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## 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 MoS _{2} 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 G _{k}. 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 G _{k} are not uniquely determined bymore »

- Authors:

- Northern Illinois Univ., DeKalb, IL (United States); Mindanao State Univ. (Philippines)
- 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.
```

```
@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}

}

Works referenced in this record:

##
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journal, September 2010

- Mak, Kin Fai; Lee, Changgu; Hone, James
- Physical Review Letters, Vol. 105, Issue 13, Article No.136805

##
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