and for its mechanical properties [51, 52]. It has been the subject of a large number of experimental and
theoretical studies including inelastic neutron scattering [53, 54] and first-principles
calculations [55–57]. The calculations show that FeTi has a Fermi level that lies in
a pseudogap in its electronic density of states (DOS) [58–60], so thermal smearing
could increase the effective density of electrons at the Fermi level.
First-principles calculations on FeTi were performed with projector augmented
wave potentials [61, 62] and the generalized gradient approximation [63] of density
functional theory (DFT) [64] using the VASP package [65, 66]. The electronic
DOS curves at various temperatures were obtained through static calculations and
constant volume ab-initio molecular dynamics (AIMD) calculations on 128-atom
supercells. Convergence with respect to kinetic energy cutoffs and sampling of k
points in the Brillouin zone was checked in all cases. The calculations show that
the pseudogap is present at the Fermi level in the 0 K electronic DOS, and is filled
as the number of electronic states at the Fermi level increases by 218% from 0 K to
1035 K, as shown in the Supporting Information.
Forces and atomic configurations in the AIMD simulations were used to compute
interatomic force constants at different temperatures with the temperature-dependent

[000]

0.5[111]

0.5[100] 0

Phonon Energy (meV)

40
30

Total DOS

Fe partial DOS

20
10
[000] 0.5[100] 0.5[110]

0.1 0.2
DOS (1/meV)

Figure 2.1: Calculated FeTi phonon dispersions at temperatures from 300 to 1500 K.
Also shown are phonon DOS curves for the motions of all atoms (total) and iron
atoms (Fe partial).
effective potential (TDEP) method [11, 67]. The quadratic and cubic force constants
were calculated from the model Hamiltonian fit to the potential energy surface at
the most probable positions of atoms in an AIMD simulation:
H = U0 +

Õ p2

2mi

1 Õ αβ α β 1 Õ αβγ α β γ
Φ u u +
Φ u u u ,
2! i jαβ i j i j 3! i j kαβγ i j k i j k

(2.1)

where pi is the momentum of atom i, Φi j and Φi j k are quadratic and cubic force
constants, and uiα is the Cartesian component α of the displacement of atom i. The
effects of the quartic force constants are included by renormalizing the quadratic
force constants [68, 69]. The cubic force constants account for cubic anharmonicity
associated with phonon-phonon interactions [70]. The thermal phonon-phonon
interaction shifts ∆(T) of a phonon mode qν calculated from the third-order force
constants were negligible compared to the calculated phonon thermal softening from
the quadratic force constants.
The force constants from Eq. 2.1 were used to obtain phonon dispersions and
phonon DOS curves at temperatures from 300 to 1500 K (Fig. 2.1). Thermal
expansion causes phonons to soften with temperature, and this was accounted for
by the quasiharmonic calculations presented in the Supporting Information section.
The AIMD calculations were performed without thermal expansion, so Fig. 2.1

Phonon Energy (meV)

27
26
25 27
26
24

25
300
300

600 900
600
900
1200
Temperature (K)

1500

Figure 2.2: Temperature dependence of the M5− phonon energy calculated from
TDEP (squares). The colors of the squares are identical to those shown in Fig. 2.1.
The green and blue markers are mean phonon energies obtained from Lorentizan
fits to the Fe NRIXS DOS and the INS DOS, respectively. The dashed line is the
thermal softening of the M5− phonon from quasiharmonicity alone. The inset shows
the agreement in the slopes of the experimental and computational phonon energies
without quasiharmonic contributions. For convenience in showing the slopes, the
NRIXS data were offset by −0.76 meV and the INS by −0.35 meV.
shows the thermal effects from pure anharmonicity and from the adiabatic EPI.
These are significantly larger than the thermal softenings from quasiharmonicity
reported in the Supporting Information.
The nonadiabatic electron-phonon interaction (EPI) is well known from conventional
superconductivity, where electrons are paired by phonons with wavevectors that
span the Fermi surface [16, 71]. With increasing temperature the effects of the
nonadiabatic EPI dissipate [16], but there can be an increase in the adiabatic EPI,
which requires excitations of both electrons and phonons [24, 72]. The adiabatic EPI
can have a significant effect on the high-temperature thermodynamics of materials
with sharp features in the electronic DOS at the Fermi level because the thermal
broadening of electronic states can change the availability of electrons to screen
atomic displacements in phonons [4, 10, 25, 26].

Phonon DOS (1/meV)

Neutron0.15 weighted DOS

300 K
523 K
750 K
1035 K

0.10
0.05
0.00

Fe partial DOS

20
30
Energy (meV)

Figure 2.3: Experimental FeTi phonon DOS curves. The neutron-weighted DOS
curves were obtained from INS measurements and the Fe partial DOS curves from
NRIXS measurements. Error bars are from counting statistics.
To help understand the EPI in FeTi, density functional perturbation theory [73]
implemented with the Quantum ESPRESSO package [74] was used to calculate 0 K
electron-phonon linewidths
2π Õ
dk ν
gmn (k, q) ( fnk − fmk+q )
(2.2)
Γqν =
~ mn
ΩBZ
× δ(εm (k + q) − εn (k) − ~ωqν ),

(2.3)

ν (k, q) is the matrix element of an EPI involving a phonon and two elecwhere gmn
tronic states |nki and |mk + qi, fkm is the Fermi-Dirac distribution for electrons,
and εn (k) is the eigenenergy of an electron. The Supporting Information shows that
many of the modes that soften with temperature in Fig. 2.1 are those with strong
electron-phonon coupling and large Γqν . Interestingly, these calculations showed
that the M5− mode has a negligible electron-phonon linewidth at 0 K, even though it
shows the strongest thermal softening in Fig. 2.1.

Figure 2.1 shows the anomalous behavior of the M5− mode, which softens increasingly rapidly with temperature as shown in Fig. 2.2. The M5− mode and modes near
it contribute strongly to the softening of the phonon DOS peak around 25–27 meV.
This mode is dominated by the motions of iron atoms, and both the experimental

(a) 523 K

(b) 750 K

0.5

(d) 523 K

X|M

0.0

1.0

0.5

1.0

Number of States

1.0

Energy (eV)

Number of States

Energy (eV)

1.5

Energy (eV)

2.0

Number of States

0.5

(e) 750 K

X|M

0.0

X|M

0.0

(f) 1035 K

Figure 2.4: (a)-(c) Approximated finite-temperature Fermi surfaces in the Brillouin
zone. The dark blue surfaces are the unshifted 0 K Fermi surfaces, and the lighter
blue surfaces are at energies shifted from the Fermi energy by factors of 1.8 kBT. (d)(f) Finite-temperature band structures from supercell AIMD calculations, produced
by BandUP. The Fermi surface and band structure at 523 K resemble those at 0 K
without any broadening.
DOS curves from inelastic neutron scattering (INS) and nuclear-resonant inelastic
x-ray scattering (NRIXS) shown in Fig. 2.3 emphasize phonon scattering from iron
atoms (Ti is a weaker scatterer of neutrons, and Ti cannot contribute to the NRIXS
spectrum). When the phonon softening from thermal expansion, obtained from
quasiharmonic calculations described in the Supporting Information is removed
from the experimental points, the agreement in slopes of the curves in Fig. 2.2 is
excellent. The magnitudes of the phonon energies show agreement between computation and experiment that is better than expected. For example, the experimental
peaks include contributions from phonons around the R point, which lie above 3
meV higher than the M5− mode. (Agreement at lower temperatures is not expected
owing to the use of classical statistical mechanics in the AIMD calculations.)
To calculate the adiabatic EPI, the effects of phonons were simulated by DFT
calculations on supercells with thermal atom displacements, obtained at random
times during the AIMD simulations. The thermal excitations of electrons were
described by a thermal smearing function from the energy derivative of the FermiDirac distribution function, which is similar to a Gaussian function with a standard
deviation of σ = 1.8 k BT. A discrete set of energies representative of this thermal

2.0

1.5

TA Branch 80
LA Branch
60

1.0

0.5

0.0
0.5[100]

Number of Vectors

Phonon Softening (meV)

0.5[110]

[000]

Figure 2.5: Histogram of spanning vectors that couple the new states at one of the
R symmetry points with the rest of the Fermi surface along the [ξξ0] and 12 ξ0
directions at 1035 K, displayed together with the changes in energies of the TA and
LA branches from 300 K to 1035 K along the same directions.
spread gave a set of Fermi levels that were used to construct the Fermi surfaces
of Fig. 2.4 (a)-(c). The BandUP code [75, 76] was used to project the supercell
band structures into the range of k-space for a standard B2 unit cell. Through
unfolding operations [77], BandUP obtains effective primitive cell representations
of the band structures of systems simulated using supercells. Results are shown in
Fig. 2.4(d)-(f). (We found no noticeable differences in the band structures when
thermal expansion was included, as reported previously in Ref. [57].)
Owing to a decrease in band energy from thermal atom displacements, but more
to the thermal smearing of the Fermi level, electronic states at the R point that lie
above the Fermi level at 0 K intersect the Fermi level at high temperatures. New
topological features appear in the Fermi surface around the R symmetry points and
along the M-R symmetry lines of the Brillouin zone, as shown in Figs. 2.4(a)-(c).
These new features grow with increasing temperature. This is a thermally-driven
electronic topological transition.
When the Fermi surface allows for many spanning vectors of phonons, the electronic screening of charge displacements can be more efficient, and phonons exhibit
softenings such as Kohn anomalies [19]. With the appearance of thermally-driven
features of the Fermi surface around the R point, new sets of spanning vectors are
available to couple electronic states across the Fermi surface. Spanning vectors
along the [ξξ0] and 12 ξ0 directions that connect these new features and the Fermi
surface feature around the X points were counted as described in the Supporting In-

Figure 2.6: A view of the displacement pattern of the M5− phonon, in which Fe
(orange) and Ti (blue) atoms move along the 11̄0 directions. The dashed lines are
the 1nn Fe–Ti interactions, and the solid lines are the 2nn Fe–Fe interactions.
formation. The numbers of vectors obtained for 1035 K are displayed in a histogram
in Fig. 2.5. This distribution overlaps well with the group of wavevectors over which
the transverse acoustic (TA) and longitudinal acoustic (LA) phonon branches soften
significantly around the M symmetry point. These new spanning vectors should
increase screening of the corresponding phonon modes by conduction electrons,
causing the large softening of phonons as seen in Fig. 2.2. The softening graphed in
Fig. 2.5 was corrected for the softening expected from phonon-phonon interactions
calculated by the TDEP method, as described in the Supporting Information.
Interatomic force constants were calculated by the TDEP method, and they showed
thermal weakening of both Fe–Ti first-nearest-neighbor (1nn) transverse force constants and Fe–Fe second-nearest-neighbor (2nn) longitudinal force constants. By
testing the sensitivity of the phonon dispersions to changes in these force constants,
we found that the thermal weakening of both the Fe–Ti 1nn transverse force constants and the Fe–Fe 2nn longitudinal force constants contribute significantly to the
thermal softening of the M5− mode. This behavior is consistent with the atomic dis
placement pattern shown in Fig. 2.6, in which (110) planes slide in opposite 11̄0
directions. (This is also a proposed displacement pattern for the structural phase
transition in B2-ordered NiTi, a shape-memory alloy [78, 79].) From the phonon
polarization vectors, we found that the magnitude of the Fe displacement is at least
twice as that of Ti. Softening of the the Fe–Ti 1nn transverse force constants and
the Fe–Fe 2nn longitudinal force constants are particularly effective for softening
the M5− mode, and these changes occur with the thermally-driven ETT.

19
It has been known for a number of years that the adiabatic EPI can alter the phonon
dynamics, often making an important contribution to the free energy of a metal
or alloy. A temperature dependence of the adiabatic EPI occurs when there is
a substantial variation in the electronic DOS at the Fermi level, for example. A
thermally-driven ETT is expected to cause more rapid and perhaps more abrupt
changes with temperature. Such effects are expected in materials with occupied or
unoccupied bands that are a few k BT away from the Fermi level at low temperatures,
so these effects are expected in many systems. Shifts and broadenings of the
electronic bands from atomic displacement disorder can enhance or diminish these
effects.
2.2

Supporting Information

Experimental
Sample Preparation
The FeTi sample for neutron scattering measurements was synthesized by arcmelting 99.98% pure Ti and 99.97% pure Fe in the equiatomic ratio under an argon
atmosphere. There was a negligible mass loss and no visible oxidation after melting.
The brittle sample was crushed into a fine powder. Samples for x-ray measurements
were synthesized in the same way, but were 96% enriched with 57 Fe. Conventional
characterization was performed with x-ray diffractometry using Cu Kα radiation.
All samples were found to have the B2 structure and no traces of other phases.
These diffraction measurements included the determination of thermal expansion
using a furnace for in situ measurements. The lattice parameters of B2 FeTi were
determined to be
2.978 Å at 300 K; 2.984 Å at 523 K; 2.991 Å at 750 K.
The coefficient of linear thermal expansion was 2.9×10−5 /K.
Scattering Measurements
Nuclear resonant inelastic x-ray scattering (NRIXS) measurements were performed
at beamline 16 ID-D at the Advanced Photon Source at Argonne National Laboratory.
NRIXS is sensitive only to the motions of 57 Fe atoms, so it provides partial phonon
density of states (DOS) curves for Fe alone. For measurements at 300, 523, 748,
and 1035 K, the sample was loosely dispersed between two Kapton polyimide films
and then accommodated in a custom-built vacuum furnace. The sample was fixed at

Figure 2.7: Neutron scattering function S (Q, E) spectrum of FeTi at 300 K.
a grazing angle to the x-ray beam and an avalanche photodiode detector (APD) was
set on top of the furnace at a right angle with the beam. The energy was scanned
from −90 to +90 meV around 14.413 keV, the resonant energy of 57 Fe, in several
scans that were combined for final analysis. The energy resolution of all NRIXS
measurements was measured to be 2.2 meV (FWHM) at the elastic line. The NRIXS
data were reduced using the PHOENIX code [80].
Inelastic neutron scattering (INS) measurements were performed with the ARCS
spectrometer [81] at the Spallation Neutron Source at Oak Ridge National Laboratory. The sample was loaded into an Al can which was then mounted in a
low-background vacuum furnace for measurements at 300, 523, and 748 K. The
nominal incident neutron energy was 80 meV. The energy resolution was 1.7 meV
at an energy transfer of 40 meV, increasing to 3.1 meV at the elastic line (FWHM).
The empty Al can was measured at all temperatures and subtracted from the measured spectra of the sample. The data reduction was performed using the DANSE
software [82], giving the neutron-weighted DOS curves gNW (ε) shown in Fig.2.8.
Our NRIXS and INS spectra had very similar energy resolutions over much of the
phonon spectra, and this permitted direct comparisons.
The neutron-weighting arises from the differences in the masses and neutron scattering cross-sections for each element and isotope. The neutron-weighted curves from
INS could be corrected using the partial phonon DOS gFe (ε) obtained from NRIXS
as described in Refs. [83–86]. Results are shown in the “neutron-weight-corrected

0.30

Neutronweighted DOS

300 K
523 K
750 K
1035 K

Phonon DOS (1/meV)

0.25
0.20
0.15

Neutron-weight
corrected DOS

Fe partial DOS

0.10
0.05
0.00

Ti partial DOS

20
30
Energy (meV)

Figure 2.8: Phonon DOS curves for FeTi at elevated temperatures. The neutronweighted DOS curves were obtained from INS measurements and the Fe partial
DOS curves from NRIXS measurements. The two data sets were combined to
obtain neutron-weight-corrected DOS curves and Ti partial DOS curves. Error bars
from counting statistics.

22
DOS" panel in Fig. 2.8, noting that
gNW (ε) ≃

σFe
σTi
gFe (ε) +
gTi (ε) ,
MFe
MTi

(2.4)

where the ratios of the neutron cross section to molar mass σd /Md are 0.208 and
0.091 barns/amu for Fe and Ti, respectively. The partial Ti phonon DOS gTi (ε) was
also obtained using this approach and is shown in the lower panel of Fig. 2.8.
Computational Details
Density Functional Theory
Density functional theory (DFT) calculations [64] were performed with VASP [65,
66] with projector augmented wave potentials with the Perdew-Burke-Ernzerhof
(PBE) exchange-correlation functional [63] and an energy cutoff of 500 eV. The
equilibrium volume of the structure was optimized to minimize the total energy.
Spin-polarized calculations gave a negligible magnetic moment, in agreement with
experimental results in Ref. [47].
The 0 K electronic band structure and Fermi surface were computed on the FeTi twoatom unit cell with a 80×80×80 grid of k-points, sampled with the Monkhorst-Pack
scheme [87]. The dense grid allowed for a greater sampling of phonon wavevectors
which may span nesting features of the Fermi surface.
To assess the effect of thermal expansion on the phonons in FeTi, phonon dispersions
shown in Fig. 2.9 were calculated under the quasiharmonic approximation using the
Parlinski-Li-Kawazoe method [88] implemented in the PHONOPY code [89]. The
calculations were performed on 128-atom 4×4×4 supercells with a 6×6×6 k-point
mesh and a grid of atom displacements of 0.01 Å for the temperatures 0, 300, 523,
750, and 1035 K. The volumes of the supercells were calculated by minimizing the
free energy F(T, V):
F(T, V) = E0 (V)
i
hε
+ k BT ln 1 − e−ε/kBT .
dε g(ε)

(2.5)

Ground-state energies E0 (V) were calculated separately for each volume, and the
DOS g(ε) were calculated with lattice parameters that produced the minimized
volume at each temperature. The calculated lattice parameters were smaller than the
experimental ones by 0.7%. Figure 2.9 shows that the softening of the M5− phonon
mode from thermal expansion alone is small compared to the large anomalous
softening discussed in the main text.

Phonon Energy (meV)

40
30

0K
300 K
523 K
750 K
1035 K

[000] 0.5[100] 0.5[110]

[000]

0.5[111]

0.5[100]

Figure 2.9: FeTi phonon dispersions calculated in the quasiharmonic model.
Density Functional Perturbation Theory
Density functional perturbation theory calculations [73] were performed with Quantum ESPRESSO [74] with ultrasoft pseudopotentials [90] and the PBE exchangecorrelation functional. The electron-phonon interaction matrix elements were first
calculated on a 20 × 20 × 20 k mesh and a 10 × 10 × 10 q mesh, and later interpolated
to a 60 × 60 × 60 q mesh through Fourier interpolation implemented in the package. The matrix elements were then used to compute the electron-phonon linewidth
plotted in Fig. 2.10. It is seen that EP linewidths of the longitudinal optical modes
along the Γ-X, Γ-M, and Γ-R symmetry lines are large compared to the negligible
linewidths of many other modes, such as the M5− mode.
Ab-Initio Molecular Dynamics
Ab-initio molecular dynamics (AIMD) calculations were performed using VASP
with the potentials given in the DFT section on 128-atom supercells using a 2 × 2 × 2
k-point mesh for 23 temperatures from 300 to 1500 K. The energy cutoff was 400
eV and the Monkhorst-Pack scheme was used to sample the Brillouin zone. Each
simulation was carried out for over 10,000 timesteps using a canonical ensemble and
the standard Nosé thermostat [91]. The electronic DOS curves shown in Fig. 2.11
were computed for each temperature from 0 to 1035 K by averaging the recomputed

0.4
0.3
0.2

EP Linewidth Γqν (meV)

Phonon Energy (meV)

0.1

20
10

0.0

Electronic DOS (states/ev/unit cell)

Figure 2.10: Calculated 0 K electron-phonon linewidths displayed over the 0 K FeTi
phonon dispersion.

0K
300 K
523 K
640 K
750 K
900 K
1035 K

0.4

0.2

0.0
E - EF (eV)

0.2

0.4

Figure 2.11: Electronic DOS for FeTi from 0 to 1035 K. Obtained from AIMD and
static DFT calculations.

25
densities of states from 20 saved configurations using a 3 × 3 × 3 k-point mesh.
For each temperature, the BandUP code [75, 76] was performed on five of these
configurations with 81 k-points along each high symmetry direction. The mean
energies e
εkm and smearing widths ∆εkm of the bands were obtained from BandUP.
Spanning Vectors
The histogram in Fig. 5 of the main text was obtained by counting spanning vectors
from one of the new topological features around the R symmetry points to the rest
of the Fermi surface. The vectors were counted in the Brillouin zone defined by
an 80 × 80 × 80 grid of k-points. For each spanning vector, the conservation of
momentum and energy was required:
εk+qn − εkm ± ~ωqν − ci σ ≤ ∆εk+qn,

(2.6)

where k+qn is a state on one of the new topological features around the R symmetry
points, km is a state from the rest of the Fermi surface, e
εk+qn is the mean energy
of the state k + qn calculated by BandUP, εkm is equal to the Fermi energy, and
~ωqν is the energy of the phonon mode with polarization ν and wavevector q. The
smearing width ∆εk+qn describes the broadening of the electron pocket induced by
the thermal disorder of the Fe and Ti atoms at higher temperatures.
The ci σ term describes the thermal excitation of electrons, where σ = 1.8 k BT
is the standard deviation of the Gaussian-like thermal smearing function. Four
histograms of spanning vectors were counted, one for each value of ci ∈ { 13 , 32 , 1, 34 }.
The histogram in Fig. 5 of the main text is an average of the four histograms, where
each histogram is weighted according to the thermal smearing function.

Phonon Energy (meV)

40
30

20
300 K
1500 K
300 K FC
Exchanged

[000] 0.5[100] 0.5[110]

[000]

0.5[111]

0.5[100]

(a) Fe–Fe 2nn Longitudinal

Phonon Energy (meV)

40
30

20
300 K
1500 K
300 K FC
Exchanged

[000] 0.5[100] 0.5[110]

[000]

0.5[111]

0.5[100]

(b) Fe–Ti 1nn Transverse

Figure 2.12: FeTi phonon dispersions calculated after individually exchanging the
(a) Fe–Fe 2nn longitudinal and (b) Fe–Ti 1nn transverse force constants at 300 K
with those at 1500 K.

27
Chapter 3

THERMAL EVOLUTION OF ELECTRON-PHONON
INTERACTIONS IN VANADIUM
3.1

Introduction

Vanadium is a body-centered cubic (bcc) metal that displays some of the strongest
electron-phonon coupling for pure elements [92]. It is a superconductor with a
transition temperature Tc = 5.3 K, one of the highest critical temperatures for pure
elements [16]. Experimental and computational studies on the superconductive,
electronic, and mechanical properties of vanadium at high pressures have revealed
anomalies in the elastic constants and a positive relationship between the superconducting transition temperature with pressure [93–96].
Elemental vanadium also displays anomalous behavior with temperature. The temperature dependence of the elastic constant C44 has two points of inflection at
approximately 800 K and 1600 K [97]. Inelastic neutron scattering (INS) experiments have shown that the increase in the vibrational entropy of vanadium from
thermal expansion is cancelled by nonharmonic thermal stiffening [98] attributed
to an adiabatic electron-phonon interaction (EPI) broadening of the sharp features
in the electronic density of states (DOS). The strength of the EPI was related to the
decrease in the electronic DOS at the Fermi level [10].
Electron-phonon interactions at low temperatures have been extensively studied
for almost a century. Advances over the past two decades have allowed us to
calculate materials properties related to these electron-phonon interactions from
first-principles [99]. These state-of-the-art methods for calculating properties from
the EPI are based on density functional perturbation theory (DFPT), which does
not adequately describe thermal effects observed at finite tempeartures, such as
anharmonic lattice dynamics. The adiabatic EPI at high temperatures had been
best understood by studying the changes in the average phonon energies and the
electronic DOS [10, 25, 26]. We are only now starting to see advances in firstprinciples computational methods for studying finite temperature electron-phonon
interactions [100].
Chapter 2 showed how the thermal phonon softening in FeTi was linked to the
appearance of new features on the Fermi surface with temperature [101]. The

28
adiabatic EPI was altered dramatically by a thermally-driven electronic topological
transition (ETT), a novel Lifshitz transition [35] that had been rarely observed with
temperature [43]. We suggested that a thermally-driven ETT may be observed in
other materials with occupied or unoccupied electronic bands that are a few k BT
from the Fermi level at low temperatures.
Delaire et al. reported that the adiabatic EPI in vanadium saturates at high temperatures owing to the complete smearing of a peak in the electronic DOS at the
Fermi level. This was in agreement with the subsequent softening of phonons in
vanadium past 1000 K. In this present work, we study the nonlinear nature of the
phonon frequency shifts from the adiabatic EPI and investigate if this behavior can
be attributed to a thermally-driven ETT using an extension of the computational
methods employed in Chapter 2.
3.2

Computation

Phonon Calculations
Phonon frequencies at elevated temperatures were calculated with a modified temperature dependent effective potential (TDEP) method [11, 102, 103]. In the TDEP
procedure, the Born-Oppenheimer surface of a material at a given temperature is
sampled with ab initio molecular dynamics (AIMD). The energies, displacements,
and forces on thermally displaced atoms are recorded over time. With these energyforce-displacement data sets, force constants are obtained with a least-squares fit of
a model Hamiltonian to the potential-energy surface:
H = U0 +

Õ p2

2mi

1 Õ αβ α β 1 Õ αβγ α β γ
Φ u u +
Φ u u u ,
2! i jαβ i j i j 3! i j kαβγ i j k i j k

(3.1)

29
ensemble. For a cell of Na atoms with mass mi , a harmonic normal-mode transformation was used to generate positions {ui } consistent with a canonical ensemble:
ui =

3Na

is hAis i −2 ln ξ1 sin (2πξ2 ),

(3.2)

s=1

where ξn are uniformly distributed numbers between 0 and 1 producing the BoxMuller transform. hAis i is the thermal amplitude of the normal mode s with
eigenvector is and frequency ωs [104, 105]:
~(2ns + 1)
1 k BT
(3.3)
hAis i =
2mi ωs
ωs
mi
| {z } | {z }
quantum

classical

where ns = (e~ωs /kBT − 1)−1 is the thermal occupation of mode s, and ~ω
k BT
denotes the classical limit at high temperatures.
These stochastically generated thermal displacements from Eqs. 3.2 and 3.3 sample the Born-Oppenheimer surface in the stochastically initialized temperaturedependent effective potential (s-TDEP) method [11, 102, 103, 106, 107]. This
method approximates the inclusion of zero-point motion not included in AIMD
simulations and connects seamlessly to the classical limit at high temperature. The
s-TDEP procedure can be used to calculate force constants capturing anomalous
high-temperature effects [106–109] to low-temperature quantum effects [110, 111]
at a much lower computational cost than what is required by AIMD. The force
constants calculated with this method are numerically converged with respect to
the number of configurations and supercell size. The convergence of the force
constants and the baseline U0 was further ensured by repeating DFT calculations
on new snapshots generated from force constants from the previous iteration of
s-TDEP. The force constants used to generate the supercells in the first iteration of
s-TDEP were generated through a model pair potential as described in Ref. [106].
The weakness of the s-TDEP method is that it relies on Gaussian distributions of
coordinates generated by Eq. 3.2.
The ab initio DFT calculations were performed with the projector augmented wave
[112] formalism as implemented in VASP [65, 66]. All calculations used a supercell
with 250 vanadium atoms, a 3×3×3 Monkhorst-Pack [87] k-point grid, and a planewave energy cutoff of 580 eV. The exchange-correlation energy was calculated with
the PBE functional [63].

30
These force constants were calculated on a grid of six temperatures, {0, 300, 550, 750,
1250, 1650} K, and six volumes. The quadratic and cubic interatomic force constants for temperatures and volumes between these grid points were obtained by
interpolation. Through three iterations of the s-TDEP procedure, we obtained the
Helmholtz free energy surface F(V, T):
F(V, T) = U0 (V, T) + Fvib (V, T).
U0 (V, T) is the baseline from Eq. 3.1. Fvib (V, T) is from lattice vibrations:

∫ ∞
~ω
~ω
dω,
Fvib =
g(ω) k BT ln 1 − exp −
k BT

(3.4)

(3.5)

where g(ω) is the phonon density of states calculated from the phonons in the first
Brillouin zone,
δ(ω − ωs ).
(3.6)
g(ω) =

We minimized the free energy to calculate the equilibrium volume at each temperature and evaluated the phonon frequencies at these conditions.
We then corrected our phonon frequencies by calculating the linewidths Γs and shifts
∆s arising from anharmonicity, or phonon-phonon interactions. This required the
many-body perturbation calculation of the real and imaginary parts of the phonon
self-energy [70, 113] Σ(Ω) = ∆(Ω) + iΓ(Ω), where E = ~Ω is a probing energy. The
imaginary component Γ(Ω) is
Γs (Ω) =

~π Õ
|Φss 0 s 00 | 2 {(ns 0 + ns 00 + 1)δ(Ω − ωs 0 − ωs 00 )
16 s 0 s 00

(3.7)

+ (ns 0 − ns 00 ) [δ(Ω − ωs 0 + ωs 00 ) − δ(Ω + ωs 0 − ωs 00 )]}
and the real component is obtained by a Kramers-Kronig transformation
Γ(ω)
∆(Ω) =
dω.
ω−Ω

(3.8)

The imaginary component of the self-energy is a sum over all possible three-phonon
interactions, where Φss 0 s 00 is the three-phonon matrix element determined from the
cubic force constants Φi j k . Γ(Ω) and ∆(Ω) were calculated with a 28 × 28 × 28
q-grid.
Anharmonic phonon DOS curves were calculated with the real and imaginary parts
of the phonon self-energy:

31
ganh (ω) =

2ωs Γs (ω)
2
ω2 − ωs2 − 2ωs ∆s (ω) + 4ωs2 Γs2 (ω)

(3.9)

If both ∆ and Γ go to zero, Eq. 3.9 reduces to Eq. 3.6.
To evaluate the effects of thermal expansion, the phonon energies predicted by
the quasiharmonic (QH) approximation were calculated by interpolation of the 0 K
quadratic force constants to volumes obtained from the minimization of the free
energy. The QH calculations exclude the anharmonic corrections provided by Eqs.
3.7 and 3.8.
Electronic Band Unfolding
The electronic band structure of vanadium at 0 K can be calculated through a DFT
calculation on a static lattice using a primitive unit cell (PC). We model finite
temperatures with supercell (SC) calculations with thermally displaced atoms. The
electronic bands from these calculations are folded into a smaller SC Brillouin zone
(SCBZ), giving rise to complicated band structures that cannot be directly compared
to 0 K electronic bands in the larger primitive cell Brillouin zone (PCBZ).
We can recover an approximation of these supercell electronic bands in the PCBZ
through band unfolding [77] as implemented with the BANDUP software package
[75, 76]. Details of the implementation of the band unfolding procedure are available
in Refs. [75] and [76]. In brief, BANDUP is used to obtain the spectral function
A(k; ε) from supercell calculations [77, 114]
PmK (k)δ(ε − εm (K))
A(k, ε) =

|hmK|nki| 2 δ(ε − εm (K)),

(3.10)

where {k} and |nki are electron wavevectors and eigenstates in the PCBZ, and
{K} and |mKi are electron wavevectors and eigenstates in the SCBZ. The spectral
weight PmK (k) is the projection of |mKi on all of the PC Bloch states |nki at the
PC wavevector k. The only pairs of wavevectors (k, K) that are included in the sum
in Eq. 3.10 are those in which K unfolds onto k:
k = K + G,

(3.11)

where G is a reciprocal lattice vector in the SBCZ.
The unfolded electronic band structure is represented as an effective PC band structure (EBS). In BANDUP, this quantity is calculated from the spectral function with

32
the infinitesimal version of the cumulative probability function Sk (ε). The quantity dSk (ε) = A(k, ε)dε represents the number of PC electronic bands at the PC
wavevector k crossing the energy interval (ε, ε + dε). We can obtain the EBS
δN(ki, ε j ) in a region of interest in the (k, ε) space with energy intervals of size δε:
∫ ε j +δε/2
δN(ki, ε j ) =
dSki (ε)
ε j −δε/2

∫ ε j +δε/2
PmK (ki )

ε j −δε/2

(3.12)
δ(ε − εm (K))dε.

The EBS gives the number of PC electronic bands crossing (ki, ε j ).
The EBS calculated from this unfolding procedure is exactly equal to the PC electronic band structure only for perfect supercells, where the atoms are in their equilibrium positions. The EBS calculated with BANDUP shows the effects of perturbations
on the electronic structure such as from crystallographic defects and atom substitutions [75, 76].
For the electronic band structure at 1100 K, we assembled an ensemble of supercells
{η} with thermal displacements {ui } generated with Eqs. 3.2 and 3.3 and phonon
frequencies at 1100 K. For each of the configurations η with displacements {ui }, we
(η)
calculated the EBS δNη (ki, ε j ), where ε j ≡ E j − EF is defined with respect to the
(η)
Fermi energy EF calculated for the supercell η. The thermal atomic displacements
from the equilibrium positions are treated as perturbations. Our calculated electronic
structure at 1100 K is the ensemble average of the EBS hδNη i. This methodology
was previously used to model finite temperature electronic bands in FeTi [101].
Density Functional Perturbation Theory
Density functional perturbation theory calculations [73] were performed with Quantum ESPRESSO [74, 115] with ultrasoft pseudopotentials [90] and the PerdewBurke-Ernzerhof (PBE) exchange-correlation functional [63]. The electron-phonon
interaction matrix elements were first calculated on a 72 × 72 × 72 k-point mesh and
a 12 × 12 × 12 q-point mesh, and later interpolated to 720 q-points along the high
symmetry lines in the bcc Brillouin zone through Fourier interpolation implemented
in the package. The matrix elements were then used to compute the scattering rates
of phonons by electrons:
2π Õ
dk ν
gmn (k, q) ( fnk − fmk+q )
τqν
~ mn
ΩBZ
(3.13)
× δ(εm (k + q) − εn (k) − ~ωqν ),

33
ν (k, q) is the electron-phonon interaction matrix element associated with
where gmn
a phonon mode ν with wavevector q and two electronic states with the eigenstates
|mki and |nk + qi, and fmk is the Fermi-Dirac distribution for electrons. The
electron-phonon coupling strength associated with this interaction is

λqν =

1/τqν
πN(EF )ωqν

(3.14)

where N(EF ) is the electronic DOS at the Fermi level.
3.3

Results

Phonons
Figure 3.1(a) shows phonon densities of states of vanadium calculated with the sTDEP method at temperatures from 0 to 1650 K. There is no significant broadening
from PPI. The high-energy longitudinal phonon modes from 26 to 30 meV stiffen
before they begin to slowly soften with temperature. This anomalous behavior is seen
more clearly in the plot of average phonon energies derived from the phonon DOS in
Fig. 3.1(b): the average phonon energy increases with temperature before it begins
to decrease starting at around 750 K. This behavior strongly deviates from what is
predicted by the QH model, where the average phonon energy decreases in the entire
temperature range. The calculated average phonon energies and their thermal trend
are in good agreement with inelastic neutron scattering (INS) measurements of the
vanadium phonon DOS [10, 84, 98].
We further verified our computational results by comparing our calculated 300 K
spectral function S(q, ω) to room temperature phonon dispersions measured with
thermal diffuse x-ray scattering (TDS) [116] and inelastic x-ray scattering (IXS)
[117], shown in Fig. 3.2. The s-TDEP spectral function agrees with both the IXS
(particularly along the H-P and Γ-P directions) and the TDS measurements (the
longitudinal branch along the Γ-H and Γ-N directions). The Kohn anomalies [19]
are more pronounced in the s-TDEP spectral function along Γ-H and Γ-N than the
TDS and IXS measurements, and there are disparities for the transverse acoustic TA1
phonon mode at the N symmetry point. We observe a crossover of the longitudinal
phonon branch with the the TA2 branch at the N symmetry point in the TDS, IXS,
and s-TDEP phonon dispersions. This anomaly and many other features seen in our
s-TDEP spectral function have been observed in the phonon dispersions calculated
with DFPT by Luo et al. [94].
From 0 to 1650 K (Fig. 3.3), many phonon modes soften with temperature, including

DOS (1/meV)

0.08

(a)

0.06
0.04
0.02
0.00
23.5

E (meV)

23.0

15
20
25
Energy (meV)

ARCS
Pharos
HFIR

QH
s-TDEP

(b)

22.5
22.0
21.5
21.0
20.5

250

500 750 1000 1250 1500 1750
Temperature (K)

Figure 3.1: (a) The phonon DOS curves of vanadium calculated with the s-TDEP
method at temperatures from 0 (dark purple) to 1650 K (orange). (b) Average phonon
energies of vanadium calculated with the s-TDEP method (identical colors to those
shown in (a)), shown together with average phonon energies from inelastic neutron
scattering measurements (Refs. [10, 84, 98]). The dashed red curve corresponds to
quasiharmonic (QH) behavior as calculated from first-principles.

35 300 K
30
25
20
15
10

Intensity (arb. units)

Energy (meV)

Phonon Energy (meV)

Figure 3.2: The 300 K spectral function calculated with s-TDEP along the highsymmetry directions, plotted together with measurements from thermal diffuse x-ray
scattering (crosses) [116] and inelastic x-ray scattering (dots) [117].

35
30
25
20
15
10

[ 00]

[ 0]

Figure 3.3: Phonon dispersion curves of vanadium calculated with the s-TDEP
method at temperatures from 0 (dark purple) to 1650 K (orange). Vector coordinates
are written in simple cubic lattice coordinates.

Intensity (arb. units)

1.0

Energy (eV)

0.5

0.0

0.5

1.0

Figure 3.4: Unfolded electronic bands at 1100 K, compared with 0 K electronic
bands in dark red. The Fermi level is represented as a spread in energies.
the transverse acoustic modes at the N symmetry point and the phonon modes at the
H and P symmetry points. A few phonon modes stiffen before they begin softening
with temperature, such as the longitudinal phonon mode at q = 13 13 31 along the
Γ-P direction. A number of phonon modes stiffen with temperature, including the
Kohn anomalies close to the Γ point along the Γ-H, Γ-N, and Γ-P directions. The
anomalous crossover of the longitudinal and TA2 phonon modes at the N symmetry
point is no longer present at high temperatures because the longitudinal phonon
mode stiffens strongly with temperature. The stiffening of the longitudinal phonon
modes at the N symmetry point and q = 31 31 13 contribute to the anomalous behavior
of the longitudinal peak in the phonon DOS.
Electronic Band Structure
Figure 3.4 shows the electronic band structure of vanadium at 1100 K calculated with
band unfolding implemented with BANDUP. The 1100 K electronic band structure is
the average of the EBS hδNη i calculated from 15 stochastically generated supercells
with thermal displacements characteristic of 1100 K. Each EBS was calculated from
-1 to 1 eV along the high-symmetry directions of the bcc Brillouin zone in (k, ε)
space.
The Fermi level is represented as a distribution of energies at 1100 K. This accounts

37
for how the occupation of electronic states is neither exactly 0 nor exactly 1 within
a few k BT around the Fermi energy, as specified by the Fermi-Dirac distribution
at finite temperatures. This is our visual representation of the thermal layer ∆ε
of thickness proportional to k BT in which there are electrons together with empty
states into which they may be scattered [15].
We do not observe significant shifts in the positions of the electronic bands from 0
to 1100 K. What we do observe is a strong broadening of these electronic bands with
thermal disorder, a high temperature phenomenon that is consistent with predictions
from the Allen-Heine-Cardona (AHC) theory [118–123]. Electronic states at the Γ
point intersect the thermal layer because of this strong broadening.
Fermi Surface
We have calculated the Fermi surface of vanadium at 0 and 1100 K through band
unfolding implemented with BANDUP. The EBS for a given supercell η was calculated for ε from −0.50 to +0.50 eV for all of the k-points that make up the bcc
irreducible Brillouin zone in a 50 × 50 × 50 k-point grid. The Fermi surface F(k, ε)
is our average EBS hδNη i unfolded from the irreducible Brillouin zone to the full
Brillouin zone. We unfold the EBS by applying the symmetry operations used to
recover the full Brillouin zone from the irreducible Brillouin zone.
For visualization, each k-point in the Brillouin zone is assigned an intensity derived
from integrating the average EBS hδNη i over our integration window:
∫ εmax
∂ f (ε, T)
dε.
(3.15)
I(k) =
δNη (k, ε) −
∂ε
εmin
The EBS is weighted against the derivative of the Fermi-Dirac distribution f (ε, T)
with respect to energy. This derivative provides us a distribution function with a
width proportional to the thermal layer [15], which was represented as the distribution of energies ∆ε around the Fermi energy in Fig. 3.4. For 0 K, this distribution
function is a Dirac delta function, yielding intensities expected from the definition
of the 0 K Fermi surface as the map of k-points where electronic bands intersect the
Fermi energy (ε = E − EF = 0).
The {100} cross sections of the Fermi surface are shown in Fig. 3.5 for 0 and
1100 K. The cross section of the 0 K Fermi surface in Fig. 3.5(a) is similar to what
was calculated by Landa et al. [96]. The broadening ∆k of the electronic states in
Fig. 3.5(b) arises from both thermal atomic displacements and the thermal layer.
Representing the finite temperature Fermi surface as an overlay of surfaces defined

38
(b) 1100 K

Intensity (arb. units)

(a) 0 K

Figure 3.5: The {100} cross sections of the Fermi surface of vanadium at (a) 0 K
and (b) 1100 K (cross section indices are expressed in simple cubic coordinates).
in the thermal layer allows us to account for all of the electronic states relevant for
thermodynamic and transport properties, such as the electronic specific heat and
conductivity [15].
The values for the intensity I(k) of the 1100 K Fermi surface are lower than the
values for the 0 K Fermi surface. The broadening ∆k of the Fermi surface washes
away sharp features of the 0 K Fermi surface, especially for the closed distorted holeellipsoids centered at the N symmetry points. The distorted octahedron closed holepocket centered at the Γ point at 0 K is no longer distinguishable from these holeellipsoids at elevated temperatures. We observe a number of additional electronic
states at the Γ point at 1100 K arising from the broadening of the triply degenerate
electronic band seen in Fig. 3.4. The formerly hollow octahedron is filled up with
these new electronic states in a thermally-driven ETT.
3.4

Discussion

Fermi Surface Nesting
The 0 K electron-phonon coupling strength λqν was calculated for vanadium and is
plotted in Fig. 3.6. The maximum value for the electron-phonon coupling strength is
observed for the transverse phonon branch near the Γ point along the Γ-H direction,
coinciding with the Kohn anomaly [19]. Peaks in the electron-phonon coupling
strength are also observed for transverse phonon branches close to the Γ point along
the Γ-N and Γ-P directions. A high value for the electron-phonon coupling strength
is observed for the longitudinal phonon mode at the N point. This is the same

[ 00]

30
25

20
15
10

Electron-Phonon Coupling Strength

Phonon Energy (meV)

Autocorrelation

0K
1100 K

Figure 3.6: (Top) 0 K electron-phonon coupling strengths of phonon modes calculated with Quantum ESPRESSO displayed over the 0 K vanadium phonon dispersion
calculated with s-TDEP. (Bottom) Autocorrelation of the Fermi surface at 0 and
1100 K.
phonon mode that crosses over with the high transverse acoustic mode. All of these
phonon modes stiffen with temperature (Fig. 3.3).
The peak in the electron-phonon coupling strength along the Γ-H direction coincides
with the peak at the wavevector q = [0.24, 0, 0] in the generalized susceptibility
calculated by Landa et al. [96]. The wavevector q = [0.24, 0, 0] spans nesting
features, pairs of parallel sheets in the Fermi surface {|nki} and {|mk0i} that are
related by k0 = k + q + g, where g is a reciprocal lattice vector. A high density
of these spanning vectors results in high numbers of nonzero terms in Eq. 3.13.
The peaks in the electron-phonon coupling strength along Γ-N, Γ-P, and at N may
correspond to additional wavevectors that span Fermi surface nesting features.

40
We can probe for spanning wavevectors by calculating the periodic autocorrelation
of the Fermi surface
1 Õ
R(q) =
I(k)I(k + q + g),
(3.16)
NF k
where I(k) is the integrated Fermi surface intensity calculated from Eq. 3.15
and NF is the number of k-points on the Fermi surface. We expect peaks in the
autocorrelation where nesting features in I(k) and I(k + q + g) overlap.
The autocorrelations of the 0 and 1100 K Fermi surfaces are plotted in the bottom
panel of Fig. 3.6. A sharp peak is observed in the 0 K autocorrelation at q =
[0.24, 0, 0] along the Γ-H direction, approximately lining up with the location of
the peak in the electron-phonon coupling strength λqν . Similar sharp peaks are
observed near the Γ point along the Γ-N and Γ-P directions and at the N symmetry
point, approximately lining up with the peaks in λqν .
We still observe a peak at the N symmetry point in the 1100 K Fermi surface
autocorrelation. We see an additional peak at the H symmetry point. No peak is
observed at q = [0.24, 0, 0]. None of the peaks in the 1100 K autocorrelation are
as narrow as the 0 K peaks, and the peaks barely stand out from the baseline. The
small population of broad peaks in the 1100 K autocorrelation may indicate that
electron-phonon interactions are reduced, as nesting features are smeared out with
the thermal broadening observed in Fig. 3.5.
Temperature Dependence of Electron-Phonon Interactions
We can get a closer look at how changes in the Fermi surface with temperature affect
electron-phonon interactions by calculating the density of specific spanning vectors
Dν (q) at 0 and 1100 K. By comparing the spanning vector densities of phonon
modes between the two temperatures, we can see if electron-phonon interactions are
strengthened or weakened with temperature. We can also see if the spanning vector
densities and electron-phonon interactions are impacted by the thermally-driven
ETT observed in Fig. 3.5.
Our calculation of the spanning vector density sums over all possible electronphonon scattering processes that may occur in our window from −0.50 to +0.50 eV
with respect to the Fermi energy, which is an overestimation of the thermal layer

41
(TL):
Dν (q) =

1 Õ
F(k, ε)F(k + qi + g, ε + ~ωqν ) + F(k + qi + g, ε)F(k, ε + ~ωqν )
NF k
ε∈TL
qi ∈Sq

× f (ε, T)(1 − f (ε + ~ωqν, T)) + f (ε + ~ωqν, T)(1 − f (ε, T)) ,
(3.17)
where qi is a vector related to q by symmetry. We are treating the Fermi surface F
as a function of both k and energy ε to take the conservation of energy into account.
In the 0 K limit, the Fermi surface function F is either a nonzero integer or zero, and
the thermal layer is localized at E = EF . At 0 K, the sum in Eq. 3.17 would simply
be a count of the number of spanning vectors q on the Fermi surface. At finite
temperatures, the Fermi surface function F is interpreted as the probability of the
presence of an electronic state at (k, ε) as a consequence of electronic broadening
from thermal atomic displacements. According to the Fermi-Dirac distribution
terms in the second bracket in Eq. 3.17, electrons at energies far from the Fermi
level are less likely to be involved in scattering processes due to low occupation or
unoccupation probabilities. This is what is also described by the derivative of the
Fermi-Dirac distribution.
We can gather insights about nesting features by looking at the summands Dν(k) (q):
Dν (q) =
Dν(k) (q).
(3.18)

The nesting strength Dν(k) (q) is the density of spanning vectors q for the electronic
state at wavevector k. Nesting features are composed of electronic states with high
values of Dν(k) (q). Fig. 3.7 shows the nesting strengths Dν(k) (q) for three phonon
modes at 0 and 1100 K: the transverse phonon mode at q = [0.24, 0, 0] (subfigures
a and b), the longitudinal N phonon mode (subfigures c and d), and the H phonon
mode (subfigures e and f).
We can attribute the high 0 K electron-phonon coupling strength λqν for the transverse phonon mode at q = [0.24, 0, 0] to the high density of vectors spanning the
flat features on the hole-ellipsoids (Fig. 3.7(a)). These flat features are the previously identified nesting features associated with the Kohn anomaly along Γ-H for
vanadium [96] and contribute to the high superconducting transition temperature Tc
of vanadium [20, 21].

10 5

10 6

10 7

Nesting Strength (arb. units)

10 4

10 5

10 6

10 7

Nesting Strength (arb. units)

10 4

10 5

10 6

10 7

Nesting Strength (arb. units)

10 4

Figure 3.7: {100} cross sections of the Fermi surface nesting strengths of vanadium
for the transverse phonon mode at q = [0.24, 0, 0] at (a) 0 K and (b) 1100 K, the
longitudinal N phonon mode at (c) 0 K and (d) 1100 K, and the H phonon mode
at (e) 0 K and (f) 1100 K. Sample [0.24, 0, 0] and [0.5, 0.5, 0] spanning vectors are
shown in (a) and (c).

43
We can attribute the peak in λqν for the longitudinal phonon mode at N to the high
density of spanning vectors spanning the hole-ellipsoids and the distorted octahedron
(Fig. 3.7(c)). The crossover of the longitudinal and transverse acoustic modes at N
can be attributed to the interaction of this phonon mode with these electronic states.
No notable peaks were observed in the 0 K values of λqν and Fermi surface autocorrelation for the H phonon mode. There are almost no nesting features for the
q = [1, 0, 0] spanning vector (Fig. 3.7(e)), such that no 0 K electron-phonon interaction is observed for the H phonon mode. We observe that the vector q = [1, 0, 0]
spans more features of the Fermi surface at 1100 K, but these features make up only
a small fraction of the number of k-points making up the Fermi surface at 1100 K.
These features also display low nesting strengths, such that the spanning vector
density Dν (q) for this phonon mode is actually reduced by a factor of 1.731 from
0 to 1100 K. This is consistent with our observation that this phonon mode softens
quasiharmonically with temperature.
The [0.24, 0, 0] and [0.5, 0.5, 0] vectors span several more k-points in the Fermi
surface at 1100 K than at 0 K, as shown in Fig. 3.7(b) and (d). The nesting strengths
for these phonon modes are also low. The spanning vector density for the transverse
mode at [0.24, 0, 0] is reduced by a factor of 2.364 from 0 to 1100 K, and the spanning
vector density for the longitudinal N phonon mode is reduced by a factor of 3.094
from 0 to 1100 K.
It appears that the reduction of the spanning vector density for the transverse mode
at [0.24, 0, 0] is not as severe as the reduction for the longitudinal N phonon mode
owing to the introduction of additional nesting features from the thermally-driven
ETT. The thermally-driven ETT does not counteract the reduction in the electronphonon interaction strength for this phonon mode. The [0.5, 0.5, 0] vector does not
span any of the new electronic states at the Γ point, such that the thermally-driven
ETT has no impact on the thermal evolution of the longitudinal N phonon mode.
We had previously hypothesized that this thermally-driven ETT would counteract
the phonon thermal stiffening, explaining the apparent saturation of the adiabatic
EPI observed by Delaire et al [10] and the softening of the longitudinal peak and the
average phonon energies past 750 K. We attribute the stiffening of the longitudinal N
phonon mode only to the weakening of the low-temperature EPI owing to the thermal
smearing of the Fermi surface. This stiffening slows down with temperature as it is
opposed by softening from thermal expansion.

Relative Resistivity T/ 300

Molybdenum
Tungsten
Tantalum
Niobium
Vanadium

500

1000
1500
Temperature (K)

2000

Figure 3.8: The electrical resistivities of bcc transition metals relative to their 300 K
values. The plotted data are from measurements on high-purity samples.
Table 3.1: Electron-phonon interaction parameters and superconducting transition
temperatures of nonmagnetic bcc transition metals.
Element
Mo
Ta
Nb

Tc (K)
0.26 [20, 124]
0.015
0.199-0.41 [20, 125]
0.92
0.6923-0.88 [126–128] 4.47
0.7-0.82 [10, 124, 129]
5.3
0.867-1.22 [125, 130]
9.25

Electrical Resistivity
The thermal reduction of the electron-phonon interaction in vanadium could affect
the temperature dependence of the electrical resistivity ρ, at least the resistivity from
electron-phonon scattering ρT . We are also interested in comparing the resistivities
ρT of vanadium to other nonmagnetic bcc transition metals, as these metals may
display different electron-phonon interaction properties.
Fig. 3.8 shows the values for the electrical resistivity relative to their room temperature values for nonmagnetic bcc transition metals. The resistivity values for
molybdenum, tungsten, tantalum, and vanadium were recommended by Desai et
al. and were derived from experimental measurements of samples with 99.9% or
higher purity (Ref. [131, 132] and references therein). Low temperature values
of the electrical resistivity of high-purity niobium were obtained from measure-

45
ments by Webb [133] while high temperature values of the electrical resistivity of
high-purity niobium were obtained from measurements by Abraham and Deviot,
Peletskii, and Maglić et al. [134–136]
If electrons in the nonmagnetic bcc metals were described by the free electron
model, the relative resitivities would change linearly with temperature past the
Debye temperature. We see in Fig. 3.8 that the five nonmagnetic bcc metals deviate
from the expected linear relationship ρT ∝ T. The electrical resistivities of the group
6 transition metals molybdenum and tungsten increase with positive curvature at
elevated temperatures, meaning electrons are increasingly scattered by phonons with
temperature. The electrical resistivities of the group 5 transition metals tantalum,
vanadium, and niobium increase with negative curvature at elevated temperatures.
While the electrical resistivity increases with temperature, the electron scattering
rate from phonons is suppressed. This is consistent with the reduction of electronphonon interactions with temperature in vanadium. Fig. 3.8 would lead us to
believe that niobium and tantalum show similar thermal trends in electron-phonon
interactions. A thermal smearing of the Fermi surface nesting features can explain
the stiffening of the longitudinal N phonon mode in niobium reported by Güthoff et
al. [137].
Molybdenum and tungsten display weak low temperature electron-phonon interactions, as evidenced by their small electron-phonon interaction parameters λ. The
low electron-phonon interaction strengths in these elements contribute to their low
superconducting transition temperatures Tc , consistent with the McMillan theory of
strong-coupled superconductors [20, 21]. Tantalum, vanadium, and niobium display
strong low temperature electron-phonon interactions that lead to superconductivity,
as evidenced by how their electron-phonon interaction parameters λ are close to or
over unity, and how they display high superconducting transition temperatures.
The curvature of the thermal evolution of electrical resistivities of high-purity bcc
transition metals follows the low temperature electron-phonon interaction strength.
ρT
of the relative electrical resistivity with temperature deThe rate of change ∂∂T
creases with increasing low temperature electron-phonon interaction strengths and
superconducting transition temperatures. The stronger the electron-phonon interaction at low temperatures, the more dramatic the thermal increase in electrical
conductivity as the adiabatic EPI is lost.

46
3.5

Conclusions

The nonlinear thermal stiffening of phonons in vanadium previously measured with
inelastic neutron scattering was reproduced with first-principles calculations. The
Fermi surface of vanadium was calculated at 0 K and high temperatures through
band unfolding procedures. The sharp features of the Fermi surface at low temperatures were drastically smeared with temperature from atomic displacements and
thermal excitations of electrons. The overall weakening of the electron-phonon
interactions in vanadium is primarily attributed to this thermal smearing. There is
a thermally-driven electronic topological transition near the Γ point, but the atomic
displacements smear its effectiveness. The phonon stiffening from this reduction in
the EPI is counteracted by quasiharmonic softening at high temperatures.
The negative curvature of the electrical resistivity of vanadium at elevated temperatures follows the reduction of electron-phonon interactions with temperatures.
Based on their negative curvatures in electrical resitivity, we suggest that the bcc
nonmagnetic transition metals niobium and tantalum display similar changes in their
electronic structure and electron-phonon interactions with temperature to vanadium.

47
Chapter 4

TEMPERATURE DEPENDENCE OF PHONONS IN PD3 FE
THROUGH THE CURIE TEMPERATURE
4.1

Introduction

Advances in the design of magnetic materials are enabled by understanding how their
properties depend on the external conditions of temperature, pressure, and magnetic
field. In particular, it is important to understand their thermodynamic properties
over a range of temperatures. This requires modeling a magnetic material not only
in its ground state at 0 K, but also in its magnetically disordered states at finite
temperatures.
Progress has been made in first-principles simulations of magnetic disorder in
materials [138]. Recent approaches to modeling the paramagnetic state of magnetic
materials include disordered local moment molecular dynamics (DLM-MD) [139,
140], spin-space averaging [33, 141–143], and spin dynamics [144–146]. With such
advances, a computational study accounting for the interaction of the magnetic and
atomic degrees of freedom, which has been demonstrated to provide a more complete
calculation of the Gibbs free energy of magnetic materials [7], is now within reach.
In the present study of the thermal excitations in Pd3 Fe, magnetic disorder is included
in the finite temperature calculations of vibrational thermodynamic properties.
Fe−Pd alloys have been a subject of numerous studies owing to their magnetic and
mechanical behavior. Properties of interest include a martensitic transformation
in Fe-rich alloys [143, 147, 148], noncollinear magnetic structures [149–152], and
Invar behavior [153]. Pd3 Fe, a ferromagnetic metallic compound with an fccbased L12 structure and a Curie temperature of approximately 500 K [154], exhibits
Invar behavior under an applied pressure [155]. This compound also exhibits
an anomalous dependence of phonon frequencies and volume with pressure as a
consequence of a magnetic transition [151, 156]. In this present work, we investigate
this interaction between lattice dynamics and magnetic excitations at temperatures
through the Curie temperature, using nuclear resonant inelastic x-ray scattering and
first-principles calculations.

48
4.2

Methods

Experiment
Measurements were performed on the L12 -ordered Pd57
3 Fe sample used in a pressureinduced Invar experiment [155], which was prepared by arc-melting Pd of 99.95%
purity and 57 Fe of 95.38% isotopic enrichment before being cold rolled to a thickness
of 25 µm. This ordered sample was further annealed with a heat treatment at 873 K
for 18 hours under vacuum, 773 K for 54 hours, and subsequent cooling to 293 K over
2 hours. X-ray diffraction confirmed the L12 structure and long-range order, and
Mössbauer spectroscopy confirmed the short-range order (shown in the Supporting
Information).
Nuclear resonant inelastic x-ray scattering (NRIXS) measurements were performed
on Pd57
3 Fe at seven temperatures from 298 to 786 K. NRIXS is a low background
technique that provides direct access to the phonon partial density of states (pDOS)
of 57 Fe [157, 158]. Measurements were performed at beamline 16ID-D of the
Advanced Photon Source at Argonne National Laboratory. The synchrotron flashes
had durations of 70 ps and were separated by 153 ns. Electronic scattering occurs
within femtoseconds of the pulse arrival at the sample. The relatively long lifetime
of the nuclear resonant state (τ = 141 ns) allowed for a clear separation of the prompt
electronic scattering from the delayed resonant scattering.
The Pd57
3 Fe foil sample was held in vacuum under active evacuation in a resistive
heating furnace with a kapton window for x-ray transmission. Errors in the values
of the temperature ranged from ±10 to ±27 K. The ambiguity comes from comparing the furnace thermocouple measurements to NRIXS-derived detailed balance
temperature calculations following procedure described in the literature [80, 159].
An avalanche photodiode was positioned at approximately 90◦ from the incident
beam to collect incoherently reradiated photons. The energy was scanned from −80
to +80 meV around 14.413 keV, the resonant energy of 57 Fe, in several scans that
were combined for final analysis. The energy resolution of all NRIXS measurements
was measured to be 2.2 meV (FWHM) at the elastic line. The PHOENIX software
package was used to extract the 57 Fe pDOS from the measured NRIXS spectra [80].
Nuclear forward scattering (NFS) measurements were collected immediately prior
to NRIXS scans. The NFS spectra provide a measure of the magnetic state of
Pd57
3 Fe, using an avalanche photodiode in the path of the forward-scattered x-ray
beam to measure the transmitted intensity as a function of time.

49
In situ synchrotron x-ray diffraction (XRD) measurements were performed concurrently with the NRIXS and NFS measurements with the same monochromatic beam
of 14.413 keV x-rays and a Mar CCD detector plate. Results from synchrotron XRD
were used to obtain lattice parameters for the quasiharmonic (QH) approximation
of the Fe pDOS.
Computational: Phonon Calculation
Phonon frequencies at elevated temperatures were calculated with the s-TDEP
method [11, 102, 103, 106], as described in Chapter 3. In brief, first-principles
calculations were performed on supercells of thermally-displaced atoms generated
by stochastic sampling of a canonical ensemble. The thermal displacements were
calculated with phonon populations and polarizations from short ab initio molecular dynamics (AIMD) simulations. The ensemble of calculated forces and energies
were used to fit an effective potential with a model Hamiltonian
H = U0 +

Õ p2

2mi

1 Õ αβ α β 1 Õ αβγ α β γ
Φ u u +
Φ u u u ,
2! i jαβ i j i j 3! i j kαβγ i j k i j k

(4.1)

where pi and ui are the momentum and displacement of atom i, respectively, and αβγ
are Cartesian components. The temperature-dependent U0 is a fit parameter for the
baseline of the potential energy surface [102]. The quadratic force constants Φi j from
the thermally-displaced atoms capture nonharmonic effects such as magnon-phonon
interactions at a given temperature, and are used to calculate phonon frequencies
ωs shifted by these effects [102], where s is an index for phonon modes. The cubic
force constants Φi j k capture phonon-phonon interactions (PPI) that contribute to the
broadening and additional shifts of phonon modes [103].
The ab initio spin-polarized density functional theory (DFT) calculations with spinorbit coupling were performed with the projector augmented wave [112] formalism
as implemented in VASP [65, 66]. All calculations used a 3×3×3 supercell with 108
atoms, a 2 × 2 × 2 Monkhorst-Pack [87] k-point grid, and a plane wave energy cutoff
of 520 eV. The exchange-correlation energy was calculated with the Perdew-BurkeErnzerhof (PBE) functional [63]. It was observed from calculations and from past
experiments [160–162] that the magnetic moment of Pd atoms in Pd3 Fe is small
compared to that of Fe atoms. The Pd moments were approximated as zero.
At a given temperature, the forces and energies of 25 stochastically-generated supercells were calculated at five volumes within −2.5 to +5% around the 0 K equilibrium
volume through three iterations of the s-TDEP procedure. For each volume, the

50
Helmholtz free energy F(V, T) was calculated:
F(V, T) = U0 (V, T) + Fvib (V, T).
U0 (V, T) is the baseline from Eq. 4.1. Fvib (V, T) is from lattice vibrations:

∫ ∞
~ω
~ω
Fvib =
g(ω) k BT ln 1 − exp −
dω,
k BT

(4.2)

(4.3)

where g(ω) is the phonon density of states calculated from the phonons in the first
Brillouin zone. At a given temperature, the equilibrium volume VT was obtained
through the minimization of the Helmholtz free energy F(T, V). The force constants
were then interpolated to this volume, giving us phonon frequencies that capture
volume expansion and temperature-dependent nonharmonic effects.
We then corrected our phonon frequencies ωs (T, VT ) by calculating the linewidths
Γs and shifts ∆s arising from anharmonicity, or phonon-phonon interactions. This
required the many-body perturbation calculation of the real and imaginary parts of
the phonon self-energy [70, 113] Σ(Ω) = ∆(Ω) + iΓ(Ω), where E = ~Ω is a probing
energy. The imaginary component Γ(Ω) is
Γs (Ω) =

~π Õ
|Φss 0 s 00 | 2 {(ns 0 + ns 00 + 1)
16 s 0 s 00

× δ(Ω − ωs 0 − ωs 00 ) + (ns 0 − ns 00 )
× [δ(Ω − ωs 0 + ωs 00 ) − δ(Ω + ωs 0 − ωs 00 )]},
and the real component is obtained by a Kramers-Kronig transformation:
Γ(ω)
∆(Ω) =
dω.
ω−Ω

(4.4)

(4.5)

The imaginary component of the self-energy is a sum over all possible three-phonon
interactions, where Φss 0 s 00 is the three-phonon matrix element determined from the
cubic force constants Φi j k . Γ(Ω) and ∆(Ω) were calculated with a 36 × 36 × 36
q-grid.
Anharmonic phonon DOS curves were calculated with the real and imaginary parts
of the phonon self-energy:
ganh (ω) =

2ωs Γs (ω)
2
ω2 − ωs2 − 2ωs ∆s (ω) + 4ωs2 Γs2 (ω)

(4.6)

51
This procedure was implemented to calculate two sets of phonon dispersions and
DOS: for Pd3 Fe maintaining complete ferromagnetic order at 0, 300, 480, 600,
and 800 K (illustrated with Fig. 4.1(a)), and for Pd3 Fe with increasing magnetic
disorder at 300, 480, and 800 K (illustrated with Fig. 4.1(b)). Pd3 Fe is expected
to be completely ferromagnetic only at 0 K. The calculations of the completely
ferromagnetic Pd3 Fe at nonzero temperatures were performed for comparison with
computations with magnetic disorder.
For comparison with the s-TDEP phonon spectra, the phonon energies predicted
by the quasiharmonic (QH) approximation were calculated by interpolation of the
0 K quadratic force constants to volumes obtained from the minimization of the free
energy. This “QH DFT model” assumes that the only temperature dependence in
Eq. 4.2 and 4.3 is from volume expansion and the Planck distribution. The “QH
DFT model” excludes the anharmonic corrections provided by Eqs. 4.4 and 4.5.
Computational: Magnetic Disorder
The magnetic disorder from thermal fluctuations was modeled with special quasirandom structures (SQSs) [163] of noncollinear Fe magnetic moments. The magnetic
SQSs mimic the most relevant local correlation functions of random magnetic structures [164], where the correlation function of the coordination shell α is
1 Õ
ei · e j ,
(4.7)
Πα =
Nα i, j∈α
where ei = mi /kmi k is a unit vector in the direction of the magnetic moment mi
on site i, and Nα is the number of magnetic moment pairs in the coordination shell
α. This use of SQSs with noncollinear Fe magnetic moments is related to the
disordered local moment (DLM) model, where magnetic disorder is modeled with
randomly oriented local magnetic moments [165–170].
A histogram of magnetic SQSs was generated by simulated annealing, where Fe
magnetic moments were flipped into random orientations until the local correlation
functions matched target correlation functions. These bins were of increasing levels
of magnetic disorder, from completely ordered to completely disordered. In the
same bin of the histogram, the set of magnetic SQSs {λ} were equivalent in their
correlation functions.
All correlation functions of magnetic SQSs modeling complete ferromagnetic order
equaled 1 (Πα = 1, ∀α), and the resulting Fe magnetic moments were aligned in
the same direction. All correlation functions of magnetic SQSs modeling complete

Figure 4.1: (a) Schematic of supercells with Fe atoms (dark blue) stochastically
displaced from their ideal positions (light blue) in the 0 K ferromagnetic calculations,
where the magnetic moments (red arrows) are aligned in the same direction. (b)
Supercells with randomly oriented magnetic moments and stochastically displaced
Fe atoms in the 800 K paramagnetic calculations. Each set of randomly oriented
magnetic moments is a magnetic special quasirandom structure (SQS). Pd atoms
are not shown for this illustration.
magnetic disorder equaled 0 (Πα = 0, ∀α), and the resulting Fe magnetic moments
were randomly oriented. The correlation functions for magnetic SQSs from a bin
that was between complete order and complete disorder had values between 0 and
1. For a given bin, the generated magnetic SQSs had an averaged normalized
magnetization
1 Õ Õ
mi ,
(4.8)
hM/M0 iSQS =
Nm λ
where Nm is a normalization constant.
For the phonon calculations of Pd3 Fe with increasing magnetic disorder, we coupled the s-TDEP procedure with these magnetic SQSs. For a given temperature T,
25 stochastically sampled supercells {κ} with thermal atomic displacements characteristic of T were generated. Separately, 25 magnetic SQSs {λ} were selected
from a bin where the SQSs have an averaged normalized magnetization hM/M0 iSQS

53
approximately equal to the normalized magnetization M(T)/M0 expected at temperature T. A spin-polarized DFT calculation was performed on a supercell κ paired
with a magnetic SQS λ. The forces from these spin-polarized DFT calculations on
the ensemble of SQS-supercell (κ, λ) pairs, illustrated in Fig. 4.1(b), are used to
obtain force constants used in the calculation of phonon dispersions and DOS, as
described previously. This computational method can be considered to be a stochastic ensemble-averaged variant of disordered local moments molecular dynamics
(DLM-MD) [139, 140] with local spin correlations.
For the calculations at 800 K, the magnetic structure was treated as a random distribution of magnetic moments characterized by the vanishing of the spin correlation
functions (Πα = 0, ∀α) so that the average magnetization hMiSQS is zero, in accordance with the DLM model. The force constants calculated at 800 K were
interpolated to a volume that minimized a modified Helmholtz free energy F(T, V)
F(T, V) = U0 (T, V) + Fvib (T, V) + Fmag (T, V),

(4.9)

where Fmag (T, V) is the magnetic free energy. Because there is no exact formulation
for the magnetic free energy, this free energy was approximated with a mean-field
term
Fmag (T, V) = −T Smag = −kBT ln(hm(V)i + 1),
(4.10)
where hm(V)i is the average magnitude of the magnetic moments in units of µB . The
magnetic entropy Smag is the maximum orientational disorder of magnetic moments
in the paramagnetic state for systems with local magnetic moments. This approach
is widely used to describe the magnetic entropy of paramagnetic systems [142, 171].
For calculations at 300 and 480 K, the magnetic structures were sampled so the
averaged normalized magnetizations hM/M0 iSQS were approximately equal to normalized hyperfine fields H(T)/H0 obtained from Mössbauer spectroscopy measurements performed in this study and by Longworth [154], as seen in Fig. 4.3. Because
there is no reliable method for calculating the magnetic free energy of Pd3 Fe at
intermediate temperatures, force constants were calculated at volumes obtained by
scaling the volume calculated at 800 K by volumes obtained from synchrotron x-ray
diffraction measurements.
4.3

Results

Nuclear Forward Scattering
The NFS spectra measured from the Pd57
3 Fe sample are shown in Fig. 4.2. The
NFS spectrum at 298 K exhibits a clear magnetic beat pattern expected from a

786 K

Log Intensity (arb. units)

627 K
535 K
485 K
459 K
417 K
298 K
30

60 75
Time (ns)

105

Figure 4.2: The 57 Fe nuclear forward scattering spectra from L12 -ordered Pd3 Fe
at several temperatures. The fits (black curves) overlay experimental data (points).
The spectra are displayed using a log scale, and offset for clarity.

1.0

30
0.8

0.6

20
15

0.4

Magnetization
Longworth
NFS

0.2
0.0

Hyperfine Field (T)

Reduced Magnetization

100 200 300 400 500 600 700 800

Temperature (K)
Figure 4.3: The magnetization curve of Pd3 Fe obtained from an empirical fit of a
magnetic shape function [172] to hyperfine magnetic fields obtained from the NFS
spectra in this study (green) and Mössbauer data from a study by Longworth [154]
(orange). The shaded region indicates the temperature range where Pd3 Fe exhibits
ferromagnetic order.
magnetically-ordered material, similar to a previous NFS measurement of Pd57
3 Fe
at ambient conditions [155]. The amplitudes and periods of the magnetic beats
diminish with temperature, and the magnetic beats disappear above 485 K, consistent
with a second-order phase transition in which the magnetic order continuously
decreases through the Curie temperature. The remaining beats above 485 K are
from the thickness of the sample.
A quantitative analysis of the NFS spectra was performed with the software package
CONUSS [80, 173]. The refined fits overlay the experimental spectra in Fig. 4.2.
Parameters extracted from these fits include the hyperfine field H, shown together
with the magnetization of Pd3 Fe in Fig. 4.3, and the Lamb-Mössbauer factor,
shown in the Supporting Information. The decrease in the hyperfine field of Pd57
3 Fe
with temperature is in agreement with the hyperfine fields measured by Longworth
[154], and the decrease in this quantity with temperature tracks the decrease in
magnetization through the Curie temperature.

0.8

57Fe Partial

Phonon DOS
298 K
0.7 786 K

DOS (1/meV)

0.6
0.5
0.4
0.3
0.2
0.1

627 K
535 K
485 K
459 K
417 K

0.0 298 K
0 5 10 15 20 25 30 35 40
Energy (meV)
Figure 4.4: The normalized 57 Fe pDOS extracted from NRIXS measurements at
various temperatures. The spectra from measurements above 298 K are offset and
compared with the 298 K pDOS (black curve). Error bars are from counting statistics.

E (meV)

21.0
20.5
20.0
19.5
19.0 300

NRIXS
QH T
QH DFT

TC
400 500 600 700
Temperature (K)

800

Figure 4.5: Average energies of the Fe pDOS from NRIXS measurements (blue
points) plotted with the average Fe phonon energies from the Grüneisen parameter
model (green line) and the QH DFT model (red line).
Phonons
The 57 Fe pDOS curves measured from the Pd57
3 Fe sample are shown in Fig. 4.4.
The pDOS do not show significant energy shifts below the Curie temperature. The
phonons begin to show a small but significant thermal softening beyond the Curie
temperature, made apparent when the 627 and 786 K pDOS are compared with the
298 K curve in Fig. 4.4. This trend is seen more clearly with the average Fe phonon
energies calculated from the 57 Fe pDOS, shown in Fig. 4.5. These phonon shifts are
similar to the experimental trends observed in the ferromagnetic cementite through
the Curie temperature [31]. It is noted that the quality of the high temperature 57 Fe
pDOS at 786 K was impacted by reduced counting statistics.
The phonon DOS curves calculated at 0, 300, 480, and 800 K with s-TDEP are
shown in Fig. 4.6. The phonon DOS curves soften with temperature, although
features like the peak at 21 meV in the Fe pDOS or at 25 meV in the Pd pDOS do not
soften until the Curie temperature. The s-TDEP phonon DOS curves also exhibit a
thermal broadening that is most prominent at 800 K, indicative of phonon-phonon
interactions (PPI) [22, 70, 113, 174].
Figure 4.7 shows thermal trends from experiment (panel a) and computation (panel
b). The difference spectra shown in both panels are in reasonable agreement with
each other, indicating that the NRIXS measurements and s-TDEP calculations capture similar thermal trends in Pd3 Fe.

Total DOS

0.30
DOS (1/mev)

0.25

Pd DOS

0.20
0.15
0K
300 K
480 K
800 K

0.10
0.05
0.00

Fe DOS

10 15 20 25 30 35 40
Energy (meV)

Figure 4.6: Total, Pd partial, and Fe partial phonon DOS curves of Pd3 Fe calculated
with the s-TDEP method from 0 to 800 K.
Nonharmonic Behavior
The experimental nonharmonic behavior was analyzed by comparing the measured
57 Fe pDOS with Fe pDOS curves predicted by quasiharmonic (QH) approximations. The experimental QH phonon frequencies ωiQH (T) were calculated with the
Grüneisen parameter model (referred to as the “QH γT Model”), a QH model using
a thermal Grüneisen parameter γ̄T , averaged for all phonon modes:
VT − V298K
QH
298K
(4.11)
ωi (T) = ωi
1 − γ̄th
V298K
with the 298 K 57 Fe phonons used for scaling the Grüneisen parameter in the QH
γT model. The volumes at elevated temperatures VT were calculated from the
lattice parameters determined from our synchrotron XRD measurements, together
with lattice parameters reported by Jääskeläinen [175]. The thermal Grüneisen
parameter was calculated from bulk properties of Pd3 Fe:
γ̄th (T) =

α(T)KT (T)ν(T)
CV (T)

(4.12)

where KT (T) is the bulk modulus, α(T) is the linear thermal expansion, ν(T) is the
crystalline volume per atom, and CV (T) is the heat capacity at constant volume. The

DOS (1/meV)

0.10
(a) NRIXS

298 K
786 K

0.05

0.00
0.04
0.00
0.04

DOS (1/meV)

0.15

15 20 25
Energy (meV)

(b) s-TDEP

300 K
800 K

0.10
0.05
0.00
0.04
0.00
0.04

15 20 25
Energy (meV)

Figure 4.7: (a) NRIXS 57 Fe pDOS curves compared at 298 and 786 K. (b) s-TDEP
Fe pDOS curves compared at 300 and 800 K. Phonon difference spectra are shown
for both NRIXS and s-TDEP.

(a) NRIXS
NRIXS
QH T

SFe
qh

QH DFT

SFe
vib

300

0.05
0.10

TC
800

800

(b) s-TDEP

s-TDEP
Ferro.
No PPI
QH DFT

SFe
vib

SFe
vib (kB /atom)

400

0.00

TC
400 600
500
600
700
Temperature (K)

SFe
qh

SFe
vib (kB /atom)

0.0
0.1
0.2
0.3

TC
0 200 400 600 800

100 200 300 400 500 600 700 800
Temperature (K)

Figure 4.8: (a) The Fe partial vibrational entropy from the NRIXS measurements
compared with the entropy from the Grüneisen parameter model (QH γT ) and the
QH DFT model. (b) The s-TDEP Fe partial vibrational entropy calculated for Pd3 Fe
with changing magnetic order (blue), ferromagnetic order (green), and the absence
of phonon-phonon interactions (orange). The red line is the entropy from the QH
DFT model. The insets in (a) and (b) show the nonharmonic contributions to the
vibrational entropy.

61
quantities α(T) and ν(T) were calculated from volumes obtained from synchrotron
XRD and from Jääskeläinen. The heat capacity was calculated by integrating
the total phonon DOS calculated by s-TDEP. The bulk modulus was previously
determined by Winterrose through energy-dispersive x-ray diffraction [155].
The average Fe phonon energies from the QH γT and QH DFT models are plotted in
Fig. 4.5. The Fe phonons calculated with the two QH models soften more strongly
with temperature than the NRIXS 57 Fe pDOS, indicating that there is a nonharmonic
stiffening in Pd3 Fe opposing the softening from thermal expansion.
Thermodynamic consequences of nonharmonic phonons were assessed by calculatFe , which contributes to the total thermodying the Fe partial vibrational entropy Svib
Fe is obtained from the Fe pDOS as
namic entropy. This Svib
Fe
Svib
(T) = 3kB

gTFe (ε){(n + 1) ln(n + 1) − n ln(n)}dε,

(4.13)

where k B is the Boltzmann constant, gTFe (E) is the Fe pDOS at temperature T, and n
is a Planck distribution evaluated at T for a given energy E. This expression provides
accurate entropy values that include both quasiharmonic effects and nonharmonic
effects [174].
Fe (T) are plotted in Fig. 4.8(a). The quasiharmonic vibrational entropy S Fe ,
The Svib
qh
also shown in the figure, was calculated by substituting the Fe pDOS from the
QH γT and QH DFT models into Eq. 4.13. The agreement is surprisingly good,
considering that the QH model neglects so many nonharmonic effects, including
those shown in Fig. 4.8(b).

Fig. 4.8(b) shows the Fe partial vibrational entropy calculated with the phonons
calculated with s-TDEP, together with the Fe partial vibrational entropy without
effects from PPI. This quantity, labeled “No PPI,” was calculated by substituting the
Fe pDOS calculated with Γ → 0 and ∆ → 0 into Eq. 4.13. Also shown in the figure
is the Fe partial vibrational entropy for ferromagnetic Pd3 Fe. These quantities were
compared with the quasiharmonic entropy calculated from the QH DFT model.
Fe of Pd Fe calculated with s-TDEP is lower than what is expected from the
The Svib
Fe of
QH DFT model by 0.08 k B /atom at 800 K. The inclusion of PPI increases Svib
Fe from ferromagnetic Pd Fe to
Pd3 Fe by 0.2 k B /atom at 800 K. The change in Svib
Pd3 Fe with magnetic disorder increases from roughly −0.03 k B /atom around the
Curie temperature to roughly 0.1 k B /atom at 800 K. This happens as the deviation

Figure 4.9: Calculated phonon dispersions for the ferromagnetic and paramagnetic
states at 800 K. The dispersions displayed do not include effects from phononphonon interactions. Displacement patterns are shown for two high-energy optical
phonon modes that soften with decreasing magnetization. The orange and green
spheres represent Fe and Pd atoms, respectively. The Fe partial phonon DOS curves
of Pd3 Fe calculated with the s-TDEP method for the ferromagnetic and paramagnetic
states at 800 K are shown in the lower left.
from quasiharmonicity sharply increases for ferromagnetic Pd3 Fe past the Curie
temperature.
4.4

Discussion

Magnon-Phonon Interaction
The increase in Fe partial vibrational entropy with decreasing magnetization at
800 K arises from the softening of Fe vibrational energies with the magnetic transition. A more in-depth analysis of this softening with decreasing magnetization is
performed by comparing the 800 K phonon dispersions for both ferromagnetic and
paramagnetic Pd3 Fe, as shown in Fig. 4.9.
A number of vibrational modes in Pd3 Fe undergo energy shifts with the randomization of the Fe magnetic moment orientations, particularly the two optical modes
at the X symmetry point highlighted in Fig. 4.9. The softening of these modes
contributes to the softening of the high-energy peak in the Fe partial phonon DOS
with decreasing magnetization. By testing the sensitivity of the phonon dispersions
to changes in each of the quadratic force constants obtained from the fit of Eq.
4.1 to the Born-Oppenheimer surface, we found that these two modes soften with
the weakening of the Fe−Fe second-nearest-neighbor (2NN) longitudinal force con-

63
stants, which weaken by about 50% from the ferromagnetic state to the paramagnetic
state at 800 K. This behavior is consistent with the atomic displacement patterns for
the optical modes, which involve the motions of adjacent (100) planes of Fe atoms
in opposite [100] directions. The softening of these modes coincides with a change
in the interactions between the closest-neighbor magnetic atoms due to the loss of
short-range magnetic order past the Curie temperature.
The average 57 Fe phonon energies from the NRIXS measurements change slowly
below the Curie temperature. The s-TDEP calculations for Pd3 Fe and ferromagnetic
Pd3 Fe show that the thermal evolution of the optical phonons depends on whether
the magnetization changes with thermal fluctuations. A thermal optical phonon
stiffening in ferromagnetic Pd3 Fe counteracts the phonon softening from thermal
expansion. This behavior is observed in the NRIXS measurements from 298 to
485 K, where Pd3 Fe still maintains short-range magnetic order. In this case, the
short-wavelength optical modes do not soften strongly with temperature, consistent
with their behavior in a material with full magnetic order. Beyond the Curie
temperature, where there is both long- and short-range magnetic disorder, the change
in Fe−Fe interactions cancels this ferromagnetic stiffening of the short-wavelength
optical phonons.
Phonon-Phonon Interaction
The s-TDEP calculations of the phonon DOS of Pd3 Fe show that anharmonicity
has significant effects on the thermodynamics of the material, as indicated by the
increase in the Fe partial vibrational entropy by up to 0.2 k B /atom at 800 K. The
anharmonic phonon shifts and broadenings can be studied in more detail with
the phonon spectral functions S(q, E), the spectra of lattice excitations that can be
interpreted as phonon modes broadened and shifted by phonon-phonon interactions.
The spectral function is calculated with ωqs , the phonon dispersion from quadratic
force constants, and the real and imaginary components of the phonon self energy
from Eq. 4.4 and 4.5:
S(q, E) ∝

2ωqs Γqs (Ω)
2 − 2ω ∆ (Ω) 2 + 4ω2 Γ2 (Ω)
Ω2 − ωqs
qs qs
qs qs

(4.14)

The spectral functions shown in Fig. 4.10 were calculated at multiple temperatures.
The 0 K spectral function is in good agreement with the phonon dispersion measured
with inelastic neutron scattering at 80 K by Stirling [176]. The significant phonon

64
35

800 K
Intensity (arb. units)

30
25
20
15
10

Figure 4.10: Pd3 Fe spectral functions (logarithmic intensity scale) calculated with
s-TDEP along the high-symmetry directions at 0, 300, and 800 K. Measurements of
the 80 K phonon dispersion by inelastic neutron scattering [176] are shown on top
of the 0 K spectral function.

Intensity (a.u.)

1.0

800 K

0.5
0.00

15 20 25
Energy (meV)

Figure 4.11: Pd3 Fe phonon lineshapes at the X high symmetry point at 800 K. The
orange and green peaks are the optical modes that shift with changing magnetic
order. The black dashed peak is the lineshape of the optical mode after the Pd−Pd
1NN cubic force constant is set to zero.
broadening and shifts in the spectral functions at higher temperatures come from
many decay channels available to the phonons in the twelve branches.
We examined the lineshapes of phonon modes at specific q points in the Brillouin
zone, including the 800 K lineshapes of the phonon modes affected by the magnetic
transition at the X symmetry point, shown in Fig. 4.11. What was unusual was a
double-peak structure of one of these high-energy optical modes. Phonon modes
with lineshapes that are characterized by a single peak broadened by a Lorentzian
are mildly anharmonic, whereas phonon modes that have unusual lineshapes are
more strongly anharmonic [177, 178].
To study the nature of the double peak of the high-energy phonon mode at X,
we analyzed the cubic irreducible force constants responsible for three-phonon

65
interactions. The lineshapes at the X point were calculated when an irreducible
force constant was set to zero. It was found that zeroing the cubic force constants
for the Fe−Pd first nearest neighbors (1NN) along the h110i directions partially
removes the double-peak structure, but zeroing the cubic force constants for the
Pd−Pd 1NN along the h110i directions fully transforms the double-peak lineshape
to a single Lorentzian peak, shown in Fig. 4.11.
These force constants are related to the movement of Fe and Pd atoms in the [100]
direction against the adjacent stationary (200) planes of Pd atoms. As shown in
the displacement pattern of the vibrational mode in Fig. 4.9, (100) planes of Fe
atoms and (100) of Pd atoms alternate in oscillating against the stationary (200)
planes of Pd atoms. The cubic interactions from the oscillation of the (100) planes
of Pd atoms against the (200) Pd planes more strongly contribute to the unusual
phonon lineshape. We suggest that the palladium atoms dominate the anharmonic
phonon-phonon interactions in L12 -ordered Pd3 Fe. This is consistent with how fcc
Pd shows strong PPI at high temperatures [179].
4.5

Conclusions

Nuclear resonant inelastic x-ray scattering was used to measure the 57 Fe partial
phonon DOS of L12 -ordered Pd3 Fe from room temperature through the Curie transition. The iron partial vibrational entropy at temperatures far from the Curie transition was observed to be approximately what was predicted by the quasiharmonic
approximation owing to a cancellation of effects. A nonharmonic phonon stiffening
opposed the expected softening from thermal expansion below the Curie temperature. Similar trends were observed from first-principles calculations that couple
the stochastically-initialized temperature dependent effective potential (s-TDEP)
method with magnetic special quasirandom structures (SQSs) of noncollinear magnetic moments.
The s-TDEP calculations showed that phonon-phonon interactions (PPI) contribute
to the softening and broadening of the phonon spectra at elevated temperatures. A
high-energy optical mode at the X symmetry point was calculated to have a doublepeak lineshape. The first-nearest-neighbor Pd−Pd cubic interactions strongly contribute to this unusual lineshape, highlighting the strong contribution of the majority
Pd atoms to the phonon anharmonicity in Pd3 Fe.
The calculations also showed that high-energy optical modes soften with decreasing
magnetization, so a ferromagnetic optical phonon stiffening is lost. This softening

66
of optical modes originates with how the randomization of orientations of the Fe
magnetic moments alters the short-range Fe−Fe interactions, softening the Fe−Fe
second-nearest-neighbor force constants. The dependence of these optical vibrational modes on the magnetic transition can be understood as how magnon-phonon
interactions alter lattice vibrations at elevated temperatures.

60
60

(321)

(222)
80
80
2 (deg)

(320)

(300)

(211)

(210)
40

100
100

(420)

(400)

(311)

(200)

(111)
(110)

Intensity (arb. units)

Pre-NRIXS
Post-NRIXS

(331)

(220)

120
120

Figure 4.12: X-ray diffraction patterns of Pd57
3 Fe (“Ordered-II”) collected on a
Cu Kα laboratory diffractometer. Measurements were performed on the sample
after the heat treatment (“Pre-NRIXS”) and after the NRIXS measurements (“PostNRIXS”). The labeled peaks include both fundamental and superlattice peaks. The
intensity is displayed in a logarithmic scale.
4.6

Supporting Information

Sample Preparation
Conventional X-Ray Diffraction
Characterization was performed on the Pd57
3 Fe sample with x-ray diffractometry
using Cu Kα radiation. The diffraction measurements were performed on the sample
after the heat treatment described in the main text and after the nuclear resonant
inelastic x-ray scattering (NRIXS) measurements at high temperature, with results
shown in Fig. 4.12.
The diffraction patterns include fundamental peaks expected from fcc and fcc-based
structures. The patterns also include superlattice peaks expected in the L12 structure,
including (110), (210), (211), (320), and (321). The presence of these peaks are
indicative of long-range L12 chemical order in the sample before and after the
NRIXS measurements. X-ray diffractometry showed that the chemical order was at
least as high after the NRIXS measurements as before.

68
Table 4.1: Hyperfine fields H and change in the hyperfine field ∆H in the Pd57
3 Fe
sample at different stages of this study, shown together with measurements performed
by Longworth [154]. For the Pd57
3 Fe sample, the hyperfine field for “Ordered-II” is
treated as a reference for ∆H. For the measurements performed by Longworth, the
hyperfine field for the ordered sample is treated as a reference for ∆H.
Sample

Hyperfine Field H
(kOe)
Ordered-I
286.402 ± 0.01
Ordered-II
292.251 ± 0.01
Post-NRIXS
290.821 ± 0.01
Ordered (Longworth)
285.5 ± 1
Disordered (Longworth)
278 ± 1

∆H
(kOe)
-5.849
-1.430
-7.5

The lattice parameter of Pd3 Fe was measured to be 3.852 ± 0.001 Å. All diffraction
measurements provided similar values of lattice parameters within the margin of
error.
Mössbauer Spectrometry
Mössbauer spectra were collected from conversion electrons in backscatter geometry
using a constant acceleration spectrometer with a 57 Co in Rh γ-ray source. Velocity
and isomer shift calibrations were performed by reference to room temperature αiron. The conversion electron Mössbauer spectra of the Pd57
3 Fe foil sample over the
course of this experiment are shown in Fig. 4.13.
The hyperfine field was measured from Mössbauer spectrometry for the previously
annealed Pd57
3 Fe sample before the second heat treatment described in the main
text (“Ordered-I”), after this heat treatment (“Ordered-II”), and after the hightemperature NRIXS measurements. These values are reported in Table 4.1, together
with the hyperfine field of the ordered Pd3 Fe and disordered Pd0.73 Fe0.27 samples
measured by Longworth [154].
From the mean hyperfine fields, it appears that there was little change in the state
of chemical order near the surface of the sample probed by the conversion electron
Mössbauer spectrometer.

Intensity

Ordered-II

2 0
Velocity (mm/s)

Intensity

Ordered-I
Ordered-II
Post-NRIXS

Figure 4.13: (Top) Room temperature Mössbauer spectrum of the annealed Pd57
3 Fe
57
foil sample before the NRIXS experiment (Ordered-II). (Bottom) Pd3 Fe Mössbauer spectra for the “Ordered-I” state, “Ordered-II” state, and after the NRIXS
experiment.

20
2 (degrees)

(210)

(200)

786 K

(111)

(110)

(c)

(100)

(a) 298 K

(b) 786 K

log(Intensity)

627 K
485 K
459 K
417 K
298 K
10

Figure 4.14: Two-dimensional synchrotron diffraction patterns of Pd3 Fe recorded on
a CCD detector plate at (a) 298 K and (b) 786 K. (c) One-dimensional synchrotron
x-ray diffraction patterns of Pd57
3 Fe from 298 to 786 K. The black dashed lines are
the locations of the 298 K diffraction peaks. The dips in intensity for 627 and 786 K
are where extraneous diffraction peaks from the aluminum foil were masked.
Synchrotron Measurements
Synchrotron X-ray Diffraction
In situ synchrotron x-ray diffraction (XRD) measurements were performed at beamline 16ID-D of the Advanced Photon Source at Argonne National Laboratory using
a monochromatic beam at 14.413 keV and a resistive heating furnace with a kapton
window for x-ray transmission. Diffraction was measured using a Mar CCD detector
plate. The two-dimensional diffraction rings on the image plate (Fig. 4.14(a) and
(b)) were integrated with the FIT2D [180] program to produce diffraction patterns
of intensity vs. 2θ (Fig. 4.14(c)).
For measurements above the Curie temperature, a 16 micron-thick piece of aluminum foil was placed over the kapton window to reflect the radiative heat from the
furnace away from the avalanche photodiodes used in the NRIXS measurements.
The aluminum foil caused the additional diffraction rings at 627 and 786 K. These
extraneous rings were masked in the FIT2D program, resulting in the dips in intensity
in the integrated diffraction patterns for 627 and 786 K.

71
The lattice parameters of the L12 -ordered Pd3 Fe were determined from synchrotron
XRD to be
3.8541 Å at 298 K; 3.8546 Å at 417 K; 3.8636 Å at 459 K;
3.8661 Å at 485 K; 3.8719 Å at 627 K; 3.8854 Å at 786 K.
These results are consistent with a prior XRD measurement of the lattice parameter
of Pd3 Fe with respect to temperature [175].
Nuclear Resonant Inelastic X-ray Scattering
NRIXS measurements were performed on Pd57
3 Fe at seven temperatures from 298
to 786 K at beamline 16ID-D of the Advanced Photon Source at Argonne National
Laboratory. For each temperature, the NRIXS spectrum was measured by scanning
the energy transfer from −80 to +80 meV around the resonant energy of 57 Fe in
several scans that were combined for final analysis. The spectra are shown in Fig.
4.15. The energy resolution of all NRIXS measurements was measured to be 2.2
meV (FWHM) at the elastic line. The instrument resolution function is also shown
in Fig. 4.15.
The PHOENIX software package was used to extract the 57 Fe partial phonon densities
of states (pDOS) from the NRIXS spectra [80]. The data reduction procedure in the
PHOENIX software package involves the removal of the resonant elastic peak and the
multi-phonon processes from the NRIXS spectra. The one-, two-, and three-phonon
contributions to the NRIXS spectra at 298 and 786 K are shown in Fig. 4.16.
Other physical quantities were calculated from the data reduction procedure in
PHOENIX. One such physical quantity is the Lamb-Mössbauer factor, plotted in
Fig. 4.17. The Lamb-Mössbauer factor was also calculated from fits to the nuclear
forward scattering (NFS) spectra with the CONUSS software package [173]. Due
to the low counts in the nuclear forward scattering spectra, the Lamb-Mössbauer
factors obtained from the NFS spectra are expected to have a greater error than those
obtained from the data reduction of the NRIXS spectra.
Figure 4.18 shows previous room temperature NRIXS measurements of the L12 ordered Pd57
3 Fe compound in its “Ordered-I” state and the disordered fcc alloy
Pd0.75 57 Fe0.25 . The 57 Fe pDOS for the disordered sample is considerably different
from the “Ordered-I” pDOS and the pDOS from this study (“Ordered-II”).

72
1200

786 K
627 K

800

535 K

600

485 K
459 K

400

12000
10000
Intensity

Counts

1000

6000
4000

417 K

200

8000

2000

298 K
60 40 20 0 20 40 60 80
Energy (meV)

0 10

Energy Transfer (meV)

Figure 4.15: (Left) Raw NRIXS scattering spectra showing 57 Fe vibrational excitations in Pd57
3 Fe as a function of scattering energy. Spectra are collected over a
range of temperatures. (Right) The elastic line of the raw scattering spectrum of
57 Fe at room temperature, used as the instrument resolution function for the NRIXS
measurements.
Inelastic Data
1 Phonon
2 Phonon
3 Phonon

101

Intensity

100

101

10 1

Inelastic Data
1 Phonon
2 Phonon
3 Phonon

100

10 2
10 3

10 1
80

20 0 20
Energy (meV)

80 60 40 20 0 20 40 60 80
Energy (meV)

Figure 4.16: The multi-phonon components of the NRIXS spectra of Pd57
3 Fe at (left)
298 K and (right) 786 K.
Table 4.2: The Fe partial vibrational heat capacity calculated from the integration
of the 57 Fe phonon DOS.
Temperature (K)
298
417
459
485
535
627
786

Fe Partial Heat Capacity (kB /atom)
2.8382 ± 0.0106
2.8355 ± 0.0376
2.9227 ± 0.0535
2.8686 ± 0.0385
2.9125 ± 0.0518
2.9702 ± 0.0408
3.0107 ± 0.0434

Lamb-Mossbauer Factor

0.7
0.6
0.5
0.4
0.3

300

NRIXS
NFS
400 500 600 700
Temperature (K)

800

Figure 4.17: Lamb-Mössbauer factors obtained from NRIXS (blue squares) and
NFS (green triangles).

Table 4.3: The vibrational kinetic energy calculated from the integration of the 57 Fe
phonon DOS.
Temperature (K)
298
417
459
485
535
627
786

Kinetic Energy (meV/atom)
13.5647 ± 0.0603
19.8122 ± 0.4101
20.6811 ± 0.5960
22.0571 ± 0.4264
23.7510 ± 0.5040
28.4609 ± 0.4977
33.1471 ± 0.5778

DOS (1/meV)

0.10
0.05

Disordered
Ordered-I
Ordered-II
(Present Study)

0.00

15 20 25
Energy (meV)

Figure 4.18: The room temperature 57 Fe pDOS of the Pd57
3 Fe sample measured in
this study (purple) compared with prior measurements of the room temperature 57 Fe
57
pDOS of ordered Pd57
3 Fe (black) and the disordered fcc alloy Pd0.75 Fe0.25 (red).

Table 4.4: The mean force constant calculated from the integration of the 57 Fe
phonon DOS.
Temperature (K)
298
417
459
485
535
627
786

Mean Force Constant (N/m)
2.435 ± 0.024
2.440 ± 0.030
2.421 ± 0.027
2.448 ± 0.043
2.360 ± 0.034
2.343 ± 0.040
2.340 ± 0.047

75
Chapter 5

CONCLUDING REMARKS AND FUTURE DIRECTIONS
This thesis research explored the temperature dependence of adiabatic electronphonon interactions in the metallic systems FeTi and vanadium, and the temperature dependence of magnon-phonon interactions through the Curie temperature of
ferromagnetic Pd3 Fe. Our ability to study these interactions and their effects on the
thermal evolution of the vibrational entropy was made possible by synchronously
pushing the limits on both experimental measurements and theoretical calculations.
5.1

Electron-Phonon Interactions

First-principles computational methods ranging from the temperature dependent
effective potential (TDEP) method to electronic band unfolding have provided a new
perspective of the relationship between lattice dynamics and the changing electronic
structure. The strength of the electronic screening of atomic vibrations is altered by
the increasingly large amplitudes of atomic displacements with temperature. The
electron-phonon interaction nesting features of the Fermi surface are smeared by
both these displacements and the thermal excitations of electrons occupying states
within the thermal layer surrounding the Fermi level. Thermal shifts in electronic
bands can sometimes change the topology of the Fermi surface. If the effects of
thermally-driven electronic topological transitions (ETT) overtake the effects of the
thermal smearing of nesting features, these ETTs bring forth dramatic changes to
the temperature dependence of adiabatic electron-phonon interactions.
We are interested in continuing this study of the temperature dependence of electronphonon interactions and possibly thermally-driven ETTs. Studies of phonon anomalies and electron-phonon interactions in transition metal carbides (TMCs) and nitrides (TMNs) [181–183] may point to future opportunities for research. Li et al.
have recently reported that group V TMCs display lattice thermal conductivities that
are nearly independent of temperature owing to Fermi surface nesting [184]. This
research direction may direct us to new insights about electronic and heat transport.
We conducted our analyses of the electron-phonon interactions in FeTi and vanadium
by quantifying changes in the Fermi surface nesting features and the spanning vector
densities. This in-depth analysis of the electronic structure of metals has proven to be

76
rich in information in comparison to analyzing only the electronic density of states.
ν (k, q),
What we had not accounted for was the electron-phonon matrix element gmn
a crucial quantity for calculating the strength of electron-phonon interactions:
gmn
(k, q) =

κα
~ Õ eqν
√ hmk + q|∂qκαV |nki.
2ωqν κα mκ

(5.1)

ν (k, q) for arbitrary q-points using arWe are currently capable of calculating gmn
bitrarily dense k-grids through a variety of software packages including EPW and
ABINIT [185, 186]. As we have stated in Chapter 3, however, the recent ab initio
methods for calculating the electron-phonon matrix element are based on density
functional perturbation theory, which does not adequately describe thermal effects
ν (k, q)
observed at finite temperatures. Temperature dependent matrix elements gmn
were calculated using phonon energies ωqν (T) and eigenvectors eqν (T) calculated
with the TDEP method in a recent study by Zhou et al. [100]. These matrix elements
were used to predict the electron mobility in SrTiO3 between 150-300 K, which were
in good agreement with experimental values. The predictive power of these calculations of temperature dependent electron-phonon matrix elements can be further
optimized with ab initio calculations of the matrix elements hmk + q|∂qκαV |nki,
which require calculations of the gradient of temperature-dependent potentials with
respect to atomic displacements and the temperature-dependent electronic structure.
This calculation still remains an open challenge to computational materials physics.

The materials physics research community is moving in the right direction towards
the prediction of the matrix elements hmk + q|∂qκαV |nki at finite temperatures.
We have seen several recent developments towards the calculation of temperaturedependent electronic band structure ranging from methods based off of perturbation
theory [118–123, 187] to nonperturbative adiabatic calculations [23, 188–193] not
very different in nature from the electronic band unfolding procedure employed
in Chapters 2 and 3. The research community is building towards a formalism
for the ab initio determination of electron-phonon interaction strengths at elevated
temperatures, and we hope that our work contributes to this research effort.
We had hypothesized at the end of Chapter 2 that materials with occupied or
unoccupied bands that are a few k BT away from the Fermi level at low temperatures
may display thermally-driven ETTs that cause rapid and abrupt changes in physical
properties with temperatures. Based on this alone, we may be tempted to reduce
this search for candidate materials with adiabatic EPIs impacted by thermally-

77
driven ETTs down to a search through 0 K electronic band structures provided in
materials databases. Merely looking at 0 K electronic band structures, however, led
us to predict that a thermally-driven ETT involving the disappearance of electron
pockets in vanadium would contribute to the thermal phonon stiffening. We did not
predict that a thermally-driven ETT that was different from what we had expected
would be cancelled by the thermal smearing of the Fermi surface in vanadium. A
strong understanding of the temperature evolution of both the lattice dynamics and
electronic bands in materials can aid our search for these candidate materials. The
ability to calculate temperature-dependent electron-phonon interaction parameters
can also allow us to more quantitatively analyze thermally-driven ETTs and their
novel impacts on high temperature thermodynamics.
5.2

Magnon-Phonon Interactions

We have performed an in-depth study of the lattice dynamics of the ferromagnetic
Pd3 Fe with both experimental measurements and advanced theoretical calculations.
By combining our experimental results with a combination of the TDEP method
and magnetic special quasirandom structures (SQSs), we proposed an atomistic
mechanism for the temperature-dependent magnon-phonon interaction in Pd3 Fe by
linking it to the reduction of the ferromagnetic optical phonon stiffening from Fe−Fe
second-nearest-neighbor interactions through the Curie transition. The computational methods performed in this study can serve as building blocks towards the
accurate modeling the finite temperature lattice dynamics of magnetic materials
[33, 34, 138–146]. We would ideally like to see these computational tools develop to the point where we can predict the thermodynamics of materials displaying
magnetism ranging from localized magnetism to itinerant magnetism across a wide
range of temperatures.
We observed that the ferromagnetic system Pd3 Fe displayed several competing
contributions to the vibrational entropy. The cancellation of magnon-phonon interactions and anharmonicity have resulted in thermal trends close to quasiharmonicity.
We have now observed cases where temperature-dependent magnon-phonon interactions can either have profound effects on the vibrational thermodynamics [7] or
have muted effects due to competing factors [31]. If we wish to continue investigating magnon-phonon interactions at elevated temperatures, we would benefit from
studying materials with significant thermal phonon anomalies arising from magnetic
transitions with very little competing factors like anharmonicity.

Figure 5.1: The frequency shifts and linewidth of the Eg 187 cm−1 Raman line
versus temperature in FeF3 , measured by Shepherd [197].
We may look to studying the temperature dependence of magnon-phonon interactions in bcc chromium with inelastic neutron scattering and first-principles calculations. Chromium displays dramatic phonon softening with temperature [194, 195]
that may be related to magnetic disorder [196]. There is, however, a possibility that
other factors contribute to the thermal phonon softening in chromium, including
electron-phonon interactions. Significant challenges would meanwhile lie in modeling its spin-density-wave state and paramagnetic state from low to intermediate
temperatures.
We may also be interested in conducting an in-depth study of the lattice dynamics
of FeF3 with first-principles calculations and scattering methods including nuclear
resonant inelastic x-ray scattering (NRIXS). Shepherd had observed an anomalous
temperature dependence of the Eg 187 cm−1 Raman peak in the vicinity of the
magnetic transition, as shown in Fig. 5.1 from Ref. [197]. A study of this
trigonal material material may provide an opportunity for the further development
of our computational methods for performing phonon calculations with diminishing
magnetic order.
Finally, we can revisit the interplay between lattice dynamics and the Curie transition

79
in bcc α-iron through both NRIXS and first-principles calculations at elevated
temperatures and pressures. Not only would this study push boundaries in the
experimental measurements of phonon frequencies, we would be able to study
the dependence of nonharmonic lattice dynamics on simultaneous pressure and
temperature that we would not see at high pressure or high temperature alone
[109, 198]. Such a study would provide us insights into the physical properties of
iron-based materials at extreme conditions. We see that there are still many open
questions regarding the vibrational thermodynamics of magnetic materials, and that
there remains much work to be done.

80
Appendix A

HARMONIC MODEL
A.1

Einstein and Debye Models

In the Einstein model, the harmonic solid is treated as N oscillators with the same
vibrational frequency ω0 = ε0 /~, such that the phonon density of states (DOS) is
a delta function δ(ε − ε0 ). While this is a great simplification of lattice dynamics,
the results from the Einstein model suggest the energies of lattice vibrations are
quantized, and that the contribution of high-energy vibrational modes to the specific
heat exponentially drops to zero as the temperature approaches 0 K.
The Debye model assumes that phonon frequencies ω obey a linear relationship
with respect to the reciprocal lattice vector q. This model treats the harmonic solid
as a repeating lattice of N atoms that vibrates as if it were an elastic continuum.
In this model, the vibrational frequencies cannot exceed a certain frequency ωD
chosen to make the total number of vibrational modes equal to the total number
of classical degrees of freedom. In a three-dimensional solid, the phonon DOS
behaves as ε 2 up to the cutoff energy. This assumption of linear phonon branches
accurately describes the long wavelength limit (very low |q|), and it recovers the
low temperature T 3 behavior observed in measured heat capacities.
A.2

Born von Kármán Model

Phonon frequencies ω and the phonon DOS can be more accurately calculated
through a normal mode analysis as presented by Born and von Kármán [5]. In a
crystal with an atomic basis described by the vectors {rκ }, the position Rlκ of a
vibrating atom κ in the unit cell l at time t is
Rlκ (t) = rl + rκ + ulκ (t),

(A.1)

where ulκ is the displacement of the atom about its equilibrium position rl + rκ . The
potential energy U of the crystal at any t is a function of the instantaneous positions
of the atoms. We can expand the potential energy in a Taylor series of the atomic
displacements about ulκ (t) = 0, the equilibrium positions:
U = U0 +

αlκ

Φαlκ uαlκ +

1Õ Õ
Φαα 0ll 0 κκ 0 uαlκ uα 0l 0 κ 0 + · · · ,
2 αlκ α 0l 0 κ 0

(A.2)

81
where U0 is a static potential term. We use the Cartesian components uαlκ for
the displacement vector, where α = {x, y, z}. The coefficients of the Taylor series
are derivatives of the potential with respect to the displacements evaluated in the
equilibrium configuration:
∂U
∂uαlκ 0
∂ 2U
Φαα 0ll 0 κκ 0 =
∂uαlκ ∂uα 0l 0 κ 0 0
Φαlκ =

(A.3)
(A.4)

Because there is no force on any atom in equilibrium, Φαlκ = 0 for all atoms. In the
harmonic model, we keep the remaining terms of the series written explicitly in Eq.
A.2 while neglecting terms of higher order. We can rewrite what remains of this
Taylor expansion in matrix form:
U = U0 +

1 Õ T
uÜ Φlκl 0 κ 0 ul 0 κ 0,
2 lκl 0 κ 0 lκ

(A.5)

where Φlκl 0 κ 0 is the force constant matrix defined for each atom pair (lκ; l 0 κ0).
The equations of motion for all atomic nuclei are
mκ ulκ (t) = −
Φlκl 0 κ 0 ulκ (t)

∀l, κ.

(A.6)

l0κ0

With periodic boundary conditions, the solution of Eq. A.6 can be in the form of
plane waves of wavevector q, angular frequency ωqν , and polarization eνκ (q), where
ν is a “branch index”:
ulκqν (t) ∝ eνκ (q)ei(q·rl −ωqν t)) .
(A.7)
The substitution of the plane wave displacements into the equations of motion is
equivalent to taking Fourier transforms, reducing the problem into diagonalizing the
“dynamical matrix” D(q):
2 κ
Dκκ 0 (q)eνκ (q) = ωqν
eν (q),
(A.8)
κ0

where each sub-matrix Dκκ 0 (q) is the Fourier transform of the force constant matrix
Φlκl 0 κ 0 . The dynamical matrix D(q) is Hermitian for any q, meaning the eigenvalues
2 are real. The phonon DOS in this normal mode analysis is the histogram of
ωqν
frequencies calculated from the diagonalization of D(q) over a large number of
points covering the first Brillouin zone.

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