Simulation of Nonlinear Electronic Transport Using Wannier Interpolation

Abstract

Introduction 1
1 Electron dynamics in electromagnetic fields 5
1.1 Boltzmann equation................................................................................. 5
1.2 Berry Curvature and anomalous velocity............................................... 6
1.2.1 Berry Curvature in Bloch states..................................................... 6
1.2.2 Anomalous velocity ...................................................................... 8
1.3 Zeeman coupling in magnetic fields........................................................ 9
2 Expansion of the current density 11
2.1 Non-equlliburum distribution function.................................................. 11
2.2 Formalism for evaluation of conductivities............................................. 12
2.2.1 Relaxation time (t) independent conductivities......................... 13
2.2.2 Conductivités ex r ........................................................................ 14
2.2.3 Conductivités ex r2........................................................................ 16
2.2.4 Conductivités ex r3........................................................................ 18
2.3 Hall-like and Ohmic-like conductivity..................................................... 19
2.4 Transport effects ...................................................................................... 22
2.5 Evaluation of resistivities ........................................................................ 27
2.6 Tensor symmetry and Jahn’s Symbols..................................................... 30
3 Wannier interpolation of the gradient of Berry-like quantities 33
3.1 Introduction.............................................................................................. 33
3.2 Statement of the problem......................................................................... 36
3.3 Gradient of the Berry curvature of an effective Hamiltonian model ... 39
3.4 Non-Abelian covariant derivative.............................................................. 44
3.5 Catalogue of geometric quantities........................................................... 47
3.5.1 Gauge-covariant matrices.............................................................. 48
3.5.2 Gauge-invariant traces................................................................... 49
3.6 Wannier interpolation.............................................................................. 51
3.6.1 Wannier functions......................................................................... 51
3.6.2 Wannier interpolation of Berry-like quantities............................ 53
3.6.3 Covariant derivatives of Berry-like quantities ............................ 57
3.7 First principles results: trigonal Tellurium .............................•.............. 60
4 WannierBerri 69
4.1 Mixed Fourier transform........................................................................... 69
4.2 Symmetriztion............................................................................................ 72
4.2.1 Rotation of atomic-like Wannier function.................................... 72
4.2.2 Symmetrizazion of Wannier matrix elements ............................ 75
4.2.3 Spin-orbit coupling and time reversal symmetry........... .. 78
4.2.4 Improvement from symmetrization............................................ 79
5 Electrical magnetochiral anisotropy (eMChA) in trigonal tellurium 85
5.1 Boltzmann-transport theory of eMChA.................................................... 86
5.2 Numerical results for p-Te.......................................................................... 88
5.3 All components of eMChA-tensor in Te.................................................. 93
5.4 Two bands effective model ..................................................................... 94
6 Conclusions and Outlook 97
Bibliography 99
Curriculum Vitae 111
Acknowledgements 113