Guang Guang - - Yu Guo ( Yu Guo ( 郭光宇 郭光宇 ) ) Physics Dept., Natl. Taiwan Univ., Taipei, Taiwan Physics Dept., Natl. Taiwan Univ., Taipei, Taiwan ( ( 國立台灣大學物理系 國立台灣大學物理系 ) ) Intrinsic spin Hall effect Intrinsic spin Hall effect in semiconductors and metals: in semiconductors and metals: Ab Ab initio initio calculations and model studies calculations and model studies (A talk in Institute of Physics, NCTU, June 21, 2007) (A talk in Institute of Physics, NCTU, June 21, 2007)
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Intrinsic spin Hall effect in semiconductors and metals · Intrinsic spin Hall effect in semiconductors and metals: Ab initio calculations and model studies (A talk in Institute of
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(2) In a 2-D electron gas in n-type semiconductor heterostructures
Rashba Hamiltonian
Universal spin Hall conductivity
(3) Significances of these theoretical discoveries of intrinsic spin Hall effects
Among other things, it would enable us to generate spin current electrically in semiconductor microstructures without applied magnetic fields or magnetic materials, and hence make possible pure electric driven spintronics in semiconductors which could be readily integated with conventional electronics.
4) Experiments on spin Hall effect
[Kato et al., Science 306, 1910 (2004)]
(a) in n-type 3D GaAs and InGaAs thin films
Attributed to extrinsic SHE because of weak crystal direction dependence.
(b) in p-type 2D semiconductor quantum wells
[Wunderlich, et al., PRL 94 (2005) 047204]
Attributed to intrinsic SHE.
(1) Will the intrinsic spin Hall effect exactly cancelled by the intrinsic orbital-angular-momentum Hall effect?
[S. Zhang and Z. Yang, PRL 2005]
for RashbaHamiltonian.
3. MotivationsIn this work, we study intrinsic spin Hall effect in semiconductors and metals by performing ab initio calculations.
(2) To go beyond the spherical 4-band Luttinger Hamiltonian.
(3) To understand the effects of epitaxial strains.
(4) To determine Rashba and Dresselhaus spin-orbit couplings in polar semiconductors, e. g., wurtzite.
↓E
(5) Spin Hall effect in transition metals
Nature 13 July 2006 Vol. 442, P. 04937
fcc AlσsH = 27~34 (Ωcm)-1
(T= 4.2 K)
( ) ( ) , λψλε nn
( ) ( )( ) ( )tidti
nn
tn eett γε
λψ −− ∫=Ψ 0/
∫ ∂∂
= t
nnn id
0
λ
λψ
λψλγ
Geometric phase:
λ
nε
Adiabatic theorem:
Parameter dependent system:
1 λ
2 λ
0 λ
t λ
(1) Berry phase[Berry, Proc. Roy. Soc. London A 392, 451 (1984)]
II. Theory and Computational Method1. Berry phase and semiclassical transport theory
ψλ
ψλ
ψλ
ψλ 1 2 2 1 ∂
∂∂∂
−∂∂
∂∂
=Ω ii
Ω= ∫∫ 21 λλγ ddn1λ
2λ
C
∫ ∂∂
=C
nnn id ψλ
ψλγ
Well defined for a closed path
Stokes theorem
Berry Curvature
ψλ
ψ∂∂i
)(λΩ )(rB
)( 2 λλψλ
ψλ Ω=∂∂
∫∫∫ did )( )( 2 rBrdrAdr ∫∫∫ =
)(rA
integer)( 2 =Ω∫∫ λλd ehBrd r /integer )( 2 =∫∫
Analogies
Berry curvature
Geometric phase
Berry connection
Chern number Dirac monopole
Vector potential
Aharonov-Bohm phase
Magnetic field
Symmetry properties
Time reversal:
Space inversion:
Both:
Violation: e.g., ferromagnets, zincblende crystal
)()( kk Ω−=−Ω
)()( kk Ω=−Ω
0)( =Ω k
(2) Semiclassical dynamics of Bloch electronsOld version [e.g., Aschroft, Mermin, 1976]
.)(
,)(1
BxrrBxEk
kkx
×−∂
∂=×−−=
∂∂
=
cc
nc
eeee ϕ
ε
Berry phase correction [e.g., Chang & Niu, PRL (1995), PRB (1996)]New version [Marder, 2000]
.||Im)(
,)(
),()(1
kkkΩ
Bxrrk
kΩkkkx
kk
∂∂
×∂∂
−=
×−∂
∂=
×−∂
∂=
nnn
c
nn
c
uu
ee ϕ
ε
(Berry curvature)
∫ −= ),()(3 krxj gekd
(3) Semiclassical transport theory
),()(),(
)(
krkkr
ΩEkkx
ffg
en
δ
ε
+=
×+∂
∂=
kkrkkΩkkEj
∂∂
−×−= ∫ ∫)(),()( 33
2nfdefde εδ
(Anomalous Hall conductance) (ordinary conductance)
Anomalous Hall conductivity [Yao, et al., PRL 2004]2
3
2' '
( ( )) ( )
2 Im | | ' ' | |( )
( )
zxy n n
n
x yzn
n n n n
e d f
n v n n v n
σ ε
ω ω≠
= Ω
Ω = −−
∑∫
∑k k
k k k
k k k kk
P.N. Dheer,Phys.Rev156, 637(1967)1032 (ohm cm)-1
G. S. Krinchik and V. S. Gushchin, Sov. Phys.-JETP 24, 984 (1969).
velocity operator v = cα , current operator j = -ecα (AHE),
(SHE),
(OHE).
α, β, Σ are 4×4 Dirac matrices.
Calculations must be based on a relativistic band theory because all the intrinsic Hall effects are caused by spin-orbit coupling.
αβ cz ,4
Σ=j
αβ cLz ,2
=j
2. Ab initio relativistic band structure method
Fully relativistic extension of linear muffin-tin orbital (LMTO) method. [Ebert, PRB 1988; Guo, Ebert, PRB 1995]
Dirac Hamiltonian
Density functional theory with generalized gradient approximation (DFT-GGA).
Scizzor’s approximation. The number of k-points in the IW (3/48 of BZ) for Si and Ge used is 49395 and for AlAl and GaAs is 98790 (6/48) (56 divisions of ΓX).
1. dc Hall conductivity
[Guo,Yao,Niu, PRL 94, 226601 (2005)]
III. Spin and orbital-angular-momentum Hall effects in p-type zincblende semicoductors
[Murakami, Nagaosa, Zhang, 2004 PRL93, 156804]Spin Hall insulators
Spin-orbit gap
cubic zincblende structure
Undoped PbSσxy
s =57.5 ( /e) Ω-1cm-1
2. Effects of lattice mismatch strains
3. ac Hall conductivity
IV. Strain dependence of spin Hall effect in n-type wurtzite semiconductors [Chang, Chen, Chen, Hong, Tsai, Chen, Guo, PRL 98, 136403 (2007);98, 239902 (2007) (Erratum)]