Classification of spin Hall effect in two-dimensional systems

doi:10.1007/s11467-023-1358-3

Front. Phys.

2024, Vol. 19

Issue (3) : 33205 https://doi.org/10.1007/s11467-023-1358-3

RESEARCH ARTICLE

Classification of spin Hall effect in two-dimensional systems

Longjun Xiang¹, Fuming Xu^1,², Luyang Wang¹(

), Jian Wang^1,³(

)

¹. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
². Quantum Science Center of Guangdong-Hongkong-Macao Greater Bay Area (Guangdong), Shenzhen 518045, China
³. Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

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Abstract

Physical properties such as the conductivity are usually classified according to the symmetry of the underlying system using Neumann’s principle, which gives an upper bound for the number of independent components of the corresponding property tensor. However, for a given Hamiltonian, this global approach usually can not give a definite answer on whether a physical effect such as spin Hall effect (SHE) exists or not. It is found that the parity and types of spin-orbit interactions (SOIs) are good indicators that can further reduce the number of independent components of the spin Hall conductivity for a specific system. In terms of the parity as well as various Rashba-like and Dresselhaus-like SOIs, we propose a local approach to classify SHE in two-dimensional (2D) two-band models, where sufficient conditions for identifying the existence or absence of SHE in all 2D magnetic point groups are presented.

Keywords spin Hall effect symmetry two-dimensional system

Corresponding Author(s): Luyang Wang,Jian Wang

About author:

Peng Lei and Charity Ngina Mwangi contributed equally to this work.

Issue Date: 10 November 2023

Cite this article:

Longjun Xiang,Fuming Xu,Luyang Wang, et al. Classification of spin Hall effect in two-dimensional systems[J]. Front. Phys. , 2024, 19(3): 33205.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1358-3
https://academic.hep.com.cn/fop/EN/Y2024/V19/I3/33205

In-plane Dresselhaus-like SOI (symmetry)		In-plane Rashba-like SOI (symmetry)

$H 2 = ℜ (k + σ +)$ (21')		$H 1 = k y σ x − k x σ y ∼ ℑ (k + σ −)$ ( $C ∞$ , $T$ )
$H 3 x = ℜ (k + 2 σ −)$ (2'mm') [33, 39], $H 4 x = ℜ (k + 2 σ +)$ (6'm'm) [33, 39]		$H 3 y = ℑ (k + 2 σ −)$ (2'm'm) [33], $H 4 y = ℑ (k + 2 σ +)$ (6'm'm) [33]
$H 2 ′ = ℜ (k + 3 σ −)$ (21'), $H 5 = ℜ (k + 3 σ +)$ (41')		$H 6 = ℑ (k + 3 σ −)$ (2mm1') [33], $H 7 = ℑ (k + 3 σ +)$ (4mm1') [33]
$H 5 ′ = ℜ (k + 5 σ −)$ (41'), $H 8 = ℜ (k + 5 σ +)$ (61')		$H 7 ′ = ℑ (k + 5 σ −)$ (4mm1'), $H 9 = ℑ (k + 5 σ +)$ (6mm1')

Out-of-plane Dresselhaus-like SOI		Out-of-plane Rashba-like SOI

$ℜ (k +) σ z$ (m1') [40]		$ℑ (k +) σ z$ (m1')
$ℜ (k + 2) σ z$ (4'm'm) [11]		$ℑ (k + 2) σ z$ (4'm'm)
$ℜ (k + 3) σ z$ (3m) [32, 41–43]		$ℑ (k + 3) σ z$ (3m) [32, 42, 44, 45]

Tab.1 SOI Hamiltonians of various orders and symmetries.

Fig.1 Schematic plots of the in-plane (a) and out-of-plane (b) spin Hall effects.

J s y / z

denotes the spin current, and

E

is the driving electric field.

MPG	$σ x y x$	$σ x y y$	$σ x y z$

$m x$ , $m x$ 1', $m x$ '	0	N	N
2'	N	0	N
$T$	N	N	N
3, 31', 6'	S	S	A
3 $m x$ , 3 $m x$ 1', 3 $m x$ ', 6' $m y$ ' $m x$	0	S	A
4mm, 4mm1', 4'm'm, 4m'm', 4, 41', 6mm1', 6mm, 6, 61', 6m'm'	0	0	A
2, 21', 2mm, 2mm1', 2'm'm, 2m'm', 4'	0	0	N

Tab.2 Results from Neumann’s principle for IP-SHE. Here “N” represents no constraint. “S” and “A” refer to symmetric relation for

σ x y x / y

and antisymmetric relation for

σ x y z

with respective to

x

and

y

, respectively. “0” stands for zero

σ x y α

. The results with

m x

are listed. For MPGs containing

m y

σ x y x

and

σ x y y

are interchanged. For example, for MPG

m y

m y

1',

m y

σ x y x

= N and

σ x y y

= 0.

MPG	$σ x y x$	$σ x y y$	$σ x y z$

$m x$	0	1	0/1
$m x$ 1', 3 $m x$ 1'	0	1	1
$T$ , 3, 31'	0/1	0/1	1
$m x$ ', 3 $m x$ , 3 $m x$ '	0	0/1	1
4mm, 4'm'm, 4m'm', 4, 6mm, 6, 6m'm', 2, 2mm, 2'm'm, 2m'm', 4'	0	0	1
2', 21', 2'mm', 2mm1', 41', 4mm1', 6', 6'm'm, 61', 6mm1'	NA	NA	1

Tab.3 Results of direct calculation. Here “0” and “1” stand for zero and nonzero

σ x y α

, respectively. “0/1” means SHE can be switched on and off and “NA” means that SOI Hamiltonian is not available for

d z ≠ 0

. The results with

m x

are listed. For MPGs containing

m y

σ x y x

and

σ x y y

are interchanged.

IP-SOI	$m x$	$m y$	$m x + y$	OP-SOI	$m x$	$m y$	$m x + y$

$R I P −$	S	S	S	$R O P −$	A	S	A
$R I P +$	A	S	A	$R O P +$	S	S	S
$D I P −$	A	A	S	$D O P −$	S	A	A
$D I P +$	S	A	A	$D O P +$	A	A	S

Table A1 Mirror symmetry of IP-SOI and OP-SOI. “A” and “S” represent antisymmetric and symmetric under different mirror operations, respectively. The SOI has the corresponding mirror symmetry if “S” is marked.

$H 1 = ℑ (k − σ +)$ ( $C ∞$ , $T$ )	$H 2 = ℜ (k + σ +)$ (21')	$H 3 x = ℜ (k + 2 σ −)$ (2'mm')	$H 3 y = ℑ (k + 2 σ −)$ (2'm'm)

$H 4 x = ℜ (k + 2 σ +)$ (6'mm')	$H 4 y = ℑ (k + 2 σ +)$ (6'm'm)	$H 5 = ℜ (k + 3 σ +)$ (41')	$H 6 = ℑ (k + 3 σ −)$ (2mm1')
$H 7 = ℑ (k + 3 σ +)$ (4mm1')	$H 8 = ℜ (k + 5 σ +)$ (61')	$H 9 = ℑ (k + 5 σ +)$ (6mm1')	$H 10 x = ℜ (k +) σ z$ (m1')
$H 10 y = ℑ (k +) σ z$ (m1')	$H 11 = ℜ (k + 2) σ z$ (4'm'm)	$H 12 = ℑ (k + 2) σ z$ (4'm'm)	$H 13 x = ℜ (k + 3) σ z$ (3m1')
$H 13 y = ℑ (k + 3) σ z$ (3m1')	$H 14 = ℜ (k + 4) σ z$ (4m'm')	$H 15 = ℑ (k + 4) σ z$ (4mm)	$H 16 = ℜ (k + 6) σ z$ (6m'm')
$H 17 = ℑ (k + 6) σ z$ (6mm)	$H 18 = H 2 + H 3$ (2')	$H 19 = H 1 + H 4 x + H 4 y$ (6')	$H 20 x = H 1 + H 3 y + H 11$ (m')
$H 20 y = H 1 + H 3 x + H 11$ (m')	$H 21 = H 2 + H 12$ (2)	$H 22 = H 1 + H 11 + H 14$ (2m'm')	$H 23 = H 5 + H 14$ (4)
$H 24 x = H 1 + H 4 y + H 16$ (3m')	$H 24 y = H 1 + H 4 x + H 16$ (3m')	$H 25 = H 4 x + H 4 y + H 13 x$ (3)	$H 26 = H 1 + H 16 + H 17$ (6)
$H 27 = H 1 + H 10 x + H 10 y$ ( $T$ )	$H 28 x = H 1 + H 3 x + H 12$ (m)	$H 28 y = H 1 + H 3 y + H 12$ (m)	$H 29 = H 1 + H 12 + H 15$ (2mm)
$H 30 = H 1 + H 11 + H 12$ (4')	$H 31 = H 1 + H 13 x + H 13 y$ (31')	$H 32 x = H 1 + H 4 x + H 13 x$ (3m)	$H 32 y = H 1 + H 4 y + H 13 y$ (3m)

$H 2 ′ = ℜ (k + 3 σ −)$ (21')	$H 4 x ′ = ℜ (k + 4 σ −)$ (6'mm')	$H 4 y ′ = ℑ (k + 4 σ −)$ (6'm'm)	$H 5 ′ = ℜ (k + 5 σ −)$ (41')
$H 7 ′ = ℑ (k + 5 σ −)$ (4mm1')

Table A2 SOI Hamiltonians and corresponding symmetries for 31 MPGs.

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