Spin Hall Effect

Guang-Yu Guo (郭光宇)

Physics Dept, National Taiwan University, Taiwan (國立臺灣大學物理系)

Spin Hall Effect

(A Colloquium Talk in Department of Physics,

National Taiwan University, 22 April 2014)

I. Introduction, overview and outlook

Plan of this Talk

II. Ab initio calculation of intrinsic spin Hall effect in solids 1. What is spin Hall effect

2. Spin Hall effect observed in semiconductors

3. Large room-temperature spin Hall effect in metals

4. Spintronics, magneto-electric devices and spin Hall effect 5. Spin-off’s: Topological insulators and spin caloritronics

1. Motivations

2. Berry phase formalism for intrinsic Hall effects.

3. Intrinsic spin Hall effect in platinum

III. Gigantic spin Hall effect in gold and multi-orbital Kondo effect 1. Gigantic spin Hall effect in gold/FePt

2. Spin Hall effect enhanced by multi-orbital Kondo effect.

3. Quantum Monte Carlo simulation IV. Summary

1) Ordinal Hall Effect

[Hall 1879]

2) Anomalous Hall Effect

Lorentz force

3) Extrinsic spin Hall Effect

Spin-orbit interaction

Spin current spin current

Charge current ( )

( )

dV r

dr s L⋅ qv B×

[Dyakonov & Perel, JETP 1971]

1. What is spin Hall effect

I. Introduction, overview and outlook

(Mott or skew scattering)

Edwin H. Hall (1855-1938)

4) Intrinsic spin Hall effect

(1) In p-type zincblende semiconductors





⋅

− +

= 2 ²

2 2 1

2

0 ) 2 ( )

2 ( 5

2 k k S

H m  

γ γ

Luttinger model γ

(hole)

i i

l il i

i

e E k

k m F

X k





 



=

+

=

λ

> 0 e

E k k

e m

X k  





 = − λ₃ ×

λ

Equation of motion

Anomalous velocity

n_h = 10¹⁹ cm^-3, μ= 50 cm /V·s, σ= eμn_h = 80 Ω^-1cm^-1; σ_s= 80 Ω^-1cm^-1

n_h = 10¹⁶ cm^-3, μ= 50 cm /V·s, σ= eμn_h = 0.6 Ω^-1cm^-1; σ_s= 7 Ω^-1cm^-1

[Science 301, 1348 (2003)]

Dirac monopole

(2) In a 2-D electron gas in n-type semiconductor heterostructures

Rashba Hamiltonian

Universal spin Hall conductivity

[PRL 92, 126603]

2. Spin Hall effect observed in semiconductors

[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

[PRL 94 (2005) 047204]

Attributed to intrinsic SHE.

(c) Spin Hall effect in strained n-type nitride semiconductors

[Chang, Chen, Chen, Hong, Tsai, Chen, Guo, PRL 98, 136403; 98, 239902 (E) (2007)]

n-type (5nm In_xGa_1-xN/3nm GaN) superlattice (x=0.15)

wurtzite

Nature 13 July 2006 Vol. 442, P. 176

fcc Al

σ_sH = 27~34 (Ωcm)^-1 (T= 4.2 K)

3. Large room-temperature spin Hall effect in metals

(direct) spin Hall effect

inverse spin Hall effect

[Saitoh, et al.,

APL 88 (2006) 182509]

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

σ_sH = 240 (Ωcm)^-1 (T= 290 K)

Assumed to be extrinsic!

[Hoffmann, IEEE Trans. Magn. 49 (2013) 5172]

4. Spintronics, magneto-devices and spin Hall effect

1) Spintronics (spin electronics)

Three basic elements: Generation, detection, & manipulation of spin current.

Ferromagnetic leads

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

semiconductor technologies.

(1) Direct spin Hall effect would allow us to generate pure spin current electrically in nonmagnetic microstructures without applied magnetic fields or magnetic materials, and make possible pure electric driven

spintronics which could be readily integated with conventional electronics.

(2) Inverse spin Hall effect would enable us to detect spin current

electrically, again without applied magnetic fields or magnetic materials.

(a) non-magnetic metals, (b) ferromagnetic metals and (c) half-metallic metals.

Usual spin current generations:

Spin-torque switching with the giant spin Hall effect of tantalum

[Liu et al., Science 336, 555 (2012)]

2) Magneto-electric devices

5. Spin-off’s: Topological insulators and spin caloritronics Quantum Hall effect in conventional 2DEG

2

BZ

2 , Chern (TKNN) number

1 ( )

2

xy

n

x x xy

n

e h

dk dk

σ ν ν

ν π

= ± =

=



[Thouless et al., PRL49, 405 (1982)]

[Laughlin, PRB23, 5632 (1981)]

Quantum Hall states are insulating with broken time-reversal symmetry.

Topological invariant is Chern number.

2D Topological insulators from quest for quantum spin Hall effect

[Kane & Mele, PRL 95 (2005) 146801]

Kane-Mele SOC Hamiltonian for graphene

† †

KM

,

i j i z ij j

i j ij

H t c c iλ c s v c

< > << >>

=





Ef

A B

y

x

no SOC SOC

SOC is too small

(<0.01 meV) to make QSHE observable!

[Chen, Xiao, Chiou, Guo, PRB 84, 165453 (2011)]

, 0.

2

s

xy xy

σ ν e σ

= π =

[Bernevig, Hughes, Zhang, Science 314, 1757 (2006)]

Quantum spin Hall effect in topological

phase in HgTe quantum well [Koenig et al., Science

318, 766 (2007)]

Evidence for quantum spin Hall effect in quantum wells

[Du et al., arXiv.1306.1925]

3D Topological insulators

[Fu, Kane, Mele, PRL98, 106803(the.)]

[Hsieh et al., Nature 460 (2009) 1101;

Xia et al., NP 5 (2009) 398]

Bi₂Te₃

Host a number of exotic phenemona, e.g., majorana fermion superconductivity, axion electrodynamics and quantum anomalous Hall effect

observed in

Cr_0.15(Bi_0.1Sb_0.9)

1.85Te₃ film

Chang et al. Science 340, 167 (2013)]

2. Spin caloritronics

Spin Nernst effect

Spin-orbit interaction

Spin current

( ) ( ) dV r

dr s L⋅

Spin Hall Effect

spin current

Spin Nernst Effect

[Cheng et al., PRB 2008]

[Bauer, Saitoh, van Wees, Nature Mater. 11 (2012) 391]

Spin Seebeck effect

[Uchida et al., Nature 455 (2008) 778]

Thus, we could have thermally driven spintronic devices, i.e., spin caloritronics.

1. Motivations

1) Will the intrinsic spin Hall effect exactly cancelled by the intrinsic orbital-angular-momentum Hall effect?

[S. Zhang and Z. Yang, cond-mat/0407704; PRL 2005]

In conclusion, we have shown that the ISHE is accompanied by the intrinsic orbital- angular-momentum Hall effect so that the total angular momenttum spin current is zero in a SOC system.

For Rashba Hamiltonian,

[Chen, Huang, Guo, PRB73 (2006) 235309]

This is confirmed for Rashba system by us. However, in Dresselhaus and Rashba systems, spin Hall conductivity would not be cancelled by the orbital Hall conductivity.

II. Ab initio studies of intrinsic spin Hall effect in solids

2) To go beyond the spherical 4-band Luttinger Hamiltonian.

3) To understand the

effects of epitaxial

strains.

4) To understand the detailed mechanism of the SHE in metals because it would lead to the material design of the large SHE

even at room temperature with the application to the spintronics.

To this end, ab initio band theoretical calculations for real metal

systems is essential.

( ) ( ) { ^ε

^λ

^ψ

^λ }

( ) ^{t = ψ}

( ) ^λ ( ) ^{t e}

^

^e

Ψ



= ^t n n

n d i

0

λ

λ ψ

ψ λ λ γ

Geometric phase:

λ

ε

Adiabatic theorem:

Parameter dependent system:

λ1

λ2

λ0

λt

1) Berry phase

[Berry, Proc. Roy. Soc. London A 392, 451 (1984)]

2. Berry phase formalism for

intrinsic Hall effects

λ ψ λ ψ

λ ψ

λ1 ψ 2 2 ∂ 1

∂

∂

− ∂

∂

∂

∂

= ∂

Ω i i

Ω

=  ^λ

^λ

γ

^{d d}

λ

λ

C

 _∂ ^∂

=

C

n n

n

d i ψ

ψ λ λ

γ

Well defined for a closed path

Stokes theorem

Berry Curvature

λ ψ ψ

i ∂

) (

Ω

) (

2 λ λ

λ ψ ψ

λ = Ω ^

∂

∂





 ^{dr A ( r  ) =}  ^d

^{r B ( r  )}

) (^r A ^

integer )

(

2

=





Analogies

Berry curvature

Geometric phase Berry connection

Chern number Dirac monopole

Vector potential

Aharonov-Bohm phase Magnetic field

2) Semiclassical dynamics of Bloch electrons Old version

) . ( ) ,

1 (

B r x

B r x

E k

k x k

×

∂ −

= ∂

×

−

−

=

∂

= ∂

c c

c n

e e

e

e 



 



 

 

ϕ ε

New version

Berry phase correction

.

|

| Im

) (

) , (

), ) (

( 1

k k k

Ω

B r x

k r

k Ω

k k x k

k k

∂

× ∂

∂

− ∂

=

×

∂ −

= ∂

×

∂ −

= ∂

n n n

c n n

c

u u

e

e 



 

 



ϕ ε

(Berry curvature)

3

( ) ( , ), d k e g

=  −

j x  r k

3) Semiclassical transport theory

n

( ) e

ε

= ∂ + ×

∂

x k E Ω

 k

 

k r k

k k

Ω k

k E

j ∂

− ∂

×

−

= ^e _

 ^d

^{f ( )} _ ^e  ^d

^{δ f ( , ) ε}

^{( )}

(Anomalous Hall conductance)

(ordinary conductance)

( , ) ( ) ( , ) g r k = f k + δ f r k

Anomalous Hall conductivity

2

3

' ' 2

( ( )) ( )

2 Im | | ' ' | |

( ) ( )

z

xy n n

n

x y

z n

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



σ_xy (S/cm) theory Exp.

bcc Fe 750^a 1030 hcp Co 477^b 480

a[Yao, et al., PRL 92 (2004) 037204]

fcc Ni -1066^c -1100

c[Fuh, Guo, PRB 84 (2011) 144427 ]

b[Wang, et al., PRB 76 (2007) 195109 ] [FLAPW (WIEN2k) calculations]

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.

{

}

4 β ^z cαⁱ

= Σ

j 

{

}

2

=  j

4) Ab initio relativistic band structure methods

Relativistic extension of linear muffin-tin orbital (LMTO) method.

[Ebert, PRB 1988; Guo & Ebert, PRB 51, 12633 (1995)]

Dirac Hamiltonian

H

= c α p ⋅ + mc

( β − + I ) v ( ) r I

3

' ' 2

( ( )) ( )

2 Im | | ' ' | |

( ) ( )

z

xy n n

n

x y

z n

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



(charge current operator) (spin current operator)

(orbital current operator)

[Guo,Yao,Niu, PRL 94, 226601 (2005)]

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

semicoductors

5) Application to intrinsic spin Hall effect in semiconductors

Strain effect

Pt: σ_sH = 2200 (Ωcm)^-1 (T = 0 K)

[Guo, Murakami, Chen, Nagaosa, PRL100, 096401 (2008)]

[Valenzuela, Tinkham, Nature 442, 176 (2006)]

Al: σ_sH (4.2 K) = 17 (Ωcm)^-1

σ_sH (exp., 4.2K) = 27, 34 (Ωcm)^-1

3. Large intrinsic spin Hall effect in platinum

Pt: σ_sH (300K) = 240 (Ωcm)^-1 σ_sH (exp., RT) = 240 (Ωcm)^-1

2

' '

( ) ( ( )) ( )

2 Im | | ' ' | |

( ) ( )

z z

xy n n

n z

x y

z n

n 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 k

k

 

[Kimura et al PRL98, 156601 (2007)]

Pt: σ_sH (0K) = 2200 (Ωcm)^-1

σ_sH (exp., 5 K) = 1700 (Ωcm)^-1

[Morota et al, PRB83, 174405 (2011)]

Pt has been widely used as a spin current generator and detector in recent novel spin current experiments, e.g.,

spin Seebeck effect,

[Uchida et al., Nature 455, 778 (2008)]

spin pumping,

[Kajiwara et al., Nature 464, 262 (2010)]

spin Hall switching

[Miron et al., Nature 476, 189 (2011)].

[Hoffmann,

IEEE Trans. Magn. 49 (2013) 5172]

Intrinsic spin Hall effect in pure Au

Au: σ_sH = 415 (Ωcm)^-1 (T = 0 K)

= 750 (Ωcm)^-1 (T = 300 K)

[Guo, JAP 105, 07C701 (2009)]

σ_sH (exp., RT) = 882 (Ωcm)^-1

[Mosendz, et al., PRB 82 (2010) 214403]

III. Giantic spin Hall effect in gold and multi-orbital

Kondo effect

spin Hall angle _s ^sH 0.1 at RT

xx

θ σ

= σ ≈

Au

[Seki, et al., Nat. Mater. 7 (2008)125]

5 1 -1

10 cm

σ

1.

= 2 ^Au exp( / )

ISHE s Au

Au

R P d

t

θ ρ λ

Δ −

What is the origin of giant spin Hall effect in gold Hall bars?

(i) Surface and interface effect? [Seki, et al., Nat. Mater. 7 (2008)125]

[Cercellier, et al., PRB73, 195413 (2006)]

(ii) Defect and impurity origin ?

Possible impurities: (a) vacancy of Au atom (b) Pt impurity

(c) Fe impurity

[Guo, Maekawa, Nagaosa, PRL 102, 036401 (2009)]

Results of FLAPW calculations

(a) the change in DOS in the 5d bands.

(b) the DOS change is near -1.5 eV.

Nonmagnetic in (a) and (b)

(c) A peak in DOS at the Fermi level and magnetic.

Proposal: Multiorbital Kondo effect in Fe impurity in gold.

2. Spin Hall effect enhanced by

multi-orbital Kondo effect

Kondo effect in metals with magnetic impurities

(a classic many-body phenomenon in condensed matter physics) (1) Resistivity abnormality in Au

with dilute magnetic impurities discovered by de Haas et al. in 1930’s. [Physica 1 (1934) 1115]

(2) Kondo proposed a (Kondo) model and solved it in the 2nd-order perturbation theory to explain the phenomenon in 1960’s.

[Prog. Theo. Phys. 32 (1964) 37]

† (0) ( 0, 1 / 2)

k f f

H c c_{σ σ} J J S

σ ε

=



k

σ S

0 1

1/5 1/5

min 1

( ) ln ,

( / 5 ) (Kondo temperature)

imp imp

imp K

T aT C C T

T a C T

ρ ρ ρ

ρ

= + −

= ≈

Extrinsic spin Hall effect due to skew scattering

[Guo, Maekawa, Nagaosa, PRL 102, 036401 (2009)]

scattering amplitudes

skewness function spin Hall angle

1 2 2

2 2

2 2 2 2

3 (cos 2 cos 2 )

9sin 4sin 3[1 cos 2( )]

s

δ δ δ

θ δ δ δ δ

+ −

+ − + −

≅ − −

+ + − − θ

≅ ≈ δ

0.1

[Guo, Maekawa, Nagaosa, PRL 102, 036401 (2009)]

0.001 ~ 0.01

θ

≈

Occupation numbers are related to the

phase shifts through generalized Friedel sum rule.

In a paper appearing in Physical Review Letters, Guo et al., propose an intriguing theory for this giant spin Hall effect.

Magnetic iron impurities have long been known to have a large effect on the low-T resistivity of gold, via the Kondo effect. If Guo et al. are right in their interpretation, the observation of a giant spin Hall effect resulting from the Kondo effect will add a curious new twist to this story. The history of the Kondo effect stretches back over seventy-five years. Despite its long history, the detailed Kondo physics of iron

remains a controversial subject.

This is a fascinating state of affairs—a wonderful example of the synergy that is possible between electronics applications and condensed-matter physics. If Guo et al. are right, the spin Hall

conductivity of gold should scale with the iron concentration, moreover, one might expect iron atoms to produce a large anomalous Hall effect. This could be a very exciting and unexpected turn in the long-standing story of the Kondo effect of iron in gold.

X-ray magnetic circular dichroism measurements

3. Quantum Monte Carlo simulation

1) problems

suggests an effective 3-channel Kondo model

( )













− +

+

+ +

+

=

↓

↑

+ +

+

σ σ σ

σ

σ σ σ

ξ ξ ξ

σ ξ

α α ξ α σ ξσ σ

ξ ξ ξσ ξσ σ

α

ε

ε

2 1

'

, 1 2 '

, , , ,

, ,

'

. . n

n J

n n U

n n

U

c h d

c V

d d

c c

H

k

k k

k

k k

k

For host band structure, α = 9 bands (6s, 6p, 5d orbitals of Au) are included.

For impurity-host hybridization, Au₂₆Fe supercell (3X3X3 primitive FCC cell) is considered. ξ = 5 (3d orbitals of Fe).

A realistic Anderson model is formulated with the host band structure and the impurity-host hybridization determined by ab initio DFT-LDA calculations.

(1) Single-impurity multi-orbital Anderson Model 2) Quantum Monte Carlo simulation

U = 5 eV, J = 0.9 eV, U’= U – 2J = 3.2 eV

For impurity Fe, one e_g orbital (z²) and one t_2g orbital (xz) are considered with the following parameters.

[Gu, Gan, Bulut, Ziman, Guo, Nagaosa, Maekawa, PRL105 (2010) 086401]

Local moment

Impurity magnetic susceptibility 3-Orbitals case

Occupation number

[Gu, Gan, Bulut, Ziman, Guo, Nagaosa, Maekawa, PRL105 (2010) 086401]

(2) Magnetic behaviors for Fe in Au from QMC simulations

Ising-type spin-orbit interaction for p-electrons: l =1, m =1,0,-1.

(3) Spin-orbit interaction within t

oribtals for Fe in Au

T = 350 K, = 75 meV

[Gu, Gan, Bulut, Ziman, Guo, Nagaosa, Maekawa, PRL105 (2010) 086401]

(4) Estimation of spin Hall angle for Fe impurity in Au

Since we consider only two t_2g orbitals with ℓ_z = ±1, the SOI within the t_2g orbitals gives rise to the difference in the occupation numbers

between the parallel (n_P) and anti-parallel (n_AP) states of the spin and angular momenta. These occupation numbers are related to the phase shifts δ_P and δ_AP, through generalized Friedel sum rule, respectively, as n_P(AP) = δ_P(AP)/π, and π < ℓ_zσ_z > = δ_P − δ_AP, π < n₂ > = π < n₃ >= δ_P + δ_AP. Putting < ℓ_zσ_z >= −0.44 for λ = 75 meV, and < n₂ > = <n₃>= 0.65,

we obtain δ_P = 1.35 and δ_AP = 2.73.

Taking into account the estimate sin δ₁ = 0.1, γ_s = 0.06 is thus obtained.

1 2 2

2 2

2 2 2 2

3 (cos 2 cos 2 )

9sin 4sin 3[1 cos 2( )]

s

δ δ δ

γ δ δ δ δ

+ −

+ − + −

≅ − −

+ + − −

[Seki, et al., Nat. Mater. 7 (2008)125]

γ_s = 0.11 (exp.)

Skew scattering θs ∼0.07 ,

independent of Fe concentration.

IV. Summary

1. Spin Hall effect, a manifestation of special relativity, is rich of fundamental physics, and is related to such classic phenomena in condensed matter physics as Kondo effect.

2. Spin Hall effect may be used to generate, detect and manipulate spin currents, and hence has important applications in spintronics and

magneto-devices.

3. Ab initio band theoretical calculations not only play an important role in revealing the mechanism of spin Hall effect, but also help in searching for and designing new spintronic materials.

4. Recent intensive research on spin Hall effect has also led to the creation of such hot fields such as topological insulators and spin caloritronics.

Discussions and Collaborations:

Qian Niu (UT Austin & PKU), Yugui Yao (BIT) Tsung-Wei Chen (Nat’l Sun Yat-sen U.)

Yang-Fang Chen and his experimental team (Taida) Naoto Nagaosa (Tokyo U.)

Shuichi Murakami (Tokyo Inst. Techno.) Bo Gu, Sadamichi Maekawa (JAEA)

Acknowledgements:

Financial Support:

National Science Council of The R.O.C.