---
author:
- Filipe Matusalem
- Lars Matthes
- Jürgen Furthmüller
- Marcelo Marques
- 'Lara K. Teles'
- Friedhelm Bechstedt
bibliography:
- '/home/filipe/Dropbox/artigos/library.bib'
title: 'Supplemental Material for: Quantization of spin Hall conductivity in two-dimensional topological insulators versus symmetry and spin orbit interaction'
---

Atomic Geometries
=================

![\[hex\] Atomic geometries in the unit cell (dashed line) of germanene, iodine-decorated germanene (GeI), stanene and fluorine-decorated stanene (fluorostanene). Besides a primitive hexagonal cell also a non-primitive rectangular one is displayed for GeI and stanene.](figures/geometries-hex.pdf){width="0.8\linewidth"}

![\[mos2\] Atomic geometries in square and rectangular unit cells (dashed line) of 1S-MoS$_2$/WS$_2$ and 1T'-MoS$_2$/WS$_2$.](figures/TS-MoS2-WS2.pdf){width="0.8\linewidth"}

\[ht\]

              germanene   GeI    GeI (rectangular)   stanene   fluorostanene   stanene (rectangular)   1S-MoS$_2$   1S-WS$_2$   1T'-MoS$_2$   1T'-WS$_2$
  ---------- ----------- ------ ------------------- --------- --------------- ----------------------- ------------ ----------- ------------- ------------
     $a$        4.06      4.31         7.47           4.68         4.94                8.10               6.34        6.36         5.72          5.71
     $b$          -        -           4.31             -            -                 4.68                -            -          3.17          3.19
   $\delta$     0.69      0.69         0.69           0.85         0.72                0.85                0            0          0.15          0.14
      d           -       2.91         2.91             -          2.82                  -                1.56        1.55         1.73          1.74

Band Structures
===============

Topological properties
======================

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - + +$ & $-$\
(0.5,0.5) & $- + - +$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - + - + + + - - - +$ & $-$\
(0.5,0.5) & $- + - + + - - + - + -$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - + +$ & $-$\
(0.5,0.5) & $- + - +$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - + - + + + - - - +$ & $-$\
(0.5,0.5) & $- + - + + - - + - + -$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - - + - - + + + - + - - - - - - + + + + + - + + + - - - + - + + + + -$ & $-$\
(0.5,0.5) & $- - + + + + - - + + + + - - - - + + + + - - - - - - - - + + + + - - + +$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - - + - - + + + + - - - - - - - + + + + + - + - + + - - - + + + + + -$ & $-$\
(0.5,0.5) & $- - + + + + - - + + - - + + - - + + + + - - - - - - - - + + + + - - + +$ & $+$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - - + - + + - - + + + - - + + - +$ & $+$\
(0.5,0.0) & $+ - - + - + - - + - + + - - - + + +$ & $-$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

<span>|c|c|c|</span>\
$k_{TRIM}$ & parity $\xi_{2m}$ of occupied bands & $\delta_i=\prod_{m}\xi_{2m}$\
(0.0,0.0) & $+ - - + - + + - - + + + - - + + - +$ & $+$\
(0.5,0.0) & $+ - - + + - - - + - + + - - + - + +$ & $-$\
& Z$_2$ invariant $\nu$: $(-1)^{\nu}=\prod_i\delta_i$ & $\nu=1$\

Static Spin Hall Conductivity
=============================

![\[doping\] Comparison of the real part of the static spin Hall conductivity $\sigma^z_{xy}$ of stanene (a, c) and GeI (b, d) computed using primitive hexagonal unit cells (a, b) and non-primitive rectangular unit cells (c, d).](figures/doping-hex-rec.pdf){width="0.7\linewidth"}

Frequency-dependent Spin Hall Conductivity
==========================================

{'source': 'arXiv_out/1909'}