GuangGuang--Yu Guo (Yu Guo (郭光宇郭光宇))Physics Dept., Natl. Taiwan Univ., Taipei, TaiwanPhysics 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:

AbAb initioinitio calculations and model studiescalculations and model studies

(A talk in Institute of Physics, NCTU, June 21, 2007)(A talk in Institute of Physics, NCTU, June 21, 2007)

I. Introduction

Plan of this Talk

II. Theory and computational method

1. Basic elements in spintronics. 2. Spin Hall effect.3. Motivations

1. Berry phase and semiclassical transport theory2. Ab initio relativistic band structure method.

III. Spin and orbital-angular-momentum Hall effects in p-type zincblende semiconductors

IV. Strain dependence of spin Hall effect in n-type wurtzitesemiconductors

V. Intrinsic spin Hall effect in platinum metal

1. Basic elements of spintronics (spin electronics)

Spin currents:

generation,detection,manipulation (control).

I. Introduction

Usual spin current generations:

Ferromagnetic leads especially half-metals

Schematic band pictures of (a) non-magnetic metals, (b) ferromagnetic metals and (c) half-metallic metals .

Spin filter[Slobodsky, et al., PRL 2003]

Problems: magnets and/or magnetic fields needed, and difficult to integrate with semiconductor technologies.

2. Spin Hall effect

1) The Hall Effect

Ordinal Hall Effect (1879)

Anomalous Hall Effect (Hall, 1880 & 1881)(Extraordinary or spontaneous Hall effect)

Zoo of the Hall EffectsOrdinary Hall effect (1879);

Anomalous Hall effect (1880 & 1881);

Integer quantum Hall effect (von Klitzing, et al., 1980);

Fractional quantum Hall effect (Tsui, et al., 1982);

(Extrinsic) spin Hall effect (Dyakonov+Perel, 1971; Hirsch, 1999);

Intrinsic spin Hall effect (Murakami, et al., 2003, Sinova, et al., 2004).

Hall effect of light (Nagaosa, et al., 2004)

Thermal Hall effect (?)

2) (Extrinsic) Spin Hall Effect

[Dyakonov & Perel, JETP 1971; Hirsch, PRL 1999; Zhang, PRL 2000]

(Extrinsic) spin Hall effect

Lσ ⋅+= )()( rVrVV soc

3) Intrinsic spin Hall effect

Luttinger Hamiltonian

(1) In p-type bulk semiconductors

[Murakami, cond-mat/0504353 (2005)]

Spin current

nh = 1019 cm-3, μ= 50 cm /V·s, σ= eμnh = 80 Ω-1cm-1;σs= 80 Ω-1cm-1

nh = 1016 cm-3, μ= 50 cm /V·s, σ= eμnh = 0.6 Ω-1cm-1;σs= 7 Ω-1cm-1

(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).

FM bcc Fe

(4) Berry phase formalism for Hall effects

3

2' '

( ( )) ( )

2 Im | | ' ' | |( )

( )

zxy n n

n

x yzn

n n n n

e d f

n j n n v n

σ ε

ω ω≠

= Ω

Ω =−

∑∫

∑k k

k k k

k k k kk

current operator jx = -evx (AHE),jx = ħσz,vx/4 (SHE), jx = ħLz,vx/4 (OHE).

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)]

n-type (5nm InxGa1-xN/3nm GaN) superlattice (x=0.15)

↓E

GaN

2 22

0 * ( )( )2 e x y y x

kH k k km

α β σ σ= + + −For wurtzite structures

2 23

* ( ), 2 e e

kE k km

α β α λ α± = + = +k ∓

←

2 2

*

1 ( )2 2

kE Em− ++ =k k

31 ( ) ( )2 eE E k kα β− +− = +k k

* *2 2

2 2

2 2 2 2[1 ( ) ] ( )8 3 8 3

sHxy e e

e m e mσ α β α βπ π

= + ≈

[PRL 98, 136403 (2007)]

Linear response Kubo formalism

zero due to any disorder [Sinova, et al., SSC138 (2006) 214]

↑

α(exp) ~ 0.06 (eVA)

V. Large intrinsic spin Hall effect in platinumNature 13 July 2006 Vol. 442, P. 04937

fcc AlσsH = 27~34 (Ωcm)-1

(T= 4.2 K)

[PRL98, 156601; 98 (2007), 139901 (Erratum) (2007)]

σsH = 240 (Ωcm)-1

(T= 290 K)

Assumed to be extrinsic!

Ab initio relativistic band structure calculations

Pt: σsH = 2200 (Ωcm)-1

(T = 0 K)

Al: σsH = 17 (Ωcm)-1

(T = 0 K)

[Guo, et al., arXiv: 0705.0409 (2007)]

Pt: σsH (300K) = 240 (Ωcm)-1

σsH (exp., RT) = 240 (Ωcm)-1

Al: σsH (4.2 K) = 17 (Ωcm)-1

σsH (300 K) = 6 (Ωcm)-1

σsH (exp., 4.2K) = 27, 34 (Ωcm)-1

2' '

( ) ( ( )) ( )

2 Im | | ' ' | |( )

( )

z zxy n n

nzx yz

nn n n n

e e f

n j n n v n

σ ε

ω ω≠

= Ω = Ω

Ω =−

∑ ∑∑

∑

k k

k k

k k k

k k k kk

Effect of impurity scattering and two band model analysis

Intrinsic SHE is robust against short-ranged impurity scattering!

Collaborators:

Qian Niu (UT Austin), Yugui Yao (CAS, Beijing)[on p-type zincblende semiconductors, PRL 94, 226601 (2005)]

Yang-Fang Chen + his exp. team (Nat’l Taiwan U.)Tsung-Wei Chen (Nat’l Taiwan U.)

[on n-type wurtzite nitride semiconductors, PRL 98, 136403 (2007)]Shuichi Murakami, Naoto Nagaosa (Tokyo U.)Tsung-Wei Chen [on platinum, arXiv: 0705.0409]

Financial Support:

Taiwan National Science Council.

Acknowledgements: