GuangYu 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 roomtemperature spin Hall effect in metals
4. Spintronics, magnetoelectric devices and spin Hall effect 5. Spinoff’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 multiorbital Kondo effect 1. Gigantic spin Hall effect in gold/FePt
2. Spin Hall effect enhanced by multiorbital Kondo effect.
3. Quantum Monte Carlo simulation IV. Summary
1) Ordinal Hall Effect
[Hall 1879]
2) Anomalous Hall Effect
[Hall, 1880 & 1881]Lorentz force
3) Extrinsic spin Hall Effect
Spinorbit 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 (18551938)
4) Intrinsic spin Hall effect
(1) In ptype 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
= − λ_{3} ×
λ
Equation of motion
Anomalous velocity
n_{h} = 10^{19} cm^{3}, μ= 50 cm /V·s, σ= eμn_{h} = 80 Ω^{1}cm^{1}; σ_{s}= 80 Ω^{1}cm^{1}
n_{h} = 10^{16} cm^{3}, μ= 50 cm /V·s, σ= eμn_{h} = 0.6 Ω^{1}cm^{1}; σ_{s}= 7 Ω^{1}cm^{1}
[Science 301, 1348 (2003)]
Dirac monopole
(2) In a 2D electron gas in ntype 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 ntype 3D GaAs and InGaAs thin films
Attributed to extrinsic SHE because of weak crystal direction dependence.
(b) in ptype 2D semiconductor quantum wells
[PRL 94 (2005) 047204]
Attributed to intrinsic SHE.
(c) Spin Hall effect in strained ntype nitride semiconductors
[Chang, Chen, Chen, Hong, Tsai, Chen, Guo, PRL 98, 136403; 98, 239902 (E) (2007)]
ntype (5nm In_{x}Ga_{1x}N/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 roomtemperature 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, magnetodevices 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) nonmagnetic metals, (b) ferromagnetic metals and (c) halfmetallic metals.
Usual spin current generations:
Spintorque switching with the giant spin Hall effect of tantalum
[Liu et al., Science 336, 555 (2012)]
2) Magnetoelectric devices
5. Spinoff’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
= ± =
=
Ω ^{k}[Thouless et al., PRL49, 405 (1982)]
[Laughlin, PRB23, 5632 (1981)]
Quantum Hall states are insulating with broken timereversal symmetry.
Topological invariant is Chern number.
2D Topological insulators from quest for quantum spin Hall effect
[Kane & Mele, PRL 95 (2005) 146801]
KaneMele 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]
Bi_{2}Te_{3}
Host a number of exotic phenemona, e.g., majorana fermion superconductivity, axion electrodynamics and quantum anomalous Hall effect
observed in
Cr_{0.15}(Bi_{0.1}Sb_{0.9})
1.85Te_{3} film
Chang et al. Science 340, 167 (2013)]
2. Spin caloritronics
Spin Nernst effect
Spinorbit 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 orbitalangularmomentum Hall effect?
[S. Zhang and Z. Yang, condmat/0407704; PRL 2005]
In conclusion, we have shown that the ISHE is accompanied by the intrinsic orbital angularmomentum 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 4band 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}
^{n}^{λ}
^{,}^{ }^{ψ}
^{n}^{λ} }
( ) ^{t} ^{=} ^{ψ}
^{n}( ) ^{λ} ( ) ^{t} ^{e}
^{−}^{i}^{}
^{t}^{dt} ^{ε}^{n}^{e}
^{−}^{i}^{γ}^{n}^{( )}^{t}Ψ
^{0} ^{/}^{}
_{∂}^{∂}= ^{t} n n
n d i
0
λ
λ ψ
ψ λ λ γ
Geometric phase:
λ
ε
nAdiabatic 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}
^{1}^{λ}
^{2}γ
n^{d} ^{d}
λ
1λ
2C
_{∂} ^{∂}
=
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 ∂
) (
_{λ}^{}Ω
^{B}^{(}^{r}^{}^{)}) (
2 λ λ
λ ψ ψ
λ = Ω ^{}
∂
∂
^{d} ^{i} ^{d} ^{dr} ^{A} ^{(} ^{r} ^{} ^{)} ^{=} ^{d}
^{2}^{r} ^{B} ^{(} ^{r} ^{} ^{)}
) (^{r} A ^{}
integer )
(
2
=
^{d} ^{λ} Ω ^{λ}^{}
^{d}^{2}^{r}^{ }^{B}^{(}^{r}^{}^{)} ^{=} ^{integer }^{h} ^{/} ^{e}Analogies
Berry curvature
Geometric phase Berry connection
Chern number Dirac monopole
Vector potential
AharonovBohm phase Magnetic field
2) Semiclassical dynamics of Bloch electrons Old version
[e.g., Aschroft, Mermin, 1976]) . ( ) ,
1 (
B r x
B r x
E k
k x k
×
∂ −
= ∂
×
−
−
=
∂
= ∂
c c
c n
e e
e
e
ϕ ε
New version
[Marder, 2000]Berry phase correction
[Chang & Niu, PRL (1995), PRB (1996)].

 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} _{}
^{2} ^{d}
^{3}^{f} ^{(} ^{)} _{} ^{e} ^{d}
^{3}^{δ} ^{f} ^{(} ^{,} ^{)} ^{ε}
^{n}^{(} ^{)}
(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 kk 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 spinorbit coupling.
{
^{,}}
4 β ^{z} cα^{i}
= Σ
j
{
^{β}^{L}^{z}^{,}^{c}^{α}}
2
= j
4) Ab initio relativistic band structure methods
Relativistic extension of linear muffintin orbital (LMTO) method.
[Ebert, PRB 1988; Guo & Ebert, PRB 51, 12633 (1995)]
Dirac Hamiltonian
H
_{D}= c α p ⋅ + mc
^{2}( 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 kk 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 ptype 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 multiorbital
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
sH ≈ Ω^{−}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
multiorbital Kondo effect
Kondo effect in metals with magnetic impurities
(a classic manybody 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 2ndorder 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} ^{k} + ⋅ > =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
_{s}≅ ≈ δ
^{1}0.1
[Guo, Maekawa, Nagaosa, PRL 102, 036401 (2009)]
0.001 ~ 0.01
θ
H≈
[Fert, et al., JMMM 24 (1981) 231]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 lowT 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 seventyfive 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 condensedmatter 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 longstanding story of the Kondo effect of iron in gold.
Xray magnetic circular dichroism measurements
3. Quantum Monte Carlo simulation
1) problems
suggests an effective 3channel 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
α α σ α σε
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 impurityhost hybridization, Au_{26}Fe 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 impurityhost hybridization determined by ab initio DFTLDA calculations.
(1) Singleimpurity multiorbital 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 (z^{2}) 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 3Orbitals case
Occupation number
[Gu, Gan, Bulut, Ziman, Guo, Nagaosa, Maekawa, PRL105 (2010) 086401]
(2) Magnetic behaviors for Fe in Au from QMC simulations
Isingtype spinorbit interaction for pelectrons: l =1, m =1,0,1.
(3) Spinorbit interaction within t
_{2g}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 antiparallel (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_{z}σ_{z} > = δ_{P} − δ_{AP}, π < n_{2} > = π < n_{3} >= δ_{P} + δ_{AP}. Putting < ℓ_{z}σ_{z} >= −0.44 for λ = 75 meV, and < n_{2} > = <n_{3}>= 0.65,
we obtain δ_{P} = 1.35 and δ_{AP} = 2.73.
Taking into account the estimate sin δ_{1} = 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
magnetodevices.
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.