Spin-Phonon Interactions and Spin Decohe

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Spin-Phonon Interactions and Spin Decoherence from First Principles
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Park, Jinsoo
(2022)
Spin-Phonon Interactions and Spin Decoherence from First Principles.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/80bd-x991.
Abstract
Developing a microscopic understanding of spin decoherence is essential to advancing quantum technologies. Electron spin decoherence due to atomic vibrations (phonons) plays a special role as it sets an intrinsic limit to the performance of spin-based quantum devices. Two main sources of phonon-induced spin decoherence, the Elliott-Yafet (EY) and Dyakonov-Perel (DP) mechanisms, have distinct physical origins and theoretical treatments. First-principles calculations of electron-phonon (
-ph) interactions combined with many-body perturbation theory are promising to study phonon-induced spin decoherence. However, predicting the spin response in materials remains an open challenge; methods for quantifying spin-dependent
-ph interactions in materials, as well as a linear response framework for spins in the presence of
-ph interaction is missing. In this thesis, we provide a first-principles framework for computing the relativistic spin-dependent electron-phonon interactions. We develop a formalism that unifies the modeling of EY and DP spin decoherence, and provide a rigorous many-body perturbation theory for obtaining the spin-spin correlation function including the vertex corrections due to
-ph interactions. We compute the phonon-dressed vertex of the spin-spin correlation function with a treatment analogous to the calculation of the anomalous electron magnetic moment in QED. We find that the vertex correction provides a giant renormalization of the electron spin dynamics in solids, greater by many orders of magnitude than the corresponding correction from photons in vacuum. We further identify the long-range quadrupole
-ph interaction in materials, and demonstrate its importance in the description of phonon-induced spin decoherence. We show first-principle calculations of spin-dependent
-ph interactions in correlated electron systems, using the framework of Hubbard-corrected density functional theory. Lastly, we provide technical details in the implementation of
ab-initio
-ph interaction in PERTURBO, a software package for first-principles calculations of charge transport, spin dynamics, and ultrafast carrier dynamics in materials. In summary, the thesis demonstrates a general approach for quantitative analysis of spin decoherence in materials, advancing the quest for spin-based quantum technologies.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
first-principles calculations; electron-phonon interactions; spin-phonon interactions; many-body techniques; spin-orbit coupling; spin dynamics; spin relaxation; spin decoherence; spintronics;
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Bernardi, Marco
Thesis Committee:
Nadj-Perge, Stevan (chair)
Alicea, Jason F.
Yeh, Nai-Chang
Bernardi, Marco
Defense Date:
27 May 2022
Non-Caltech Author Email:
jinsoop412 (AT) gmail.com
Record Number:
CaltechTHESIS:06052022-215214933
Persistent URL:
DOI:
10.7907/80bd-x991
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DOI
Article adapted for Chapter 2.
arXiv
Article adapted for Chapter 3.
DOI
Article adapted for Chapter 4.
DOI
Article adapted for Chapter 5.
DOI
Article adapted for Chapter 6.
ORCID:
Author
ORCID
Park, Jinsoo
0000-0002-1763-5788
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
14944
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Deposited By:
Jinsoo Park
Deposited On:
06 Jun 2022 22:29
Last Modified:
20 Feb 2025 21:13
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Spin-phonon interactions and spin decoherence from first
principles

Thesis by

Jinsoo Park

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

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2022
Defended May 27, 2022

Jinsoo Park
ORCID: 0000-0002-1763-5788

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to express my sincere gratitude towards my advisor,
Prof. Marco Bernardi, for his guidance, advice, and encouragement throughout
my doctoral studies. If I had been able to become a better researcher, scientist,
or communicator, it would be entirely due to the efforts of my wonderful advisor,
Marco.
I would like to thank my thesis committee members, Prof. Stevan Nadj-Perge, Prof.
Jason Alicea, and Prof. Nai-Chang Yeh, for their invaluable advice and helpful
guidance.
I would like to thank the Korea Foundation for Advanced Studies for their kind
support during my doctoral studies. I want to thank my collaborators, Dr. Iurii
Timrov, Dr. Andrea Floris, Dr. Matteo Cococcioni, and Dr. Nicola Marzari,
for whom I am grateful for the insights into correlated electron systems they have
shared with me, and Dr. Cyrus E. Dreyer, for fruitful collaborations on quadrupole
interactions. I thank Dr. Jin-Jian Zhou for sharing his vast knowledge of physics
and computational techniques with me. I want to thank Dr. I-Te Lu for being both
a friend and a patient mentor ever since our first meeting. I thank all members of
the Bernardi group, including Dr. Luis Agapito, Dr. Vatsal Jhalani, Dr. Nien-En
Lee, Dr. Hsiao-Yi Chen, Xiao Tong, Benjamin K. Chang, Dr. Raffaello Bianco, Dr.
Shiyuan Gao, Dr. Ivan Maliyov, Dhruv Desai, Dr. Sergio Pineda Flores, Yao Luo,
Kelly Yao, David Abramovitch, and other members who have overlapped with me.
I would like to thank Joy Kim and my relatives in Los Angeles for their warm
company and kindly inviting me to Thanksgiving dinner every year. I would also
like to thank my friends who helped to stay together in the uncertain times of the
global pandemic. Last but not least, I would like to thank my family members, my
father, mother, and my younger brothers, for their unconditional support and love.

ABSTRACT

Developing a microscopic understanding of spin decoherence is essential to advancing quantum technologies. Electron spin decoherence due to atomic vibrations
(phonons) plays a special role as it sets an intrinsic limit to the performance of spinbased quantum devices. Two main sources of phonon-induced spin decoherence,
the Elliott-Yafet (EY) and Dyakonov-Perel (DP) mechanisms, have distinct physical
origins and theoretical treatments. First-principles calculations of electron-phonon
(𝑒-ph) interactions combined with many-body perturbation theory are promising to
study phonon-induced spin decoherence. However, predicting the spin response in
materials remains an open challenge; methods for quantifying spin-dependent 𝑒-ph
interactions in materials, as well as a linear response framework for spins in the
presence of 𝑒-ph interaction is missing. In this thesis, we provide a first-principles
framework for computing the relativistic spin-dependent electron-phonon interactions. We develop a formalism that unifies the modeling of EY and DP spin decoherence, and provide a rigorous many-body perturbation theory for obtaining the
spin-spin correlation function including the vertex corrections due to 𝑒-ph interactions. We compute the phonon-dressed vertex of the spin-spin correlation function
with a treatment analogous to the calculation of the anomalous electron magnetic
moment in QED. We find that the vertex correction provides a giant renormalization
of the electron spin dynamics in solids, greater by many orders of magnitude than
the corresponding correction from photons in vacuum. We further identify the longrange quadrupole 𝑒-ph interaction in materials, and demonstrate its importance in
the description of phonon-induced spin decoherence. We show first-principle calculations of spin-dependent 𝑒-ph interactions in correlated electron systems, using
the framework of Hubbard-corrected density functional theory. Lastly, we provide
technical details in the implementation of ab-initio 𝑒-ph interaction in PERTURBO,
a software package for first-principles calculations of charge transport, spin dynamics, and ultrafast carrier dynamics in materials. In summary, the thesis demonstrates
a general approach for quantitative analysis of spin decoherence in materials, advancing the quest for spin-based quantum technologies.

PUBLISHED CONTENT AND CONTRIBUTIONS
[1] Dhruv C. Desai∗ , Jinsoo Park∗ , Jin-Jian Zhou, and Marco Bernardi. Transport
in 3D topological Dirac semimetal Na3 Bi from first principles. In Preparation., 2022.
J.P contributed to developing the theory and computational method, performed calculations, analyzed the data, and participated in the writing of
the manuscript.
∗ These authors contributed equally to this work.
[2] Vatsal A. Jhalani∗ , Jin-Jian Zhou∗ , Jinsoo Park, Cyrus E. Dreyer, and Marco
Bernardi. Piezoelectric electron-phonon interaction from ab initio dynamical
quadrupoles: Impact on charge transport in wurtzite GaN. Phys. Rev. Lett.,
125(13):136602, 2020. doi: 10.1103/PhysRevLett.125.136602.
J.P contributed to developing the theory and computational method, performed calculations, analyzed the data, and participated in the writing of
the manuscript.
∗ These authors contributed equally to this work.
[3] I-Te Lu, Jinsoo Park, Jin-Jian Zhou, and Marco Bernardi. Ab initio electrondefect interactions using Wannier functions. npj Comput. Mater., 6(1):1–7,
2020. doi: 10.1038/s41524-020-0284-y.
J.P contributed to developing the theory and computational method and participated in the writing of the manuscript.
[4] I-Te Lu, Jin-Jian Zhou, Jinsoo Park, and Marco Bernardi. First-principles
ionized-impurity scattering and charge transport in doped materials. Phys. Rev.
Mater., 6(1):L010801, 2022. doi: 10.1103/PhysRevMaterials.6.L010801.
J.P contributed to theory development and participated in the writing of the
manuscript.
[5] Ivan Maliyov, Jinsoo Park, and Marco Bernardi. Ab initio electron dynamics
in high electric fields: Accurate prediction of velocity-field curves. Phys. Rev.
B, 104(10):L100303, 2021. doi: 10.1103/PhysRevB.104.L100303.
J.P contributed to performing calculations and participated in the writing of
the manuscript.
[6] Jinsoo Park, Jin-Jian Zhou, and Marco Bernardi. Spin-phonon relaxation
times in centrosymmetric materials from first principles. Phys. Rev. B, 101(4):
045202, 2020. doi: 10.1103/PhysRevB.101.045202.
J.P developed the theory and computational method, performed calculations,
analyzed the data, and participated in the writing of the manuscript.
[7] Jinsoo Park, Jin-Jian Zhou, Vatsal A. Jhalani, Cyrus E. Dreyer, and Marco
Bernardi. Long-range quadrupole electron-phonon interaction from first principles. Phys. Rev. B, 102(12):125203, 2020. doi: 10.1103/PhysRevB.102.

vi
125203.
J.P contributed to developing the theory and computational method, performed calculations, analyzed the data, and participated in the writing of
the manuscript.
[8] Jinsoo Park, Yao Luo, and Marco Bernardi. Many-body theory of phononinduced spin relaxation and decoherence. In Preparation., 2022.
J.P developed the theory and computational method, performed calculations,
analyzed the data, and participated in the writing of the manuscript.
[9] Jinsoo Park, Jin-Jian Zhou, and Marco Bernardi. Predicting electron spin
decoherence with a many-body first-principles approach. arXiv preprint
arXiv:2203.06401, 2022.
J.P developed the theory and computational method, performed calculations,
analyzed the data, and participated in the writing of the manuscript.
[10] Jin-Jian Zhou, Jinsoo Park, I-Te Lu, Ivan Maliyov, Xiao Tong, and Marco
Bernardi. Perturbo: A software package for ab initio electron–phonon interactions, charge transport and ultrafast dynamics. Comput. Phys. Commun., 264:
107970, 2021. doi: 10.1016/j.cpc.2021.107970.
J.P contributed in developing the PERTURBO code, the tutorial for the PERTURBO code, and participated in the writing of the manuscript.
[11] Jin-Jian Zhou∗ , Jinsoo Park∗ , Iurii Timrov, Andrea Floris, Matteo Cococcioni,
Nicola Marzari, and Marco Bernardi. Ab initio electron-phonon interactions
in correlated electron systems. Phys. Rev. Lett., 127(12):126404, 2021. doi:
10.1103/PhysRevLett.127.126404.
J.P contributed to developing the computational method, performed calculations, analyzed the data, and participated in the writing of the manuscript.
∗ These authors contributed equally to this work.

vii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 First-Principles Computational Methods . . . . . . . . . . . . . . . 2
1.3 Many-Body Perturbation Theory . . . . . . . . . . . . . . . . . . . 8
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter II: Spin-phonon relaxation times in centrosymmetric materials from
first principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . 27
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter III: Predicting phonon-induced spin decoherence and colossal spin
renormalization in condensed matter . . . . . . . . . . . . . . . . . . . . 40
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter IV: Long-range quadrupole electron-phonon interaction from first
principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter V: Ab initio electron-phonon interactions in correlated electron systems 82
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

viii
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Chapter VI: PERTURBO: a software package for ab initio electron-phonon
interactions, charge transport and ultrafast dynamics . . . . . . . . . . . . 101
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Capabilities and workflow . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Technical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Parallelization and performance . . . . . . . . . . . . . . . . . . . . 124
6.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 125
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Chapter VII: Summary and Future Directions . . . . . . . . . . . . . . . . . 131
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

LIST OF ILLUSTRATIONS

Number
Page
1.1 Feynman diagrams and Kubo formula . . . . . . . . . . . . . . . . . 9
2.1 First-principles spin-phonon relaxation times . . . . . . . . . . . . . 21
2.2 Phonon dispersions in silicon and diamond . . . . . . . . . . . . . . 24
2.3 Spin relaxation time and momentum relaxation time . . . . . . . . . 25
2.4 Elliott approximation in silicon and diamond . . . . . . . . . . . . . 26
2.5 Importance sampling method . . . . . . . . . . . . . . . . . . . . . 28
2.6 Conduction band in silicon and diamond . . . . . . . . . . . . . . . 29
2.7 Phonon dispersions overlaid with spin-flip matrix elements . . . . . . 30
2.8 Momentum relaxation times in diamond . . . . . . . . . . . . . . . . 31
2.9 Interpolated spin-flip 𝑒-ph matrix elements . . . . . . . . . . . . . . 32
2.10 Phonon dispersions and spin-flip 𝑒-ph matrix elements in silicon . . . 33
2.11 Phonon dispersions and spin-flip 𝑒-ph matrix elements in diamond . . 34
3.1 Feynman diagrams for spin decoherence . . . . . . . . . . . . . . . . 42
3.2 Spin relaxation times . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Microscopic spin decoherence . . . . . . . . . . . . . . . . . . . . . 45
4.1 Schematic of the dipole and quadrupole charge configurations . . . . 63
4.2 Mode-resolved 𝑒-ph coupling strength in silicon . . . . . . . . . . . 68
4.3 Mode-resolve 𝑒-ph coupling strength in tetragonal PbTiO3 . . . . . . 70
4.4 Scattering rate versus electron energy in silicon and PbTiO3 . . . . . 72
4.5 Electron mobility in silicon and tetragonal PbTiO3 . . . . . . . . . . 74
5.1 𝑒-ph matrix elements and phonon dispersions in CoO . . . . . . . . . 86
5.2 Contributions to 𝑒-ph coupling . . . . . . . . . . . . . . . . . . . . . 88
5.3 Kohn-Sham and Hubbard contribution to 𝑒-ph self-energy . . . . . . 89
5.4 Electron spectral function in CoO . . . . . . . . . . . . . . . . . . . 90
5.5 CoO phonon dispersion . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Workflow of the PERTURBO code . . . . . . . . . . . . . . . . . . 107
6.2 Absolute value of the gauge-invariant 𝑒-ph matrix elements . . . . . 111
6.3 The 𝑒-ph deformation potential in GaAs and monolayer MoS2 . . . . 115
6.4 Mean free paths, 𝑒-ph scattering rates, and relaxation times in GaAs 116
6.5 Crystal structure and mode-resolved 𝑒-ph scattering rates in naphthalene117
6.6 Electron mobility and Seebeck coefficient of GaAs and MoS2 . . . . 119

6.7
6.8

Hot carrier cooling in silicon . . . . . . . . . . . . . . . . . . . . . . 120
Performance scaling of PERTURBO . . . . . . . . . . . . . . . . . . 122

Chapter 1

INTRODUCTION
1.1

Overview

Decoherence is a major unresolved challenge in quantum technology as it limits the
manipulation of quantum information. Solid state devices based on the electron’s
spin, such as spin qubits and spintronic technologies, require long spin coherence
times for optimal performance. There are various sources of spin decoherence in
solids — noise, nuclear spins, etc. — but atomic thermal vibrations (phonons) are
the only mechanism that is always present and that sets an intrinsic limit to the
performance of any spin-based quantum technology.
Consequently, predicting spin decoherence due to phonons with a high accuracy
remains a high priority for basic and applied quantum science. Previous work has
identified two key mechanisms of phonon-induced spin decoherence in the presence
of spin-orbit coupling. In the Elliott-Yafet (EY) mechanism, spin decoherence
occurs as each 𝑒-ph collision has a finite probability to alter the spin direction. This
mechanism dominates decoherence in centrosymmetric materials. In the DyakonovPerel (DP) mechanism, spin decoherence occurs as a result of spin precessions
in-between electron scattering events. This mechanism dominates decoherence in
materials which do not have inversion symmetry. The interplay between these two
spin decoherence mechanisms is crucial in systems ranging from semiconductors
for spintronics to defects for quantum technologies.
Despite decades of progress, theory and computational methods for quantitative
predictions of spin decoherence remain an open problem. All existing approaches
require empirical fitting of the spin interactions to construct the model, and of the
dynamics to extract the spin relaxation times. Much research in this area has resorted
to the momentum relaxation dynamics to indirectly estimate spin relaxation, usually
by assuming a direct proportionality between the spin and momentum-scattering
interactions or relaxation times for the EY theory, and an inverse proportionality
between the two for the DP theory. A general scheme to compute spin-phonon
interactions and spin relaxation times with quantitative accuracy is still missing;
if developed, they could have significant impact and enable progress on many
unresolved questions in spin physics and devices. Accurately predicting EY and DP

spin decoherence can establish the intrinsic limit to the performance of spin-based
devices and provide a detailed understanding of spin interactions.
1.2

First-Principles Computational Methods

Understanding the dynamical processes involving electrons, spins, and lattice vibrations (phonons) in the solid state is the key first step to investigate spin decoherence.
Due to the increasing complexity of functional materials, there is a critical need
for computational tools that can take into account the atomic and electronic structure of materials and make quantitative predictions on their physical properties. A
generally applicable first-principles method to predict and microscopically understand spin-phonon interactions, and capture atomistic details such as the electronic
wave function, spin texture, phonon modes and their mode-dependent spin-flip interactions, is necessary. We review the first-principles methods that have became a
standard for calculating 𝑒-ph interactions in condensed matter.
Electrons, phonons, and 𝑒-ph interactions
Density functional theory (DFT) has become the mode widely accepted approach
for computational modeling of the electronic structure in materials. DFT places its
roots in the Hohenberg-Kohn theorem, which states that the total energy of a system
is a unique functional of the electron density, and that the ground state energy can be
obtained through a variational principle with respect to the electron density. Shortly
after its introduction, Kohn and Sham proposed an effective Hamiltonian consisted
of fictitious non-interacting particles to reproduce the original electron density of
the interacting system. The corresponding Kohn-Sham (KS) Hamiltonian consists
of the Kohn-Sham kinetic energy operator of the effective orbitals, the external
potential, the Hartree potential, and the exchange-correlation potential.
The formalism of DFT has been extensively expanded since its initial introduction.
It is implemented in numerous open-source codes such as Quantum ESPRESSO [1]
and Abinit [2]. Modern research employing DFT focuses on excited state and
microscopic electron interactions. For instance, the phonon dispersion is computed
with density functional perturbation theory (DFPT), which is a linear response
extension of DFT. In principle, the 𝑒-ph matrix elements can also be computed
with these methods and used directly for carrier dynamics calculations. However,
to converge transport and ultrafast dynamics, the 𝑒-ph matrix elements and the
scattering processes need to be computed on ultra-dense k- and q-point BZ grids
with roughly 100 × 100 × 100 or more points. Therefore, the computational cost is

prohibitive for direct DFPT calculations, and we resort to interpolation techniques to
obtain the 𝑒-ph matrix elements and other relevant quantities on fine grids, starting
from DFT and DFPT calculations on coarser BZ grids, typically of order 10×10×10.
Wannier interpolation of the electronic structure
We use Wannier interpolation to compute efficiently the electron energy and band
velocity on ultra-fine k-point grids [3]. We first perform DFT calculations on a
regular coarse grid with points k𝑐 , and obtain the electron energies 𝜀 𝑛k𝑐 and Bloch
wavefunctions |𝜓𝑛k𝑐 ⟩. We construct maximally localized Wannier functions |𝑛R𝑒 ⟩,
with index 𝑛 and centered in the cell at R𝑒 , from the Bloch wavefunctions using the
Wannier90 code [4, 5]:
|𝑛R𝑒 ⟩ =

1 ∑︁ −𝑖k𝑐 ·R𝑒
U𝑚𝑛 (k𝑐 ) |𝜓𝑚 k𝑐 ⟩,
𝑁𝑒

(1.1)

𝑚 k𝑐

where 𝑁 𝑒 is the number of k𝑐 -points in the coarse grid, and U (k𝑐 ) are the unitary
matrices transforming the Bloch wavefunctions to a Wannier gauge [6],
∑︁
|𝜓𝑛(𝑊)
(1.2)
U𝑚𝑛 (k𝑐 ) |𝜓𝑚 k𝑐 ⟩.

For entangled band structures, U (k𝑐 ) are not in general square matrices since
they are also used to extract a smooth subspace from the original DFT Bloch
eigenstates [7].
We compute the electron Hamiltonian in the Wannier function basis,
ˆ ′ R𝑒 ⟩
𝐻𝑛𝑛 ′ (R𝑒 ) = ⟨𝑛0| 𝐻|𝑛
1 ∑︁ −𝑖k𝑐 ·R𝑒 †
U (k𝑐 )𝐻 (k𝑐 )U (k𝑐 ) 𝑛𝑛 ′ ,
𝑁𝑒

(1.3)

k𝑐

where 𝐻 (k𝑐 ) is the Hamiltonian in the DFT Bloch eigenstate basis, 𝐻𝑛𝑚 (k𝑐 ) =
𝜀 𝑛k𝑐 𝛿𝑛𝑚 . The Hamiltonian in the Wannier basis, 𝐻𝑛𝑛 ′ (R𝑒 ), can be seen as an ab
initio tight-binding model, with hopping integrals from the Wannier orbital 𝑛′ in the
cell at R𝑒 to the Wannier orbital 𝑛 in the cell at the origin. Due to the localization
of the Wannier orbitals, the hopping integrals decay rapidly with |R𝑒 |, so a small
set of R𝑒 vectors is sufficient to represent the electronic structure of the system.
Starting from 𝐻𝑛𝑛 ′ (R𝑒 ), we obtain the band energy 𝜀 𝑛k and band velocity v𝑛k at any
desired k-point. We first compute the Hamiltonian matrix 𝐻 (𝑊) (k) in the basis of

Bloch sums of Wannier functions using an inverse discrete Fourier transform, and
then diagonalize it through a unitary rotation matrix 𝑈 (k) satisfying
∑︁
𝐻 (𝑊) (k) =
𝑒𝑖k·R𝑒 𝐻 (R𝑒 ) = 𝑈 (k)𝐻 (𝐻) (k)𝑈 † (k) ,
(1.4)
R𝑒
(𝐻)
where 𝐻𝑛𝑚
(k) = 𝜀 𝑛k 𝛿𝑛𝑚 , and 𝜀 𝑛k and 𝑈 (k) are the eigenvalues and eigenvectors
(𝑊)
of 𝐻 (k), respectively. One can also obtain the corresponding interpolated Bloch
eigenstates as
∑︁
∑︁
∑︁
|𝜓𝑛k ⟩ =
𝑈𝑚𝑛 (k) |𝜓𝑚(𝑊)
(k)
𝑒𝑖k·R𝑒 |𝑚R𝑒 ⟩.
(1.5)
𝑚𝑛

R𝑒

The band velocity in the Cartesian direction 𝛼 is computed as
ℏv𝑛𝛼k = [𝑈 † (k)𝐻𝛼(𝑊) (k)𝑈 (k)] 𝑛𝑛 ,

(1.6)

where 𝐻𝛼(𝑊) (k) is the k-derivative of 𝐻 (𝑊) (k) in the 𝛼-direction, evaluated analytically using
∑︁
𝐻𝛼(𝑊) (k) = 𝜕𝛼 𝐻 (𝑊) (k) =
𝑒𝑖k·R𝑒 𝐻 (R𝑒 ) · (𝑖𝑅𝑒𝛼 ) .
(1.7)
R𝑒

An appropriate extension of Eq. (1.6) is used for degenerate states [8].
Interpolation of the phonon dispersion
The lattice dynamical properties are first obtained using DFPT on a regular coarse
q𝑐 -point grid. Starting from the dynamical matrices 𝐷 (q𝑐 ), we compute the interatomic force constants (IFCs) 𝐷 (R 𝑝 ) (here, without the mass factor) through a
Fourier transform [9, 10],
𝐷 (R 𝑝 ) =

1 ∑︁ −𝑖q𝑐 ·R 𝑝
𝐷 (q𝑐 ),
𝑁𝑝 q

(1.8)

where 𝑁 𝑝 is the number of q𝑐 -points in the coarse grid. If the IFCs are short-ranged,
a small set of 𝐷 (R 𝑝 ), obtained from dynamical matrices on a coarse q𝑐 -point grid,
is sufficient to obtain the dynamical matrix at any desired q-point with an inverse
Fourier transform,
∑︁
𝐷 (q) =
𝑒𝑖q ·R 𝑝 𝐷 (R 𝑝 ).
(1.9)
R𝑝

We obtain the phonon frequencies 𝜔 𝜈q and displacement eigenvectors e𝜈q by diagonalizing 𝐷 (q).

Wannier interpolation of the 𝑒-ph matrix elements
The key quantities for 𝑒-ph scattering are the 𝑒-ph matrix elements 𝑔𝑚𝑛𝜈 (k, q).
They are given by
√︄
𝜅𝛼
ℏ ∑︁ e𝜈q
𝑔𝑚𝑛𝜈 (k, q) =
𝜓𝑚 k+q 𝜕q,𝜅𝛼𝑉 𝜓𝑛k ,
(1.10)
2𝜔 𝜈q 𝜅𝛼 𝑀𝜅
where |𝜓𝑛k ⟩ and |𝜓𝑚 k+q ⟩ are the wavefunctions of the initial and final Bloch states,
respectively, and 𝜕q,𝜅𝛼𝑉 is the perturbation potential due to lattice vibrations, computed as the variation of the Kohn-Sham potential 𝑉 with respect to the atomic
displacement of atom 𝜅 (with mass 𝑀𝜅 ) in the Cartesian direction 𝛼:
𝜕q,𝜅𝛼𝑉 =

∑︁
R𝑝

𝑒 𝑖 q ·R 𝑝

𝜕𝑉
𝜕𝑅 𝑝𝜅𝛼

(1.11)

We obtain this perturbation potential as a byproduct of the DFPT lattice dynamical
calculations at a negligible additional cost.
We compute the bra-ket in Eq. (1.10) directly, using the DFT Bloch states on a
coarse k𝑐 -point grid and the perturbation potentials on a coarse q𝑐 -point grid,
𝜅𝛼
(k𝑐 , q𝑐 ) = 𝜓𝑚 k𝑐 +q𝑐 𝜕q𝑐 ,𝜅𝛼𝑉 𝜓𝑛k𝑐 ,
𝑔˜ 𝑚𝑛

(1.12)

from which we obtain the 𝑒-ph matrix elements in the Wannier basis [11, 12],
𝑔˜𝑖𝜅𝛼
𝑗 R𝑒 , R 𝑝 , by combining Eq. (1.1) and the inverse transformation of Eq. (1.11):
𝑔˜𝑖𝜅𝛼

R𝑒 , R 𝑝

𝜕𝑉
= 𝑖0
𝑗R𝑒
𝜕𝑅 𝑝,𝜅𝛼
1 ∑︁ −𝑖(k𝑐 ·R𝑒 +q𝑐 ·R 𝑝 ) 𝜅𝛼,(𝑊)
(k𝑐 , q𝑐 ) ,
g̃𝑖 𝑗
𝑁𝑒 𝑁 𝑝

(1.13)

k 𝑐 ,q 𝑐

where
g̃ 𝜅𝛼,(𝑊) (k𝑐 , q𝑐 ) = U † (k𝑐 + q𝑐 ) g̃ 𝜅𝛼 (k𝑐 , q𝑐 ) U (k𝑐 )
are the matrix elements in the Wannier gauge. Similar to the electron Hamiltonian
in the Wannier basis, 𝑔˜𝑖𝜅𝛼
𝑗 R𝑒 , R 𝑝 can be seen as a hopping integral between two
localized Wannier functions, one at the origin and one at R𝑒 , due to a perturbation
caused by an atomic displacement at R 𝑝 . If the interactions are short-ranged in
real space, 𝑔˜ decays rapidly with |R𝑒 | and |R 𝑝 |, and computing it on a small set of
R𝑒 , R 𝑝 lattice vectors is sufficient to fully describe the coupling between electrons
and lattice vibrations.

The 𝑒-ph matrix elements at any desired pair of k- and q-points can be computed
efficiently using the inverse transformation in Eq. (1.13),
∑︁
𝜅𝛼
(k, q) =
𝑔˜ 𝑚𝑛
𝑈𝑚𝑖
(k + q)𝑈 𝑗𝑛 (k)
𝑖, 𝑗

∑︁

𝑒𝑖(k·R𝑒 +q ·R 𝑝 ) 𝑔˜𝑖𝜅𝛼
𝑗 R𝑒 , R 𝑝 ,

(1.14)

R𝑒 , R 𝑝

where 𝑈 (k) is the matrix used to interpolate the Bloch states in Eq. (1.5).
The main requirement of this interpolation approach is that the 𝑒-ph interactions
are short-ranged and the 𝑒-ph matrix elements in the local basis decay rapidly.
Therefore, the 𝑒-ph interpolation works equally well with localized orbitals other
than Wannier functions, as we have shown recently using atomic orbitals [13].
Polar corrections for phonons and 𝑒-ph interactions
The assumption that the IFCs and 𝑒-ph interactions are short-ranged does not hold
for polar semiconductors and insulators. In polar materials, the displacement of
ions with a non-zero Born effective charge creates dynamical dipoles, and the longwavelength longitudinal optical (LO) phonon mode induces a macroscopic electric
field [9]. The dipole-dipole interactions introduce long-range contributions to the
IFCs and dynamical matrices [14], resulting in the well-known LO-TO splitting in
the phonon dispersion at q → 0. For this reason, the dynamical matrix interpolation
scheme in Eqs. (1.8)-(1.9) cannot provide correct phonon dispersions at small q for
polar materials. To address this issue, a polar correction is typically used [10], in
which the dynamical matrix is separated into two contributions: a short-range part
that can be interpolated using the Fourier transformation in Eqs. (1.8)-(1.9), and a
long-range part evaluated directly using an analytical formula involving the Born
effective charges and the dielectric tensor [10].
Similar to the IFCs, the 𝑒-ph interactions possess both short-range and long-range
components. A multipole expansion of the e-ph matrix elements by Vogl has shown
that the long-range part consists of a dipole and a quadrupole contribution. The
field due to the dynamical dipoles introduces long-range 𝑒-ph contributions — in
particular, the Fröhlich interaction [15], a long-range coupling between electrons
and LO phonons. The strength of the Fröhlich 𝑒-ph interaction diverges as 1/𝑞
for q → 0 in bulk materials. As a result, the Wannier interpolation is impractical
and usually fails to correctly reproduce the DFPT 𝑒-ph matrix elements at small q.
Using a scheme analogous to the polar correction for phonon dispersion, one can

split the 𝑒-ph matrix elements into a long-range part due to the dipole field and a
short-range part [16, 17].
In this thesis, we first focus on a computational approach for the long-range 𝑒ph matrix elements by replacing the perturbation potential in Eq. (1.12) with the
potential of the dipole field,
−𝑖τ ·(q+G)
2 ∑︁
(q
G)
𝑒 𝜅
𝑖𝑒
𝜅𝛼,𝐿
(k, q) =
𝑔˜ 𝑚𝑛
𝜖0 Ω
(q + G) · ϵ · (q + G)
(1.15)
G≠−q
× 𝜓𝑚 k+q 𝑒𝑖(q+G)·r 𝜓𝑛k ,
where Z𝜅∗ and τ𝜅 are the Born effective charge and position of atom 𝜅 in the unit cell,
respectively, while Ω is the unit cell volume and ϵ the dielectric tensor. In practice,
the summation over G is performed using the Ewald method, by introducing a
decay factor 𝑒 −(q+G)·ϵ·(q+G)/4𝛬 with convergence parameter 𝛬, and multiplying
each term in the summation by this factor. It is convenient to evaluate the braket in Eq. (1.15) in the Wannier gauge, in which one can apply the smooth phase
approximation ⟨𝑢 𝑚 k+q |𝑢 𝑛k ⟩ (𝑊) = 𝛿𝑚𝑛 , where 𝑢 𝑛k is the periodic part of the Bloch
function. Combining Eq. (1.2) and the smooth phase approximation in the Wannier
gauge, we obtain
𝜓𝑚 k+q 𝑒𝑖(q+G)·r 𝜓𝑛k = [U (k + q)U † (k)] 𝑚𝑛 .
(1.16)
Using the analytical formula in Eq. (1.15), with the bra-ket computed using Eq. (1.16),
the long-range part of the 𝑒-ph matrix elements can be evaluated directly for any
desired values of k and q. Only the short-range part is computed using Wannier
interpolation, and the full 𝑒-ph matrix elements are then obtained by adding together
the short- and long-range contributions.
To extend the phonon and 𝑒-ph polar correction schemes to 2D materials, only small
changes to the long-range parts are needed, as discussed in detail in Refs. [18–20].
In particular, the 2D extension of the long-range 𝑒-ph matrix elements is obtained
by replacing in Eq. (1.15) the dielectric tensor ϵ with the effective screening ϵeff (|q|)
of the 2D system and the unit cell volume Ω with 2𝐴, where 𝐴 is the area of the 2D
unit cell.
Limitations of the current approach
Note that the current ab-initio approach is incomplete as it lacks the quadrupole
interaction, which is essential to accurately describe e-ph interactions in all materials — both polar and nonpolar — and is particularly important for piezoelectric

materials. As a result, the e-ph interactions cannot be described correctly from first
principles even in materials as simple as silicon, and currently e-ph calculations on
piezoelectric materials (such as wurtzite and titanate crystals) lead to large errors.
The current approach also remains an open challenge in correlated electron systems
(CES), where density functional theory fails to describe the ground state. As a
result, reliable 𝑒-ph calculations remain out of reach for broad classes of strongly
correlated materials such as high-temperature superconductors, Mott insulators,
transition metal oxides, f-electron systems, planetary materials, and multiferroics.
Widely used first-principles approaches to compute the ground state of CES include
Hubbard-corrected DFT (DFT+U), hybrid functionals, and dynamical mean-field
theory. Yet, calculations of 𝑒-ph interactions are currently not possible in any of
these methods. Developing accurate 𝑒-ph calculations in CES and understanding
their spin-dependent nature thus remains an important open challenge – if fulfilled,
it would advance investigations of transport, high-temperature superconductivity,
charge-density waves and metal-insulator transitions in broad classes of strongly
correlated materials relevant for quantum technology.
1.3

Many-Body Perturbation Theory

First-principles approaches combined with many-body perturbation theory are particularly promising to tackle the problem of spin decoherence. Linear response theory is a powerful framework that can broaden our understanding of the microscopic
correlation functions and the response of the system with respect to external perturbations [21–26]. Modern ab-initio microscopic electron interactions [11, 13, 27–31]
complement linear response theory, allowing precise predictions of material properties without resorting to empirical fitting parameters. We provide a brief summary
of the many-body techniques employed in this thesis.
Interacting Green’s functions
We consider an unperturbed Hamiltonian 𝐻0 diagonal in a Bloch basis, ⟨𝑛′k| 𝐻0 |𝑛k⟩ =
𝜀 𝑛k 𝛿𝑛𝑛 ′ . The interacting imaginary-time Green’s function G(𝑖𝜔𝑎 ) is written using
the Dyson equation [21]
G(𝑖𝜔𝑎 ) −1 = G (0) (𝑖𝜔𝑎 ) −1 − Σ(𝑖𝜔𝑎 ),

(1.17)

where 𝜔𝑎 is the fermionic Matsubara frequency of the electron, G (0) (𝑖𝜔𝑎 ) is the
non-interacting Green’s function, and Σ(𝑖𝜔𝑎 ) is the lowest order (Fan-Migdal) 𝑒-ph

(a)

(b)

(c)

Figure 1.1: (a) Bare bubble diagram without the vertex correction. (b) Bubble
diagram including the vertex correction. (c) Bethe-Salpeter equation for the vertex
corrections Λ from electron-phonon interactions within the ladder approximation.
The wavy line corresponds to the phonon propagator.
self-energy [21, 32, 33], whose band- and k-dependent expression is
Σ𝑛𝑛 ′k (𝑖𝜔𝑎 ) = −

∑︁
[𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q)] ∗ 𝑔𝑛𝑚𝜈 (k, q)D𝜈q (𝑖𝑞 𝑐 )G𝑚𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 ).
𝛽𝑁 𝑞 𝑉 𝑚𝑚 ′q 𝜈,𝑖𝑞

(1.18)
Above, 𝛽 = 1/𝑘 𝐵𝑇 at temperature 𝑇, 𝑁 𝑞 is the number of q-points, 𝑉 is the
volume of the unit cell, 𝑞 𝑐 is the bosonic Matsubara frequency of the phonon, and
D𝜈q (𝑖𝑞 𝑐 ) = 2𝜔q 𝜈 /((𝑖𝑞 𝑐 ) 2 − 𝜔2q 𝜈 ) is the non-interacting phonon Green’s function
for a phonon of mode index 𝜈 and wave-vector q.
The key quantity is the 𝑒-ph matirx elements, 𝑔𝑛𝑚𝜈 (k, q), which quantify the probability amplitude for an electron in a Bloch state |𝜓𝑛k ⟩, with band index 𝑛 and crystal
momentum k, to scatter into a final state |𝜓𝑚 k+q ⟩ by emitting or absorbing a phonon
with mode index 𝜈, wave-vector q, and energy ℏ𝜔 𝜈q [21, 32],
𝑔𝑛𝑚𝜈 (k, q) = ⟨𝜓𝑚 k+q | 𝑑q 𝜈𝑉ˆ |𝜓𝑛k ⟩,

(1.19)

where 𝑑q 𝜈𝑉ˆ is the 𝑒-ph perturbation due to the change of the potential acting on an
electron from a phonon with mode index 𝜈 and crystal momentum q.

10
Kubo formula and correlation function
ˆ with matrix elements in the direction 𝛼
One considers a complex vector operator 𝐴,
= ⟨𝑚k| 𝐴ˆ 𝛼 |𝑛k⟩.
given as 𝐴𝑛𝑚
Working in imaginary time coordinate and frequency, the retarded correlation function for the operator 𝐴ˆ can be obtained from the Kubo formula [21]
∫ 𝛽
𝜒𝛼𝛽 (p, 𝑖𝜈 𝑏 ) =
𝑑𝜏𝑒𝑖𝜈𝑏 𝜏 𝑇𝜏 𝐴ˆ 𝛼 (p, 𝜏) 𝐴ˆ 𝛽 (−p, 0) ,
(1.20)

where p is the wave-vector, 𝜈 𝑏 is the bosonic Matsubara frequency, 𝜏 is imaginary
time ranging from 0 to 𝛽 = 1/𝑘 𝐵𝑇 at temperature 𝑇, and 𝑇𝜏 is the imaginary time
ordering operator. Here we focus on the p → 0 limit, and thus drop p from the
equations. This correlation function can be expressed as a sum of bubble diagrams
𝑃 as
1 ∑︁
𝑃(𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ).
(1.21)
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =
𝛽 𝑖𝜔

For the simplest case, one considers the bare bubble diagram that includes electron
self-energy only in the electron propagator G as shown Fig. 1.1(a):
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =

1 ∑︁
Tr G(𝑖𝜔𝑎 ) 𝐴ˆ 𝛼 G(𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) 𝐴ˆ 𝛽 ,
𝛽𝑉 𝑖𝜔

(1.22)

where the trace is evaluated over the band and momentum indices. In this expression,
the operator 𝐴ˆ can be regarded as the bare vertex of the correlation function. For the
velocity operator, Eq. (1.22) leads to the well-known Drude conductivity [21, 24].
In this work, the corrections to the vertex originates from 𝑒-ph interactions, which
couple electronic states with different band and crystal momenta. Figure 1.1(b)
shows the correlation function including the vertex correction Λ,
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =

1 ∑︁
Tr G(𝑖𝜔𝑎 ) 𝐴ˆ 𝛼 G(𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) 𝐴ˆ 𝛽 Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) ,
𝛽𝑉 𝑖𝜔

(1.23)

where 𝐴ˆ 𝛽 Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 +𝑖𝜈 𝑏 ) is the phonon-dressed vertex for the operator 𝐴ˆ along the
Cartesian direction 𝛽. Note that the vertex correction Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 +𝑖𝜈 𝑏 ) is a complex
vector, and it contains information about the renormalized operator dynamics due
to the 𝑒-ph interactions. In the ladder approximation, the vertex correction Λ
satisfies a self-consistent Bethe-Salpeter equation (BSE), shown diagrammatically
in Fig. 1.1(c).

11
1.4

Thesis Outline

In this thesis, we focus on three objectives: (i) developing a many-body framework
for investigating phonon-induced spin decoherence, (ii) identifying and expanding
the limitations of the first-principles 𝑒-ph interactions relevant for spin decoherence,
and (iii) presenting an open-source software package for studying 𝑒-ph interactions
from first principles.
In Chapter 2, we focus on spin-phonon interaction and relaxation. Spin-phonon
interactions limit the performance of spin-based quantum technologies, but are
challenging to compute with quantitative accuracy. We demonstrate a precise firstprinciples method to compute spin-phonon interactions and spin relaxation times.
Our approach can predict spin relaxation times in silicon in excellent agreement
with experiment, without using empirical or fitting parameters. We also predict
intrinsic-limit spin relaxation times in diamond, a key material for spin-based quantum technologies. Our results further show that the widely used proportionality
between spin and momentum relaxation times is inaccurate, highlighting the need
for atomistic spin relaxation calculations.
In Chapter 3, we present a novel many-body first-principles approach for phononinduced spin decoherence. Electron spin decoherence from atomic vibrations
(phonons) limits the performance of spin-based quantum technologies but is challenging to model quantitatively. We present a many-body theory of phonon-induced
spin decoherence that can capture both spin relaxation and precession. Our approach
can predict with a high accuracy the intrinsic spin relaxation times in key materials
for quantum devices — Si, GaAs, and WSe2 — without using empirical or fitting
parameters. Using this formalism, we find a colossal phonon-induced renormalization of the spin dynamics, and discover that spin decoherence times in condensed
matter and the anomalous electron magnetic moment share a similar origin.
In Chapter 4, we develop a first-principles approach to model piezoelectric 𝑒-ph
interaction, a long-range scattering mechanism due to acoustic phonon in noncentrosymmetric polar materials, which has not been accurately described at present.
The accuracy of the approach is demonstrated by comparing with direct density
functional perturbation theory calculations. We apply our method to silicon as
a case of a nonpolar semiconductor and tetragonal PbTiO3 as a case of a polar
piezoelectric material. In both materials we find that the quadrupole term strongly
impacts the 𝑒-ph matrix elements.
In Chapter 5, we demonstrate how spin-dependent 𝑒-ph interaction in correlated

12
electron systems (CES) can be captured from first-principles calculations. Accurate
studies of electron-phonon (e-ph) interactions in CES remain an open challenge as
DFT often fails to describe their ground state. Here we show a broadly applicable and
affordable approach for quantitative studies of e-ph interactions in CES, using the
framework of Hubbard-corrected DFT. The accuracy of our approach is showcased
on a prototypical Mott insulator, cobalt oxide (CoO), by carrying out a detailed
investigation of its e-ph interactions and electron spectral functions. The new
method enables investigations of transport, polarons, superconductivity, chargedensity waves, and metal-insulator transitions in broad classes of strongly correlated
materials.
In Chapter 6, we develop PERTURBO, a software package for first-principles calculations of charge transport, spin dynamics, and ultrafast carrier dynamics in
materials. PERTURBO uses results from density functional theory and density
functional perturbation theory calculations as input, and employs Wannier interpolation to reduce the computational cost. It supports norm-conserving and ultrasoft
pseudopotentials, spin–orbit coupling, and polar electron–phonon interactions for
bulk and 2D materials. Hybrid MPI plus OpenMP parallelization is implemented
to enable efficient calculations on large systems (up to at least 50 atoms) using
high-performance computing.
In Chapter 7, we summarize the main advancements made in this thesis and outline
possible future research directions.

13
References
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Guido Fratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco
Mauri, Riccardo Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo
Paulatto, Carlo Sbraccia, Sandro Scandolo, Gabriele Sclauzero, Ari P. Seitsonen, Alexander Smogunov, Paolo Umari, and Renata M. Wentzcovitch.
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CO;2-2.

17
Chapter 2

SPIN-PHONON RELAXATION TIMES IN
CENTROSYMMETRIC MATERIALS FROM FIRST PRINCIPLES
2.1

Introduction

Accurately predicting spin-phonon relaxation processes remains an open problem,
particularly due to the challenge of quantifying spin-flip 𝑒-ph interactions [1]. Calculations of EY spin relaxation have mainly relied on empirical models [2, 3] and
symmetry analysis [4–7], yet these approaches are laborious even for simple materials and not geared toward quantitative predictions. Attempts have also been
made to study spin relaxation from first principles by assuming a direct proportionality between spin-flip and momentum-scattering 𝑒-ph interactions [8], or between
spin-flip and momentum relaxation times [9]. However, these assumptions hold
only for simple model potentials [10–12] as the spin-flip and momentum-scattering
processes can differ greatly depending on the electronic wave function, spin texture,
and phonon perturbation [7, 13].
Recently developed first-principles methods for computing 𝑒-ph interactions and
relaxation times [14] are promising for studying EY spin-phonon relaxation. Their
typical workflow [15] involves density functional theory (DFT) calculations of the
ground state and electronic band structure, combined with density functional perturbation theory (DFPT) [16] to compute the phonon dispersions and 𝑒-ph perturbation
potentials, followed by interpolation of the 𝑒-ph coupling matrix elements to fine
Brillouin zone (BZ) grids. However, this workflow cannot be applied as is to investigate spin-flip 𝑒-ph interactions because the spin information is lost when one
computes the e-ph matrix elements. For example, the electronic states in centrosymmetric crystals are at least two-fold degenerate, and their spin points in an arbitrary
direction due to the freedom in describing the degenerate subspace. Computing
spin-phonon interactions ab initio, especially in the presence of spin-orbit coupling
(SOC) and spinor wave functions, remains an open challenge.
Here we present a first-principles method for computing the spin-flip 𝑒-ph coupling
matrix elements and the 𝑇1 spin-phonon relaxation times (SRTs). Our approach
assumes no relationship between the matrix elements for spin-flip and momentum scattering, and treats spinor wave functions and SOC through fully-relativistic

18
DFT and DFPT calculations [17]. These advances enable accurate calculations of
SRTs and shed light on microscopic details of spin-phonon interactions. We apply
our method to investigate SRTs in two key materials for spintronic and quantum
technologies, silicon and diamond. Our predicted SRTs in silicon are in excellent
agreement with experiment between 50−300 K, while in diamond, where SRT measurements are missing, we predict intrinsic-limit SRTs of roughly 0.5 ms at 77 K and
2 𝜇s at 300 K. In both materials, we find that spin-flip and momentum-scattering
𝑒-ph interactions differ widely and are not directly proportional, and the temperature
dependence of the spin-flip and momentum relaxation times also differ greatly. Our
work demonstrates a precise first-principles approach for computing SRTs, highlighting the limits of widely used simplified analyses and opening new avenues for
microscopic understanding of spin dynamics.
2.2

Theoretical Framework

Spin-flip interactions
In centrosymmetric materials, the Bloch states with band index 𝑛 and crystal momentum k can be decomposed into effective up and down spin states, denoted as ⇑
and ⇓, which diagonalize the spin operator 𝑆ˆ𝛼 (where 𝛼 is the Cartesian direction
of the spin quantization axis) in the Kramers degenerate subspace [1, 18, 19]:
⟨𝑛k⇑| 𝑆ˆ𝛼 |𝑛k⇑⟩ = − ⟨𝑛k⇓| 𝑆ˆ𝛼 |𝑛k⇓⟩ ,
⟨𝑛k⇓| 𝑆ˆ𝛼 |𝑛k⇑⟩ = 0.

(2.1)

The key ingredients for computing the SRTs are the spin-flip 𝑒-ph matrix elements [1],
flip
𝑔𝑚𝑛𝜈 (k, q) = ⟨𝑚k + q⇓|Δ𝑉ˆ𝜈q |𝑛k⇑⟩ ,
(2.2)
which quantify the probability amplitude to scatter from an initial Bloch state |𝑛k⇑⟩
to a final state |𝑚k + q⇓⟩ with opposite effective spin, by emitting or absorbing
a phonon with mode index 𝜈 and wave vector q due to the Kohn-Sham potential
perturbation Δ𝑉ˆ𝜈q [14], which is a 2 × 2 matrix in spin space in the presence of SOC.
To compute the SRTs, we obtain the effective spin states and from them the spin-flip
flip
𝑒-ph matrix elements 𝑔𝑚𝑛𝜈 (k, q) on fine BZ grids. We calculate the effective spin
states from the spin matrix 𝑆(k), which provides a matrix representation of the spin
operator 𝑆ˆ𝛼 in the wave function basis [20], 𝑆 𝑚𝑠 ′,𝑛𝑠 (k) = ⟨𝑚k𝑠′ | 𝑆ˆ𝛼 |𝑛k𝑠⟩, where 𝑠
and 𝑠′ denote the spin. We diagonalize separately each degenerate subspace in the
spin matrix at each k-point, obtaining the unitary matrices 𝐷 k that make each of the

19
subspaces in 𝐷 k 𝑆(k)𝐷 †k diagonal, with eigenvalues equal to the effective spin 1.
flip
The spin-flip 𝑒-ph matrix elements, 𝑔𝑚𝑛𝜈 (k, q), are then computed using Eq. (2.2)
for all pairs of states with opposite effective spin.
Interpolation
Since DFPT calculations of Δ𝑉ˆ𝜈q on the fine BZ grids needed to converge the SRTs
are prohibitively expensive, we interpolate the spin-flip 𝑒-ph matrix elements and
flip
spin matrices using Wannier functions [21–23]. To obtain 𝑔𝑚𝑛𝜈 (k′, q′) at a desired
pair of k′ and q′ points in the BZ, we first apply the usual Wannier interpolation
𝑠𝑠 ′ (k′, q ′) between states
workflow [15, 24] to obtain the 𝑒-ph matrix elements 𝑔𝑚𝑛𝜈
𝜎𝜎 ′ (k′, q ′) coupling states
with arbitrary spins 𝑠 and 𝑠′. The 𝑒-ph matrix elements 𝑔𝑚𝑛𝜈
with effective spins 𝜎 and 𝜎′ are then computed using the unitary matrix 𝐷 k ′ (the
latter is obtained from Wannier interpolation of the spin matrix [20]):
h ′
ih
𝜎𝜎 ′ ′ ′
𝑠𝑠
′ ′
𝑔𝑚𝑛𝜈 (k , q ) = 𝐷 k ′+q ′
𝑔𝑚𝑛𝜈 (k , q ) 𝐷 k ′ ′ ′ .
(2.3)
𝑛𝑠 ,𝑛𝜎

𝑚𝜎,𝑚𝑠

The spin-flip 𝑒-ph matrix elements are finally computed between all pairs of electronic states with opposite sign of the effective spin. Our interpolation scheme can
accurately reproduce spin-flip 𝑒-ph matrix elements obtained by combining effective spin states with perturbation potentials computed directly with DFPT (see the
Supplemental Material), thus enabling precise calculations of SRTs.
Spin relaxation times
flip

The band- and k-dependent spin-flip 𝑒-ph relaxation times, 𝜏𝑛k , are computed using
lowest-order perturbation theory [1],

flip

𝜏𝑛k

4𝜋 ∑︁ flip
𝑔𝑚𝑛𝜈 (k, q)
ℏ 𝑚𝜈q

(2.4)

[(𝑁 𝜈q + 1 − 𝑓𝑚 kq )𝛿(𝜀 𝑛k − 𝜀 𝑚 kq − ℏ𝜔 𝜈q )
+ (𝑁 𝜈q + 𝑓𝑚 kq )𝛿(𝜀 𝑛k − 𝜀 𝑚 kq + ℏ𝜔 𝜈q )],

where 𝜀 𝑛k and ℏ𝜔 𝜈q are the electron and phonon energies, respectively, and 𝑓𝑛k and
𝑁 𝜈q the corresponding temperature-dependent occupations.
flip

Converging the band- and k-dependent spin-flip 𝑒-ph relaxation times, 𝜏𝑛k , is of
paramount importance for precise predictions of 𝑇1 spin-phonon relaxation times.
1 When only the two-fold degeneracy due to time-reversal plus inversion symmetry is present,

the diagonal elements of 𝑆(k) naturally determine the effective spin value. For states with additional
degeneracies, 𝐷 k diagonalizes the degenerate subspace, giving multiple pairs of states with opposite
effective spin.

20
flip

Computing 𝜏𝑛k involves a sum over the q-point grid, which is typically performed by
flip
random sampling of the BZ [25]. However, the spin-flip matrix elements 𝑔𝑚𝑛𝜈 (k, q)
entering the summation vary by several orders of magnitude throughout the BZ, so
flip
converging 𝜏𝑛k is challenging.
Here we develop and employ an importance sampling approach for efficiently converging the BZ q-point summation in Eq. (2.4). We first sample on a regular q-point
BZ grid the quantity
𝑃𝑛k (q) =

∑︁

flip

𝑔𝑚𝑛𝜈 (k, q)

𝑚𝜈

[(𝑁 𝜈q + 1 − 𝑓𝑚 kq )𝛿(𝜀 𝑛k − 𝜀 𝑚 kq − ℏ𝜔 𝜈q )

(2.5)

+ (𝑁 𝜈q + 𝑓𝑚 kq )𝛿(𝜀 𝑛k − 𝜀 𝑚 kq + ℏ𝜔 𝜈q )].
From 𝑃𝑛k (q), we construct a three-dimensional probability density function, 𝑃˜𝑛k (q),
by nearest neighbor interpolation, and then perform an importance sampling integraflip
tion for 𝜏𝑛k by drawing samples from the probability density function 𝑃˜𝑛k (q). The
convergence rate of the importance sampling approach is orders of magnitude faster
than random sampling (see the Supplemental Material for the importance sampling
approach used in Eq. (2.4), the states chosen for the comparison in Fig. 2.2, additional
comparison of spin-flip and momentum-scattering matrix elements, momentumscattering processes in diamond, and convergence of the interpolation scheme with
respect to the coarse grid size). The considerable time-saving afforded by our
importance sampling method allows us to fully converge the SRTs.
The temperature-dependent SRT, 𝜏𝑠 (𝑇), is the main physical observable computed in
this work. It is obtained as an ensemble average of the spin-flip relaxation times [1]
by tetrahedron integration [31]:
−1
∑︁ 1 𝑑𝑓
𝑛k
𝑑k ª
+ −1 ©
flip
𝑑𝐸
𝑛k 𝜏𝑛k
= ∑︁
𝜏𝑠 (𝑇) = flip
® .
𝑑𝑓𝑛k
𝜏𝑛k 𝑇
𝑑k ®
𝑑𝐸
« 𝑛k

(2.6)

Numerical methods
We apply our approach to investigate spin relaxation in silicon and diamond. We obtain their ground state and band structure using DFT with a plane-wave basis with the
Quantum ESPRESSO code [32]. Briefly, we use relaxed lattice constants of 5.43 Å
for silicon and 3.56 Å for diamond, together with a kinetic energy cutoff of 60 Ry

21
1 0 4

S p in r e la x a tio n tim e ( n s )

(a )

(b )
1 0 6

S ilic o n
~ T

1 0

~ T

D ia m o n d

-2

-3

1 0 5
1 0 2

1 0 1

~ T

1 0 4

T h is w o r k
A p p e lb a u m e t a l.
L e p in e
L a n c a s te r e t a l.

-5 .5

1 0 3

1 0 0

S c a tte r in g r a te ( n s -1 )

1 0 -3
1 0

-1

1 0

-2

1 0

-3

o ta l

In tra
o c
g p r

v a ll

e y

ƒp r

o c e

s s

T o ta l
1 0 -5
In tr a v a lle y

e s s

ƒp ro c e s s
g p ro c e s s

1 0 -7

1 0 -4
5 0

1 0 0

2 0 0

T e m p e ra tu re (K )

3 0 0

4 0 0

5 0

1 0 0

2 0 0

3 0 0

4 0 0

T e m p e ra tu re (K )

Figure 2.1: Computed spin-phonon relaxation times as a function of temperature in
(a) silicon and (b) diamond. The experimental data in (a) are taken from Refs. [26–
30]. The lower panels show the process-resolved spin-flip 𝑒-ph scattering rates,
defined as the inverse of 𝜏𝑠 . Shown are the contributions from intravalley processes
(blue line), 𝑓 processes (red line) and 𝑔 processes (green line), which add up to the
total (gray line). The inset in (a) is a schematic of the intravalley and intervalley
processes.

for silicon and 120 Ry for diamond. We employ the PBEsol exchange-correlation
functional [33] and fully-relativistic norm-conserving pseudopotentials [17] from
Pseudo Dojo [34], which correctly include the SOC. We use DFPT [16] to compute
the phonon dispersions and the perturbation potential, Δ𝑉ˆ𝜈q in Eq. (2.2), on coarse
q-point grids; our in-house developed perturbo code (see Chapter 6) is employed
to compute the spin-dependent 𝑒-ph matrix elements on coarse BZ grids 2. The
DFPT calculations are done only in the irreducible q-point grid, following which
we extend the coarse-grid 𝑒-ph matrix elements to the full q-point grid in perturbo
by rotating the spinor wave functions with 𝑆𝑈 (2) matrices. The Wannier functions
and spin matrices are computed with the Wannier90 code [20] and employed in
perturbo to interpolate the spin-flip 𝑒-ph matrix elements on fine BZ grids with
up to 2003 k-points to converge the SRTs. The spin quantization axis is chosen
2 The DFPT calculations are carried out on an 8 × 8 × 8 q-point grid in diamond and a 10 × 10 × 10

q-point grid in silicon. The spin-flip 𝑒-ph matrix elements are computed on 16 × 16 × 16 k-point
and 8 × 8 × 8 q-point grids in diamond and 10 × 10 × 10 k-point and q-point grids in silicon.

22
as the [001] direction 3. We employ a non-degenerate electron concentration of
7.4 × 1014 cm−3 for silicon, which is identical to the experimental value in Ref. [29],
and 1.0 × 1017 cm−3 for diamond; in each case, the Fermi energy is computed from
the carrier concentration. The carrier concentration dependence of the SRTs is
negligible in this non-degenerate regime.
2.3

Results

Temperature-dependent spin relaxation times
Figure 2.1(a) shows our calculated SRT as a function of temperature in silicon,
which is in excellent agreement with experiments [26–30] (see also Ref. [35]) at
all temperatures between 50−300 K. For example, our calculated SRT at roomtemperature is 4.9 ns, versus a 6.0 ns value measured by Lancaster et al. 4. The
SRT in silicon exhibits an approximate 𝑇 −3 temperature dependence; to explain its
origin, we analyze in Fig. 2.1(a) the contributions from the three valley-dependent
scattering processes, including the intravalley and so-called 𝑔 and 𝑓 intervalley
processes, which correspond to scattering between valleys along the same direction
(𝑔 processes) or along different directions ( 𝑓 processes). We find that the SRTs
are comparable in magnitude for the three processes at all temperatures. The intravalley processes govern spin relaxation below 60 K, while 𝑓 intervalley scattering
dominates at higher temperatures.
In the conventional theory of spin relaxation, the power-law temperature behavior
of the SRT is typically attributed to a specific physical origin. For example, Yafet’s
prediction of a 𝑇 −2.5 temperature trend for the SRTs in silicon [1] took into account
only acoustic phonons and intravalley processes. In our quantitative approach, all
phonon modes and valley processes are taken into account on the same footing
and enter the scattering rate in Eq. (2.4). Each phonon mode has its own energy
dispersion and population factor, and the spin-flip coupling strength depends on the
electronic states and phonon modes considered in the scattering process. As a result,
the temperature trend emerges not from a unique origin, but due to a combination of
factors due to all electronic-state and phonon-mode dependent quantities in Eq. (2.4).
Therefore one cannot attribute a single physical origin to the approximate 𝑇 −3 power
3 Our computed SRTs are nearly independent of the choice of the spin quantization axis, which
changes the SRTs by only 2−5% in our temperature range of interest, due to a symmetry breaking
introduced by the Wannier interpolation.
4 We have verified that the results are nearly unchanged when using a different exchangecorrelation functional. Using the same settings, the calculated SRT at 300 K is 4.8 ns with PBE and
4.5 ns with LDA

23
law dependence of the SRTs.
Due to its weak SOC and correspondingly long SRT, diamond is a promising material
for spintronics and spin-based quantum technologies. However, SRT measurements
have not yet been reported in diamond due to challenges related to spin injection [36].
Figure 2.1(b) shows our computed SRT in diamond as a function of temperature.
We find SRTs of 540 𝜇s at 77 K and 2.3 𝜇s at 300 K; these values set an intrinsic
limit due to phonons to the SRTs in diamond. The SRT exhibits a 𝑇 −2 temperature
dependence below ∼170 K and a stronger 𝑇 −5.5 trend above 170 K. This trend is in
contrast with a previous prediction [8] of a 𝑇 −5 temperature dependence throughout
the entire temperature range and of an order-of-magnitude smaller SRT of 180 ns
at room temperature. Ref. [8] assumed a direct proportionality between the spinflip and momentum-scattering 𝑒-ph matrix elements, but, as we show below, this
assumption is in general incorrect and can lead to inaccurate phonon contributions
to the SRT. We analyze the valley scattering processes in diamond in Fig. 2.1(b),
and find that the intravalley processes dominate below 170 K, while the intervalley
𝑓 processes dominate above 170 K.
Spin-flip versus momentum scattering
Our quantitative approach reveals stark differences between the spin-flip and the
momentum-scattering interactions. Figure 2.2 compares the spin-flip coupling maflip
trix elements, 𝑔𝜈 (q) , with the spin-flip plus spin-conserving (i.e., momentumscattering) 𝑒-ph matrix elements, 𝑔𝜈tot (q) , and resolves their ratio for different
phonon modes. Depending on the phonon branch, we find that the spin-flip and
momentum matrix elements can differ by several orders of magnitude, as we find for
the longitudinal acoustic (LA) and longitudinal optical (LO) branches along Γ−X
and for the LO and for specific transverse optical (TO-1) and transverse acoustic
(TA-2) branches along X−K−Γ. For other phonon modes and BZ directions, the two
quantities exhibit smaller, yet quantitatively important, differences. Only in specific
cases are the spin-flip and momentum-scattering interactions nearly identical, as we
find for the TO-2, TA-1 and LA branches along X−K−Γ. These trends are common
to silicon and diamond. Analogous results are found when analyzing various initial
and final electronic states.
Lastly, we compare the spin-phonon and momentum relaxation times. The momentum relaxation time 𝜏𝑝 is defined as the usual (spin-independent) 𝑒-ph relaxation
time [14], thermally averaged using Eq. (2.6) to make the comparison meaningful.

P h o n o n e n e rg y (m e V )

S ilic o n

L O

6 0

T O -1
L O

4 0

0 .8
0 .6

L A

0 .4

T A -2

2 0

0 .2

T A -1

D ia m o n d
P h o n o n e n e rg y (m e V )

| g flip / g to t|

T O -2

L O

T O -2

1 5 0

| g flip / g to t|

T O -1

0 .8

L A
0 .6

1 0 0
L A

0 .4
0 .2

T A -2

5 0

T A -1

Figure 2.2: Phonon dispersions in silicon and diamond, overlaid with a color map of
flip
the ratio 𝑔𝜈 (q)/𝑔𝜈tot (q) between the spin-flip and the momentum-scattering 𝑒-ph
matrix elements. The two matrix elements differ by orders of magnitude for the
branches shown in red. The data shown are the square root of the gauge-invariant
trace of |𝑔| 2 for a low-energy spin-degenerate conduction band. The initial electron
momentum is set to the Γ point and we plot the ratio for phonon wave vectors q
along a high-symmetry BZ line.

The conventional wisdom is that spin and momentum relaxation times are directly
proportional [10, 18], an assumption that has been widely used to analyze spin relaxation mechanisms in experimental data [37–42]. Figure 2.3 shows the temperature
dependent spin and momentum relaxation times in silicon and diamond. In silicon,

25
the SRT follows a 𝑇 −3 temperature dependence, whereas the momentum relaxation
time follows a 𝑇 −2 trend. In diamond, the SRT makes a sharp transition from a
𝑇 −2 trend at low temperature to a stronger 𝑇 −5.5 trend above 170 K. In contrast,
the momentum relaxation time exhibits a much weaker temperature dependence,
roughly 𝑇 −1.5 at low temperature and 𝑇 −2.5 near room temperature.
1 0 5

1 0 4

~ T

1 0 3

-3

1 0 1

1 0 2

S p in
M o m e n tu m
1 0 0

5 0

1 0 0

2 0 0

1 0 5

~ T

3 0 0 4 0 0

T e m p e ra tu re (K )

1 0 4

-1 .5

1 0 4

~ T

-2 .5

5 0

1 0 0

2 0 0

1 0 3

-5 .5

S p in
M o m e n tu m

1 0 3
1 0 1

-2

r e la x a tio n tim e ( fs )

1 0

-2

r e la x a tio n tim e ( fs )

~ T

1 0 5

M o m e n tu m

1 0 3

D ia m o n d

1 0 6

S p in r e la x a tio n tim e ( n s )

S ilic o n

M o m e n tu m

S p in r e la x a tio n tim e ( n s )

1 0 4

1 0 2
3 0 0 4 0 0

T e m p e ra tu re (K )

Figure 2.3: Comparison between the temperature dependence of the SRT (gray
squares) and the momentum relaxation time (red circles) in silicon and diamond.
The labels give the exponent 𝑛 of the SRT temperature dependence, 𝑇 −𝑛 , separately
for each of the spin and momentum relaxation times. Note that the SRTs are in ns
units, and the momentum relaxation times in fs units.
There is no discernible direct proportionality between the spin and momentum
relaxation times — rather, they both exhibit an approximate 𝑇 −𝑛 temperature dependence, but with different values of the exponent 𝑛 (see Fig. 2.3). These differences
originate from the different coupling strengths and phonon mode contributions, as
we illustrate in Fig. 2.2. For example, we find that for momentum scattering in
diamond the intravalley processes dominate over the entire temperature range up to
400 K, as opposed to just below 170 K as we show above for spin relaxation (see
the Supplemental Material).
Simple formulas such as the Elliott approximation [18], 𝜏𝑠 = 𝜏𝑝 /4 𝑏 2 𝑇 , where
𝑏 2 is the spin-mixing parameter [12, 43], also fail in both materials, as we show

26
in Fig. 2.4. For silicon, the average spin-mixing parameter 𝑏 2 changes only by a
small factor of 2 throughout the entire temperature range, whereas for diamond the
change is less than 10 %. As a result, the temperature dependence of the SRTs from
the Elliott approximation is weaker than the temperature dependence of the SRTs
from first principles, and the SRTs computed with the Elliott approximation exhibit
a trend similar to the momentum-relaxation times. We conclude that a reliable
analysis of SRTs needs atomistic calculations that take into account the different
nature of the spin-phonon and momentum-scattering 𝑒-ph interactions, by using
accurate spin-flip 𝑒-ph matrix elements as we show in this work.

S ilic o n

1 0 3

1 0 2

1 0 1

T h is w o r k
E llio tt a p p r o x .
1 0 0

5 0

1 0 0

D ia m o n d

1 0 6

S p in r e la x a tio n tim e ( n s )

1 0 4

2 0 0

1 0 5

1 0 4

T h is w o r k
E llio tt a p p r o x .

1 0 3
3 0 0 4 0 0

T e m p e ra tu re (K )

5 0

1 0 0

2 0 0

3 0 0 4 0 0

T e m p e ra tu re (K )

Figure 2.4: Comparison between the temperature dependence of the SRT from first
principles (gray squares) and the Elliott approximation (blue circles) in silicon and
diamond.

2.4

Discussion

Since SRT calculations involve a subtle interplay between spin-flip 𝑒-ph matrix
elements and phonons and electronic states, the relative magnitude of the spinphonon interactions for different phonon modes is of paramount importance for
accurate predictions. Our results show that the widely used proportionality between
spin and momentum relaxation times can be inaccurate, highlighting the need for
atomistic details such as the electronic wave function, spin texture, phonon modes,

27
and their mode-dependent spin-flip interactions. When these microscopic details
are captured, as we have shown above, one can predict the SRTs within ∼10−20%
of experiment over a wide temperature range, and predict which phonon modes
govern spin relaxation. While computing 𝑒-ph interactions and carrier relaxation has
become a main effort in first-principles calculations [25, 44–47], SRT calculations
are still in their infancy, and more work is needed to expand their scope beyond the
EY mechanism discussed here.
2.5

Conclusion

In summary, we have developed a quantitatively accurate approach for computing
spin-flip 𝑒-ph interactions and SRTs due to the EY mechanism. Our calculations
can predict accurately (within 10-20 of experiment) the measured spin relaxation
times in silicon between 50-300 K, without using any empirical or fitting parameter.
In diamond, where spin relaxation times have not yet been measured in spite of its
wide use in spin-based qubits, we predict intrinsic-limit spin relaxation times over
a wide temperature range, and reveal their microscopic origin. Our work further
shows that the spin- and momentum-relaxation mechanisms are governed by distinct microscopic processes. We demonstrate that the widely used proportionality
between spin and momentum relaxation times is inaccurate, highlighting the limits
of simplified models and the need for atomistic spin relaxation calculations. Our
work enables first-principles modeling of spin-phonon dynamics in broad classes of
materials. It can advance microscopic understanding of spin dynamics in semiconductors, localized spins, ions, or quantum dots and other systems of fundamental
and technological relevance.
2.6

Supplemental Material

Importance sampling approach for computing spin-flip relaxation times
flip

Figure 2.5 compares the convergence of 𝜏𝑛k with a random sampling approach and
with our importance sampling method. The convergence rate of the importance
sampling approach is orders of magnitude faster than random sampling. For example, the required number of q-points to reach a 1% error for the importance
sampling method is 30, 000, versus a much larger value of 65 million points for the
random sampling method. The considerable time saving afforded by our importance
sampling method allows us to fully converge the spin relaxation times.

28
1 .0 5

τn f l i k p / τn f l i k p *

R a n d o m S a m p lin g
Im p o r ta n c e S a m p lin g

1 .0 0

0 .9 5
2 x 1 0 7

4 x 1 0 7

6 x 1 0 7

8 x 1 0 7

1 x 1 0 8

N u m b e r o f q - p o in ts u s e d

Figure 2.5: Comparison between the convergence of the random sampling method
(orange line) and the importance sampling method (blue line). The initial electron
band 𝑛 and momentum k are set to the conduction band minimum, and only the
𝑓 process is considered to illustrate the convergence trend. Shown is the ratio
flip
flip ∗
between the computed 𝜏𝑛k and the converged value 𝜏𝑛k , which is computed with a
1000 × 1000 × 1000 q-point BZ grid. The 𝑃k (q) distribution used in the importance
sampling method is computed using a 100 × 100 × 100 q-point grid.

E le c tr o n e n e r g y ( e V )

S ilic o n

E le c tr o n e n e r g y ( e V )

D ia m o n d

Figure 2.6: The low-energy spin-degenerate conduction band in silicon and diamond
chosen to compute the ratio between the spin-flip and the momentum-scattering 𝑒-ph
matrix elements in Fig. 2.2 of the main text. The blue circle is the initial electronic
state.

P h o n o n e n e rg y (m e V )

S ilic o n
6 0

L O

| g flip / g to t|

T O

0 .8

4 0

0 .6
0 .4

L A

0 .2

2 0

T A

P h o n o n e n e rg y (m e V )

D ia m o n d

L O

| g flip / g to t|

1 5 0

T O

0 .8

L A

1 0 0

0 .6

T A

0 .4
0 .2

5 0

Figure 2.7: Phonon dispersions in silicon and diamond, overlaid with a color map
flip
of the ratio 𝑔𝜈 (q)/𝑔𝜈tot (q) between the spin-flip and the total (i.e. momentumscattering) 𝑒-ph matrix elements; the two matrix elements differ by orders of magnitude for the branches shown in red. The data shown are the square root of the
gauge-invariant trace of |𝑔| 2 over a low-energy spin-degenerate conduction band
shown in Fig. 2.6. The initial electron momentum is set to the X point, which is
different from the Γ point used in Fig. 2.2 of the main text, and we plot the ratio for
phonon wave vectors q along Γ−X.

r e la x a tio n tim e ( fs )

1 0 4

S c a tte r in g r a te ( fs -1 ) M o m e n tu m

1 0 3

D ia m o n d

1 0 2

T o ta l

1 0 -3
1 0

-4

1 0

-5

1 0

-6

In tr a v a lle y
f p ro c e s s
g p ro c e s s

1 0 -7

5 0

1 0 0

2 0 0

3 0 0

4 0 0

T e m p e ra tu re (K )
Figure 2.8: Computed momentum relaxation times as a function of temperature
in diamond. The lower panel shows the process-resolved momentum-scattering
𝑒-ph scattering rates, defined as the inverse of the momentum relaxation time 𝜏𝑝
(see the main text). Shown are the contributions from intravalley processes (blue
line), 𝑓 processes (red line), and 𝑔 processes (green line), which add up to the
total (gray line). It is seen that for momentum scattering in diamond, intravalley
scattering dominates over the entire temperature range, while for spin relaxation the
intravalley processes are dominant only below 170 K (see the main text).

S ilic o n

1 2 0

D F T + D F P T
2 ×2 ×2
4 ×4 ×4
8 ×8 ×8
1 0 ×1 0 ×1 0

| g flip ( q ) | ( m e V )

1 0 0

8 0

6 0

4 0

2 0

flip

Figure 2.9: Comparison between the spin-flip 𝑒-ph matrix elements 𝑔𝜈 (q) computed with DFPT (gray squares) and those obtained from our Wannier interpolation
scheme discussed in the main text. For the latter, we show results obtained with
coarse q-point grids of 2 × 2 × 2 (purple dots), 4 × 4 × 4 (blue dots), 8 × 8 × 8 (green
dots), and 10 × 10 × 10 (red dots). It is seen that 8 × 8 × 8 or finer q-point grids,
as we used in our work, are sufficient for accurately interpolating the DFPT 𝑒-ph
matrix elements. The data shown are for silicon, and we plot the square root of the
gauge-invariant trace of |𝑔| 2 over the lowest spin-degenerate conduction band. The
initial electron momentum is set to the Γ point and we plot the matrix elements for
phonon wave vectors q along a high-symmetry BZ line.

S ilic o n
P h o n o n e n e rg y (m e V )

T o ta l

L O

6 0

| g to t|

T O -2

1 8 0

T O -1
L O

1 5 0
1 2 0

4 0

L A

9 0

L A

6 0

T A -2

3 0

2 0

T A -1

P h o n o n e n e rg y (m e V )

S p in - f lip

| g flip |

6 0

1 8 0
1 5 0
1 2 0

4 0

9 0
6 0
3 0

2 0

Figure 2.10: Phonon dispersions in silicon, overlaid with a color map of the total
flip
(i.e. momentum-scattering) 𝑒-ph matrix elements 𝑔𝜈 (q) (upper panel) and the
spin-flip 𝑒-ph matrix elements 𝑔𝜈tot (q) (lower panel). The data shown are the square
root of the gauge-invariant trace of |𝑔| 2 for a low-energy spin-degenerate conduction
band (see Fig. 2.6). The initial electron momentum is set to the Γ point and we plot
the values for phonon wave vectors q along a high-symmetry BZ line.

D ia m o n d
P h o n o n e n e rg y (m e V )

T o ta l

L O

T O -2

1 5 0

| g to t|

T O -1

8 0 0
6 0 0

L A

1 0 0

4 0 0

L A

2 0 0

T A -2

5 0

T A -1

P h o n o n e n e rg y (m e V )

S p in - f lip

| g flip |

1 5 0

8 0 0
6 0 0

1 0 0

4 0 0
2 0 0

5 0

Figure 2.11: Phonon dispersions in diamond, overlaid with a color map of the total
flip
(i.e. momentum-scattering) 𝑒-ph matrix elements 𝑔𝜈 (q) (upper panel) and the
spin-flip 𝑒-ph matrix elements 𝑔𝜈tot (q) (lower panel). The data shown are the square
root of the gauge-invariant trace of |𝑔| 2 for a low-energy spin-degenerate conduction
band (see Fig. 2.6). The initial electron momentum is set to the Γ point and we plot
the values for phonon wave vectors q along a high-symmetry BZ line.

35
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40
Chapter 3

PREDICTING PHONON-INDUCED SPIN DECOHERENCE AND
COLOSSAL SPIN RENORMALIZATION IN CONDENSED
MATTER
3.1

Introduction

Spin decoherence from phonons is a pressing question in quantum technology − it
governs spin transport [1–6] and limits the manipulation of quantum information [7–
13] and the realization of reliable quantum devices [14–16]. Previous work has
identified two key sources of phonon-induced spin decoherence in the presence
of spin-orbit coupling (SOC): the Elliott-Yafet (EY) mechanism [17, 18], whereby
electron-phonon (𝑒-ph) collisions change the spin direction, and the Dyakonov-Perel
(DP) mechanism [19] originating from spin precession between 𝑒-ph collisions.
Historically, these two mechanisms have been described with distinct theoretical
models [17–21], but significant efforts have been made to unify them, for example
using real-time evolution of spin ensembles [22–24] or analyzing quasiparticle
broadening in model systems [25–27].
However, formulating a theory that encompasses both the EY and DP mechanisms,
and developing corresponding quantitative calculations of spin decoherence in real
materials, are still outstanding challenges. Many-body approaches combined with
density functional theory (DFT) and related first-principles calculations are particularly promising to tackle this problem. These ab initio methods have become
a gold standard for calculations of 𝑒-ph interactions and transport phenomena in
solids [28–36]. Recent work has extended this framework to compute spin-flip
processes due to 𝑒-ph interactions, leading to predictions of EY spin decoherence
within the spin relaxation time approximation (sRTA) [37]. It is widely accepted
that the sRTA neglects spin precession, and thus a different formalism is needed to
capture the DP mechanism [20, 24].
Inspired by the work of Kim et al. [38], which rigorously proved that the Boltzmann
equation is equivalent to the ladder vertex correction to the conductivity, we ask if a
similar many-body approach can be used to study spin dynamics. The development
of this framework, and of corresponding first-principles calculations, would provide
a viable tool to study phonon-induced spin decoherence, mimicking the progress

41
of first-principles studies of charge transport [28–36]. In turn, accurate predictions
of spin decoherence would advance both condensed matter theory and spin-based
quantum technology.
Here we present a many-body theory of spin relaxation and develop precise ab
initio calculations of phonon-induced spin decoherence in semiconductors. Our
approach calculates the 𝑒-ph vertex corrections to the spin susceptibility, with an
accurate account of electronic and vibrational states, SOC, and 𝑒-ph interactions.
We compute the spin relaxation times (SRTs) of electron and hole carriers in Si and
GaAs — two key candidates for spin-based quantum computing — and in monolayer
WSe2 , a 2D semiconductor with strong SOC. Our predicted SRTs are in excellent
agreement with experiments over a wide temperature range. We demonstrate that
our formalism can calculate both spin relaxation and spin precession, and capture EY
and DP decoherence on equal footing; we contrast these results with the sRTA, which
lacks DP decoherence and gives unphysical SRTs near the band gap. Our analysis
shows that the 𝑒-ph interactions lead to a colossal renormalization of the electron
spin dynamics in solids, significantly modifying the SRTs and spin precession rates
(SPRs). The theory and computational method developed in this work pave the way
for a deeper understanding of electron spin decoherence, with broad implications
for quantum materials and devices.
3.2

Methods

To describe phonon-induced spin decoherence, we consider the Kubo formula for
the spin-spin correlation function [39], and include the ladder vertex correction [38]
from 𝑒-ph interactions (see Fig. 3.1(a)). We derive a Bethe-Salpeter equation
for the phonon-dressed spin vertex (in short, spin-phonon BSE), as discussed in
the Supplemental Material. Our spin-phonon BSE is shown diagrammatically in
Fig. 3.1(b), and can be written as:
s𝚲k (𝜀) = sk +

1 ∑︁ † 𝐴
g𝜈kq G s𝚲G 𝑅 k+q, g𝜈kq 𝐹± (𝑇)
𝑉 𝜈q±
𝜀±𝜔 𝜈 q

(3.1)

where all bolded quantities are matrices in Bloch basis. Above, s𝚲k (𝜀) = s𝑛𝑛 ′k 𝚲𝑛𝑛 ′k (𝜀)
is the phonon-dressed spin vertex, Λ𝑛𝑛
′ k (𝜀) is the vertex correction at energy 𝜀 for
the Cartesian direction 𝛼, and s𝑛𝑛 ′k = ⟨𝑛k| 2ℏ 𝜎|𝑛
ˆ ′k⟩ is the bare spin vertex; G 𝑅/𝐴 are
the retarded/advanced interacting Green’s functions [39], 𝑉 is the system volume,
𝐹± (𝑇) is a thermal occupation factor at temperature 𝑇, and g𝜈kq 𝑛𝑚 = 𝑔𝑛𝑚𝜈 (k, q)
are 𝑒-ph matrix elements [29].

Figure 3.1: Feynman diagrams for spin decoherence. (a) Bubble diagram for the
spin-spin correlation function including the vertex correction. (b) Bethe-Salpeter
equation for the phonon-dressed spin vertex in the ladder approximation.
The vertex correction Λ governs the spin dynamics by renormalizing the microscopic
SRTs and SPRs (see Supplemental Material). The macroscopic SRTs are obtained
as the thermal average
e-ph 𝛽
𝑑𝑓𝑛k
𝑛k 𝑠 𝑛𝑛k 𝑠 𝑛𝑛k 𝜏𝑛k Λ𝑛𝑛k (𝜀 𝑛k )(− 𝑑𝜀 )
(𝑠)
(3.2)
𝜏𝛼𝛽 =
𝑑𝑓𝑛k
𝛼 𝑠𝛽
𝑛k 𝑛𝑛k 𝑛𝑛k
𝑑𝜀
e-ph

where 𝜏𝑛k are 𝑒-ph collision times [28, 39]. For 𝛼 = 𝛽 along the external magnetic
field, Eq. (3.2) gives the longitudinal SRT, usually called 𝑇1 . The renormalized
𝛼 ) and SPRs (𝜔 𝛼 ), which are matrices in Bloch basis, are
microscopic SRTs (𝜏𝑛𝑛
′k
𝑛𝑛 ′ k
computed from the vertex correction using

𝜏𝑛𝑛
′ k (𝜀)

+ 𝑖𝜔𝑛𝑛
′ k (𝜀)

Λ𝑛𝑛
′ k (𝜀)

𝑖(Σ𝑛𝑅k − Σ𝑛𝐴′k ) + 𝑖(𝜀 𝑛k − 𝜀 𝑛 ′k )

(3.3)

with Σ 𝐴/𝑅 the advanced/retarded 𝑒-ph self-energy [28]. The diagonal components
e-ph 𝛽
with 𝑛 = 𝑛′ give the renormalized microscopic SRTs, 𝜏𝑛𝑛k = 𝜏𝑛k Λ𝑛𝑛k (𝜀 𝑛k ) entering
Eq. (3.2). We implement and solve Eqs. (3.1)-(3.3) in our perturbo code [29] (see
Supplemental Material).
The ground state, band structures, and phonon dispersions are obtained using
Quantum ESPRESSO [40]. We employ perturbo [29] to compute and interpolate
the 𝑒-ph matrix elements and spin matrices, using a method described in Ref. [37],
starting from spinor Wannier functions from the wannier90 code [41]. We model all
materials in the intrinsic (i.e., undoped) limit, and accordingly compare our results
with experiments carried out on undoped samples.
3.3

Results

Using this formalism, in Fig. 3.2 we compute the macroscopic SRTs in Eq. (3.2)
as a function of temperature for Si, GaAs, and monolayer WSe2 (see Methods).

1 0

(c )

S i ( e l)

1 0 0

2 0 0

3 0 0

E r s f e ld e t a l.
L i e t a l.
Y a n e t a l.
G o r y c a e t a l.

S p in r e la x a tio n tim e ( n s )

1 ,0 0 0
1 0 0
1 0
0 .1

2 0

S e 2 (h )

4 0 0

( e ) 1 .2

1 0 0

O e r te l e t a l.
K im e l e t a l.
D z h io e v e t a l.
L a i e t a l.

1 0

(d ) 100

T e m p e ra tu re (K )

2 0 0

3 0 0

2 0 0

3 0 0

H ilt o n e t a l.

1 0

4 0 0

( f ) 1 .3
1 .2

1 .1

1 .0

0 .9
0 .9

0 .8

0 .1

1 0 0

G a A s ( e l)

G a A s ( e l)
G a A s (h )
S ilic o n
W S e 2

1 .1

S p i n r e l a x a t i o n t i m e ( τs )

1 0 0

( b )1 0 0 0
S p in r e la x a tio n tim e ( p s )

S p in - p h o n o n B S E
s R T A
A p p e lb a u m e t a l.
L e p in e
L a n c a s t e r e t a l.

S p in r e la x a tio n tim e ( p s )

S p in r e la x a tio n tim e ( n s )

(a )

5 0

G a A s (h )

1 0 0

2 0 0

T e m p e ra tu re (K )

3 0 0

4 0 0

0 .8

0 .9

1 .0

1 .1

e - p h c o llis io n t im e

1 .2

0 .7

0 .8

0 .9

S O C

1 .0

1 .1

b a n d s p lit t in g

1 .2

Figure 3.2: Spin relaxation times. (a)-(d) Computed spin relaxation times as a
function of temperature, for (a) electrons in Si, (b) electrons in GaAs, (c) holes in
monolayer WSe2 , and (d) holes in GaAs. Results obtained from the spin-phonon
BSE (black solid line) are compared with sRTA calculations (red dashed line).
Experimental results from Refs. [5, 42–51] are shown for comparison. (e)-(f) The
SRTs at room temperature for these four cases are recomputed by artificially varying
(e) the 𝑒-ph collision time and (f) the SOC band splitting entering the spin-phonon
BSE. In all cases, the axes are referenced to the real system values. The conventional
DP spin relaxation trend (black dotted line) is also shown for comparison.

In Si, a centrosymmetric material where spin decoherence is governed by the EY
mechanism, the results are in excellent agreement with experiments [5, 42, 43] in
the 100−300 K temperature range. For example, the SRT computed at 300 K is
6.1 ns, in remarkable agreement with the 6.0 ns value measured in Ref. [43]. Due to
the dominant EY mechanism, in this case the sRTA, which neglects spin precession,
also gives accurate SRTs.
In GaAs, the SOC induces a small (∼1 meV) splitting in the conduction band, so
spin relaxation is dominated by the DP mechanism [21]. Figure 3.2(b) shows our
calculated SRTs for electrons in GaAs as function of temperature; the excellent
agreement with experiments [44–47] is a strong evidence that the spin-phonon BSE
describes correctly the DP mechanism. By contrast, the sRTA, which captures only
the EY mechanism, clearly fails in GaAs, predicting SRTs an order of magnitude
greater than experiments.
Our spin-phonon BSE achieves a similar accuracy for calculations on hole carriers.
In Fig. 3.2(c), we compute the SRTs for hole spins in monolayer WSe2 , obtaining
excellent agreement with all available experimental results between 20−90 K [48–
51]. Note that the valence band of WSe2 has a large (∼0.4 eV) splitting due to SOC,

44
leading to a precession rate far greater than the hole 𝑒-ph collision rates; in this
strong precession regime, the spin dynamics is controlled by the diagonal part of
the spin vertex and the DP mechanism becomes irrelevant, so EY spin decoherence
dominates the SRTs. Conversely, for heavy holes in GaAs (see Fig. 3.2(d)) both
EY and DP spin decoherence are important. The agreement with experiment is
noteworthy in this regime where both mechanisms are relevant: our computed SRT
for holes in GaAs at 300 K is 200 fs, versus a 110 fs value measured by Hilton et
al. [52].
A key distinction between the EY and DP mechanisms is their dependence on the
e-ph
e-ph
𝑒-ph collision time, 𝜏𝑛k in Eq. (3.2): the SRT is proportional to 𝜏𝑛k for EY, and
e-ph
inversely proportional to 𝜏𝑛k for DP. Our spin-phonon BSE can capture both of
e-ph
these trends, as we show in Fig. 3.2(e) by artificially increasing 𝜏𝑛k (via the 𝑒-ph
coupling strength |𝑔| 2 ) and recomputing the SRTs at 300 K for all four cases. In Si
and WSe2 , where EY spin decoherence is dominant, we find that the recomputed
SRTs are directly proportional to the 𝑒-ph collision time, consistent with the EY
mechanism [17, 18]. Conversely, for electron spins in GaAs, the SRTs are nearly
inversely proportional to the 𝑒-ph collision time (see Fig. 3.2(e)), in agreement with
the DP mechanism [19]. (Note that the computed trend slightly deviates from the
conventional DP inverse proportionality because EY decoherence, although weak,
is still present.) For hole spins in GaAs, the recomputed SRTs exhibit a trend
intermediate between pure EY and DP, further supporting our conclusion that both
mechanisms are important for hole spins in GaAs [24, 53].
Spin precession in the DP mechanism is induced by the SOC field, which is proportional to the band splitting for each electronic state. To examine the role of DP
spin decoherence, we artificially vary the SOC band splitting Δ𝐸 and for each new
value we recompute the SRTs (see Fig. 3.2(f)). For WSe2 , varying the SOC band
splitting has no effect on the SRTs, showing that spin decoherence is controlled by
the EY mechanism. For electrons in GaAs, the SRTs are highly sensitive to the
SOC splitting, a clear evidence that our formalism can capture the dominant DP
mechanism. This dependence is weaker than in the conventional trend for pure DP,
𝜏 (𝑠) ∝ 1/(Δ𝐸) 2 , due to the coexistence of EY decoherence. For hole carriers in
GaAs, the SRTs are less sensitive to the SOC splitting than for electrons, as the decoherence originates from a balanced combination of both EY and DP mechanisms.
This analysis also shows that band structure calculations accurately describing the
SOC band splitting are essential to predict spin precession and DP decoherence.

Figure 3.3: Microscopic spin decoherence. (a) Microscopic electron SRTs in Si as
a function of conduction band energy, computed with the spin-phonon BSE (black)
and sRTA (red). (b) Zoom-in of the spin-phonon BSE results in (a). (c) Vertex
corrections Λ𝑛𝑛k in Si (black dots) compared with the inverse 𝑒-ph collision times
(green crosses). (d) Microscopic electron SRTs in GaAs from the spin-phonon BSE,
shown as a function of conduction band energy and overlaid with a color map of the
expectation value of 𝑆 𝑧 for each electronic state; the sRTA results (red) are given for
comparison. (e) Microscopic off-diagonal SRTs, 𝜏𝑛𝑛 ′k in Eq. (3.3), overlaid with a
color map of the SOC band splitting Δ𝐸. (f) Renormalized electron SPRs in GaAs,
𝜔𝑛𝑛 ′k in Eq. (3.3), plotted as a function of SOC band splitting and overlaid with
a color map of the conduction band energy; the bare electron SPRs (black dashed
line) are given for comparison. All results are computed at 300 K, and the zero of
the energy axis is the conduction band minimum.

The phonon-induced renormalization greatly modifies the microscopic spin dynamics. Figure 3.3(a) compares sRTA and spin-phonon BSE calculations of the
e-ph 𝑧
= 𝜏𝑛k Λ𝑛𝑛
(𝜀 ) defined below Eq. (3.3), in Si
microscopic electron SRTs, 𝜏𝑛(𝑠)
k 𝑛k
at 300 K for energies near the conduction band minimum. The sRTA results are
strongly energy dependent, with an unphysical divergence at low energy. By contrast, the results from the spin-phonon BSE are nearly energy independent. The
vertex correction makes spins with similar energy relax on the same time scale (a
constant value of 6.1 ns nearly equal to the macroscopic SRT) and overcomes the
limitations of the sRTA. A closer examination of the SRTs from the spin-phonon
BSE (see Fig. 3.3(b)) reveals an oscillatory pattern with a period of 𝜔𝑂 ≈ 60 meV,
the energy of an optical phonon with strong 𝑒-ph coupling; this pattern disappears
when optical phonons are neglected. This oscillation is a manifestation of the selfconsistency of the spin-phonon BSE and its ability to capture strong coupling effects

46
beyond lowest-order perturbation theory. We observe the same energy dependence
and SRT oscillations due to optical phonons for hole spins in WSe2 .
Figure 3.3(c) shows the computed vertex correction Λ𝑛𝑛
(𝜀 ) as a function of enk 𝑛k
ergy in Si. The vertex correction from the 𝑒-ph interactions is colossal; relative to
the bare spin, it is of order Λ − 1 ≈ 105 , and thus eight orders of magnitude greater
than the corresponding vertex correction due to photons in vacuum [54, 55] (with
value Λ − 1 ≈ 1.16 · 10−3 , which corrects the electron magnetic moment). The
energy dependence of the vertex correction is nearly identical to that of the inverse
𝑒-ph collision times, thus explaining the origin of the constant trend with energy of
the microscopic SRTs. We find large vertex correction values (102 − 105 ) also in
GaAs and WSe2 . These giant values account for the large differences between 𝑒-ph
collision times (femtoseconds) and SRTs (nanoseconds) in condensed matter, and
are key to accurately predicting long spin coherence times of interest in quantum
technologies.
In GaAs, due to the Dresselhaus SOC band splitting, the bare spin vertex 𝑠𝑛𝑛 ′k
acquires large off-diagonal (𝑛 ≠ 𝑛′) components that precess in the effective SOC
magnetic field with a bare SPR of 𝜀 𝑛k−𝜀 𝑛 ′k . While the macroscopic SRTs in Eq. (3.2)
are determined only by the band diagonal components 𝑠𝑛𝑛k , the spin-phonon BSE
couples the diagonal and off-diagonal components via Eq. (3.1), so spin precession
modifies the SRTs. The microscopic SRTs for electrons in GaAs (see Fig. 3.3(d))
exhibit trends similar to Si, with renormalized SRTs nearly energy independent near
the band edge, in contrast with the rapidly varying SRTs predicted by the sRTA; an
oscillating pattern is evident with period equal to the 30 meV longitudinal optical
(LO) phonon energy, a signature of strong coupling with LO phonons [32]. Yet,
due to the spin precession, we also observe unique trends not found in Si. The SRTs
decrease at higher energies due to the increasing spin precession (the SOC band
splitting increases with energy), a manifestation of DP spin decoherence. In addition, the SRTs are strongly state dependent as states with a smaller spin component
along the quantization axis, shown with lighter colors in Fig. 3.3(d), are subject to
stronger precession.
The relaxation of the off-diagonal spin components, quantified by the off-diagonal
SRTs 𝜏𝑛𝑛 ′k in Eq. (3.3), reveals additional signatures of the DP mechanism. Figure 3.3(e) shows these off-diagonal electron SRTs for GaAs and highlights their
correlation with the SOC band splitting. When the band splitting is small (black),
precession is negligible and the SRTs are identical to the diagonal SRTs in Fig. 3.3(d).
However, for increasing values of the band splitting (lighter colors), spin preces-

47
sion significantly enhances the SRTs. These intriguing microscopic phenomena
are encoded in the vertex correction Λ in Eq. (3.3), which suppresses the real part
1/𝜏𝑛𝑛 ′k in the denominator, thus slowing down spin relaxation. Similarly, the vertex
correction significantly slows down spin precession, as shown in Fig. 3.3(f) for
GaAs. Electrons with a bare SPR of 1 meV drop to a ∼10−2 meV precession rate
after renormalization due to phonons. These renormalized SPRs are strongly energy
dependent, with higher electron energies leading to faster precession for spins with
the same bare SPRs. This microscopic dynamics reveals the rich interplay between
spin relaxation and precession in materials.

3.4

Conclusion

In conclusion, our findings highlight the dramatic effects of phonon-induced renormalization on electron spins in solids. Our spin-phonon BSE can capture renormalized spin dynamics beyond relaxation, shedding light on the interplay between the
EY and DP spin decoherence mechanisms, and describing their diverse physics on
the same footing. This formalism reveals that the long intrinsic spin coherence times
in condensed matter are due to the colossal vertex correction from 𝑒-ph interactions.
3.5

Supplemental Material

Bethe-Salpeter equation for the phonon-dressed spin vertex
We provide a detailed derivation of the spin-phonon BSE, which computes the
vertex correction to the spin operator due to 𝑒-ph interactions. We use atomic units
and set ℏ = 1.
Interacting Green’s function
We consider an unperturbed Hamiltonian 𝐻0 diagonal in a Bloch basis, ⟨𝑛′k| 𝐻0 |𝑛k⟩ =
𝜀 𝑛k 𝛿𝑛𝑛 ′ . The interacting imaginary-time Green’s function G(𝑖𝜔𝑎 ) is written using
the Dyson equation [39]
G −1 (𝑖𝜔𝑎 ) = G (0)−1 (𝑖𝜔𝑎 ) − Σ(𝑖𝜔𝑎 ),

(3.4)

where 𝜔𝑎 is the fermionic Matsubara frequency of the electron, G (0) (𝑖𝜔𝑎 ) is the
non-interacting Green’s function, and Σ(𝑖𝜔𝑎 ) is the lowest-order (Fan-Migdal) 𝑒-ph
self-energy [28, 39, 56], whose band- and k-dependent expression is
∑︁
[𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q)] ∗ 𝑔𝑛𝑚𝜈 (k, q)D𝜈q (𝑖𝑞 𝑐 ) G𝑚𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 ).
Σ𝑛𝑛 ′k (𝑖𝜔𝑎 ) = −
𝛽𝑉 𝑚𝑚 ′ 𝜈q,𝑖𝑞

(3.5)

48
Above, 𝛽 = 1/𝑘 𝐵𝑇 at temperature 𝑇, 𝑞 𝑐 is the bosonic Matsubara frequency of
the phonon, 𝑉 is the system volume, and D𝜈q (𝑖𝑞 𝑐 ) = 2𝜔q 𝜈 /((𝑖𝑞 𝑐 ) 2 − 𝜔2q 𝜈 ) is the
non-interacting phonon Green’s function for a phonon with mode index 𝜈 and wavevector q.

Spin-spin correlation function
We consider the spin operator 𝑠ˆ, with matrix elements in the direction 𝛼 given by
= ⟨𝑚k| 𝑠ˆ𝛼 |𝑛k⟩. We derive the spin-spin correlation function with a procedure
𝑠𝑛𝑚
analogous to the derivation of the dc electrical conductivity in the ladder approximation [38], where the operator of interest is the velocity operator 𝑣 𝑛k , which is
diagonal in Bloch basis. Here, due to the spin-orbit coupling, the spin operator 𝑠ˆ
is in general non-diagonal in the band index, leading to matrix elements 𝑠𝑛𝑚 k , so
the derivation for the diagonal case given in Ref. [38] needs to be extended to a
non-diagonal operator and vertex correction.
The calculation of the spin-spin correlation function is first carried out in the imaginary time domain and then extended to real frequencies by analytic continuation.
Working in imaginary time and frequency, the retarded spin-spin correlation function for the operator 𝑠ˆ can be obtained from the Kubo formula as [39]
∫ 𝛽
𝜒𝛼𝛽 (p, 𝑖𝜈 𝑏 ) =
𝑑𝜏 𝑒𝑖𝜈𝑏 𝜏 𝑇𝜏 𝑠ˆ𝛼 (p, 𝜏) 𝑠ˆ 𝛽 (−p, 0) ,
(3.6)

where p is the wave-vector, 𝜈 𝑏 is the bosonic Matsubara frequency, 𝜏 is imaginary
time ranging from 0 to 𝛽 = 1/𝑘 𝐵𝑇 at temperature 𝑇, and 𝑇𝜏 is the imaginary time
ordering operator. Here we focus on the p → 0 limit, and thus drop p from the
equations. This correlation function can be expressed as a sum of bubble diagrams,
where in the simplest case one considers the bare bubble diagram that includes the
electron self-energy only in the electron propagator G:
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =

1 ∑︁
Tr G(𝑖𝜔𝑎 ) 𝑠ˆ𝛼 G(𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) 𝑠ˆ 𝛽 ,
𝛽𝑉u𝑐 𝑖𝜔

(3.7)

where 𝑉u𝑐 is the volume of the unit cell, and the trace is evaluated over the band
and momentum indices. In this expression, the operator 𝑠ˆ can be regarded as the
bare vertex of the correlation function (the left vertex of the diagram in Fig. 3.1(a)
in the main text). For the velocity operator, Eq. (3.7) leads to the well-known Drude
conductivity [38, 39].
In this work, the corrections to the spin vertex originate from the 𝑒-ph interactions,

49
which couple electronic states with different bands and crystal momenta. Figure 3.1(a) in the main text shows the spin-spin correlation function including the
vertex correction Λ,
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =

1 ∑︁
Tr G(𝑖𝜔𝑎 ) 𝑠ˆ𝛼 G(𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) 𝑠ˆ 𝛽 Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) ,
𝛽𝑉u𝑐 𝑖𝜔

(3.8)

where 𝑠ˆ 𝛽 Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) is the phonon-dressed vertex for the operator 𝑠ˆ along
the cartesian direction 𝛽. Note that the vertex correction Λ 𝛽 (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ) is a
complex vector, and it contains information about the renormalized spin dynamics
due to the 𝑒-ph interactions.
Bethe-Salpeter equation for the vertex correction
The leading correction to the spin vertex is obtained by summing over the ladder
diagrams, which can be viewed as an abstract form of charge conservation during
the 𝑒-ph scattering process [38, 39]. The vertex correction Λ𝑛𝑛
′ k satisfies a selfconsistent Bethe-Salpeter equation (BSE), shown diagrammatically in Fig. 3.1(b) of
the main text and written as
𝑠𝑛𝑛
′ k Λ𝑛𝑛 ′ k (𝑖𝜔 𝑎 , 𝑖𝜔 𝑎 + 𝑖𝜈 𝑏 )
∑︁
[ 𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q)] ∗ 𝑔𝑛𝑚𝜈 (k, q)D𝜈q (𝑖𝑞 𝑐 )
= 𝑠𝑛𝑛
′k −
𝛽𝑉 𝑚𝑚 ′𝑙𝑙 ′ 𝜈q,𝑖𝑞

× G𝑚𝑙 k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 )G𝑙 ′ 𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝑞 𝑐 )𝑠𝑙𝑙𝛼 ′k Λ𝑙𝑙𝛼 ′k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝑞 𝑐 ).
(3.9)
The kernel of this BSE [57] is the 𝑒-ph interaction 𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q) ∗ 𝑔𝑛𝑚𝜈 (k, q)D𝜈q (𝑖𝑞 𝑐 );
this kernel is the origin of the Elliott-Yafet spin decoherence [37], whereas the
electron propagators in Eq. (3.9), G𝑚𝑙 k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 )G𝑙 ′ 𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝑞 𝑐 ), are
the origin of the Dyakonov-Perel spin decoherence.
Following Mahan [39] and Ref. [38], we first sum over the bosonic Matsubara
frequency 𝑖𝑞 𝑐 in Eq. (3.9). This summation, defined as 𝑆(𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 ), reads
𝑆(𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 )
∑︁
𝑆𝑙𝑙 ′ (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 )
𝑙𝑙 ′

1 ∑︁
D𝜈q (𝑖𝑞 𝑐 )Λ𝑙𝑙𝛼 ′k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝑞 𝑐 )G𝑚𝑙 k+q (𝑖𝜔𝑎 + 𝑖𝑞 𝑐 )G𝑙 ′ 𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝑞 𝑐 ).
𝛽 𝑙𝑙 ′𝑖𝑞

(3.10)

50
As usual, the summation is done by a constructing a contour integral along a circle
at infinity,
𝑑𝑧
𝑛 𝐵 (𝑧)D𝜈q (𝑧)Λ𝑙𝑙𝛼 ′k+q (𝑖𝜔𝑎 + 𝑧, 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑧)G𝑚𝑙 k+q (𝑖𝜔𝑎 + 𝑧)G𝑙 ′ 𝑚 ′k+q (𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑧).
2𝜋𝑖
(3.11)
The integrand has poles at 𝑧 = 𝑖𝑞 𝑐 and 𝑧 = ±𝜔 𝜈q , and branch cuts along 𝑧 = −𝑖𝜔𝑎
and 𝑧 = −𝑖𝜔𝑎 − 𝑖𝜈 𝑏 [38, 39]. The leading contribution to 𝑆𝑙𝑙 ′ (𝑖𝜔𝑎 , 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 )
comes from the combination of retarded and advanced Green’s functions, 𝐺 𝑅 and
𝐺 𝐴 respectively, while terms of 𝑂 ([𝐺 𝑅 ] 2 , [𝐺 𝐴 ] 2 ) can be neglected at low electron
density [38, 39]. Therefore, after the analytic continuations 𝑖𝜔𝑎 → 𝜀 − 𝑖𝜂 and
= P 1𝑥 − 𝑖𝜋𝛿(𝑥), we obtain
𝑖𝜔𝑎 + 𝑖𝜈 𝑏 → 𝜀 + 𝜈 + 𝑖𝜂, and using the identity 𝑥+𝑖𝜂
𝑆𝑙𝑙 ′ (𝜀 − 𝑖𝜂, 𝜀 + 𝑖𝜂) in the limit of 𝜈 → 0:
𝑆𝑙𝑙 ′ (𝜀−𝑖𝜂, 𝜀 + 𝑖𝜂)
= −[𝑁 𝜈q + 𝑓 (𝜀 + 𝜔 𝜈q )]Λ𝑙𝑙𝛼 ′k+q (𝜀 + 𝜔 𝜈q )𝐺 𝑚𝑙
k+q (𝜀 + 𝜔 𝜈 q )𝐺 𝑙 ′ 𝑚 ′ k+q (𝜀 + 𝜔 𝜈 q )
−[𝑁 𝜈q + 1 − 𝑓 (𝜀 − 𝜔 𝜈q )]Λ𝑙𝑙𝛼 ′k+q (𝜀 − 𝜔 𝜈q )𝐺 𝑚𝑙
k+q (𝜀 − 𝜔 𝜈 q )𝐺 𝑙 ′ 𝑚 ′ k+q (𝜀 − 𝜔 𝜈 q ),

(3.12)
where R(A) stands for retarded (advanced) functions, Λ𝛼 (𝜀) ≡ Λ𝛼 (𝜀 − 𝑖𝜂, 𝜀 + 𝑖𝜂),
𝜔 𝜈q are phonon energies, and 𝑁 𝜈q ≡ 𝑛 𝐵 (𝜔 𝜈q ) are the corresponding temperaturedependent Bose-Einstein occupations.
Using this result, we write the self-consistent BSE for the phonon-dressed vertex
𝑠ˆΛ as
𝑠𝑛𝑛
′ k Λ𝑛𝑛 ′ k (𝜀)
1 ∑︁
[𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q)] ∗ 𝑔𝑛𝑚𝜈 (k, q)𝑠𝑙𝑙𝛼 ′k+q
=𝑠𝑛𝑛
′k +
𝑉 𝑚𝑚 ′𝑙𝑙 ′ 𝜈q
× (𝑁 𝜈q + 𝑓 (𝜀 + 𝜔 𝜈q ))Λ𝑙𝑙𝛼 ′k+q (𝜀 + 𝜔 𝜈q )𝐺 𝑚𝑙
k+q (𝜀 + 𝜔 𝜈 q )𝐺 𝑙 ′ 𝑚 ′ k+q (𝜀 + 𝜔 𝜈 q )

+ (𝑁 𝜈q + 1 − 𝑓 (𝜀 − 𝜔 𝜈q ))Λ𝑙𝑙𝛼 ′k+q (𝜀 − 𝜔 𝜈q )𝐺 𝑚𝑙
k+q (𝜀 − 𝜔 𝜈 q )𝐺 𝑙 ′ 𝑚 ′ k+q (𝜀 − 𝜔 𝜈 q )

(3.13)
In matrix form and using a more compact notation, Eq. (3.13) becomes
1 ∑︁ † 𝐴
s𝚲k (𝜀) = sk +
g𝜈kq G s𝚲G 𝑅 k+q, g𝜈kq 𝐹± (𝑇),
𝑉 𝜈q±
𝜀±𝜔 𝜈 q

(3.14)

which is the spin-phonon BSE in Eq. (3.1) of the main text, with the thermal
occupation factor defined as 𝐹± (𝑇) = 𝑁 𝜈q + 21 ± [ 𝑓 (𝜀 ± 𝜔 𝜈q ) − 12 ]. By solving

51
Eq. (3.13), we obtain the phonon-dressed spin vertex s𝚲k (𝜀) and its dependence on
band, crystal momentum k and energy 𝜀.
In the weak scattering regime, where the electron spectral function has a well-defined
quasiparticle peak [58] and the off-diagonal self-energy can be neglected [59, 60],
the Green’s function becomes band-diagonal and the self-energies are evaluated
on-shell. The product of the retarded and advanced Green’s functions, 𝐺 𝑅 𝐺 𝐴 , can
be recast [61] and approximated as
𝐺𝑚
k+q (𝜀)𝐺 𝑚 ′ k+q (𝜀)

(𝜀)
𝐺 𝑚𝐴 ′k+q (𝜀) − 𝐺 𝑚
k +q
𝐺𝑚
(𝜀) −1 − 𝐺 𝑚𝐴 ′k+q (𝜀) −1
k +q

(3.15)

𝜋𝛿(𝜀 − 𝜀 𝑚 ′k+q ) + 𝜋𝛿(𝜀 − 𝜀 𝑚 k+q ) − 𝑖P 𝜀−𝜀 𝑚1′k+q + 𝑖P 𝜀−𝜀1𝑚k+q
𝑖(Σ𝑚𝑅 k+q − Σ𝑚𝐴 ′k+q ) + 𝑖(𝜀 𝑚 k+q − 𝜀 𝑚 ′k+q )

a function that is strongly peaked at electron energies 𝜀 = 𝜀 𝑚 k+q and 𝜀 = 𝜀 𝑚 ′k+q .
Therefore, we can further simplify the spin-phonon BSE from its full-frequency
form in Eq. (3.13) to the following double-pole ansatz:
2𝜋 ∑︁
[𝑔𝑛 ′ 𝑚 ′ 𝜈 (k, q)] ∗ 𝑔𝑛𝑚𝜈 (k, q)𝑠𝑚𝑚
𝑠𝑛𝑛
(𝜀)
′ k 𝑛𝑛 ′ k
′ k +q
𝑛𝑛 ′ k
𝑉 𝑚𝑚 ′ 𝜈q
× [(𝑁 𝜈q + 𝑓𝑚 k+q )(𝛿(𝜀+𝜔 𝜈q −𝜀 𝑚 k+q ) − P
𝜋 𝜀+𝜔 𝜈q −𝜀 𝑚 ′k+q
Λ𝛼𝑚𝑚 ′k+q (𝜀 𝑚 k+q )
)] ×
+ (𝑁 𝜈q +1− 𝑓𝑚 k+q )(𝛿(𝜀−𝜔 𝜈q −𝜀 𝑚 k+q ) − P
𝜋 𝜀−𝜔 𝜈q −𝜀 𝑚 ′k+q
𝑖(Σ𝑚𝑅 k+q −Σ𝑚𝐴 ′k+q ) +𝑖(𝜀 𝑚 k+q −𝜀 𝑚 ′k+q )
+ [(𝑁 𝜈q + 𝑓𝑚 ′k+q )(𝛿(𝜀+𝜔 𝜈q −𝜀 𝑚 ′k+q ) + P
𝜋 𝜀 + 𝜔 𝜈q −𝜀 𝑚 k+q
Λ𝛼𝑚𝑚 ′k+q (𝜀 𝑚 ′k+q )
)] ×
+ (𝑁 𝜈q +1− 𝑓𝑚 ′k+q )(𝛿(𝜀−𝜔 𝜈q −𝜀 𝑚 ′k+q ) + P
𝜋 𝜀−𝜔 𝜈q −𝜀 𝑚 k+q
𝑖(Σ𝑚𝑅 k+q −Σ𝑚𝐴 ′k+q ) +𝑖(𝜀 𝑚 k+q −𝜀 𝑚 ′k+q )
(3.16)
where 𝜀 stands for either 𝜀 𝑛k or 𝜀 𝑛 ′k , and 𝑓𝑚 k+q ≡ 𝑓 (𝜀 𝑚 k+q ). Note that the spinphonon BSE in Eq. (3.16) should not be confused with the widely-used BSE for
excitons and optical spectra [62], which is entirely unrelated.
The dressed vertex and its interpretation
We focus on the dressed spin operator divided by the band energy difference, which
we introduced in Eq. (3.16):
𝑠𝑚𝑚
′ k+q Λ𝑚𝑚 ′ k+q (𝜀 𝑚 ′ k+q )
(3.17)
𝑖(Σ𝑚𝑅 k+q −Σ𝑚𝐴 ′k+q ) +𝑖(𝜀 𝑚 k+q −𝜀 𝑚 ′k+q )

52
This ratio describes the renormalized spin magnetization in the presence of 𝑒-ph
interactions. The effective renormalized dynamics of the spin operator is obtained
by dividing Eq. (3.17) by the bare spin expectation value 𝑠𝑚𝑚
′ k+q , which gives
Λ𝛼𝑚𝑚 ′k+q (𝜀 𝑚 ′k+q )
(3.18)

𝑖(Σ𝑚𝑅 k+q −Σ𝑚𝐴 ′k+q ) +𝑖(𝜀 𝑚 k+q −𝜀 𝑚 ′k+q )

The physical meaning of these ratios can be understood by analyzing the simpler
case of the velocity operator, 𝑣 𝛼𝑚𝑚 ′k+q = 𝑣 𝛼𝑚 k+q 𝛿𝑚𝑚 ′ . As the velocity operator
is band-diagonal, the band energy difference in the denominator vanishes, and
the denominator is purely real because Σ𝑚𝐴 ′k+q = (Σ𝑚𝑅 ′k+q ) ∗ . Thus the analog of
Eq. (3.17) for the velocity operator becomes
𝑣 𝛼𝑚 k+q Λ𝛼𝑚𝑚 k+q (𝜀 𝑚 k+q )
𝑖(Σ𝑚𝑅 k+q − Σ𝑚𝐴 k+q )

e-ph

= 𝑣 𝛼𝑚 k+q 𝜏𝑚 k+q Λ𝛼𝑚𝑚 k+q (𝜀 𝑚 k+q ),

(3.19)

e-ph

where we used 𝜏𝑚 k+q = 1/|2ℑΣ𝑚 k+q | for the 𝑒-ph collision time. Equation (3.19)
gives the renormalized mean free path, and dividing by the bare velocity we obtain
the renormalized relaxation time, also known as the transport relaxation time [38],
e-ph
𝜏𝑚𝛼(tr)
≡ 𝜏𝑚 k+q Λ𝛼𝑚𝑚 k+q =
k +q

Λ𝛼𝑚𝑚 k+q (𝜀 𝑚 k+q )
(3.20)

e-ph
𝜏𝑚k+q

For a non-diagonal operator, both the vertex correction and the operator expectation
value are complex, so the ratio in Eq. (3.18) cannot be represented by a single real
quantity with units of time as in Eq. (3.20). We thus extend this formalism by
defining the renormalized microscopic relaxation times 𝜏𝑚𝑚
′ k+q (𝜀) and precession
frequencies 𝜔𝛼𝑚𝑚 ′k+q (𝜀):
Λ𝛼𝑚𝑚 ′k+q (𝜀)
𝑖(Σ𝑚𝑅 k+q −Σ𝑚𝐴 ′k+q ) +𝑖(𝜀 𝑚 k+q −𝜀 𝑚 ′k+q )

+ 𝑖𝜔𝛼𝑚𝑚 ′k+q (𝜀)
𝜏𝑚𝑚
′ k+q (𝜀)

(3.21)

where 𝜀 stands for either 𝜀 𝑚 k+q or 𝜀 𝑚 ′k+q . This way, without the vertex correction
e-ph
the renormalized relaxation time reduces to the 𝑒-ph collision time, 𝜏𝑚𝑚 ′k+q =
1/|ℑΣ𝑚 k+q + ℑΣ𝑚 ′k+q |, and the renormalized precession frequency reduces to the
bare precession frequency, equal to the energy difference (𝜀 𝑚 k+q + ℜΣ𝑚 k+q ) −
(𝜀 𝑚 ′k+q + ℜΣ𝑚 ′k+q ).

53
Renormalized spin relaxation times from linear response theory
We use linear response theory to derive the macroscopic spin relaxation times
inclusive of vertex corrections. We focus on the spin magnetic moment 𝑚ˆ = 𝑔𝜇 𝐵 𝑠ˆ,
where 𝑔 is the electron Landé 𝑔-factor and 𝜇 𝐵 is the Bohr magneton. The magnetic
spin susceptibility 𝜒M is defined as the response function in
𝑀 𝛼 (𝜈) = 𝜒𝛼𝛽
(𝜈)𝐵 𝛽 (𝜈),

(3.22)

where 𝐵 𝛽 is the external magnetic field along the direction 𝛽, and 𝑀 𝛼 is the magnetization of the system along 𝛼 generated in response to the applied magnetic field.
Note that the magnetic susceptibility is proportional to the spin-spin correlation
M (𝜈) = (𝑔𝜇 ) 2 𝜒 (𝜈). To study spin relaxation, we
function 𝜒 introduced above, 𝜒𝛼𝛽
𝛼𝛽
rewrite the magnetization as [63]
(𝜈) 𝐵¤ 𝛽 (𝜈).
𝑀 𝛼 (𝜈) = 𝜎𝛼𝛽

(3.23)

Thus, expressing it in terms of 𝜎 M , the magnetic susceptibility to the spin injection
field at frequency 𝜈, 𝐵¤ 𝛽 (𝜈) = −𝑖𝜈𝐵 𝛽 (𝜈) [63]. This field produces a nonequilibrium
spin density with a spin injection rate equal to the inverse spin relaxation time
1/𝜏 (𝑠) [63]. From Eqs. (3.22)-(3.23), we obtain
𝜎𝛼𝛽
(𝜈) =

𝑀 (𝜈)
𝜒𝛼𝛽

−𝑖𝜈

(3.24)

We write the spin-spin correlation function with ladder vertex correction [see
Eq. (3.8)] as a contour integral along a circle at infinity,
𝑑𝑧
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) = −
𝑓 (𝑧)Tr G(𝑧) 𝑠ˆ𝛼 G(𝑧 + 𝑖𝜈 𝑏 ) 𝑠ˆ 𝛽 Λ 𝛽 (𝑧, 𝑧 + 𝑖𝜈 𝑏 ) ,
(3.25)
𝑉u𝑐
2𝜋𝑖
which has branch cuts along 𝑧 = 0 and 𝑧 = −𝑖𝜈 𝑏 , and poles at 𝑧 = 𝑖𝜔𝑎 , such that
𝑑𝜀
𝜒𝛼𝛽 (𝑖𝜈 𝑏 ) =
𝑓 (𝜀)Tr − G(𝜀 + 𝑖𝜂) 𝑠ˆ𝛼 G(𝜀 + 𝑖𝜈 𝑏 ) 𝑠ˆ 𝛽 Λ 𝛽 (𝜀 + 𝑖𝜂, 𝜀 + 𝑖𝜈 𝑏 )
𝑉u𝑐
2𝜋𝑖
+ G(𝜀 − 𝑖𝜂) 𝑠ˆ𝛼 G(𝜀 + 𝑖𝜈 𝑏 ) 𝑠ˆ 𝛽 Λ 𝛽 (𝜀 − 𝑖𝜂, 𝜀 + 𝑖𝜈 𝑏 )
− G(𝜀 − 𝑖𝜈 𝑏 ) 𝑠ˆ𝛼 G(𝜀 + 𝑖𝜂) 𝑠ˆ 𝛽 Λ 𝛽 (𝜀 − 𝑖𝜈 𝑏 , 𝜀 + 𝑖𝜂)

+ G(𝜀 − 𝑖𝜈 𝑏 ) 𝑠ˆ G(𝜀 − 𝑖𝜂) 𝑠ˆ Λ (𝜀 − 𝑖𝜈 𝑏 , 𝜀 − 𝑖𝜂) .
(3.26)

54
After the analytic continuation 𝑖𝜈 𝑏 → 𝜈 + 𝑖𝜂, we obtain the spin-spin correlation
function to leading order by neglecting terms of order 𝑂 ([𝐺 𝑅 ] 2 , [𝐺 𝐴 ] 2 ) [38, 39]:
𝑑𝜀
𝜒𝛼𝛽 (𝜈) =
( 𝑓 (𝜀) − 𝑓 (𝜀 + 𝜈))Tr 𝐺 𝑅 (𝜀) 𝑠ˆ𝛼 𝐺 𝐴 (𝜀 + 𝜈) 𝑠ˆ 𝛽 Λ 𝛽 (𝑖𝜔𝑎 − 𝑖𝜂, 𝑖𝜔𝑎 + 𝑖𝜈 𝑏 + 𝑖𝜂)
𝑉uc ∫2𝜋𝑖
1 ∑︁
𝑑𝜀
( 𝑓 (𝜀) − 𝑓 (𝜀 + 𝜈))𝑠𝑛𝑚
𝑠𝑚𝑛k Λ𝑚𝑛k (𝜀 − 𝑖𝜂, 𝜀 + 𝜈 + 𝑖𝜂)
2𝜋𝑖
𝑛𝑚 k

𝜋𝛿(𝜀 − 𝜀 𝑛k ) + 𝜋𝛿(𝜀 + 𝜈 − 𝜀 𝑚 k )
𝑖(𝜀 𝑛k + 𝜈 − 𝜀 𝑚 k ) − 𝑖(Σ𝑛𝐴k − Σ𝑚𝑅 k )
(3.27)

where we used Eq. (3.15) to obtain the second line. Equation (3.27) completely
determines the frequency-dependent spin response of the system.
Using this result, we derive an analytic expression for the average macroscopic spin
relaxation times. We focus on the dc limit 𝜈 → 0, where the external driving field
is static, and compute the relaxation of the non-precessing (band-diagonal) spins by
setting 𝑚 = 𝑛 in Eq. (3.27). The the magnetic susceptibility to the spin injection
field in the dc limit becomes
lim 𝜎𝛼𝛽
(𝜈)| 𝑚=𝑛 = −(𝑔𝜇 𝐵 ) 2 lim ℑ𝜒𝛼𝛽 (𝜈) 𝑚=𝑛
𝜈→0
𝜈→0 𝜈
∑︁
(𝑔𝜇 𝐵 )
𝑑𝑓𝑛k
e-ph 𝛽
𝑠𝑛𝑛
),
𝑠𝑛𝑛k 𝜏𝑛k Λ𝑛𝑛k (𝜀 𝑛k )(−
𝑑𝜀

(3.28)

𝑛k

e-ph

which gives renormalized microscopic spin relaxation times 𝜏𝑛k Λ𝑛𝑛k (𝜀 𝑛k ), consistent with Eq. (3.21) for non-precessing spins. The corresponding average macroscopic spin relaxation times are
(𝑠)
𝜏𝛼𝛽

e-ph 𝛽
𝑑𝑓𝑛k
𝑛k 𝑠 𝑛𝑛k 𝑠 𝑛𝑛k 𝜏𝑛k Λ𝑛𝑛k (𝜀 𝑛k )(− 𝑑𝜀 )
𝑑𝑓𝑛k
𝑛k 𝑠 𝑛𝑛k 𝑠 𝑛𝑛k − 𝑑𝜀

(3.29)

55
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61
Chapter 4

LONG-RANGE QUADRUPOLE ELECTRON-PHONON
INTERACTION FROM FIRST PRINCIPLES
4.1

Introduction

Electron-phonon (𝑒-ph) interactions are key to understanding electrical transport,
nonequilibrium dynamics, and superconductivity [1]. First-principles calculations
can provide microscopic insight into 𝑒-ph scattering processes and are rapidly emerging as a quantitative tool for investigating charge transport and ultrafast carrier
dynamics in materials [2–13]. The typical workflow combines density functional
theory (DFT) [14] calculations of the ground state and band structure with density functional perturbation theory (DFPT) [15] for phonon dispersions and 𝑒-ph
perturbation potentials. As DFPT can compute the electronic response to periodic
lattice perturbations (phonons) with arbitrary wave-vector q, the DFPT framework
can capture both short- and long-range 𝑒-ph interactions.
However, a key challenge is that DFPT is too computationally demanding to be
carried out on the fine Brillouin zone grids needed to compute electron scattering
rates and transport properties. The established approach in first-principles 𝑒-ph
studies [9] is to carry out DFPT calculations on coarse Brillouin zone grids with
of order 10×10×10 q-points, followed by interpolation of the 𝑒-ph matrix elements
with a localized basis set such as Wannier functions or atomic orbitals [16]. As
the perturbation potential can be non-analytic near q = 0 [16] or even exhibit a
divergence for certain phonon modes, interpolation is particularly challenging and
less reliable in the region between q = 0 and its nearest-neighbor q-points in the
coarse DFPT grid. This small-q region is critical as it is dominated by long-range
𝑒-ph interactions, whose treatment can affect the quality of the interpolation even at
larger values of 𝑞 in the Brillouin zone.
A multipole expansion of the 𝑒-ph perturbation potential shows that in the longwavelenght limit (phonon wave-vector q → 0) the long-range dipole Fröhlich term
diverges as 1/𝑞, the quadrupole term approaches a constant value and the shortrange octopole and higher terms vanish [17, 18]. These trends in momentum space
are due to the spatial decay of the 𝑒-ph interactions, with 1/𝑟 2 trend for the dipole,
1/𝑟 3 for the quadrupole, and 1/𝑟 4 or faster for the short-range part. The open question is how one can carry out the 𝑒-ph matrix element interpolation in the region

62
near q = 0 using analytical expressions for the long-range dipole and quadrupole
terms. These expressions have been obtained by Vogl [18], but need to be rewritten
in the ab initio formalism and computed with first-principles quantities such as the
atomic dynamical dipoles [15] and quadrupoles [19–21] induced by lattice vibrations, which can be computed with DFPT. For each atom 𝜅, one can obtain Born
charge (Z𝜅 ) and dynamical quadrupole (Q𝜅 ) tensors, which, once contracted with
the phonon eigenvector, give the atomic contributions to the dipole and quadrupole
𝑒-ph interactions.
The dipole Fröhlich term has been derived following this strategy [22, 23] and employed in electron scattering rate and transport calculations [4, 7]. The quadrupole
term has not yet been derived or implemented in first-principles calculations and its
important effect on the 𝑒-ph matrix elements has been overlooked. To understand
the role of long-range dipole and quadrupole 𝑒-ph interactions, it is useful to consider separately the interactions for different phonon modes in the long-wavelength
limit, discerning the effect of longitudinal and transverse, and acoustic and optical
modes. Analytical models of 𝑒-ph interactions rely on such an intuition for the role
of different phonon modes in various materials [24].
In ionic and polar covalent crystals (here and below, denoted as polar materials),
the dipole Fröhlich term is dominant as q → 0 due to its 1/𝑞 trend. This 𝑒-ph
interaction is due to longitudinal optical (LO) phonons and it dominates small-q
scattering. For other phonon modes in polar materials, the dipole term vanishes, and
the dominant long-range 𝑒-ph interaction is the quadrupole term, which is particularly important for acoustic phonons in piezoelectric (polar noncentrosymmetric)
materials. In nonpolar semiconductors such as silicon and germanium, the dipole
Fröhlich interaction vanishes and the quadrupole term is a key contribution for all
modes in the long-wavelength limit. The quadrupole 𝑒-ph interaction is thus expected to play an important role in many classes of materials, making a compelling
case for its inclusion in the first-principles framework.
Here we show ab initio calculations of the long-range quadrupole 𝑒-ph interaction
and an approach to include it in the 𝑒-ph matrix elements. The accuracy of our
method is confirmed by comparing the 𝑒-ph matrix elements with direct DFPT calculations. We find that the quadrupole contribution is significant for most phonon
modes in both nonpolar and polar materials. In silicon, a nonpolar semiconductor,
the quadrupole term has a large effect on the 𝑒-ph coupling for optical modes, but
is negligible for acoustic modes in the long-wavelength limit. In tetragonal PbTiO3
(a polar piezoelectric material) the quadrupole corrections are substantial for all

63
phonon modes and particularly important for acoustic modes, which contribute to
the piezoelectric 𝑒-ph interaction. Including only the long-range Fröhlich interaction and neglecting the quadrupole term leads to large errors in PbTiO3 , while
adding the quadrupole term leads to 𝑒-ph matrix elements that accurately reproduce
the DFPT benchmark results for all phonon modes in the entire Brillouin zone. We
investigate the impact of the quadrupole 𝑒-ph interaction on the electron scattering
rates and mobility in silicon and PbTiO3 , finding mobility corrections of order 10%
in silicon and 20% in PbTiO3 at 100 K (and smaller corrections at 300 K) when the
quadrupole term is included. The correction on the scattering rate at low electron
energy in PbTiO3 is substantial. Taken together, our results highlight the need to
include the quadrupole term in all materials to correctly capture the long-range 𝑒-ph
interactions. In turn, this development enables more precise calculations of electron
dynamics and scattering processes from first principles.

Figure 4.1: Schematic of the dipole and quadrupole charge configurations giving
rise to long-range 𝑒-ph interactions.

4.2

Theory

The electron distribution changes in response to a displacement of an atom from
its equilibrium position. The cell-integrated charge response to a displacement of
atom 𝜅 due to a phonon with wave-vector q → 0 can be written as a multipole
expansion [21]:
𝐶𝜅,𝛼
= −𝑖𝑍 𝜅,𝛼𝛽 𝑞 𝛽 − 𝑄 𝜅,𝛼𝛽𝛾 𝑞 𝛽 𝑞 𝛾 + . . . ,
(4.1)
where summation over the Cartesian indices 𝛽 and 𝛾 is implied. This polarization
response defines the Born effective charge Z𝜅 , a rank-2 tensor associated with the
dipole term, and the dynamical quadrupole Q𝜅 , the rank-3 tensor in the quadrupole
term; both tensors can be computed in the DFPT framework [15, 19]. Each of the

64
dipole and quadrupole responses generates macroscopic electric fields and corresponding long-range 𝑒-ph interactions in semiconductors and insulators [18, 25],
while in metals they are effectively screened out.
In a field-theoretic treatment of the 𝑒-ph interactions, one computes the dipole and
quadrupole perturbation potentials Δ𝑉𝜈q due to a phonon with mode index 𝜈 and
wave-vector q, and the corresponding 𝑒-ph matrix elements [2]

𝑔𝑚𝑛𝜈 (k, q) =
2𝜔 𝜈q

⟨𝑚k + q| Δ𝑉𝜈q |𝑛k⟩ ,

(4.2)

which quantify the probability amplitude of an electron in a Bloch state |𝑛k⟩ with
band index 𝑛 and crystal momentum k to scatter into a final state |𝑚k + q⟩ by
emitting or absorbing a phonon with energy ℏ𝜔 𝜈q .
Dipole and quadrupole 𝑒-ph interactions
To derive the dipole and quadrupole perturbation potentials, we consider a Born-von
Karman (BvK) crystal [26] with 𝑁 unit cells and volume 𝑁Ω. The potential due to
a dipole configuration with dipole moment p centered at position τ in the crystal
(see Fig. 4.1) can be written as [22, 23]
Δ𝑉 dip (r; τ ) = 𝑖

𝑒 ∑︁ ∑︁ p · (q + G)𝑒𝑖(q+G)·(r−τ )
𝑁Ω 𝜀0 q G≠−q (q + G) · ϵ · (q + G)

(4.3)

where ϵ is the dielectric tensor of the material, the phonon wave-vector q belongs
to a regular Brillouin zone grid with 𝑁 points, and G are reciprocal lattice vectors.
This result is derived by adding together the potentials generated in the crystal by two
point charges of opposite sign with distance u → 0, resulting in a dipole p [4, 23].
The potential in Eq. (4.3) is readily extended to the case of an atomic dynamical
dipole p𝜅,R from atom 𝜅 in the unit cell at Bravais lattice vector R, due to the displacement induced by a phonon with mode index 𝜈 and wave-vector q. The resulting
𝑖q·R , where the phonon eigenvector
atomic dynamical dipole is p𝜅,R = (𝑒Z𝜅 ) ẽ𝜈(𝜅)
q𝑒
(𝜅) √
projected on atom 𝜅 is defined as ẽ𝜈(𝜅)
q = e𝜈 q / 𝑀𝜅 , with e𝜈 q the eigenvector of
the dynamical matrix at q and 𝑀𝜅 the mass of atom 𝜅. Summing over the contributions from all atoms 𝜅 at lattice vectors R with positions τ𝜅R = τ𝜅 + R in
the BvK supercell, the total 𝑒-ph dipole interaction due to the phonon mode is
dip
Δ𝑉𝜈q (r) = 𝜅R Δ𝑉 dip (r; τ𝜅R ). Using the identity 𝑁1 R 𝑒𝑖q·R = 𝛿q,0 , we obtain:
dip
Δ𝑉𝜈q (r) = 𝑖

(𝜅)
𝑒 2 ∑︁ −1/2 ∑︁ (Z𝜅 e𝜈q ) · (q + G)𝑒𝑖(q+G)·(r−τ )
𝑀𝜅
Ω𝜀0 𝜅
(q + G) · ϵ · (q + G)
G≠−q

(4.4)

65
The ab initio Fröhlich 𝑒-ph coupling is obtained by evaluating the matrix elements
with this potential:

d𝑖 𝑝
𝑔𝑚𝑛𝜈 (k, q) = 𝑖

1/2 ∑︁
(Z𝜅 e𝜈(𝜅)
𝑒 2 ∑︁
q ) · (q + G)
⟨𝑚k + q| 𝑒𝑖(q+G)·(r−τ ) |𝑛k⟩ .
Ω𝜀0 𝜅 2𝜔 𝜈q 𝑀𝜅
(q
G)
(q
G)
G≠−q
(4.5)

The potential due to the dynamical quadrupole response can be derived with a
similar strategy. We first consider the potential generated by a quadrupole charge
configuration consisting of two equal and oppositely oriented dipoles p and −p,
centered at positions τ ± u2 respectively (see Fig. 4.1). The configuration, with
quadrupole moment [27] 𝑀𝛼𝛽 = 𝑝 𝛼 𝑢 𝛽 , gives a potential:
u
u i
Δ𝑉 quad (r; τ ) = lim Δ𝑉 dip r; τ + −Δ𝑉 dip r; τ −
u→0
∑︁
∑︁
(4.6)
(q + G) · M · (q + G) 𝑖(q+G)·(r−τ )
𝑁Ω 𝜀0 q G≠−q (q + G) · ϵ · (q + G)
where to obtain the second line we used Δ𝑉 dip (r; τ ) in Eq. (4.3) and expanded the
first line to first order in u.
Similar to the dipole case, the potential from atomic quadrupoles (M𝜅,R )𝛼𝛽 =
(𝜅)
𝑖q·R due to the displacement induced by a phonon is obtained
2 (𝑒Q𝜅 ) 𝛼𝛽𝛾 e𝜈 q ,𝛾 𝑒
quad
as Δ𝑉𝜈q (r) = 𝜅R Δ𝑉 quad (r; τ𝜅R ). Following steps analogous to the dipole case,
we find:
(𝜅)
𝑒 2 ∑︁ −1/2 ∑︁ 1 (q + G) · (Q𝜅 e𝜈q ) · (q + G)
quad
𝑀𝜅
Δ𝑉𝜈q (r) =
Ω𝜀0 𝜅
(q + G) · ϵ · (q + G)
(4.7)
G≠−q
× 𝑒𝑖(q+G)·(r−τ𝜅 ) .
The corresponding 𝑒-ph matrix elements due to the quadrupole perturbation potential are:
21 ∑︁
(𝜅)
𝑒 2 ∑︁
1 (q + G)𝛼 (𝑄 𝜅,𝛼𝛽𝛾 e𝜈q,𝛾 )(q + G) 𝛽
quad
𝑔𝑚𝑛𝜈 (k, q) =
⟨𝑚k + q| 𝑒𝑖(q+G)·(r−τ𝜅 ) |𝑛k⟩ .
Ω𝜀0 𝜅 2𝜔 𝜈q 𝑀𝜅 G≠−q 2
(q + G)𝛼 𝜖 𝛼𝛽 (q + G) 𝛽
(4.8)
Note that in the q → 0 limit the Fröhlich 𝑒-ph matrix elements are of order 1/𝑞 and
the quadrupole matrix elements of order 𝑞 0 , thus approaching a constant value; both
quantities are non-analytic as q → 0. Octopole and higher electronic responses in
Eq. (4.1) lead to potentials that vanish as q → 0 and can be grouped together into a
short-range 𝑒-ph interaction, commonly referred to as the “deformation potential”
in analytic 𝑒-ph theories [18].

66
Interpolation scheme for 𝑒-ph interactions
The total 𝑒-ph matrix elements 𝑔 (here we omit the band and mode indices) can be
formed by adding together the short-range part 𝑔 S and the dipole and quadrupole
interactions, which can be combined into a long-range part 𝑔 L . Therefore,
𝑔 = 𝑔S + 𝑔L
= 𝑔 S + 𝑔 dip + 𝑔 quad .

(4.9)

We start from a set of 𝑒-ph matrix elements 𝑔(k, q) computed with DFPT on a regular
coarse grid of k- and q-points [15]. The short-range part is obtained by subtracting
the long-range terms on the coarse grid, 𝑔 S (k, q) = 𝑔(k, q)−𝑔 dip (k, q)−𝑔 quad (k, q).
The short-range 𝑒-ph matrix elements decay rapidly in real space, and thus are ideal
for interpolation using a localized basis set such as Wannier functions [28] or atomic
orbitals [16]. After interpolating the short-ranged part [9] on fine k- and q-point
grids, we add back the long-range dipole and quadrupole matrix elements, computed
using Eqs. (4.5) and (4.8) directly at the fine-grid k and q-points.
As DFPT accurately captures the long-range dipole and quadrupole 𝑒-ph interactions [15], the matrix elements obtained from DFPT can be used as a benchmark
for the interpolated results. For this comparison, following Ref. [22] we compute
𝜈 (q), which is proportional to the
the gauge-invariant 𝑒-ph coupling strength, 𝐷 tot
absolute value of the 𝑒-ph matrix elements:
2𝜔 𝜈q 𝑀uc ∑︁ |𝑔𝑚𝑛𝜈 (k = Γ, q)| 2
(4.10)
𝐷 tot (q) =
ℏ2
𝑚𝑛
where 𝑀uc is the mass of the unit cell and the band indices 𝑛 and 𝑚 run over the 𝑁 𝑏
bands selected for the analysis.
Computational details
We investigate the effect of the quadrupole 𝑒-ph interaction in silicon, a nonpolar
semiconductor, and tetragonal PbTiO3 , a polar piezoelectric material. Calculations
on GaN are shown in the companion work [29]. The ground state and band structure
are obtained using DFT in the local density approximation with a plane-wave basis
using the Quantum ESPRESSO code [30]. Kinetic energy cutoffs of 40 Ry for
silicon and 76 Ry for PbTiO3 are employed, together with scalar-relativistic normconserving pseudopotentials from Pseudo Dojo [31]. We have verified that spinorbit coupling has a negligible effect. The calculations employ relaxed lattice
constants of 10.102 bohr for silicon and 7.275 bohr (with aspect ratio 𝑐/𝑎 = 1.046)

67
for PbTiO3 . We use the dynamical quadrupole tensors computed in Ref. [21] [32].
The phonon dispersions and 𝑒-ph perturbation potentials on coarse q-point grids are
computed with DFPT [15]. We employ the perturbo code [9] to compute the 𝑒-ph
matrix elements on coarse Brillouin zone grids with 10 × 10 × 10 k- and q-points
for silicon and 8 × 8 × 8 k- and q-points for PbTiO3 . The Wannier functions are
computed with the Wannier90 code [28] and employed in perturbo [9] to interpolate
the short-range 𝑒-ph matrix elements.
We compute the scattering rates and electron mobility using the perturbo code [9].
Briefly, the band- and k- dependent 𝑒-ph scattering rate Γ𝑛k is obtained as
Γ𝑛k =

2𝜋 ∑︁
|𝑔𝑚𝑛𝜈 (k, q)| 2
ℏ 𝑚𝜈q
[(𝑁 𝜈q + 1 − 𝑓𝑚k+q )𝛿(𝜀 𝑛k − 𝜀 𝑚k+q − ℏ𝜔 𝜈q )

(4.11)

+ (𝑁 𝜈q + 𝑓𝑚k+q )𝛿(𝜀 𝑛k − 𝜀 𝑚k+q + ℏ𝜔 𝜈q )],
where 𝜀 𝑛k and ℏ𝜔 𝜈q are the electron and phonon energies, respectively, and 𝑓𝑛k
and 𝑁 𝜈q the corresponding temperature-dependent occupations. The scattering
rate can be further divided into the long-range part [4], Γ𝑛𝐿k , by replacing |𝑔| 2
in Eq. (4.11) with 𝑔 L . The carrier mobility is computed using 𝜇 = 𝜎/(𝑛𝑐 𝑒),
where 𝜎 is the electrical conductivity and 𝑛𝑐 is the carrier concentration. The
electrical conductivity 𝜎 is computed within the relaxation time approximation of
the Boltzmann transport equation [9, 33]:
∫ +∞
𝜎𝛼𝛽 = 𝑒
𝑑𝐸 (−𝜕 𝑓 /𝜕𝐸)Σ𝛼𝛽 (𝐸, 𝑇),
(4.12)
−∞

where Σ𝛼𝛽 (𝐸, 𝑇) is the transport distribution function at energy 𝐸,
𝑠 ∑︁
𝜏𝑛k (𝑇)𝑣 𝑛𝛼k 𝑣 𝑛k 𝛿(𝐸 − 𝜀 𝑛k ),
Σ𝛼𝛽 (𝐸, 𝑇) =
Nk Ω

(4.13)

𝑛k

which is computed in perturbo using the tetrahedron integration method [34]. Above,
𝑠 is the spin degeneracy, Nk is the number of k-points, 𝑣 𝑛k is the band velocity, and
𝜏𝑛k = (Γ𝑛k ) −1 is the relaxation time. The mobility is computed with non-degenerate
electron concentrations of 1015 cm−3 for silicon and 1017 cm−3 for PbTiO3 . To fully
converge the scattering rates and mobility, we use 𝑒-ph matrix elements evaluated
on fine Brillouin zone grids with 200 × 200 × 200 k-points and 8 × 106 random
q-points 1.
1 Uniform grids and random sampling can both be used to converge the scattering rate as long

as a sufficiently large number of q-points is employed. In our experience, random sampling, which

S ilic o n
L O

) (e V /Å )

T O

L A

to t( q

L A

D F P T
W a n n . o n ly
W a n n . + q u a d .
0 .5 L

0 .5 K

Figure 4.2: Mode-resolved 𝑒-ph coupling strength [see Eq. (4.10)] in silicon, computed using the lowest valence band. The electron momentum k is fixed at the
Γ point and the phonon wave-vector q is varied along high-symmetry lines in the
Brillouin zone. Benchmark results from DFPT (black circles) are compared with
Wannier interpolation with the quadrupole 𝑒-ph interaction included (orange line)
or neglected (blue line). The coarse-grid q-points are indicated with vertical lines.
4.3

Results

Quadrupole effect on the 𝑒-ph matrix elements
The long-range quadrupole 𝑒-ph interaction is present in a wide range of semiconductors and insulators, where the atomic dynamical quadrupoles are in general
non-zero. We illustrate this point by studying silicon, a simple nonpolar semiconductor in which the Born charges — and thus the Fröhlich interaction — vanish
and the presence of long-range interactions is not immediately obvious. Figure 4.2
𝜈 (q) in Eq. (4.10), computed directly using
shows the 𝑒-ph coupling strength, 𝐷 tot
DFPT as a benchmark and compared with Wannier interpolation with and without
corresponds to Monte Carlo integration, usually leads to slightly faster convergence than uniform
grids. It also has the advantage that one can systematically improve the convergence by running
additional calculations, thus improving the sampling. For comparison, checking convergence with
uniform grids requires using increasingly denser uniform meshes, which is computationally more
expensive and inconvenient.

69
inclusion of the quadrupole term. The DFPT benchmark 𝑒-ph matrix elements for
optical modes approach a constant value as q → 0, as we show for the LO mode
in the Γ − 𝐿 direction and the transverse optical (TO) mode along Γ − 𝐾. This
trend is distinctive of the quadrupole 𝑒-ph interaction, which is of order 𝑞 0 in the
long-wavelength limit.
If the quadrupole term is neglected and all 𝑒-ph interactions are treated as shortranged, the 𝑒-ph matrix elements for optical modes in silicon incorrectly vanish
as q → 0. The interpolated values for optical modes are underestimated between
the Γ point, where the error is greatest, and its nearest-neighbor q-points in the
coarse grid, where the error vanishes. Outside this q-point region close to Γ, the
interpolated matrix elements without the quadrupole interaction still deviate from
the DFPT result, although the error is smaller than near Γ. When the quadrupole
term is included, the long-range 𝑒-ph interactions for the optical modes are captured
correctly, as can be seen for the Wannier plus quadrupole curves in Fig. 4.2. The
𝜈 (q) from DFPT, for the optical branches shown
root-mean-square deviation of 𝐷 tot
in Fig. 4.2, is 0.78 eV/Å when the quadrupole term is neglected versus 0.03 eV/Å
when the quadrupole term is included in the interpolation. This result highlights the
importance of the quadrupole term to correctly capture long-range 𝑒-ph interactions
in nonpolar semiconductors.
Observe also how for acoustic modes in silicon the quadrupole term has a nearly
negligible effect, as we show for the longitudinal acoustic (LA) mode in Fig. 4.2. As
contracting the dynamical quadrupoles Q𝜅 with a rigid shift of the lattice leads to a
vanishing quadrupole contribution [18], one can obtain the quadrupole acoustic sum
rule 𝛼 𝑄 𝜅,𝛼𝛽𝛾 = 0 for nonpolar materials [18]. This sum rule, which is satisfied by
the dynamical quadrupole values we employ for silicon [21], leads to a negligible
quadrupole correction for acoustic modes in the long-wavelength limit. Though we
focus on silicon in this work, on the basis of our results we expect sizable quadrupole
contributions for optical modes, and negligible for acoustic modes, in all nonpolar
semiconductors.
The quadrupole 𝑒-ph interaction is particularly critical in piezoelectric materials, as discussed here for tetragonal PbTiO3 , a prototypical piezoelectric insulator.
Piezoelectric materials are polar noncentrosymmetric systems with non-zero Born
charges. As a result, the dipole Fröhlich interaction is dominant for LO modes near
q → 0 due to its 1/𝑞 divergence. The quadrupole contribution is expected to be
important for TO and acoustic modes (the quadrupole acoustic sum rule does not
hold for polar noncentrosymmetric crystals).

L O

P b T iO

) (e V /Å )

L A

to t( q

T A

0 .2 M

L A

Γ Γ

D F P T
D ip o le o n ly
D ip . + q u a d .

T A

0 .2 X

0 .2 Z

Figure 4.3: Mode-resolved 𝑒-ph coupling strength [see Eq. (4.10)] in tetragonal
PbTiO3 , computed using the lowest conduction band. The initial electron momentum is fixed at the Γ point and the phonon wave-vector q is varied along highsymmetry lines in the Brillouin zone. Benchmark results from DFPT (black circles)
are compared with Wannier interpolation plus the Fröhlich interaction (blue line)
and Wannier interpolation plus the Fröhlich and quadrupole interactions (orange
line). Note that two LO branches with coupling strength exceeding the y-axis limit
are not shown.
𝜈 (q) in Eq. (4.10), for the DFPT
Figure 4.3 shows the 𝑒-ph coupling strength, 𝐷 tot
benchmark in tetragonal PbTiO3 , and compares it with interpolated results that include only the Fröhlich dipole interaction or both the Fröhlich and the quadrupole
interactions. The short-range interactions are included through Wannier interpolation in both cases. When only the Fröhlich dipole interaction is included, the 𝑒-ph
matrix elements deviate dramatically from the DFPT results. The values are either
overestimated or underestimated depending on the phonon mode considered, with
deviations from DFPT that depend strongly on the direction in which q approaches
Γ due to the non-analytic character of the long-range 𝑒-ph interactions. When the
quadrupole 𝑒-ph interaction is taken into account, the interpolated 𝑒-ph coupling
strength matches the DFPT result very accurately for all phonon modes. For LO
modes, the quadrupole correction is moderate due to the dominant Fröhlich term
near q = 0. For other optical and acoustic modes with a finite 𝑒-ph coupling at

71
q = 0, the quadrupole term removes the large error in the dipole-only results (up to
an order of magnitude) and gives 𝑒-ph matrix elements in nearly exact agreement
with DFPT. For the branches shown in Fig. 4.3, the root-mean-square deviation of
𝜈 (q) from DFPT is 0.46 eV/Å for dipole-only results versus 0.03 eV/Å for our
𝐷 tot
dipole plus quadrupole interpolation scheme. It is clear that the quadrupole term is
essential in piezoelectric materials for all phonon modes.
Contrary to silicon and nonpolar materials, the quadrupole term has a large effect
for acoustic modes in piezoelectric materials, where it is one of the two contributions to the so-called piezoelectric 𝑒-ph interaction [24]. Expanding the phonon
(1)
eigenvectors at q → 0 as e𝜈q ≈ e𝜈(0)
q + 𝑖q · e𝜈 q , one finds two contributions of order
𝑞 0 [18]. One is from the Born charges, Z𝜅 e𝜈(1)
q , and is a dipole-like interaction generated by atoms with a net charge experiencing different displacements due to strain
from an acoustic mode. The other is from the dynamical quadrupoles, Q𝜅 e𝜈(0)
q , and
is associated with a clamped-ion electronic polarization [35]. The ab initio Fröhlich interaction includes only the former term, namely the strain component of the
piezoelectric 𝑒-ph interaction, and thus the dipole-only scheme leads to large errors
for acoustic phonons in PbTiO3 (see Fig. 4.3) as it neglects the important electronic
quadrupole contribution. Until now, the ab initio Fröhlich term has been mistakenly
thought to fully capture piezoelectric 𝑒-ph interactions. Our results demonstrate
that both dipole and quadrupole terms are essential for accurate acoustic mode 𝑒-ph
interactions in piezoelectric materials [36]. The relative magnitude of the strain and
quadrupole contributions is material dependent; the two terms can nearly cancel
each other out, as we have shown elsewhere for GaN [29], or their ratio can be mode
and phonon wave-vector dependent, as we find in PbTiO3 .

Quadrupole contribution to the scattering rate
Because the quadrupole interaction has a significant effect on the 𝑒-ph matrix elements, we expect that it also plays a role in calculations of the 𝑒-ph scattering rate
and mobility. Figure 4.4(a) shows both the quadrupole contribution and the total
𝑒-ph scattering rate in silicon at 300 K for electron energies near the conduction
band minimum. We find that the quadrupole contribution to the scattering rate is
about 1% of the total scattering rate at temperatures between 100−400 K. At electron
energies below the optical phonon emission threshold in silicon (ℏ𝜔O ≈ 65 meV
relative to the conduction band minimum), absorption and emission of acoustic
phonons dominate the scattering processes, and thus we find a small correction due

Figure 4.4: Room temperature scattering rate versus electron energy (referenced
to the conduction band minimum) in (a) silicon and (b) PbTiO3 . For silicon, we
plot the quadrupole contribution multiplied by 100 (orange) and the total scattering
rate (black), which includes the short-range and the quadrupole contributions. For
PbTiO3 , we show the long-range scattering rate computed using only the Fröhlich
interaction (blue) or both the Fröhlich and quadrupole interactions (orange).

to the quadrupole interaction, which minimally affects acoustic modes in silicon.
Since the quadrupole acoustic sum rule holds only in the long wavelength limit, the
quadrupole interaction can still contribute to finite-q acoustic scattering, as is shown
by the fact that the quadrupole scattering rate at energy below ℏ𝜔O is proportional
to the total scattering rate. The quadrupole contribution increases sharply above the
optical emission threshold because the quadrupole term is greater for optical modes
in silicon. For the same reason, the relative contribution of the quadrupole term
increases slightly with temperature in the 100−400 K range, varying from 1% of the
total scattering rate at 100 K to 1.5% at 400 K.
The effect of the quadrupole interaction on the scattering rates is greater in PbTiO3 .
Our analysis focuses on the 𝑒-ph scattering rate due to the long-range 𝑒-ph interactions, although similar conclusions hold for the total scattering rate. Figure 4.4(b)
shows the long-range 𝑒-ph scattering rate in PbTiO3 at 300 K as a function of
electron energy, comparing results that include only the dipole Fröhlich interaction

73
with results from our approach including both the dipole and quadrupole terms.
The scattering rate from the long-range 𝑒-ph interactions is lower at all energies
when the quadrupole term is taken into account. The difference is greatest near the
band edge, where the scattering rate due to the dipole interaction alone is 0.075 fs−1
versus a 50% smaller value of 0.050 fs−1 for dipole plus quadrupole.
These trends can be understood on the basis of the 𝑒-ph matrix element analysis in
𝜈 (q), which
Fig. 4.3. The errors found when neglecting the quadrupole term in 𝐷 tot
is proportional to the absolute value of the matrix elements [see Eq. (4.10)], are
amplified in calculations of the scattering rate, which is proportional to the square
𝜈 (q) are in the q → 0
of the matrix elements. The largest errors we find for 𝐷 tot
limit, especially for the acoustic modes. For example, for the LA mode in the
𝜈 (q) from the dipole-only calculation
Γ − 𝑀 and Γ − 𝑋 directions, the value of 𝐷 tot
is 0.17 eV/Å compared to a twice-greater value of 0.40 eV/Å when the quadrupole
term is included. This leads to a four-fold increase of the LA mode scattering rate
upon including the quadrupole interaction. Opposite to the silicon case, in PbTiO3
the relative magnitude of the quadrupole correction is greater at lower temperatures
because the quadrupole interaction is stronger for acoustic modes. Near the band
edge, we find quadrupole corrections to the long-range scattering rate ranging from
97% at 100 K to 38% at 400 K. Given that low-energy electronic states near the
band edge govern transport properties, including the quadrupole term is critical to
accurately computing electronic transport.

Quadrupole contribution to the mobility
The effect of the quadrupole 𝑒-ph interaction on the mobility is noteworthy. Figure 4.5(a) shows the temperature dependent electron mobility in silicon computed
with and without the quadrupole term. Including the quadrupole interaction reduces the computed mobility by approximately 5−10% due to the increased 𝑒-ph
coupling strength and scattering rates. For example, the computed mobility at
300 K is 1390 cm2 /Vs when including the quadrupole interaction versus a value
of 1473 cm2 /Vs with the conventional interpolation approach in which all 𝑒-ph
interactions in silicon are treated as short-ranged. This discrepancy is due to the
underestimation of the 𝑒-ph coupling strength for optical modes in the conventional
approach, especially at small values of q as shown in Fig. 4.2.
In silicon, 𝑒-ph scattering mediates both intravalley and intervalley processes. The
quadrupole interaction affects mainly small-q intravalley processes associated with

74
(a )

(b ) 2 5 0

S ilic o n

1 5 0 0 0

W a n n ie r o n ly
W a n n ie r + q u a d .

/V s )

M o b ility ( c m

P b T iO

1 0 0 0 0

2 0 0

D ip o le o n ly
D ip o le + q u a d .

1 5 0

1 0 0
5 0 0 0
5 0

1 0 0

2 0 0

3 0 0

T e m p e ra tu re (K )

4 0 0

1 0 0

2 0 0

3 0 0

4 0 0

T e m p e ra tu re (K )

Figure 4.5: Computed temperature-dependent electron mobility in (a) silicon and (b)
tetragonal PbTiO3 . The plot compares the mobility obtained when the quadrupole
𝑒-ph interaction is included (orange squares) or neglected (blue circles). The PbTiO3
results are for transport in the basal 𝑥𝑦 plane.

optical phonons. However, intravalley processes — particularly those associated
with acoustic phonons — are dominant only at low temperature, while at higher
temperatures intervalley processes mediated by large-q phonons are dominant [37].
As a result, the intravalley optical phonon scattering processes mediated by the
quadrupole interaction are active mainly at low temperature and are overall weaker
than other scattering contributions in silicon, including acoustic intravalley scattering at low temperature. The contribution of the quadrupole correction to the
mobility is thus maximal at low temperature and overall relatively small.
Although we focus on silicon, we expect that these trends apply in general to nonpolar
semiconductors because small-q optical 𝑒-ph coupling will consistently be underestimated without the quadrupole term. The long-range quadrupole 𝑒-ph interaction
is thus surprisingly manifest in the transport properties of nonpolar materials.
We find an opposite trend in PbTiO3 , in which including the quadrupole interaction
increases the mobility by 10−25% between 100−400 K, as seen in Fig. 4.5(b). The
quadrupole term gives a larger correction at lower temperatures, reaching values up

75
to ∼25% at 100 K. This result is due to the dominant acoustic mode contribution at
low temperatures together with the large quadrupole correction for acoustic modes
in piezoelectric materials. At higher temperatures, where optical mode scattering
is dominant and acoustic scattering less important, the quadrupole contribution is
smaller, only about 10% at 400 K. Due to differences in the quadrupole interaction
for different phonon modes and to varying mode contributions to the mobility as
a function of temperature, including the quadrupole term corrects the temperature
dependence of the mobility [29] and is essential in piezoelectric materials.

4.4

Discussion

We briefly discuss a technical aspect of the 𝑒-ph matrix element interpolation. The
treatment of long-wavelength perturbations with wave-vector q → 0 in DFPT is
critical in semiconductors and insulators [16, 38]. The lattice-periodic part of the
phonon perturbation potential, Δ𝑣 q (r), is the sum of a Coulomb and an exchangecorrelation contribution,
Δ𝑣 q (r) = Δ𝑣 q,𝐶 (r) + Δ𝑣 q,𝑋𝐶 (r).

(4.14)

The Coulomb contribution Δ𝑣 q,𝐶 (r) combines the variation of the Hartree and
electron-nuclei interactions. Its integral over the unit cell [38],
𝑑r 𝑣 q,𝐶 (r),
(4.15)
Δ(q) =
Ω Ω
is well-behaved for insulators (and semiconductors) at finite q values, but is illdefined at q = 0. First-principles codes such as Quantum ESPRESSO [30] subtract
Δ(q) from the perturbation potential at q = 0, thus making it discontinuous at q = 0.
Therefore, due to both the discontinuity at q = 0 and the non-analytic behavior
near q = 0, the 𝑒-ph matrix elements are challenging to interpolate in the longwavelength limit.
In our scheme, we identify the quadrupole interaction as the key long-range term in
nonpolar materials, and remove the non-analytic behavior near q = 0 on the coarse
grid by subtracting the quadrupole term. This strategy improves the interpolation
near q = 0 in nonpolar materials, at once capturing the correct physics and smoothing
the coarse-grid matrix element to be interpolated. Due to the non-analytic behavior,
denser DFPT grids cannot fully remove the interpolation error if the quadrupole term
is not subtracted on the coarse grid. For polar materials such as PbTiO3 , the nonanalytic behavior is due to both the dipole (Fröhlich) and quadrupole long-range 𝑒-ph

76
interactions. By subtracting both terms in our scheme in polar materials, the coarsegrid matrix elements to be interpolated are made smooth and the interpolation
approach more reliable. The non-analytic behavior of the Coulomb potential is
correctly reconstructed by adding back the dipole (in polar materials) and quadrupole
(in all insulators) contributions after interpolation.
4.5

Conclusion

In summary, we developed an accurate approach for computing the quadrupole 𝑒-ph
interaction from first principles. This advance resolves the outstanding problem of
correctly quantifying long-range 𝑒-ph interactions for all phonon modes in semiconductors and insulators. Our results clearly show that the quadrupole interactions are
crucial for obtaining accurate 𝑒-ph matrix elements, scattering rates and electronic
transport properties. The quadrupole effect is particularly apparent in piezoelectric
materials such as wurtzite GaN [29] and PbTiO3 , in which neglecting the quadrupole
interaction leads to large and uncontrolled errors. The method introduced in this
work enables accurate calculations of electrical transport, thermoelectric properties,
and superconductivity in a wide range of materials.
Note added. Recently, we became aware of a related work by another group that
reaches similar conclusions about the importance of the dynamical quadrupole term
to obtain an accurate physical description of e-ph interactions [39, 40].

77
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82
Chapter 5

AB INITIO ELECTRON-PHONON INTERACTIONS IN
CORRELATED ELECTRON SYSTEMS
5.1

Introduction

Strongly correlated materials are at the center of exciting advances in condensed
matter physics. These correlated electron systems (CES) can host states of matter ranging from high-temperature superconductivity [1] to Mott transitions [2, 3],
colossal magnetoresistance [4] and multiferroicity [5]. Electron-phonon (𝑒-ph) interactions play an important role in these phenomena, often governing their origin
and temperature dependence. A promising direction to study quantitatively 𝑒-ph
interactions in CES is using first-principles calculations, where one employs density functional theory (DFT) to compute the electronic structure, density functional
perturbation theory [6] (DFPT) for the lattice dynamics, and their combination to
obtain the 𝑒-ph coupling [7, 8]. This approach can successfully describe 𝑒-ph interactions and electron dynamics in a wide range of materials [7–20]. Recent work
has extended this framework by combining the GW and DFPT approaches to reveal
correlation-enhanced e-ph interactions in metallic systems [21, 22].
However, computing 𝑒-ph interactions in CES remains challenging as standard DFT
usually fails to describe their ground state, mainly due to self-interaction errors
in open subshells of localized d or 𝑓 electrons. In addition, correlated transition
metal oxides (TMOs) often exhibit strong 𝑒-ph coupling and polaron effects, requiring treatments beyond lowest-order perturbation theory [12]. Widely used firstprinciples approaches to compute the ground state of CES include DFT+U [23–26],
hybrid functionals [27], and dynamical mean-field theory [28, 29]. Developing accurate 𝑒-ph calculation in any of these frameworks is an important open challenge; if
fulfilled, it would advance investigations of the rich physics of CES and significantly
expand the scope of first-principles studies of 𝑒-ph interactions.
The DFT+U method [23–26] is particularly promising to mitigate the self-interaction
error of DFT, using the Hubbard correction to better capture the physics of localized
d electrons [30, 31]. It can predict the ground state of various families of correlated
TMOs, including Mott insulators [23], high-temperature superconductors [32] and
multiferroics [33, 34]. Its linear response variant, DFPT+U, has been employed
successfully to study the lattice dynamics of TMOs [35–38]. As the Hubbard-U

83
value can be computed ab initio [39], as we do here, the framework is entirely free
of empirical parameters.
In this chapter, we show calculations of 𝑒-ph interactions in the framework of
DFT+U, focusing on a prototypical Mott insulator, cobalt oxide (CoO), as a case
study. While DFT predicts CoO to be a dynamically unstable metal, DFT+U correctly predicts its antiferromagnetic insulating ground state [36, 40, 41]. We thus find
that the long-range Fröhlich 𝑒-ph interaction is restored in DFPT+U, and unphysical
divergences of the 𝑒-ph coupling due to spurious soft modes are removed. With the
correct Fröhlich interaction in hand, we study the electron spectral function with a
cumulant approach, revealing the formation of a polaron state with sharp quasiparticle and satellite peaks at low temperature that broaden and disappear entirely at
room temperature. The Hubbard U-derived 𝑒-ph perturbation, missing in DFPT, is
found to act primarily on the partially filled d bands of each spin channel, showing
the impact of the 𝑑 electron Coulomb repulsion on 𝑒-ph interactions. The DFT+U
𝑒-ph calculations developed in this work are poised to advance the understanding of
𝑒-ph coupling, transport and superconductivity in strongly correlated materials.

5.2

Methods

For quantitative studies of 𝑒-ph interactions, of key interest are the 𝑒-ph matrix
𝜎 (k, q), which quantify the probability amplitude for an electron in a
elements, 𝑔𝑚𝑛𝜈
Bloch state |𝜓𝑛k𝜎 ⟩, with band index 𝑛, spin 𝜎 and crystal momentum k, to scatter
into a final state |𝜓𝑚 k+q 𝜎 ⟩ by emitting or absorbing a phonon with mode index 𝜈,
wave-vector q, energy ℏ𝜔 𝜈q , and displacement eigenvector e𝜈q [7, 42, 43],
𝑔𝑚𝑛𝜈
(k, q) =

1
ℏ 2 ∑︁ e𝜈q
⟨𝜓𝑚 k+q 𝜎 | 𝑑q 𝐼 𝑉ˆ 𝜎 |𝜓𝑛k𝜎 ⟩,
2𝜔 𝜈q
𝑀𝐼

(5.1)

where 𝑑q 𝐼 𝑉ˆ 𝜎 ≡ 𝑝 𝑒𝑖q ·R 𝑝 𝑑 𝑝𝐼 𝑉ˆ 𝜎 is the 𝑒-ph perturbation due to the change of the
potential acting on an electron with spin 𝜎 from a unit displacement of atom 𝐼 (with
mass 𝑀𝐼 and located in the unit cell at R 𝑝 ).
In DFPT+U, besides the usual Kohn-Sham (KS) perturbation potential [42], there
is an additional term from the perturbation of the Hubbard potential [36]:
𝑑𝑉ˆ 𝜎 = 𝑑𝑉ˆKS
+ 𝑑𝑉ˆHub

(5.2)

84
This Hubbard perturbation potential is the sum of projector and occupation-matrix
derivative terms [36],
∑︁
𝐼 𝛿𝑚1 𝑚2
𝐼𝜎
𝑑𝑉Hub =
− 𝑛𝑚1 𝑚2 𝜕 𝑃ˆ𝑚𝐼 1 𝑚2
𝐼𝑚 1 𝑚 2
(5.3)
∑︁
𝐼𝜎
𝑈 𝐼 (𝑑𝑛𝑚
𝑚1 𝑚2
1 𝑚2
𝐼𝑚 1 𝑚 2

where 𝑚 1 and 𝑚 2 are magnetic quantum numbers of the 3d orbitals, 𝑈 𝐼 is the
𝐼𝜎
effective Hubbard parameter for atom 𝐼, and 𝑛𝑚
is the occupation matrix for
1 𝑚2
orbitals with magnetic quantum numbers 𝑚 1 and 𝑚 2 on atom 𝐼,
∑︁
𝐼𝜎
𝑛𝑚
⟨𝜓𝑛k𝜎 | 𝑃ˆ𝑚𝐼 2 𝑚1 |𝜓𝑛k𝜎 ⟩ .
(5.4)
1 2
𝑛k
𝐼 ,
Here, 𝑃ˆ is the generalized projector on the space of the localized atomic orbitals 𝜑𝑚
𝑃ˆ𝑚𝐼 2 𝑚1 = 𝑆ˆ |𝜑𝑚
⟩ ⟨𝜑𝑚
| 𝑆,

(5.5)

and 𝑆ˆ is the overlap operator of the ultrasoft pseudopotential [44]. In Eq. 5.3, the projector derivative term is efficiently computed with an analytical formula [36] while
𝐼𝜎
the occupation-matrix derivative 𝑑𝑛𝑚
is computed with DFPT and includes con1 𝑚2
tributions from the response of the wave functions to the atomic displacements [36]:
∑︁
𝐼𝜎
𝑑𝑛𝑚
⟨𝜓𝑛k𝜎 | 𝜕 𝑃ˆ𝑚𝐼 2 𝑚1 |𝜓𝑛k𝜎 ⟩
1 2
∑︁ 𝑛k
⟨𝑑𝜓𝑛k𝜎 | 𝑃ˆ𝑚𝐼 2 𝑚1 |𝜓𝑛k𝜎 ⟩ + ⟨𝜓𝑛k𝜎 | 𝑃ˆ𝑚𝐼 2 𝑚1 |𝑑𝜓𝑛k𝜎 ⟩ .

(5.6)

𝑛k

We apply this framework to investigate the 𝑒-ph interactions and electron spectral
functions in CoO, focusing on the effects of the Hubbard U correction. The ground
state electronic structure of CoO is obtained with collinear spin-polarized DFT+U
calculations in a plane-wave basis using the Quantum ESPRESSO code [45]. We use
the PBEsol exchange-correlation functional [46] and ultrasoft pseudopotentials [44]
from the GBRV library [47]. We employ a 4-atom rhombohedral unit cell with
relaxed lattice constants (𝑎 = 5.206 Å, 𝑏 = 3.019 Å, 𝑐 = 3.009 Å and angle
𝛽 = 125.05◦ ) and kinetic energy cutoffs of 60 Ry for the wave functions and 720 Ry
for the charge density. Results for the 4-atom rhombohedral and 8-atom monoclinic
unit cells [36] are nearly identical, as we have verified, so we choose the 4-atom unit
cell for simplicity. Leveraging a recent implementation of DFPT+U [36, 43], we
compute the lattice dynamics and 𝑒-ph perturbation potentials on coarse irreducible

85
q-point grids. We then rotate the KS and Hubbard perturbation potentials with
the Perturbo code to obtain the 𝑒-ph matrix elements in the full Brillouin zone
(BZ), using coarse grids with 8 × 8 × 8 k and q points. The Wannier functions
are obtained with the Wannier90 code [48] and used in Perturbo [42] to interpolate
the 𝑒-ph matrix elements to finer grids. We use atomic orbitals as the basis for
the Hubbard manifold. Our method is free of adjustable parameters, including the
Hubbard U value, 𝑈 = 4.55 eV for Co 3𝑑 states, which is determined ab initio with
a linear response approach [36, 39, 49].
Using these quantities, we compute the lowest-order (Fan-Migdal) 𝑒-ph self-energy,
Σ𝑛k𝜎 (𝜔, 𝑇), at temperature 𝑇 and electron energy 𝜔, as implemented in Perturbo [12,
42]; the imaginary part is computed off-shell on a fine energy grid while the real part
is evaluated on-shell at the band energy 𝜀 𝑛k𝜎 . To capture strong 𝑒-ph interactions
beyond the lowest-order, we use the finite-temperature cumulant approach described
in Ref. [12]. The latter allows us to obtain the temperature-dependent retarded
Green’s function 𝐺 𝑛𝑅k𝜎 (𝜔) and the resulting electron spectral function, 𝐴𝑛k𝜎 (𝜔) =
−Im𝐺 𝑛𝑅k𝜎 (𝜔)/𝜋, which includes polaron effects such as band renormalization and
satellite peaks [12, 50–52]. Our framework therefore captures two key aspects of
the physics of correlated TMOs, the effects of the localized Coulomb repulsion
through DFT+𝑈 and the strong 𝑒-ph coupling and its temperature dependence with
the finite-temperature cumulant approach.

5.3

Results and discussion

The 𝑒-ph matrix elements from DFPT+U, which include effects from both the KS
and Hubbard perturbations, are computed for the 𝑑 bands of CoO in Fig. 5.1(a) and
compared with results from standard DFPT [53]. The Hubbard U correction has
a dramatic effect: the two sets of 𝑑 band 𝑒-ph matrix elements differ widely, for
all phonon modes and everywhere in the BZ. The largest difference occurs near the
zone center, where the DFPT+U results show the presence of the Fröhlich interaction [54], whereby the 𝑒-ph matrix elements diverge as 𝑞 → 0 for the longitudinal
optical (LO) modes [10], whereas in plain DFPT they approach a finite value. The
reason for this difference is subtle: although CoO is a semiconductor with a 2.5 eV
band gap [55], DFT fails to properly describe its d electrons due to self-interaction
errors and incorrectly predicts CoO to be a metal, so the Born effective charges
and the Fröhlich interaction vanish in DFPT. When the Hubbard U correction is
included, the self-interaction errors are mitigated and CoO is correctly predicted to

Figure 5.1: (a) Comparison between the 𝑒-ph matrix elements in CoO computed with DFPT+U (orange) and standard DFPT (blue). Shown is 𝑔𝜈𝜎 (q) ≡
𝜎 (k = 0, q) 2 /𝑁 ) 1/2 for all phonon modes 𝜈, where the phonon wave( 𝑚𝑛 𝑔𝑚𝑛𝜈
vector q is varied along high-symmetry BZ lines and the summation runs over the
𝑁 𝑏 = 10 spin-up Co 3d bands [53]. (b) CoO phonon dispersions overlaid with a
log-scale color map of 𝑔𝜈𝜎 (q) computed with DFPT+U (colored line). The CoO
phonon dispersions from standard DFPT (gray line) are given for comparison, with
imaginary frequencies shown as negative values.

be a polar semiconductor with divergent 𝑒-ph coupling for LO phonons near the
zone center. This hallmark of the Fröhlich interaction is of critical importance for
studies of transport and carrier dynamics in polar materials [10–12].

87
Figure 5.1(b) highlights the dramatic differences in the phonon dispersions computed
with DFPT+U and plain DFPT [36]. In the latter, the ground state is dynamically
unstable and the phonon dispersions exhibit soft phonon modes with imaginary
frequencies. These errors are propagated to the 𝑒-ph interactions, resulting in 𝑒-ph
matrix elements with unphysical divergences — near the 𝐹, 𝑀, 𝐿, and 𝐻 points of
the BZ in Fig. 5.1(a) — corresponding to zero-frequency phonon modes [11]. In
DFPT+U, the ground state is stabilized to the correct antiferromagnetic phase, and
the phonon dispersions are significantly improved [36] and in very good agreement
with experiments (see Supplemental Material); the soft phonon modes are removed
entirely and the 𝑒-ph matrix elements are well behaved throughout the BZ, without
spurious divergences. These results underscore the importance of the Hubbard U
correction for describing the electronic ground state and the resulting 𝑒-ph interactions in correlated TMOs.
In CoO, correcting the wave functions and charge density with DFT+U provides
the main improvement to the 𝑒-ph coupling. To illustrate this point, Fig. 5.2 shows
that the 𝑒-ph matrix elements computed with the KS perturbation alone but with
DFT+U wave functions, 𝑔˜ KS ∝ ⟨𝜓H𝑢𝑏 | 𝑑𝑉ˆKS |𝜓H𝑢𝑏 ⟩, can capture both the Fröhlich
interaction and the main trends in the 𝑒-ph coupling. Yet, the Hubbard perturbation
potential 𝑑𝑉ˆHub , which describes the effect of the lattice dynamics on the Hubbard U
correction, also gives an important contribution. Figure 5.2 compares 𝑔˜ K𝑆 with the
total DFPT+U 𝑒-ph matrix elements, 𝑔t𝑜𝑡 ∝ ⟨𝜓H𝑢𝑏 | 𝑑𝑉ˆKS + 𝑑𝑉ˆHub |𝜓H𝑢𝑏 ⟩, showing
that both the KS and Hubbard terms are needed for quantitative accuracy. Direct
DFPT+U calculations, shown in Fig. 5.2 as a benchmark, confirm this point and
also validate our interpolation procedure.
Further analysis reveals that the contribution of the Hubbard 𝑒-ph perturbation is
strongly band dependent and acts primarily on the partially filled 3𝑑 states of each
spin channel [56]. To demonstrate this result, we compute the imaginary part of the
𝑒-ph self-energy [42] with contribution from only the Hubbard 𝑒-ph perturbation,
and map it on the electronic spin-up band structure in Fig. 5.3(a). The plot shows the
selective contribution of the Hubbard perturbation to 𝑒-ph processes in the partially
filled 3d bands and the nearly negligible contribution in the completely filled 3d
bands. The situation is analogous for the spin-down bands.
This trend is confirmed by studying the 𝑒-ph matrix elements in the Wannier basis,
𝑔𝑖 𝑗 (r 𝑝 ) ∝ ⟨𝜙𝑖 (0)| 𝑑𝑉ˆ (r 𝑝 ) |𝜙 𝑗 (0)⟩ [42], computed using Co 3d Wannier functions
𝜙𝑖 and 𝜙 𝑗 located on the same Co atom. These 𝑒-ph matrix elements decay exponentially with perturbation distance |r 𝑝 | due to the localized nature of the 3d

D F P T + U

C o O

g to t
g K S

n s

to t( q

) (e V /Å )

1 0

Figure 5.2: Comparison between the 𝑒-ph coupling from the KS potential contribution alone (cyan line) and the total result including the Hubbard correction (orange
line). In each case, we show the gauge-invariant 𝑒-ph coupling strength [42],
𝜈𝜎 (q) = (2𝜔 𝑀 Í
1/2 , computed respectively with 𝑒𝐷 tot
𝜈 q u𝑐 𝑛𝑚 𝑔𝑛𝑚𝜈 (k = 0, q) /𝑁 𝑏 )
ph matrix elements 𝑔˜ K𝑆 and 𝑔t𝑜𝑡 , summing over all 𝑁 𝑏 = 13 spin-up valence bands.
The BZ labeling refers to an equivalent (distorted) rocksalt structure [36]. Direct
DFPT+ 𝑈 calculations (circles), shown as a benchmark, validate the Wannier interpolation. The arrows indicate the divergence due to the Fröhlich interaction.

Wannier functions. For these local 𝑒-ph interactions, we find that the KS and Hubbard contributions are nearly identical for the Co atom with partially filled spin-up
3d orbitals [Fig. 5.3(b)], whereas for the Co atom with completely filled spin-up
3d orbitals the Hubbard contribution is orders of magnitude smaller than the KS
contribution [Fig. 5.3(c)] [56].
In TMOs, due to the polar bonds, electrons typically couple strongly with LO
phonons via the Fröhlich interaction. In this common scenario, the 𝑒-ph interactions are strong enough to form large polarons, which can dominate transport and
electron dynamical processes. The dominant coupling of electrons with LO phonons
is clearly seen in Fig. 5.1(b), and thus we expect significant polaron effects in CoO.
To investigate them, we compute the electron spectral function with our recently
developed finite-temperature cumulant approach, using the DFPT+U 𝑒-ph matrix

Figure 5.3: (a) Band structure of CoO overlaid with the Hubbard contribution to
the imaginary part of the 𝑒-ph self-energy, Im(ΣHub ), for the representative case of
the spin-up bands. The right panel shows the projected density of states (PDOS)
of the partially and completely filled 3𝑑 orbitals. (b),(c) Comparison between the
spatial decay of the real-space 𝑒-ph matrix elements for (b) the partially filled and
(c) the completely filled 3𝑑 orbitals. Shown are the contributions from the KS
(black squares) and Hubbard (red circles) 𝑒-ph perturbations [see Eq. (5.2)] to the
maximum value of the Wannier basis matrix elements [8], 𝑔(r 𝑝 ) = max𝑖 𝑗 𝑔𝑖 𝑗 (r 𝑝 ) ,
normalized using the KS contribution. The inset in (b) is a schematic of the 𝑒-ph
matrix elements in the Wannier basis, showing the atomic displacement perturbation
at distance |r 𝑝 |.

elements as input [12].
Figure 5.4 shows the computed electron spectral functions at three temperatures
between 100 − 300 K, for an electronic state near the top of the valence band. The
spectral function at 100 K shows a sharp quasiparticle (QP) peak and two promi-

90
10

T = 100 K

2ωLO

A(ω) (eV−1)

-0.2

ωLO

QP peak

T = 200 K

T = 300 K

0.0

0.2

Electron energy ω (eV)

0.4

Figure 5.4: Electron spectral function in CoO, computed at three temperatures from
100 K to 300 K, for the highest valence band at crystal momentum k = F. In each
panel, the zero of the electron energy 𝜔 is set to the band energy obtained from
DFT+U calculations.
nent sideband peaks, respectively at energies 𝜔L𝑂 and 2𝜔L𝑂 below the QP peak,
where 𝜔L𝑂 ≈ 65 meV is the energy of the zone center LO phonon with strongest
𝑒-ph coupling [see Fig. 5.1(b)]. These phonon sidebands are a hallmark of strong
𝑒-ph coupling and polaron effects [12]. Note that our calculations are performed
with the Fermi energy lying above the valence band edge (a situation corresponding
to lightly 𝑝-doped CoO) so the QP peak corresponds to a holelike QP excitation.
Accordingly, the phonon sidebands appear at energy lower than the QP peak [57]
and are associated with the simultaneous excitation of a holelike QP plus one or two
LO phonons, respectively.
Due to a well-known sum rule, the spectral function integrates to one over energy,
and thus the phonon sidebands transfer spectral weight from the QP peak. In CoO,
the QP spectral weight is strongly renormalized to a value of 0.2 at 100 K, with
significant weight transfer to the phonon sidebands due to the strong 𝑒-ph interactions. As the temperature increases from 100 to 200 K, the QP peak becomes
broader and overlaps with the phonon sidebands. At 300 K and higher temperatures,
the peaks merge into a continuous background and the QP peak representing the

91
original electronic state melts entirely into a polaron excitation. As the Fröhlich
interaction making up the large polaron is entirely missing in DFT, our study of
polaron effects in TMOs is enabled by the correct account of 𝑒-ph interactions in
the DFT+U framework developed in this work.

5.4

Conclusion

In summary, we introduced an ab initio approach enabling quantitative calculations
of 𝑒-ph interactions and polarons in correlated systems. Our method can be applied
broadly to various families of strongly correlated materials with localized d or f
electrons, leveraging the framework of parameter-free DFT+U. As shown in this
work, our formalism can capture the strong coupling of electron, spin, and lattice
degrees of freedom in CES and their combined effect on the 𝑒-ph interactions,
paving the way for quantitative studies of the rich physics of various families of
strongly correlated materials.
5.5

Supplemental Material

Additional derivations on the projector derivatives
𝐼 is
The generalized projector 𝑃ˆ on the space of the localized atomic orbitals 𝜑𝑚
defined as
ˆ 𝑚
𝑃ˆ𝑚𝐼 2 𝑚1 = 𝑆|𝜑
⟩⟨𝜑𝑚
| 𝑆,

(5.7)

where 𝑚 1 and 𝑚 2 are magnetic quantum numbers, 𝐼 is the atomic index, and 𝑆ˆ
is the overlap operator in the ultrasoft (US) or projector-augmented-wave (PAW)
framework, which is defined as
∑︁
𝑆ˆ = 1 +
𝑞 𝐽𝜇𝜈 |𝛽 𝐽𝜇 ⟩⟨𝛽𝜈𝐽 |.
(5.8)
𝐽 𝜇𝜈

In Eq. (5.8), 𝛽 𝐽𝜇 and 𝛽𝜈𝐽 are the localized atom-centered projector functions for the
US or PAW schemes, labeled by an atomic (𝐽) and a state (greek letter) index, and
the coefficients 𝑞 𝐽𝜇𝜈 are integrals of the augmentation functions 𝑄 𝐽𝜇𝜈 (r):
𝑞 𝜇𝜈 =
𝑄 𝐽𝜇𝜈 (r)𝑑r.
(5.9)
The bare derivative of the generalized projector 𝜕𝐼𝛼 𝑃ˆ𝑚𝐼 2 𝑚1 with respect to a unit
displacement of atom 𝐼 in the direction 𝛼 becomes [36]
ˆ 𝑚
ˆ 𝑚
ˆ 𝑚
𝜕𝐼𝛼 𝑃ˆ𝑚𝐼 2 𝑚1 = |𝜕𝐼𝛼 ( 𝑆𝜑
)⟩⟨𝜑𝑚
| 𝑆ˆ + 𝑆|𝜑
⟩⟨𝜕𝐼𝛼 ( 𝑆𝜑
)|.

(5.10)

92
𝐼 ) can be further expanded as
ˆ 𝑚
In Eq. (5.10), the derivatives 𝜕𝐼𝛼 ( 𝑆𝜑
ˆ 𝑚
ˆ |𝜑𝑚
|𝜕𝐼𝛼 ( 𝑆𝜑
)⟩ = 𝜕𝐼𝛼 ( 𝑆)
⟩ + 𝑆ˆ |𝜕𝐼𝛼 (𝜑𝑚
)⟩,

(5.11)

where [see Eq. (5.8)]
ˆ =
𝜕𝐼𝛼 ( 𝑆)

∑︁

𝑞 𝐽𝜇𝜈 |𝜕𝐼𝛼 (𝛽 𝐽𝜇 )⟩⟨𝛽𝜈𝐽 | + |𝛽 𝐽𝜇 ⟩⟨𝜕𝐼𝛼 (𝛽𝜈𝐽 )| .

(5.12)

𝜇𝜈
𝐼 ) and 𝜕 (𝛽 𝐽 ) can be computed efficiently in reciprocal
The derivatives 𝜕𝐼𝛼 (𝜑𝑚
𝐼𝛼 𝜇
space, as discussed in Ref. [36].
In the case of norm-conserving pseudopotentials, 𝑆ˆ = 1 and thus Eqs. (5.7) and
(5.10) simplify to
𝑃ˆ𝑚𝐼 2 𝑚1 = |𝜑𝑚
⟩⟨𝜑𝑚
|,
(5.13)

and
𝜕𝐼𝛼 𝑃ˆ𝑚𝐼 2 𝑚1 = |𝜕𝐼𝛼 (𝜑𝑚
)⟩⟨𝜑𝑚
| + |𝜑𝑚
⟩⟨𝜕𝐼𝛼 (𝜑𝑚
)|.

(5.14)

7 0

P h o n o n e n e rg y (m e V )

6 0
5 0
4 0
3 0
2 0
1 0

Figure 5.5: CoO phonon dispersion in an equivalent (distorted) rock-salt cell. Shown
are the DFPT+𝑈 results from our work (black solid line, 8 × 8 × 8 coarse q-point
grid), DFPT+𝑈 results from Ref. [36] (red dashed line, 4×4×4 coarse q-point grid),
and experimental results from Ref. [58] (blue circles) and Ref. [59] (green squares).
Experimental data along the Γ-T direction were folded to account for the doubled
periodicity of the four-atoms rhombohedral unit cell along the [111] direction [36].

94
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Chapter 6

PERTURBO: A SOFTWARE PACKAGE FOR AB INITIO
ELECTRON-PHONON INTERACTIONS, CHARGE
TRANSPORT AND ULTRAFAST DYNAMICS
6.1

Introduction

PERTURBO is a software package for first-principles calculations of charge transport, spin dynamics, and ultrafast carrier dynamics in materials [1, 2]. The vision
behind PERTURBO is to provide a unified platform and a validated code that can
be applied broadly to compute the interactions, transport and ultrafast dynamics of
electrons and excited states in materials [3]. The goal is to facilitate basic scientific
discoveries in materials and devices by advancing microscopic understanding of
carrier dynamics, while creating a sustainable software element able to address the
demands of the computational physics community.
PERTURBO builds on established first-principles methods. It uses density functional theory (DFT) and density functional perturbation theory (DFPT) [4] as a
starting point for computing electron dynamics. It reads the output of DFT and
DFPT calculations, for now from the Quantum ESPRESSO (QE) code [5, 6], and
uses this data to compute electron interactions, charge transport and ultrafast dynamics. The current distribution focuses on electron-phonon (𝑒-ph) interactions
and the related phonon-limited transport properties [7], including the electrical conductivity, mobility and the Seebeck coefficient. It can also simulate the ultrafast
nonequilibrium electron dynamics in the presence of 𝑒-ph interactions. The developer branch, which is not publicly available yet, also includes routines for computing
spin [8], electron-defect [9, 10], and electron-photon interactions [11], as well as
advanced methods to compute the ultrafast dynamics of electrons and phonons in
the presence of electric and magnetic fields. These additional features will be made
available in future releases.
The transport module of PERTURBO enables accurate calculations of charge transport in a wide range of functional materials. In its most basic workflow, PERTURBO
computes the conductivity and mobility as a function of temperature and carrier concentration, either within the relaxation time approximation (RTA) or with an iterative
solution of the linearized Boltzmann transport equation (BTE) [12, 13]. The ultrafast dynamics module explicitly evolves in time the electron BTE (while keeping the

102
phonon occupations fixed), enabling investigations of the ultrafast electron dynamics starting from a given initial electron distribution [14]. Our routines can carry
out these calculations in metals, semiconductors, insulators, and 2D materials. An
efficient implementation of long-range 𝑒-ph interactions is employed for polar bulk
and 2D materials. Materials with spin-orbit coupling (SOC) are treated using fully
relativistic pseudopotentials [8, 13]. Both norm-conserving and ultrasoft pseudopotentials are supported. Quantities related to 𝑒-ph interactions can be easily obtained,
stored and analyzed.
PERTURBO is implemented in modern Fortran with a modular code design. All
calculations employ intuitive workflows. The code is highly efficient thanks to
its hybrid MPI (Message Passing Interface) and OpenMP (Open Multi-Processing)
parallelization. It can run on record-large unit cells with up to at least 50 atoms [15],
and its performance scales up to thousands of CPU cores. It conveniently writes files
using the HDF5 format, and is suitable for both high-performance supercomputers
and smaller computer clusters.
Target users include both experts in first-principles calculations and materials theory
as well as experimental researchers and teams in academic or national laboratories
investigating charge transport, ultrafast spectroscopy, advanced functional materials, and semiconductor or solid-state devices. PERTURBO will equip these users
with an efficient quantitative tool to investigate electron interactions and dynamics
in broad families of materials, filling a major void in the current software ecosystem.
The chapter is organized as follows: Sec. 6.2 discusses the theory and numerical
methods implemented in the code; Sec. 6.3 describes the code capabilities and
workflows; Sec. 6.4 delves deeper into selected technical aspects; Sec. 6.5 shows
several example calculations provided as tutorials in the code; and Sec. 6.6 discusses the parallelization strategy and the scaling of the code on high performance
supercomputers. We conclude in Sec. 6.7 by summarizing the main features of
PERTURBO.
6.2

Methodology

Boltzmann transport equation
The current release of PERTURBO can compute charge transport and ultrafast
dynamics in the framework of the semiclassical BTE. The BTE describes the flow
of the electron occupations 𝑓𝑛k (r, 𝑡) in the phase-space variables of relevance in a

103
periodic system, the crystal momentum k and spatial coordinate r:
𝜕 𝑓𝑛k (r, 𝑡)
= − ∇r 𝑓𝑛k (r, 𝑡) · v𝑛k + ℏ−1 ∇k 𝑓𝑛k (r, 𝑡) · F
𝜕𝑡
+ I [ 𝑓 𝑛k ] ,

(6.1)

where 𝑛 is the band index and 𝑣 𝑛k are band velocities. The time evolution of the
electron occupations is governed by the so-called drift term due to external fields F
and the collision term I [ 𝑓𝑛k ], which captures electron scattering processes due to
phonons or other mechanisms [16]. In PERTURBO, the fields are assumed to be
slowly varying and the material homogeneous, so 𝑓𝑛k does not depend on the spatial
coordinates and its spatial dependence is not computed explicitly.
The collision integral I [ 𝑓𝑛k ] is a sum over a large number of scattering processes
in momentum space, and it is very computationally expensive because it involves
Brillouin zone (BZ) integrals on fine grids. Most analytical and computational
treatments simplify the scattering integral with various approximations. A common
one is the RTA, which assumes that the scattering integral is proportional to the
deviation 𝛿 𝑓𝑛k of the electron occupations from the equilibrium Fermi-Dirac distribution, I [ 𝑓𝑛k ] = −𝛿 𝑓𝑛k /𝜏; the relaxation time 𝜏 is either treated as a constant
empirical parameter [17, 18] or as a state-dependent quantity, 𝜏𝑛k .
PERTURBO implements the first-principles formalism of the BTE, which employs materials properties obtained with quantum mechanical approaches, using
the atomic structure of the material as the only input. The electronic structure is
computed using DFT and the lattice dynamical properties using DFPT. The scattering integral is computed only for 𝑒-ph processes in the current release, while other
scattering mechanisms, such as electron-defect and electron-electron scattering, are
left for future releases. The scattering integral due to 𝑒-ph processes can be written
as
2𝜋 1 ∑︁
|𝑔𝑚𝑛𝜈 (k, q)| 2
I 𝑒−ph [ 𝑓𝑛k ] = −
ℏ Nq 𝑚 q 𝜈
(6.2)
× [𝛿 𝜀 𝑛k − ℏ𝜔 𝜈q − 𝜀 𝑚 k+q × 𝐹em
+ 𝛿 𝜀 𝑛k + ℏ𝜔 𝜈q − 𝜀 𝑚 k+q × 𝐹abs ],
where Nq is the number of q-points used in the summation, and 𝑔𝑚𝑛𝜈 (k, q) are
𝑒-ph matrix elements (see Sec. 1.2) quantifying the probability amplitude for an
electron to scatter from an initial state |𝑛k⟩ to a final state |𝑚k + q⟩, by emitting
or absorbing a phonon with wavevector q and mode index 𝜈; here and below, 𝜀 𝑛k
and ℏ𝜔 𝜈q are the energies of electron quasiparticles and phonons, respectively. The

104
phonon absorption (𝐹abs ) and emission (𝐹em ) terms are defined as
𝐹abs = 𝑓𝑛k 1 − 𝑓𝑚 k+q 𝑁 𝜈q − 𝑓𝑚 k+q (1 − 𝑓𝑛k ) 𝑁 𝜈q + 1 ,
𝐹em = 𝑓𝑛k 1 − 𝑓𝑚 k+q 𝑁 𝜈q + 1 − 𝑓𝑚 k+q (1 − 𝑓𝑛k ) 𝑁 𝜈q

(6.3)

where 𝑁 𝜈q are phonon occupations.
Ultrafast carrier dynamics
PERTURBO can solve the BTE numerically to simulate the evolution in time 𝑡 of
the electron occupations 𝑓𝑛k (𝑡) due to 𝑒-ph scattering processes, starting from an
initial nonequilibrium distribution. In the presence of a slowly varying external
electric field, and assuming the material is homogeneous, we rewrite the BTE in
Eq. (6.1) as
𝜕 𝑓𝑛k (𝑡) 𝑒E
(6.4)
· ∇k 𝑓𝑛k (𝑡) + I 𝑒−ph [ 𝑓𝑛k ] ,
𝜕𝑡
where 𝑒 is the electron charge and E the external electric field. We solve this
non-linear integro-differential equation numerically (in the current release, only for
E = 0), using explicit time-stepping with the 4th-order Runge-Kutta (RK4) or Euler
methods. The RK4 solver is used by default due to its superior accuracy. The Euler
method is much faster than RK4, but it is only first-order accurate in time, so it
should be tested carefully and compared against RK4.
Starting from an initial nonequilibrium electron distribution 𝑓𝑛k (𝑡0 ), we evolve in
time Eq. (6.4) with a small time step (typically in the order of 1 fs) to obtain 𝑓𝑛k (𝑡) as
a function of time. One application of this framework is to simulate the equilibration
of excited electrons [14, 19], in which we set E = 0 and simulate the time evolution
of the excited electron distribution as it approaches its equilibrium Fermi-Dirac
value through 𝑒-ph scattering processes. The phonon occupations are kept fixed
(usually, to their equilibrium value at a given temperature) in the current version
of the code. Another application, which is currently under development, is charge
transport in high electric fields.
Charge transport
In an external electric field, the drift and collision terms in the BTE balance out
at long enough times; the field drives the electron distribution out of equilibrium,
while the collisions tend to restore equilibrium. At steady state, a nonequilibrium
electron distribution is reached, for which 𝜕 𝑓𝑛k /𝜕𝑡 = 0. The BTE for transport at

105
steady state becomes

𝑒E
· ∇k 𝑓𝑛k (𝑡) = I 𝑒−ph [ 𝑓𝑛k ] .

(6.5)

When the electric field is relatively weak, the steady-state electron distribution
deviates only slightly from its equilibrium value. As is usual, we expand 𝑓𝑛k around
the equilibrium Fermi-Dirac distribution, 𝑓𝑛0k , and keep only terms linear in the
electric field:

𝑓𝑛k = 𝑓𝑛0k + 𝑓𝑛1k + O 𝐸 2
(6.6)

𝜕 𝑓𝑛0k
= 𝑓𝑛k + 𝑒E · F𝑛k
+O 𝐸 ,
𝜕𝜀 𝑛k
where F𝑛k characterizes the first-order deviation from equilibrium of the electron
distribution. We substitute Eq. (6.6) into both sides of Eq. (6.5), and obtain a
linearized BTE for the distribution deviation F𝑛k keeping only terms up to firstorder in the electric field:
𝜏𝑛k ∑︁
F𝑚 k+q 𝑊𝑛𝜈kq,𝑚 k+q ,
(6.7)
F𝑛k = 𝜏𝑛k v𝑛k +
Nq 𝑚,𝜈q
where 𝜏𝑛k is the electron relaxation time, computed as the inverse of the scattering
rate, 𝜏𝑛k = Γ𝑛−1
. The scattering rate Γ𝑛k is given by
Γ𝑛k =

1 ∑︁ 𝜈q
Nq 𝑚,𝜈q 𝑛k,𝑚 k+q

(6.8)

The scattering probability 𝑊𝑛𝜈kq,𝑚 k+q involves both phonon emission and absorption
processes:
2𝜋
|𝑔𝑚𝑛𝜈 (k, q)| 2

× [𝛿 𝜀 𝑛k − ℏ𝜔 𝜈q − 𝜀 𝑚 k+q 1 + 𝑁 𝜈q − 𝑓𝑚 k+q

+ 𝛿 𝜀 𝑛k + ℏ𝜔 𝜈q − 𝜀 𝑚 k+q 𝑁 𝜈0q + 𝑓𝑚0 k+q ],

𝑊𝑛𝜈kq,𝑚 k+q =

(6.9)

where 𝑁 𝜈0q are equilibrium Bose-Einstein phonon occupations. Note that since 𝜏𝑛k
is an electron quasiparticle lifetime, it can be written equivalently as the imaginary
𝑒−ph
part of the 𝑒-ph self-energy [20], 𝜏𝑛−1
= 2ImΣ𝑛k /ℏ.
In the RTA, we neglect the second term in Eq. (6.7) and obtain F𝑛𝑘 = 𝜏𝑛k v𝑛k . In
some cases, the second term in Eq. (6.7) cannot be neglected. In metals, a commonly
used scheme to approximate this term is to add a factor of (1−cos 𝜃 k,k+q ) to Eq. (6.9),
where 𝜃 k,k+q is the scattering angle between k and k + q. The resulting so-called

106
“transport relaxation time” is then used to compute the transport properties [20].
PERTURBO implements a more rigorous approach and directly solves Eq. (6.7)
using an iterative method [21, 22], for which we rewrite Eq. (6.7) as
𝜏𝑛k ∑︁ 𝑖
F𝑛𝑖+1
F𝑚 k+q 𝑊𝑛𝜈kq,𝑚 k+q .
(6.10)
𝑛𝑘
Nq 𝑚,𝜈q
0 = 𝜏 v , and then compute
In the iterative algorithm, we choose in the first step F𝑛𝑘
𝑛k 𝑛k
− F𝑛𝑖 k | is within the
the following steps using Eq. (6.10) until the difference |F𝑛𝑖+1
convergence threshold.
Once F𝑛𝑘 has been computed, either within the RTA or with the iterative solution
of the BTE in Eq. (6.10), the conductivity tensor is obtained as
𝑆 ∑︁
𝜎𝛼𝛽 =
−𝑒v𝑛𝛼k · 𝑓𝑛1k
ΩE 𝛽 Nk
𝑛k
(6.11)
𝜕 𝑓𝑛0k
𝑒 2 𝑆 ∑︁ 𝛼 𝛽
v𝑛 k F𝑛 k −
Nk Ω
𝜕𝜀 𝑛k
𝑛k

where 𝛼 and 𝛽 are Cartesian directions, Ω is the volume of the unit cell, and 𝑆 is the
spin degeneracy. We also compute the carrier mobility tensor, 𝜇𝛼𝛽 = 𝜎𝛼𝛽 /(𝑒 𝑛𝑐 ),
by dividing the conductivity tensor through the carrier concentration 𝑛𝑐 .
In our implementation, we conveniently rewrite Eq. (6.11) as
𝜎𝛼𝛽 = 𝑒
𝑑𝐸 (−𝜕 𝑓 0 /𝜕𝐸)Σ𝛼𝛽 (𝐸),
(6.12)
where Σ𝛼𝛽 (𝐸) is the transport distribution function (TDF) at energy 𝐸,
𝑆 ∑︁ 𝛼 𝛽
v𝑛k F𝑛𝑘 𝛿(𝐸 − 𝜀 𝑛k ),
Σ𝛼𝛽 (𝐸) =
Nk Ω

(6.13)

𝑛k

which is computed in PERTURBO using the tetrahedron integration method [23].
The integrand in Eq. (6.12) can be used to characterize the contributions to transport
as a function of electron energy [12, 13]. The code can also compute the Seebeck
coefficient S from the TDF, using
[σS] 𝛼𝛽 =
𝑑𝐸 (−𝜕 𝑓 0 /𝜕𝐸)(𝐸 − 𝜇)Σ𝛼𝛽 (𝐸),
(6.14)
where 𝜇 is the chemical potential and 𝑇 is the temperature.
6.3

Capabilities and workflow

Code organization and capabilities
PERTURBO contains two executables: a core program perturbo.x and the program qe2pert.x, which interfaces the QE code and perturbo.x, as shown in

107
𝜙i

bands and wavefunctions
Wannier90
on coarse grid kc
rotation matrices U(kc)

DFT

g(Re, Rp)

e-ph
gnm⌫ (kc , q c )

Quantum Espresso

mk+q

dV
ultra-fine kf, qf grids

Compute & store
g(Re, Rp) in WF basis

AAACHHicbVDLSgMxFM3UV62vqks3wVaoIGWmLnRZdOOygn1AZxgymUwbmmTGJCOUYT7Ejb/ixoUiblwI/o3pY6GtB0IO59zLvfcECaNK2/a3VVhZXVvfKG6WtrZ3dvfK+wcdFacSkzaOWSx7AVKEUUHammpGeokkiAeMdIPR9cTvPhCpaCzu9DghHkcDQSOKkTaSXz6vDvxMcFekec0NYhaqMTdfNsr9DOdn8Ld2P9VOq365YtftKeAyceakAuZo+eVPN4xxyonQmCGl+o6daC9DUlPMSF5yU0UShEdoQPqGCsSJ8rLpcTk8MUoIo1iaJzScqr87MsTVZEFTyZEeqkVvIv7n9VMdXXoZFUmqicCzQVHKoI7hJCkYUkmwZmNDEJbU7ArxEEmEtcmzZEJwFk9eJp1G3bHrzm2j0ryax1EER+AY1IADLkAT3IAWaAMMHsEzeAVv1pP1Yr1bH7PSgjXvOQR/YH39AGmvoso=

DFPT

𝜈q
nk

MLWFs

coarse kc, qc grids
phonons & ΔVscf
on coarse grid qc

𝜙j

Interpolate
gnm⌫ (kf , q f )

Transport
Dynamics

AAACLnicbVDLSgMxFM3UV62vUZdugq1QQcpMN7osiuCygn1AZxgymUwbmsmMSUYsw3yRG39FF4KKuPUzTB8L23og5HDOvcm9x08Ylcqy3o3Cyura+kZxs7S1vbO7Z+4ftGWcCkxaOGax6PpIEkY5aSmqGOkmgqDIZ6TjD6/GfueBCEljfqdGCXEj1Oc0pBgpLXnmdaXvZTxyeJpXHT9mgRxF+sqGuZc5ijzqJ7Mwz8/gX/N+3jyteGbZqlkTwGViz0gZzND0zFcniHEaEa4wQ1L2bCtRboaEopiRvOSkkiQID1Gf9DTlKCLSzSbr5vBEKwEMY6EPV3Ci/u3IUCTHk+rKCKmBXPTG4n9eL1XhhZtRnqSKcDz9KEwZVDEcZwcDKghWbKQJwoLqWSEeIIGw0gmXdAj24srLpF2v2VbNvq2XG5ezOIrgCByDKrDBOWiAG9AELYDBE3gBH+DTeDbejC/je1paMGY9h2AOxs8v1jqrcA==

qe2pert.x

perturbo.x
PERTURBO

Figure 6.1: Workflow of the PERTURBO code. Before running PERTURBO, the
user performs DFT calculations on the system of interest to obtain the Bloch states
on a coarse k𝑐 -point grid and DFPT calculations to obtain the lattice dynamical
properties and perturbation potentials on a coarse q𝑐 -point grid. The qe2pert.x
executable of PERTURBO computes the 𝑒-ph matrix elements on coarse k𝑐 - and
q𝑐 -point grids, from which the 𝑒-ph matrix elements in the localized Wannier basis
are obtained with the rotation matrices from Wannier90. The core executable
perturbo.x is then employed to interpolate the band structure, phonon dispersion,
and 𝑒-ph matrix elements on ultra-fine k 𝑓 - and q 𝑓 -point grids, and to perform charge
transport and carrier dynamics calculations.

Fig. 6.1. The current release supports DFT and DFPT calculations with normconserving or ultrasoft pseudopotentials, with or without SOC; it also supports the
Coulomb truncation for 2D materials [41]. The current features include calculations
of:
1 Band structure, phonon dispersion, and 𝑒-ph matrix elements on arbitrary BZ
grids or paths.
2 The 𝑒-ph scattering rates, relaxation times, and electron mean free paths for
electronic states with any band and k-point.
3 Electrical conductivity, carrier mobility, and Seebeck coefficient using the
RTA or the iterative solution of the BTE.
4 Nonequilibrium dynamics, such as simulating the cooling and equilibration
of excited carriers via interactions with phonons.
Several features of PERTURBO, including the nonequilibrium dynamics, are unique
and not available in other existing codes. Many additional features currently being
developed or tested will be added to the list in future releases.

108
PERTURBO stores most of the data and results in HDF5 file format, including all the
main results of qe2pert.x, the TDF and other files generated in the transport calculations, and the nonequilibrium electron distribution computed in perturbo.x.
The use of the HDF5 file format improves the portability of the results between
different computing systems and is also convenient for post-processing using highlevel languages, such as Python and Julia.
Installation and usage of PERTURBO and an up-to-date list of supported features
are documented in the user manual distributed along with the source code package,
and can also be found on the code website [42].

Computational workflow
Figure 6.1 summarizes the workflow of PERTURBO. Before running PERTURBO,
the user needs to carry out DFT and DFPT calculations with the QE code, and Wannier function calculations with Wannier90. In our workflow, we first carry out DFT
calculations to obtain the band energies and Bloch wavefunctions on a coarse grid
with points k𝑐 . A regular Monkhorst-Pack (MP) k𝑐 -point grid centered at Γ and
Bloch states for all k𝑐 -points in the first BZ are required. We then construct a set of
Wannier functions from the Bloch wavefunctions using the Wannier90 code. Only
the rotation matrices U that transform the DFT Bloch wavefunctions to the Wannier
gauge and the center of the Wannier functions are required as input to PERTURBO.
We also perform DFPT calculations to obtain the dynamical matrices and 𝑒-ph perturbation potentials on a coarse MP grid with points q𝑐 . In the current version, the
electron k𝑐 -point grid and phonon q𝑐 -point grid need to be commensurate. Since
DFPT is computationally demanding, we carry out the DFPT calculations only for
q𝑐 -points in the irreducible wedge of the BZ, and then obtain the dynamical matrices
and perturbation potentials in the full BZ grid using space group and time reversal
symmetries (see Sec. 6.4).
The executable qe2pert.x reads the results from DFT and DFPT calculations,
including the Bloch states |𝜓𝑛k𝑐 ⟩, dynamical matrices 𝐷 (q𝑐 ) and 𝑒-ph perturbation potentials 𝜕q𝑐 ,𝜅𝛼𝑉, and then computes the electron Hamiltonian in the Wannier
basis [Eq. (1.3)] and the IFCs [Eq. (1.8)]. It also computes the 𝑒-ph matrix elements on coarse grids and transforms them to the localized Wannier basis using
Eqs. (1.12)-(1.13). To ensure that the same Wannier functions are used for the
electron Hamiltonian and 𝑒-ph matrix elements, we use the same DFT Bloch states
|𝜓𝑛k𝑐 ⟩ and U (k𝑐 ) matrices for the calculations in Eqs. (1.3) and (1.12)-(1.13). Fol-

109
lowing these preliminary steps, qe2pert.x outputs all the relevant data to an HDF5
file, which is the main input for perturbo.x.
The executable perturbo.x reads the data computed by qe2pert.x and carries
out the transport and dynamics calculations discussed above. To accomplish these
tasks, perturbo.x interpolates the band structure, phonon dispersion, and 𝑒-ph
matrix elements on fine k- and q-point BZ grids, and uses these quantities in the
various calculations.
6.4

Technical aspects

𝑒-ph matrix elements on coarse grids
As discussed in Sec. 1.2, we compute directly the 𝑒-ph matrix elements in Eq. (1.12)
using the DFT states on a coarse k𝑐 -point grid and the perturbation potentials on a
coarse q𝑐 -point grid. It is convenient to rewrite Eq. (1.12) in terms of lattice-periodic
quantities,
𝜅𝛼
(k𝑐 , q𝑐 ) = 𝑢 𝑚 k𝑐 +q𝑐 𝜕q𝑐 ,𝜅𝛼 𝑣 𝑢 𝑛k𝑐 ,
(6.15)
𝑔˜ 𝑚𝑛
where |𝑢 𝑛k𝑐 ⟩ is the lattice periodic part of the Bloch wavefunction and 𝜕q𝑐 ,𝜅𝛼 𝑣 =
𝑒 −𝑖q𝑐 ·r 𝜕q𝑐 ,𝜅𝛼𝑉 is the lattice-periodic perturbation potential. Since we compute only
|𝑢 𝑛k𝑐 ⟩ on the coarse grid of the first BZ, |𝑢 𝑚 k𝑐 +q𝑐 ⟩ may not be available because
k𝑐 + q𝑐 may be outside of the first BZ. However, by requiring the q𝑐 -point grid
to be commensurate with and smaller than (or equal to) the k𝑐 -point grid, we can
satisfy the relationship k𝑐 + q𝑐 = k′𝑐 + G0 , where k′𝑐 is on the coarse grid and G0
is a reciprocal lattice vector. Starting from |𝑢 𝑚 k𝑐′ ⟩ = G 𝑐 k𝑐′ (G)𝑒𝑖G·r , we can thus
obtain |𝑢 𝑚 k𝑐 +q𝑐 ⟩ with negligible computational cost as
∑︁
𝑐 k𝑐′ (G)𝑒𝑖(G−G0 )·r .
(6.16)
|𝑢 𝑚 k𝑐 +q𝑐 ⟩ = 𝑒 −𝑖G0 ·r |𝑢 𝑚 k𝑐′ ⟩ =

The lattice-periodic perturbation potential 𝜕q𝑐 ,𝜅𝛼 𝑣 consists of multiple terms, and
can be divided into a local and a non-local part [43]. The non-local part includes
the perturbation potentials due to the non-local terms of pseudopotentials, which
typically includes the Kleinman-Bylander projectors and SOC terms. The local part
includes the perturbations to the local part of the pseudopotentials as well as the
self-consistent potential contribution. The latter, denoted as 𝜕q𝑐 ,𝜅𝛼 𝑣 sc (r), accounts
for the change in the Hartree and exchange-correlation potentials in response to the
atomic displacements. While the pseudopotential contributions, both local and nonlocal, can be evaluated efficiently for all q𝑐 -points with the analytical formula given
in Ref. [43], the self-consistent contribution 𝜕q𝑐 ,𝜅𝛼 𝑣 sc (r) is computed and stored in
real space using expensive DFPT calculations. This step is the main bottleneck of

110
the entire 𝑒-ph computational workflow.
To improve the efficiency, we compute the self-consistent contribution with DFPT
only for q𝑐 -points in the irreducible BZ wedge, and then unfold it to the equivalent
points Sq𝑐 in the full BZ using symmetry operations [34, 44]. For non-magnetic
systems, 𝜕q𝑐 ,𝜅𝛼 𝑣 s𝑐 (r) is a scalar function, so we can obtain the self-consistent
contribution at Sq𝑐 by rotating 𝜕q𝑐 ,𝜅𝛼 𝑣 s𝑐 (r) [34]:
∑︁
𝜕S q𝑐 ,𝜅𝛼 𝑣 s𝑐 (r) =
𝑒𝑖q𝑐 ·τ𝜅 ′ −𝑖S q𝑐 ·τ𝜅
𝜅′𝛽

× S

−1

𝜕 ′ 𝑣
𝛽𝛼 q𝑐 ,𝜅 𝛽 s𝑐

−1

(6.17)

{S|t} r ,

where {S|t} is a space group symmetry operation of the crystal. A detailed derivation of Eq. (6.17) can be found in Appendix C of Ref. [34]. In addition to space
group operations, time reversal symmetry is also used for non-magnetic systems via
𝜕−q𝑐 ,𝜅𝛼 𝑣 s𝑐 (r) = [𝜕q𝑐 ,𝜅𝛼 𝑣 s𝑐 (r)] ∗ ,

(6.18)

since the time reversal symmetry operator for a scalar function is the complex
conjugation operator. We emphasize that Eq. (6.17) is only used to unfold the
self-consistent contribution of the perturbation potential, while all the terms due
to the pseudopotentials are computed directly, without using symmetry, for all the
q𝑐 -points in the coarse grid.
An alternative approach to obtain the 𝑒-ph matrix elements in the full BZ starting
from results in the irreducible wedge is to rotate the wavefunctions instead of the
perturbation potential:
⟨𝑢 𝑚 k𝑐 +S q𝑐 |𝜕S q𝑐 ,𝜅𝛼 𝑣|𝑢 𝑛k𝑐 ⟩ =
⟨𝑢 𝑚 k𝑐 +S q𝑐 |D{S|
𝑣 D{S|t} |𝑢 𝑛k𝑐 ⟩,
t} q𝑐 ,𝜅𝛼

(6.19)

where D{S|t} is the symmetry operator acting on the wavefunction. In this approach
(not employed in PERTURBO), the perturbation potentials are needed only for q𝑐
in the irreducible wedge [32]. It is important to keep in mind that the wavefunctions
are spinors in non-collinear calculations, so the symmetry operators D{S|t} should
act both on the spatial coordinate and in spin space. Neglecting the rotation in
spin space would lead to significant errors in the computed 𝑒-ph matrix elements,
especially in calculations with SOC.
To benchmark our implementation, we compare the 𝑒-ph matrix elements obtained
using symmetry operations to those from direct DFPT calculations. The absolute

111

(a)

0.4
0.4

|g(k,q)| (meV)

| g(k = X/2,k + q = X) | (meV)

0.5
0.5

0.3
0.3

0.1
0.1

DFPT

Perturbo
Perturbo

55
44
|g(k,q)| (meV)

| g(k = K, k + q = M) | (meV)

0.2
0.2

0.0
0.0

(b)

X /2

33
22

K′

EPW
EPW

K′

DFPT

11
00

Perturbo
Perturbo

EPW
EPW

Figure 6.2: Absolute value of the gauge-invariant 𝑒-ph matrix elements, |𝑔 (k, q)|
in Eq. (6.20), computed with PERTURBO (orange) and EPW (green) in (a) silicon
and (b) monolayer MoS2 . SOC is included in both cases. The six bars are |𝑔 (k, q)|
values for six equivalent (k, q) pairs connected by symmetry, which are shown as
blue arrows in the inset. The result with q in the irreducible wedge (labeled with
a blue star) is computed directly with the DFPT perturbation potential, while the
others are obtained by applying symmetry operations either on the perturbation
potentials (PERTURBO) or on the wavefunctions (EPW). The red horizontal line
shows the benchmark values computed directly from DFPT for all (k, q) pairs.
value of the 𝑒-ph matrix elements, |𝑔 (k, q)|, is computed in gauge-invariant form
for each phonon mode (with index 𝜈) by summing over bands:
√︄∑︁
|𝑔𝑚𝑛𝜈 (k, q)| 2 /𝑁 𝑏 ,
|𝑔𝜈 (k, q)| =
(6.20)
𝑚𝑛

112
where 𝑚, 𝑛 are band indices for the selected 𝑁 𝑏 bands. We perform the comparison
with direct DFPT calculations for silicon and monolayer MoS2 as examples of a
bulk and a 2D material, respectively. We include SOC in both cases. For silicon,
we choose k = 𝑋/2, k + q = 𝑋 and compute |𝑔 (k, q)| in Eq. (6.20) using the four
highest valence bands; for monolayer MoS2 , we choose k = 𝐾, k + q = 𝑀 and
compute |𝑔 (k, q)| for the two lowest conduction bands with 2D Coulomb truncation
turned off. For both silicon and monolayer MoS2 , we compute |𝑔 (k, q)| for all the
six equivalent (k, q) pairs connected by space group and time reversal symmetry
[see the inset of Fig. 6.2(a, b)]. As a benchmark, DFPT calculations are carried
out to evaluate directly |𝑔 (k, q)| for all the six (k, q) pairs. The results shown in
Fig. 6.2 are for the lowest acoustic mode, though the results for the other modes
show similar trends. The |𝑔 (k, q)| values computed with DFPT are identical for
the six equivalent (k, q) pairs (see the red horizontal line in Fig. 6.2), which is
expected based on symmetry. In the PERTURBO calculation, only the (k, q) pair
with q in the irreducible wedge is computed directly with the perturbation potential
from DFPT, while results for the other five equivalent (k, q) pairs are obtained by
rotating the self-consistent perturbation potential. The results obtained with this
approach match to a high accuracy the DFPT benchmark results, which validates
the perturbation potential rotation approach implemented in PERTURBO.
For comparison, we show in Fig. 6.2 the results from the alternative approach of
rotating the wavefunctions, as implemented in the EPW code (version 5.2) [45].
Surprisingly, using the EPW code only the |𝑔 (k, q)| value for the (k, q) pair
containing the irreducible q-point agrees with the DFPT benchmark, while all other
|𝑔 (k, q)| values for q-points obtained using symmetry operations show significant
errors. We stress again that all the results in Fig. 6.2 are computed with SOC. We
have verified that, in the absence of SOC, both PERTURBO and EPW produce
results in agreement with DFPT. The likely reason for the failure of EPW in this
test is that in EPW the wavefunctions are rotated as scalars even in the presence
of SOC, rather than as spinors as they should (that is, the rotation in spin space is
missing). Further investigation of this discrepancy is critical, since the large errors
in EPW for the coarse-grid 𝑒-ph matrix elements will propagate to the interpolated
matrix elements on fine grids, giving incorrect results in calculations including SOC
carried out with EPW [46].

113
Wigner-Seitz supercell for Wannier interpolation
As discussed in Sec. 1.2, the DFT Bloch states obtained at k𝑐 -points on a regular
BZ grid are used to construct the Wannier functions. A discrete BZ grid implies a
Born-von Karman (BvK) boundary condition in real space, so that an 𝑁 × 𝑁 × 𝑁
k𝑐 -point grid corresponds (for simple cubic, but extensions are trivial) to a BvK
supercell of size 𝑁𝑎 × 𝑁𝑎 × 𝑁𝑎, where 𝑎 is the lattice constant of the unit cell. If we
regard the crystal as made up by an infinite set of BvK supercells at lattice vectors
T𝑒 , we can label any unit cell in the crystal through its position T𝑒 + R𝑒 , where R𝑒 is
the unit cell position in the BvK supercell. Because of the BvK boundary condition,
the Bloch wavefunctions are truly periodic functions over the BvK supercell, and
the Wannier function |𝑛R𝑒 ⟩ obtained using Eq. (1.1) is actually the superposition of
images of the Wannier function in all the BvK supercells. Therefore, we can write
|𝑛R𝑒 ⟩ = T𝑒 |𝑛, R𝑒 + T𝑒 ⟩ 0 , where |𝑛, R𝑒 + T𝑒 ⟩ 0 denotes the image of the Wannier
function in the BvK supercell at T𝑒 . Similarly, the electron Hamiltonian computed
using Eq. (1.3) can be expressed as
∑︁
𝐻𝑛𝑛 ′ (R𝑒 ) =
𝐻𝑛𝑛
(6.21)
′ (R𝑒 + T𝑒 ).
T𝑒
0 (R + T ) usually decay rapidly as the distance
The hopping matrix elements 𝐻𝑛𝑛
between two image Wannier functions increases. The BvK supercell should be
large enough to guarantee that only the hopping term between the two Wannier
function images with the shortest distance is significant, while all other terms in the
summation over T𝑒 in Eq. (6.21) are negligible.
We use this “least-distance” principle to guide our choice of a set of R̃𝑒 vectors
for the Wannier interpolation. For each Hamiltonian matrix element labeled by
(𝑛, 𝑛′, R𝑒 ) in Eq. (6.21), we compute the distance 𝑑 = |T𝑒 + R𝑒 + τ𝑛 ′ − τ𝑛 |, with τ𝑛
the position of the Wannier function center in the unit cell, and find the vector T𝑒0
giving the minimum distance. The set of vectors R̃𝑒 = R𝑒 + T𝑒0 is then selected to
construct the Wigner-Seitz supercell used in the Wannier interpolation. We compute
𝐻𝑛𝑛 ′ ( R̃𝑒 ) using Eq. (1.3) and use it to interpolate the band structure with Eq. (1.4).
Note that the same strategy to construct the Wigner-Seitz supercell is also used in
the latest version of Wannier90 [26].
Similarly, we choose a set of least-distance R̃ 𝑝 vectors for the interpolation of the
phonon dispersion, and separately determine least-distance R̃𝑒 and R̃ 𝑝 pairs for the
interpolation of the 𝑒-ph matrix elements in Eqs. (1.13)-(1.14).

114
Brillouin zone sampling and integration
Several computational tasks carried out by PERTURBO require integration in the
BZ. Examples include scattering rate and TDF calculations, in Eqs. (6.8) and (6.13),
respectively, and the iterative BTE solution in Eq. (6.10). In PERTURBO, we adopt
different BZ sampling and integration approaches for different kinds of calculations.
For transport calculations, we use the tetrahedron method [23] for the integration over
k in Eq. (6.13). We sample k-points in the first BZ using a regular MP grid centered
at Γ, and divide the BZ into small tetrahedra by connecting neighboring k-points.
The integration is first performed inside each tetrahedron, and the results are then
added together to compute the BZ integral. To speed up these transport calculations,
we set up a user-defined window spanning a small energy range (typically ∼0.5
eV) near the band edge in semiconductors or Fermi level in metals, and restrict
the BZ integration to the k-points with electronic states in the energy window.
Since only states within a few times the thermal energy 𝑘 𝐵𝑇 of the band edge in
semiconductors (or Fermi energy in metals) contribute to transport, including in the
tetrahedron integration only k-points in the relevant energy window greatly reduces
the computational cost.
To compute the 𝑒-ph scattering rate for states with given bands and k-points, we use
the Monte Carlo integration as the default option. We sample random q-points in
the first BZ to carry out the summation over q in Eq. (6.8) and obtain the scattering
rate. One can either increase the number of random q-points until the scattering rate
is converged or average the results from independent samples. Note that the energy
broadening parameter used to approximate the 𝛿 function in Eq. (6.9) is important for
the convergence of the scattering rate, so the convergence with respect to both the qpoint grid and broadening needs to be checked carefully [12]. PERTURBO supports
random sampling of the q-points with either a uniform or a Cauchy distribution; a
user-defined q-point grid can also be specified in an external file and used in the
calculation.
In the carrier dynamics simulations and in the iterative BTE solution in Eq. (6.10),
the k- and q-points should both be on a regular MP grid centered at Γ. The two grids
should be commensurate, with the size of the q-point grid smaller than or equal to
the size of the k-point grid. This way, we satisfy the requirement in Eq. (6.10) that
each (k+q)-point is also on the k-point grid. To perform efficiently the summation
in Eq. (6.10) for all the k-points, we organize the scattering probability in Eq. (6.9)
using (k, q) pairs. We first determine a set of bands and k-points for states inside
the energy window. We then find all the possible scattering processes in which both

115
the initial and final states are in the selected set of k-points, and the phonon wave
vector connecting the two states is on the q-point grid. The scattering processes are
indexed as (k, q) pairs; their corresponding 𝑒-ph matrix elements are computed and
stored, and then retrieved from memory during the iteration process.

GaAs
15
15

1010
55
ΓΓ

(b) 1010
88

ΓΓ

DFPT
Perturbo

MoS2

|g(k=Κ,k')| (eV/Å)

| D(k = K, q) | (eV/Å)

DFPT
Perturbo

|g(k=Γ,q)| (eV/Å)

| D(k = Γ, q) | (eV/Å)

(a) 2020

66
44
22

k' q
K+

Figure 6.3: The 𝑒-ph deformation potential [see Eq. (6.22)] in (a) GaAs and (b)
monolayer MoS2 computed using PERTURBO and compared with direct DFPT
calculations for benchmarking.

116

(a) 5

Mean free
path
4 (nm)2
>2
10

Scattering Rate (THz)

10
10

-1

103

102

10 1
10

10 0
10

100

00
10

0 <-1
10−1

Relaxation time (fs)

Scattering rate (THz)

1033
(b) 10

10
11

101

0.0

0.5
0.5
1.0
E - ECBM (eV)

Relaxation time (fs)

E - ECBM (eV)

1.5
1.5

Figure 6.4: (a) Band structure of GaAs overlaid with a log-scale color map of the
electron mean free paths computed at 300 K. (b) The 𝑒-ph scattering rates and their
inverse, the relaxation times, for the electronic states in (a) given as a function of
electron energy. The energy zero is the CBM.
6.5

Examples

In this section, we demonstrate the capabilities of the PERTURBO code with a few
representative examples, including calculations on a polar bulk material (GaAs),
a 2D material with SOC (monolayer MoS2 ), and an organic crystal (naphthalene)
with a relatively large unit cell with 36 atoms.
The ground state and band structure are computed using DFT with a plane-wave
basis with the QE code; this is a preliminary step for all PERTURBO calculations,

117
as discussed above. For GaAs, we use the same computational settings as in
Ref. [12], namely a lattice constant of 5.556 Å and a plane-wave kinetic energy
cutoff of 72 Ry. For monolayer MoS2 , we use a 72 Ry kinetic energy cutoff,
an experimental lattice constant of 3.16 Å and a layer-normal vacuum spacing of
17 Å. All calculations are carried out in the local density approximation of DFT
using norm-conserving pseudopotentials. For MoS2 , we use fully relativistic normconserving pseudopotentials from Pseudo Dojo [47] to include SOC effects.
Lattice dynamical properties and the 𝑒-ph perturbation potential [see Eq. (1.11)] are
104

(b)
Scattering rate (THz)

(a)

102
100
mode 1
mode 2
mode 3

10−2
10−4

100

200
Hole energy ( meV )

mode 20
mode 50
mode 80
300

Figure 6.5: (a) Crystal structure of naphthalene, a prototypical organic molecular
semiconductor with 36 atoms in the unit cell. (b) Mode-resolved 𝑒-ph scattering
rates in naphthalene, computed at 300 K. Results are shown for three acoustic modes
(1-3) associated with inter-molecular vibrations and three optical modes (20, 50,
80) associated with intra-molecular vibrations. The black dashed line represents
the conductivity integrand in Eq. (6.12) and quantifies the relative contribution to
transport as a function of electron energy. It is seen that only states within about
100 meV of the valence band maximum contribute to hole transport.
computed on coarse q-point grids of 8 × 8 × 8 (GaAs) and 24 × 24 × 1 (MoS2 ) using
DFPT as implemented in the QE code. We use the 2D Coulomb cutoff approach
in DFPT for MoS2 to remove the spurious interactions between layers [39]. For
naphthalene, we use the same computational settings as in Ref. [15] for the DFT
and DFPT calculations. Note that we only perform the DFPT calculations for
the irreducible q-points in the BZ grid, following which we extend the dynamical
matrices to the entire BZ grid using space group and time reversal symmetry [see
Eq. (6.17)] with our qe2pert.x routines.
The Wannier90 code is employed to obtain localized Wannier functions in each
material. Only the centers of the Wannier functions and the rotation matrices U
[see Eq. (1.2)] are read as input by qe2pert.x. The Wannier functions for GaAs
are constructed using a coarse 8 × 8 × 8 k-point grid and 𝑠𝑝 3 orbitals centered

118
at the Ga and As atoms as an initial guess. For MoS2 , we construct 22 Wannier
functions using a coarse 24 × 24 × 1 k-point grid and an initial guess of 𝑑 orbitals
on Mo and 𝑠 orbitals on S atoms, using both spin up and spin down orbitals. For
naphthalene, we construct two Wannier functions for the two highest valence bands
using a coarse k-point grid of 4 × 4 × 4. The selected columns of density matrix
(SCDM) approach [48] is employed to generate an initial guess automatically.
Following the workflow in Fig. 6.1, we compute the 𝑒-ph matrix elements on the
coarse k- and q-point grids and obtain the 𝑒-ph matrix elements in the localized
Wannier basis using qe2pert.x.
Interpolation of the 𝑒-ph matrix elements
It is important to check the accuracy of all interpolations before performing transport
and carrier dynamics calculations. PERTURBO provides routines and calculation
modes to compute and output the interpolated band structure, phonon dispersion,
and 𝑒-ph matrix elements, which should be compared to the same quantities computed using DFT and DFPT to verify that the interpolation has worked as expected.
The comparison is straightforward for the band structure and phonon dispersion.
Here we focus on validating the interpolation of the 𝑒-ph matrix elements, a point
often overlooked in 𝑒-ph calculations. Since the 𝑒-ph matrix elements are gauge
dependent, a direct comparison of 𝑔𝑚𝑛𝜈 (k, q) from PERTURBO and DFPT calculations is not meaningful. Instead, we compute the absolute value of the 𝑒-ph matrix
elements, |𝑔𝜈 (k, q)|, in the gauge-invariant form of Eq. (6.20), and the closely
related “deformation potential”, which following Ref. [37] is defined as
√︁
(6.22)
𝐷 𝜈 (k, q) = 2𝜔 𝜈q 𝑀𝑡𝑜𝑡 |𝑔𝜈 (k, q)| /ℏ,
where 𝑀𝑡𝑜𝑡 is the total mass of the atoms in the unit cell.
We show in Fig. 6.3 the deformation potentials for the bulk polar material GaAs
and the 2D polar material MoS2 . For GaAs, we choose the Γ-point as the initial
electron momentum k, and vary the phonon wave vector q along a high-symmetry
path, computing 𝐷 𝜈 (k, q) for the highest valence band. For MoS2 , we choose the
𝐾-point as the initial electron momentum and vary the final electron momentum
𝐾 + q along a high-symmetry path, computing 𝐷 𝜈 (k, q) by summing over the two
lowest conduction bands. In both cases, results are computed for all phonon modes.
The respective polar corrections are included in both materials, which is crucial to
accurately interpolate the 𝑒-ph matrix elements at small q. The results show clearly
that the 𝑒-ph interpolation works as expected, giving interpolated matrix elements
in close agreement with direct DFPT calculations.

119
(b)
1tTX
_h
Ah

JQ#BHBiv U+K2 o 1 b 1 V

GaAs

104

R8y

kyy

k8y
T UEV

jyy

j8y

Mc 4Ry17 +K 3
Mc 4Ry18 +K 3

103
JQ#BHBiv U+K2 o 1 b 1 V

105

a22#2+F *Q2{+B2Mi UµofEV

(a)

250
200
150
100

GaAs
50
R8y

kyy

k8y
T UEV

jyy

j8y

2H2+i`QM@_h
?QH2@_h

2H2+i`QM@Ah
?QH2@Ah

102

MoS2
101
R8y

kyy

k8y
T UEV

jyy

j8y

Figure 6.6: (a) Electron mobility in GaAs as a function of temperature, computed
using the RTA and ITA methods and compared with experimental data [49, 50].
(b) Temperature dependent Seebeck coefficient in GaAs, computed for two different
carrier concentrations. (c) Electron and hole mobilities in monolayer MoS2 as a
function of temperature, computed using the RTA and ITA.
Scattering rates and electron mean free paths
The perturbo.x routines can compute the 𝑒-ph scattering rate (Γ𝑛k ), relaxation
time (𝜏𝑛k = Γ𝑛−1
), and electron mean free path (𝐿 𝑛k = 𝜏𝑛k 𝑣 𝑛k ) for electronic states
with any desired band and crystal momentum k. Figure 6.4 shows the results for
GaAs, in which we compute these quantities for the lowest few conduction bands
and for k-points along a high-symmetry path.
Carefully converging the scattering rate in Eq. (6.8) is important, but it can be far
from trivial since in some cases convergence requires sampling millions of q-points
in the BZ. A widely applicable scheme is to sample random q-points uniformly, and
compute the scattering rates for increasing numbers of q-points until the scattering
rates are converged. However, uniform sampling can be nonoptimal in specific
scenarios in which importance sampling can be used to speed up the convergence.
For polar materials such as GaAs, the scattering rate for electronic states near the
conduction band minimum (CBM) is dominated by small-q (intravalley) LO phonon
scattering. In this case, importance sampling with a Cauchy distribution [12] that
more extensively samples small q values can achieve convergence more effectively
than uniform sampling. On the other hand, for electronic states in GaAs farther in
energy from the band edges, the contributions to scattering are comparable for all
phonon modes and momenta, so uniform sampling is effective because it avoids bias
in the sampling. To treat optimally both sets of electronic states in polar materials
like GaAs, we implement a polar split scheme in PERTURBO, which is detailed in
Ref. [12]. The scattering rates shown in Fig 6.4(b) are obtained with this approach.
To analyze the dominant scattering mechanism for charge transport, it is useful to
resolve the contributions to the total scattering rate from different phonon modes.

120
Figure 6.5 shows the mode-resolved scattering rates for holes in naphthalene computed at 300 K [15]. A unit cell of naphthalene includes two molecules, for a total
of 36 atoms [see Fig. 6.5(a)] and 108 phonon modes. Figure 6.5(b) shows the contribution to the scattering rate from the three acoustic phonon modes (modes 1−3),
which are associated with inter-molecular vibrations. The contributions from three
optical modes (modes 20, 50, 80) associated with intra-molecular atomic vibrations
are also shown. In the relevant energy range for transport, as shown by the dashed
line in Fig. 6.5(b), the inter-molecular modes dominate hole carrier scattering. This
analysis is simple to carry out in PERTURBO because the code can output the TDF
and mode-resolved scattering rates.
Holes

0.015

0.010

-0.2

-0.1
E - EVBM (eV)

0.020

Population (arb. units)

0.020

0.015

0.0

0.010

0.005

0.000

0.020

0.015

0.0

0.010

0.005

- 0.6

-0.6

- 0.5

- 0.4

-0.4

- 0.3

E - EVBM (eV)

- 0.2

-0.2

- 0.1

0.0

-0.0

0.000

time (fs)

0.1

0.025

Hole occupation

t = 3 ps

Electrons

0.025

0.1

Eq. at 300 K

Electron occupation

0.025

Eq. at 300 K

500
500

t = 3 ps

450

0.1
E - ECBM (eV)

0.005

0.0

0.000

0.0
0.1

0.1
0.2

0.2

0.20.3

0.3 0.4

0.4
E - ECBM (eV)

0.4

0.5

y1
y2
y3
y4
y5
y6
y7
y8
0.2 y9
y10
y11
y12
y13
y14
y15
y16
y17
0.6
0.6y18
0.6y19
y20
y21
y22
y23
y24
y25
y26
y27
y28
y29
y30
y31
y32
y33
y34
y35
y36
y37
y38
y39
y40
y41
y42
y43
y44
y45
y46
y47
y48
y49
y50
y51
y52
y53
y54
y55
y56
y57
y58
y59
y60
y61
y62
y63
y64
y65
y66
y67
y68
y69
y70
y71
y72
y73
y74
y75
y76
y77
y78
y79
y80
y81
y82
y83
y84
y85
y86
y87
y88
y89
y90
y91
y92
y93
y94
y95
y96
y97
y98
y99
y100
y101

400
400
350
300
300
250
200
200
150
100
100
50
00

Figure 6.7: Hot carrier cooling simulation in silicon, for holes (left panel) and
electrons (right panel). The time evolution of the carrier population 𝑝(𝐸, 𝑡) in
Eq. (6.23) is shown, with the simulation time 𝑡 color-coded. The carrier occupations
𝑓𝑛k (𝑡) at 𝑡 = 3 ps are compared in the inset with the equilibrium Fermi-Dirac
distribution at 300 K (red curve) for the same carrier concentration.

Charge transport

We present charge transport calculations for GaAs and monolayer MoS2 as examples of a bulk and 2D material, respectively. Since both materials are polar, the
polar corrections to the 𝑒-ph matrix elements are essential, as shown in Fig. 6.3.
For MoS2 , we include SOC in the calculation since it plays an important role. In
MoS2 , SOC splits the states near the valence band edge, so its inclusion has a
significant impact on the 𝑒-ph scattering rates and mobility for hole carriers. The
transport properties can be computed in PERTURBO either within the RTA or with
the iterative solution of the linearized BTE. In the RTA, one can use pre-computed
state-dependent scattering rates (see Sec. 6.5) or compute the scattering rates on the
fly during the transport calculation using regular MP k- and q-point grids. In some
materials, the RTA is inadequate and the more accurate iterative solution of the
BTE in Eq. (6.10) is required. PERTURBO implements an efficiently parallelized

121
routine to solve the BTE iteratively (see Sec. 6.6).
Figure 6.6(a) shows the computed electron mobility in GaAs as a function of temperature and compares results obtained with the RTA and the iterative approach (ITA).
The calculation uses a carrier concentration of 𝑛𝑐 = 1017 cm−3 ; we have checked
that the mobility is almost independent of 𝑛𝑐 below 1018 cm−3 . We perform the
calculation using an ultra-fine BZ grid of 600 × 600 × 600 for both k- and q-points,
and employ a small (5 meV) Gaussian broadening to approximate the 𝛿 function in
the 𝑒-ph scattering terms in Eq. (6.9). To reduce the computational cost, we select
a 200 meV energy window near the CBM, which covers the entire energy range
relevant for transport. In GaAs, the ITA gives a higher electron mobility than the
RTA, as shown in Fig. 6.6(a), and this discrepancy increases with temperature. The
temperature dependence of the electron mobility is also slightly different in the RTA
and ITA results. The RTA calculation is in better agreement with experimental
data [49, 50] than results from the more accurate ITA. Careful analysis, carried out
elsewhere [51], shows that correcting the band structure [52] to obtain an accurate
effective mass and including two-phonon scattering processes [51] are both essential
to obtain ITA results in agreement with experiment.
The Seebeck coefficient can be computed at negligible cost as a post-processing step
of the mobility calculation. Figure 6.6(b) shows the temperature dependent Seebeck
coefficient in GaAs, computed using the ITA at two different carrier concentrations.
The computed value at 300 K and 𝑛𝑐 = 1018 cm−3 is about 130 𝜇V/K, in agreement
with the experimental value of ∼150 𝜇V/K [53]. Our results also show that the
Seebeck coefficient increases with temperature and for decreasing carrier concentrations (here, we tested 𝑛𝑐 = 1017 cm−3 ), consistent with experimental data near
room temperature [53, 54]. Note that the phonon occupations are kept fixed at their
equilibrium value in our calculations, so the phonon drag effect, which is particularly
important at low temperature, is neglected, and only the diffusive contribution to
the Seebeck coefficient is computed. To include phonon drag one needs to include
nonequilibrium phonon effects in the Seebeck coefficient calculation. While investigating the coupled dynamics of electrons and phonons remains an open challenge
in first-principles calculations [22, 55], we are developing an approach to time-step
the coupled electron and phonon BTEs, and plan to make it available in a future
version of PERTURBO.
Figure 6.6(c) shows the electron and hole mobilities in monolayer MoS2 , computed for a carrier concentration of 2 × 1012 cm−2 and for temperatures between
150−350 K. A fine BZ grid of 180 × 180 × 1 is employed for both k- and q-points.

122
Different from GaAs, the RTA and ITA give very close results for both the electron
and hole mobilities, with only small differences at low temperature. The computed
electron mobility at room temperature is about 168 cm2 /V s, in agreement with the
experimental value of 150 cm2 /V s [56]; the computed hole mobility at 300 K is
about 20 cm2 /V s.
Ultrafast dynamics
We demonstrate nonequilibrium ultrafast dynamics simulations using hot carrier
cooling in silicon as an example. Excited (so-called “hot”) carriers can be generated
in a number of ways in semiconductors, including injection from a contact or excitation with light. In a typical scenario, the excited carriers relax to the respective
band edge in a sub-picosecond time scale by emitting phonons. While ultrafast
carrier dynamics has been investigated extensively in experiments, for example with
ultrafast optical spectroscopy, first-principles calculations of ultrafast dynamics, and
in particular of hot carriers in the presence of 𝑒-ph interactions, are a recent development [14, 19, 57, 58].
We perform a first-principles simulation of hot carrier cooling in silicon. We use a
lattice constant of 5.389 Å for DFT and DFPT calculations, which are carried out
within the local density approximation and with norm-conserving pseudopotentials.
Following the workflow in Fig. 6.1, we construct 8 Wannier functions using 𝑠𝑝 3
orbitals on Si atoms as an initial guess, and obtain the 𝑒-ph matrix elements in the
Wannier basis using coarse 8 × 8 × 8 k- and q-point grids.
Starting from the 𝑒-ph matrix elements in the Wannier basis, the electron Hamil-

103

102

perturbo.x: dynamics
Wall time (minutes)

Wall time (minutes)

qe2pert.x

128 256 512 1024 2048
Number of CPU cores

102

128 256 512 1024 2048
Number of CPU cores

Figure 6.8: Performance scaling with the number of CPU cores of qe2pert.x
and perturbo.x. The black dashed line indicates the ideal linear scaling. For
perturbo.x, we show the performance of the carrier dynamics simulation, which
is the most time-consuming task of perturbo.x.
tonian, and interatomic force constants computed by qe2pert.x, we carry out hot

123
carrier cooling simulations, separately for electron and hole carriers, using a fine
k-point grid of 100 × 100 × 100 for the electrons (or holes) and a 50 × 50 × 50
q-point grid for the phonons. A small Gaussian broadening of 8 meV is employed
to approximate the 𝛿 function in the 𝑒-ph scattering terms in Eq. (6.9). The phonon
occupations are kept fixed at their equilibrium value at 300 K. We set the initial
electron and hole occupations to hot Fermi-Dirac distributions at 1500 K, each with
a chemical potential corresponding to a carrier concentration of 1019 cm−3 . We
explicitly time-step the BTE in Eq. (6.4) (with external electric field set to zero)
using the RK4 method with a small time step of 1 fs, and obtain nonequilibrium
carrier occupations 𝑓𝑛k (𝑡) as a function of time 𝑡 up to 3 ps, for a total of 3,000 time
steps. The calculation takes only about 2.5 hours using 128 CPU cores due to the
efficient parallelization (see Sec. 6.6). To visualize the time evolution of the carrier
distributions, we compute the BZ averaged energy-dependent carrier population,
∑︁
𝑝(𝐸, 𝑡) =
𝑓𝑛k (𝑡)𝛿(𝜀 𝑛k − 𝐸),
(6.23)
𝑛k

using the tetrahedron integration method. The carrier population 𝑝(𝐸, 𝑡) characterizes the time-dependent energy distribution of the carriers.
Figure 6.7 shows the evolution of the electron and hole populations due to 𝑒-ph
scattering. For both electrons and holes, the high-energy tail in the initial population decays rapidly as the carriers quickly relax toward energies closer to the band
edge within 500 fs. The carrier concentration is conserved during the simulation,
validating the accuracy of our RK4 implementation to time-step the BTE. The hole
relaxation is slightly faster than the electron relaxation in silicon, as shown in Fig. 6.7.
The sub-picosecond time scale we find for the hot carrier cooling is consistent with
experiment and with previous calculations using a simplified approach [19]. Although the hot carriers accumulate near the band edge in a very short time, it takes
longer for the carriers to fully relax to a 300 K thermal equilibrium distribution, up
to several picoseconds in our simulation. We show in the inset of Fig. 6.7 that the
carrier occupations 𝑓𝑛k at 3 ps reach the correct long-time limit for our simulation,
namely an equilibrium Fermi-Dirac distribution at the 300 K lattice temperature.
Since we keep the phonon occupations fixed and neglect phonon-phonon scattering, hot-phonon effects are ignored in our simulation. Similarly, electron-electron
scattering, which is typically important at high carrier concentrations or in metals,
is also not included in the current version of the code. Electron-electron scattering
and coupled electron and phonon dynamics (including phonon-phonon collisions)
are both under development.

124
6.6

Parallelization and performance

Transport calculations and ultrafast dynamics simulations on large systems can be
computationally demanding and require a large amount of memory. Efficient parallelization is critical to run these calculations on high-performance computing (HPC)
systems. To fully take advantage of the typical HPC architecture, we implement
a hybrid MPI and OpenMP parallelization, which combines distributed memory
parallelization among different nodes using MPI and on-node shared memory parallelization using OpenMP.
For small systems, we observe a similar performance by running PERTURBO on
a single node in either pure MPI or pure OpenMP modes, or in hybrid MPI plus
OpenMP mode. However, for larger systems with several atoms to tens of atoms
in the unit cell, running on multiple nodes with MPI plus OpenMP parallelization
significantly improves the performance and leads to a better scaling with CPU core
number. Compared to pure MPI, the hybrid MPI plus OpenMP scheme reduces
communication needs and memory consumption, and improves load balance. We
demonstrate the efficiency of our parallelization strategy using two examples: computing 𝑒-ph matrix elements on coarse grids, which is the most time-consuming
task of qe2pert.x, and simulating nonequilibrium carrier dynamics, the most
time-consuming task of perturbo.x.
For the calculation of 𝑒-ph matrix elements on coarse grids [see Eq. (6.15)], our
implementation uses MPI parallelization for q𝑐 -points and OpenMP parallelization
for k𝑐 -points. We distribute the q𝑐 -points among the MPI processes, so that each
MPI process either reads the self-consistent perturbation potential from file (if q𝑐
is in the irreducible wedge) or computes it from the perturbation potential of the
corresponding irreducible point using Eq. (6.17). Each MPI process then computes
𝑔(k𝑐 , q𝑐 ) for all the k𝑐 -points with OpenMP parallelization.
For carrier dynamics simulations, the k-points on the fine grid are distributed among
MPI processes to achieve an optimal load balance. Each process finds all the possible scattering channels involving its subset of k-points, and stores these scattering
processes locally as (k, q) pairs. One can equivalently think of this approach as
distributing the (k, q) pairs for the full k-point set among MPI processes. After
this step, we use OpenMP parallelization over local (k, q) pairs to compute the 𝑒-ph
matrix elements and perform the scattering integral. The iterative BTE in Eq. (6.10)
is also parallelized with the same approach. This parallelization scheme requires
minimum communication among MPI processes; for example, only one MPI reduction operation is required to collect the contributions from different processes for

125
the integration in Eq. (6.10).
We test the parallelization performance of qe2pert.x and perturbo.x using calculations on MoS2 and silicon as examples (see Sec. 6.5 for the computational
details). We run the carrier dynamics simulation on silicon using the Euler method
with a time step of 1 fs and a total simulation time of 50 ps, for a total of 50,000
steps. The tests are performed on the Cori system of the National Energy Research
Scientific Computing center (NERSC). We use the Intel Xeon “Haswell” processor
nodes, where each node has 32 CPU cores with a clock frequency of 2.3 GHz.
Figure 6.8 shows the wall time of the test calculations using different numbers of
CPU cores. Both qe2pert.x and perturbo.x show rather remarkable scaling
that is close to the ideal linear-scaling limit up to 1,024 CPU cores. The scaling of
qe2pert.x for 2,048 CPU cores is less than ideal mainly because each MPI process
has an insufficient computational workload, so that communication and I/O become
a bottleneck. The current release uses the serial HDF5 library, so after computing
𝑔(k𝑐 , q𝑐 ), the root MPI process needs to collect 𝑔(k𝑐 , q𝑐 ) from different processes
and write them to disk (or vice versa when loading data), which is a serial task that
costs ∼14% of the total wall time when using 2048 CPU cores. Parallel I/O using the
parallel HDF5 library could further improve the performance and overall scaling in
this massively parallel example. We plan to work on this and related improvements
to I/O in future releases.
6.7

Conclusions and outlook

In conclusion, we present our software, PERTURBO, for first-principles calculations of charge transport properties and simulation of ultrafast carrier dynamics in
bulk and 2D materials. The software contains an interface program to read results
from DFT and DFPT calculations from the QE code. The core program of PERTURBO performs various computational tasks, such as computing 𝑒-ph scattering
rates, electron mean free paths, electrical conductivity, mobility, and the Seebeck
coefficient. The code can also simulate the nonequilibrium dynamics of excited
electrons. Wannier interpolation and symmetry are employed to greatly reduce the
computational cost. SOC and the polar corrections for bulk and 2D materials are
supported and have been carefully tested. We demonstrate these features with representative examples. We also show the highly promising scaling of PERTURBO
on massively parallel HPC architectures, owing to its effective implementation of
hybrid MPI plus OpenMP parallelization.

126
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Chapter 7

SUMMARY AND FUTURE DIRECTIONS

In summary, this thesis provided a many-body first-principles framework for studying phonon-induced spin relaxation and decoherence in condensed matter, extended
the current capability of ab-initio 𝑒-ph interaction by addressing quadrupolar interactions and correlated electron systems, and developed an open-source platform for
investigating 𝑒-ph interactions from first principles.
In Chapter 2, we focused on spin-phonon interaction and relaxation. We first present
a first-principles approach for computing the phonon-limited 𝑇1 spin relaxation time
due to the Elliott-Yafet mechanism. Our scheme combines fully-relativistic spinflip electron-phonon interactions with an approach to compute the effective spin
of band electrons in materials with inversion symmetry. We apply our method to
silicon and diamond, for which we compute the temperature dependence of the
spin relaxation times and analyze the contributions to spin relaxation from different
phonons and valley processes. The computed spin relaxation times in silicon are
in excellent agreement with experiment in the 50−300 K temperature range. In
diamond, we predict intrinsic spin relaxation times of 540 𝜇s at 77 K and 2.3 𝜇s
at 300 K. We show that the spin-flip and momentum relaxation mechanisms are
governed by distinct microscopic processes. Our work enables precise predictions
of spin-phonon relaxation times in a wide range of materials, providing microscopic
insight into spin relaxation and guiding the development of spin-based quantum
technologies.
In Chapter 3, we developed a microscopic understanding of spin decoherence, which
is essential to advancing quantum technologies. Electron spin decoherence due to
atomic vibrations (phonons) plays a special role as it sets an intrinsic limit to the
performance of spin-based quantum devices. Two main sources of phonon-induced
spin decoherence, the Elliott-Yafet (EY) and Dyakonov-Perel (DP) mechanisms,
have distinct physical origins and theoretical treatments. In this work, we show a
rigorous theoretical framework: the spin-phonon Bethe-Salpeter equation to treat
the two main spin decoherence mechanisms due to phonons on the same footing,
and develop corresponding first-principles calculations. Our framework allows us
to predict spin relaxation times in semiconductors with a high accuracy, without

132
using any free or empirical parameters, and to shed new light on microscopic
spin dynamics, such as the interplay between spin precession and relaxation. Our
approach is general and applicable well beyond the examples in this work: it enables
precise predictions of spin decoherence in systems of fundamental and technological
relevance, including materials with band and localized spins, ions, heterostructures
and nanostructures, and quantum dot qubits.
In Chapter 4, the thesis focused on the long-range quadrupolar 𝑒-ph interaction.
Lattice vibrations in materials induce perturbations on the electron dynamics in the
form of long-range (dipole and quadrupole) and short-range (octopole and higher)
potentials. The dipole Fröhlich term can be included in current first-principles
electron-phonon (𝑒-ph) calculations and is present only in polar materials. The
quadrupole 𝑒-ph interaction is present in both polar and nonpolar materials, but
currently it cannot be computed from first principles. We have shown an approach
to compute the quadrupole 𝑒-ph interaction and include it in ab initio calculations
of 𝑒-ph matrix elements. Analysis of 𝑒-ph interactions for different phonon modes
reveals that the quadrupole term mainly affects optical modes in silicon and acoustic
modes in PbTiO3 , although the quadrupole term is needed for all modes to achieve
quantitative accuracy. The effect of the quadrupole 𝑒-ph interaction on electron
scattering processes and transport is shown to be important. Our approach enables accurate studies of 𝑒-ph interactions in broad classes of nonpolar, polar and
piezoelectric materials.
In Chapter 5, the thesis focused on correlated electron systems. 𝑒-ph interactions
are pervasive in condensed matter, governing phenomena such as transport, superconductivity, charge-density waves, polarons, and metal-insulator transitions.
First-principles approaches enable accurate calculations of 𝑒-ph interactions in a
wide range of solids. However, they remain an open challenge in correlated electron systems, where density functional theory often fails to describe the ground
state. Therefore reliable 𝑒-ph calculations remain out of reach for many transition metal oxides, high-temperature superconductors, Mott insulators, planetary
materials, and multiferroics. We have shown first-principles calculations of 𝑒-ph
interactions in CES, using the framework of Hubbard-corrected density functional
theory (DFT+U) and its linear response extension (DFPT+U), which can describe
the electronic structure and lattice dynamics of many CES. We showcase the accuracy of this approach for a prototypical Mott system, CoO, carrying out a detailed
investigation of its 𝑒-ph interactions and electron spectral functions. While standard

133
DFPT gives unphysically divergent and short-ranged 𝑒-ph interactions, DFPT+U is
shown to remove the divergences and properly account for the long-range Fröhlich
interaction, allowing us to model polaron effects in a Mott insulator. Our work
establishes a broadly applicable and affordable approach for quantitative studies of
𝑒-ph interactions in CES, a novel theoretical tool to interpret experiments in this
broad class of materials.
In Chapter 6, we have presented PERTURBO, a software package for first-principles
calculations of charge transport and ultrafast carrier dynamics in materials. The
current version focuses on electron–phonon interactions and can compute phononlimited transport properties such as the conductivity, carrier mobility and Seebeck
coefficient. It can also simulate the ultrafast nonequilibrium electron dynamics in the
presence of electron–phonon scattering. Taken together, Perturbo provides efficient
and broadly applicable ab initio tools to investigate electron–phonon interactions
and carrier dynamics quantitatively in metals, semiconductors, insulators, and 2D
materials.
Several future directions are possible for the newly enabled research areas discussed
above. The workflow proposed in Chapter 2 can be adapted to different perturbation
potentials, including perturbations from defects [1, 2], through which one could
study spin-flip and other defect-induced spin scattering processes. Our approach
can be applied broadly to study spin relaxation in materials for spintronics and
magnetism, and in topological materials. It can also be extended to treat spin states
localized at ions or defect, using calculations with large supercells that at present
are still technically challenging.
The currently developed formalism of the phonon-dressed vertex in Chapter 3 is quite
general; it can be readily applied to a broad range of systems as long as the operator
expectation values and the 𝑒-ph interactions are known. Chapter 3 applied the
formalism to spin decoherence, starting from the computational advancements made
in Chapter 2, and computing simultaneously the spin vertex and 𝑒-ph interaction in
the presence of SOC through joint interpolation of the 𝑒-ph matrix elements and
spin matrices through Wannier functions [3]. The formalism extends well beyond
spins, and our work opens the avenue for theoretical studies of linear response of
the system to various external field in the presence of 𝑒-ph interactions, which
includes charge transport including the effect of interband coherence, spin-current
inter-conversion, and operator decoherence in systems of interest for spintronics,
magnetism, in topological materials, and quantum information science.

134
The capabilities of PERTURBO in Chapter 6 can be significantly expanded. Many
features are currently under development for the next major release, including spinphonon relaxation times and spin dynamics [3], transport calculations in the large
polaron regime using the Kubo formalism [4], and charge transport and ultrafast
dynamics for coupled electrons and phonons, among others. As an alternative to
Wannier functions, interpolation using atomic orbitals [5] will also be supported in
a future release. We will extend the interface program to support additional external
codes, such as the TDEP software for temperature-dependent lattice dynamical
calculations [6].
In conclusion, the novel and computationally affordable methods presented in this
thesis enable precise predictions of spin relaxation and decoherence, with broad
implications for spin-based quantum technologies and for advancing microscopic
understanding of spin dynamics in condensed matter. The advancements made in
this thesis enable new avenues for computational research of 𝑒-ph interactions in
both condensed matter physics and quantum technologies.

135
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