skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Effective dynamics of two-dimensional Bloch electrons in external fields derived from symmetry

Abstract

We develop a comprehensive theory for the effective dynamics of Bloch electrons based on symmetry. We begin with a scheme to systematically derive the irreducible representations (IRs) characterizing the Bloch eigenstates in a crystal. Starting from a tight-binding (TB) approach, we decompose the TB basis functions into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves. Each of these two subproblems is independent of the details of a particular crystal structure and it is largely independent of the relevant aspects of the other subproblem, hence permitting for each subproblem an independent universal solution. Taking monolayer MoS2 and few-layer graphene as examples, we tabulate the symmetrized p and d orbitals as well as the symmetrized plane-wave spinors relevant for these crystal structures. The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector Gk. For many crystal structures, it is possible to define multiple distinct coordinate systems such that for wave vectors k at the border of the Brillouin zone the IRs characterizing the Bloch states depend on the coordinate system, i.e., these IRs of Gk are not uniquely determined by the symmetry ofmore » a crystal structure. The different coordinate systems are related by a coordinate shift that results in a rearrangement of the IRs of Gk characterizing the Bloch states. We illustrate this rearrangement with three coordinate systems for MoS2 and trilayer graphene. The freedom to choose different distinct coordinate systems can simplify the symmetry analysis of the Bloch states. Given the IRs of the Bloch states in one coordinate system, a rearrangement lemma yields immediately the IRs of the Bloch states in the other coordinate systems. The rearrangement of the IRs in different coordinate systems does not affect observable physics such as selection rules or the effective Hamiltonians for the dynamics of Bloch states in external fields. Using monolayer MoS2 as an example, we combine the symmetry analysis of its bulk Bloch states with the theory of invariants to construct a generic multiband Hamiltonian for electrons near the K point of the Brillouin zone. Here, the Hamiltonian includes the effect of spin-orbit coupling, strain, and external electric and magnetic fields. Invariance of the Hamiltonian under time reversal yields additional constraints for the allowed terms in the Hamiltonian and it determines the phase (real or imaginary) of the prefactors.« less

Authors:
 [1]; ORCiD logo [2]
  1. Northern Illinois Univ., DeKalb, IL (United States); Mindanao State Univ. (Philippines)
  2. Northern Illinois Univ., DeKalb, IL (United States); Argonne National Lab. (ANL), Argonne, IL (United States)
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
National Science Foundation (NSF); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; USDOE
OSTI Identifier:
1562926
Alternate Identifier(s):
OSTI ID: 1560316
Grant/Contract Number:  
AC02-06CH11357
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 100; Journal Issue: 12; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Fajardo, E. A., and Winkler, R. Effective dynamics of two-dimensional Bloch electrons in external fields derived from symmetry. United States: N. p., 2019. Web. doi:10.1103/PhysRevB.100.125301.
Fajardo, E. A., & Winkler, R. Effective dynamics of two-dimensional Bloch electrons in external fields derived from symmetry. United States. doi:10.1103/PhysRevB.100.125301.
Fajardo, E. A., and Winkler, R. Tue . "Effective dynamics of two-dimensional Bloch electrons in external fields derived from symmetry". United States. doi:10.1103/PhysRevB.100.125301. https://www.osti.gov/servlets/purl/1562926.
@article{osti_1562926,
title = {Effective dynamics of two-dimensional Bloch electrons in external fields derived from symmetry},
author = {Fajardo, E. A. and Winkler, R.},
abstractNote = {We develop a comprehensive theory for the effective dynamics of Bloch electrons based on symmetry. We begin with a scheme to systematically derive the irreducible representations (IRs) characterizing the Bloch eigenstates in a crystal. Starting from a tight-binding (TB) approach, we decompose the TB basis functions into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves. Each of these two subproblems is independent of the details of a particular crystal structure and it is largely independent of the relevant aspects of the other subproblem, hence permitting for each subproblem an independent universal solution. Taking monolayer MoS2 and few-layer graphene as examples, we tabulate the symmetrized p and d orbitals as well as the symmetrized plane-wave spinors relevant for these crystal structures. The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector Gk. For many crystal structures, it is possible to define multiple distinct coordinate systems such that for wave vectors k at the border of the Brillouin zone the IRs characterizing the Bloch states depend on the coordinate system, i.e., these IRs of Gk are not uniquely determined by the symmetry of a crystal structure. The different coordinate systems are related by a coordinate shift that results in a rearrangement of the IRs of Gk characterizing the Bloch states. We illustrate this rearrangement with three coordinate systems for MoS2 and trilayer graphene. The freedom to choose different distinct coordinate systems can simplify the symmetry analysis of the Bloch states. Given the IRs of the Bloch states in one coordinate system, a rearrangement lemma yields immediately the IRs of the Bloch states in the other coordinate systems. The rearrangement of the IRs in different coordinate systems does not affect observable physics such as selection rules or the effective Hamiltonians for the dynamics of Bloch states in external fields. Using monolayer MoS2 as an example, we combine the symmetry analysis of its bulk Bloch states with the theory of invariants to construct a generic multiband Hamiltonian for electrons near the K point of the Brillouin zone. Here, the Hamiltonian includes the effect of spin-orbit coupling, strain, and external electric and magnetic fields. Invariance of the Hamiltonian under time reversal yields additional constraints for the allowed terms in the Hamiltonian and it determines the phase (real or imaginary) of the prefactors.},
doi = {10.1103/PhysRevB.100.125301},
journal = {Physical Review B},
number = 12,
volume = 100,
place = {United States},
year = {2019},
month = {9}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Electronic structure of a single MoS2 monolayer
journal, May 2012


Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group-VI Dichalcogenides
journal, May 2012


Colloquium : Excitons in atomically thin transition metal dichalcogenides
journal, April 2018


The Application of Group theory to the Quantum Dynamics of Monatomic Systems
journal, July 1930


Dependence of the Symmetry Labelling of Electron and Phonon States in Crystals on Orientation
journal, July 1972


The indirect to direct band gap transition in multilayered MoS 2 as predicted by screened hybrid density functional theory
journal, December 2011

  • Ellis, Jason K.; Lucero, Melissa J.; Scuseria, Gustavo E.
  • Applied Physics Letters, Vol. 99, Issue 26
  • DOI: 10.1063/1.3672219

Simplified LCAO Method for the Periodic Potential Problem
journal, June 1954


Symmetry of Electron States in GaP
journal, September 1968


Band Structures of Transition-Metal-Dichalcogenide Layer Compounds
journal, October 1973


Motion of Electrons and Holes in Perturbed Periodic Fields
journal, February 1955


2D transition metal dichalcogenides
journal, June 2017

  • Manzeli, Sajedeh; Ovchinnikov, Dmitry; Pasquier, Diego
  • Nature Reviews Materials, Vol. 2, Issue 8
  • DOI: 10.1038/natrevmats.2017.33

Electromagnetic coupling of spins and pseudospins in bilayer graphene
journal, May 2015


Quantum resonances in the valence bands of germanium. I. Theoretical considerations
journal, May 1974


Intervalley-Scattering Selection Rules in III-V Semiconductors
journal, May 1966


Invariant expansion for the trigonal band structure of graphene
journal, December 2010


Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory
journal, May 1956


Emerging Device Applications for Semiconducting Two-Dimensional Transition Metal Dichalcogenides
journal, January 2014

  • Jariwala, Deep; Sangwan, Vinod K.; Lauhon, Lincoln J.
  • ACS Nano, Vol. 8, Issue 2
  • DOI: 10.1021/nn500064s

Cyclotron Resonance of Electrons and Holes in Silicon and Germanium Crystals
journal, April 1955


k · p theory for two-dimensional transition metal dichalcogenide semiconductors
journal, April 2015


Topological quantum chemistry
journal, July 2017

  • Bradlyn, Barry; Elcoro, L.; Cano, Jennifer
  • Nature, Vol. 547, Issue 7663
  • DOI: 10.1038/nature23268

Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides
journal, August 2013


Band connectivity for topological quantum chemistry: Band structures as a graph theory problem
journal, January 2018


The origin dependence of Wyckoff site description of a crystal structure
journal, July 1973


Angular Momentum Theory and Localized States in Solids. Investigation of Shallow Acceptor States in Semiconductors
journal, December 1970


Atomically Thin MoS2 A New Direct-Gap Semiconductor
journal, September 2010


Origin Dependence of the Symmetry Labelling of Electron and Phonon States in Crystals
journal, February 1971


Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer MoS 2
journal, August 2013


Effect of Time-Reversal Symmetry on Energy Bands of Crystals
journal, August 1937


Two-dimensional van der Waals materials
journal, September 2016

  • Ajayan, Pulickel; Kim, Philip; Banerjee, Kaustav
  • Physics Today, Vol. 69, Issue 9
  • DOI: 10.1063/PT.3.3297

Building blocks of topological quantum chemistry: Elementary band representations
journal, January 2018


The electronic properties of graphene
journal, January 2009

  • Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.
  • Reviews of Modern Physics, Vol. 81, Issue 1, p. 109-162
  • DOI: 10.1103/RevModPhys.81.109

Quantum resonances in the valence bands of germanium. II. Cyclotron resonances in uniaxially stressed crystals
journal, May 1974


Quantum resonances in the valence bands of zinc-blende semiconductors. I. Theoretical aspects
journal, July 1979


Monolayer MoS 2 : Trigonal warping, the Γ valley, and spin-orbit coupling effects
journal, July 2013


Quantum resonances in the valence band of zinc-blende semiconductors. II. Results for p -InSb under uniaxial stress
journal, July 1979