Intrinsic Spin Hall Effect

Abstract

A brief review is given on the spin Hall effect, where an external electric field induces a transverse spin current. It has been recognized over 30 years that such effect occurs due to impurities in the presence of spin-orbit coupling. Meanwhile, it was proposed recently that there is also an intrinsic contribution for this effect. We explain the mechanism for this intrinsic spin Hall effect. We also discuss recent experimental observations of the spin Hall effect.

Full text

Page 1

arXiv:cond-mat/0504353v2 [cond-mat.mes-hall] 21 Apr 2005
Intrinsic Spin Hall Effect
Shuichi Murakami1
Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo
113-8656, Japan
Abstract. A brief review is given on a spin Hall effect, where an external electric
field induces a transverse spin current. It has been recognized over 30 years that such
effect occurs due to impurities in the presence of spin-orbit coupling. Meanwhile,
it was proposed recently that there is also an intrinsic contribution for this effect.
We explain the mechanism for this intrinsic spin Hall effect. We also discuss recent
experimental observations of the spin Hall effect.
1 Introduction
In the emerging field of spintronics [1], it is important to understand the na-
ture of spins and spin current inside semiconductors. There have been many
proposals for semiconductor spintronics devices, whereas their realization re-
mains elusive. One of the largest obstacles is an efficient spin injection into
semiconductors. One way is to make semiconductors ferromagnetic, such as
(Ga, Mn)As [2]. The Curie temperature is, however, still lower than the room
temperature, and there are still rooms for improvement towards practical use.
Spin Hall effect (SHE) can be an alternative way for efficiently injecting
spin current into semiconductors. In the first proposal of the spin Hall effect
by D’yakonov and Perel [3], followed by several papers [4,5], the SHE has
been considered as an extrinsic effect, due to impurities in the presence of
spin-orbit (SO) coupling. Nevertheless, quantitative estimate for this extrinsic
SHE is difficult, and this extrinsic effect is not easily controllable.
In 2003 two groups independently proposed an intrinsic spin Hall effect in
different systems. Murakami, Nagaosa, and Zhang [6] proposed it in p-type
semiconductors like p-GaAs. On the other hand, Sinova et al. [7] proposed a
spin Hall effect in n-type semiconductors in two-dimensional heterostructures.
This induced spin current is dissipationless, and can flow even in nonmagnetic
materials. It shares some features in common with the quantum Hall effect.
Because the predicted effect is large enough to be measured, even at room
temperature in principle, this intrinsic SHE attracted much attention, and
many related works, have been done. Nevertheless there remain several issues,
relevant also for experiments. One of the important questions is disorder
effect. While there are a lot of works on disorder effect, the most striking
one is by Inoue et al. [8,9]. They considered dilutely distributed impurities
with short-ranged potentials, and calculate the SHE, incorporating the vertex

Page 2

2Shuichi Murakami
correction in the ladder approximation. Remarkably the resulting spin Hall
conductivity is exactly zero in the clean limit. This work made many people
to consider that the SHE is “fragile” to impurities; namely, only a small
amount of impurities will completely kill the intrinsic SHE. However, this is
not in general true. In fact, the spin Hall conductivity is in general nonzero
even in the presence of disorder, as we see later.
In such circumstances, two seminal experiments on the SHE have been
done. Kato et al. [10] observed spin accumulation in n-type GaAs by means
of Kerr rotation. Wunderlich et al. [11] observed a circularly polarized light
emitted from a light-emitting diode (LED) structure, confirming the SHE in
p-type semiconductors. Separation between the intrinsic and extrinsic SHE
for these experimental data is not straightforward, and is still under debate.
The paper is organized as follows. In Sect. 2 we explain basic mechanisms
and features for the intrinsic SHE. Section 3 is devoted to a disorder effect
on the SHE. In Section 4 we collect a number of recent interesting topics on
the SHE. In Section 5 we introduce two recent experimental reports on the
SHE. We conclude the paper in Sect. 6.
2Intrinsic spin Hall effect
In this section we explain the two theoretical proposals for the intrinsic SHE.
2.1 Spin Hall effect in p-type semiconductors
We begin with semiclassical description of the SHE, and apply it to the p-
type semiconductors [6]. In this description, we introduce a “Berry phase in
momentum space”. The Berry phase [12,13,14] is a change of a phase of a
quantum state caused by an adiabatic change of some parameters. As we
explain later, Berry phase in momentum space gives rise to the Hall effect, as
first demonstrated for the quantum Hall effect [15,16,17]. Here, the wavevec-
tor k is regarded as adiabatically changing due to a small external electric
field. In two-dimensional systems, for example, the Hall conductivity σxyin
a clean system is calculated from the Kubo formula as
σxy= −e2
2πh
?
n
?
BZ
d2k nF(ǫn(k))Bnz(k), (1)
where n is the band index, and the integral is over the entire Brillouin zone.
Bnz(k) is defined as a z component of Bn(k) = ∇k× An(k), where
Ani(k) = −i
?
nk
????
∂
∂ki
????nk
?
≡ −i
?
unit cell
u†
nk
∂unk
∂ki
d2x,(2)
and unk(x) is the periodic part of the Bloch wavefunction φnk(x) = eik·xunk(x).
This Bnz(k) represents the effect of Berry phase in momentum space. nF(ǫn(k))

Page 3

Intrinsic Spin Hall Effect3
is the Fermi distribution function for the n-th band. This intrinsic Hall con-
ductivity (1) was first recognized in the paper by Karplus and Luttinger [18].
This Berry phase in momentum space has been studied in the recent works
on anomalous Hall effect (AHE) [19,20,21,22,23,24], as well as those on the
SHE.
By incorporating the effect of B(k), the Boltzmann-type semiclassical
equation of motion (SEOM) acquires an additional term [17]:
˙ x =1
¯ h
∂En(k)
∂k
+˙k × Bn(k),¯ h˙k = −e(E + ˙ x × B(x)). (3)
The term˙k ×Bn(x) represents the effect of Berry phase, and it is called an
anomalous velocity. Under the external electric field, the anomalous velocity
becomes perpendicular to the field, and gives rise to the Hall effect. This Hall
current is distinct from the usual Ohmic current, which comes from the shift
of the Fermi surface from its equilibrium. This Hall effect comes from all the
occupied states, not only from the states on the Fermi level. By summing
up the anomalous velocity over the filled states, one can reproduce the Kubo
formula result (1). Given the Hamiltonian, the vector field B(k) is calculable,
and we can get the intrinsic Hall conductivity, as in the ab initio calculation
of the AHE in [23,24]. Due to remarkable similarity of the two equations in
Eq. (3), Bn(k) can be regarded as a “magnetic field in k-space. Bn(k) can
have monopoles, and such monopoles can give nontrivial topological structure
for magnetic superconductors [25].
-0.4
-0.3
-0.2
-0.1
0
-1 -0.50 0.51
1.5
1.6
1.7
Wavenumber k(nm )
-1
Energy E (eV)
CB
HH
LH
SO
-E
F
kL
F
kH
F
Fig.1. Schematic band structure for GaAs. CB, HH, LH, SO represent the con-
duction, heavy-hole, light-hole and split-off bands, respectively.

Page 4

4Shuichi Murakami
This anomalous velocity leads to the SHE in semiconductors with dia-
mond structure (e.g. Si) or zincblende structure (e.g. GaAs). The valence
bands consist of two doubly degenerate bands: heavy-hole (HH) and light-
hole (LH) bands. They are degenerate at k = 0 as shown in Fig. 1. Near
k = 0, the valence bands are described by the Luttinger Hamiltonian [26]
H =¯ h2
2m
??
γ1+5
2γ2
?
k2− 2γ2(k · S)2
?
, (4)
where S is the spin-3/2 matrices representing the total angular momentum.
For simplicity, we employed the spherical approximation for the Luttinger
Hamiltonian, while a calculation without it is also possible [27]. In this Hamil-
tonian, a helicity λ =k·S
k
is a good quantum number, and can be used as a
label for eigenstates. The HH and LH bands have λH = ±3
respectively. The SEOM reads as
2and λL= ±1
2,
˙ x =1
¯ h
∂Eλ(k)
∂k
+˙k × Bλ(k),¯ h˙k = eE. (5)
Because we are considering holes, the sign of the charge has been changed.
By straightforward calculation, we get Bλ(k) = λ(2λ2−7
anomalous velocity due to Berry phase is along the direction E × k. By
integration in terms of the time t, we get a trajectory of the holes as shown
in Fig. 2. This shows the motion projected on a plane perpendicular to E.
We note that a semiclassical trajectory can be calculated directly from the
Heisenberg equation of motion; the resulting trajectory agree well with the
one from (5), but with small rapid oscillations [28]. This justifies validity of
the SEOM (5) for adiabatic transport, which was questioned in [29].
2)k/k3. Hence, the
λ<0
λ>0
E
k
Fig.2. Trajectory of holes from the semiclassical equation of motion with Berry-
phase terms. This is a projection on the plane perpendicular to the electric field E.
The transverse shift of the trajectory is to the opposite direction, depending on the
sign of the helicity λ =ˆk · S. The bold arrows represent the direction of spin S
Due to the anomalous velocity, the motion of the holes is deflected from
an otherwise straight motion along k (dashed line). The shift of the motion
is opposite for the signs of the helicity λ, referring to whether the spin S and
the wavevector k are parallel or antiparallel. This shift amounts to the SHE.
By summing up this shift over the occupied states, we can calculate a spin

Page 5

Intrinsic Spin Hall Effect5
current in response to the electric field. If the electric field is along l-axis, the
spin current with Sispin flowing toward the j-direction is [6]
ji
j=
e
12π2(3kH
F− kL
F)ǫijlEl, (6)
where kH
respectively. This is schematically shown in Fig. 3. Nominal values obtained
spin Hall conductivity for p-GaAs are of the similar order of magnitude as the
condutivity at room temperature [6]. In GaAs, the nominal energy difference
of the two bands is larger than the room temperature, and the effect can in
principle survive even at room temperature.
Fand kL
Fare the Fermi wavenumber for the HH and LH bands,
E
p-GaAs
spin
Fig.3. Schematic of the spin current induced by an electric field
2.2 Spin Hall effect in n-type semiconductors in heterostructure
In n-type semiconductors with diamond or zincblende structure, the SO cou-
pling is small. On the other hand, however, if they are incorporated into
two-dimensional heterostructure, the inversion symmetry is broken, and the
SO coupling becomes relevant. The Hamiltonian is approximated as
H =k2
2m+ λ(σ × k)z, (7)
where σi is the Pauli matrix. The second term is called the Rashba term
[30,31], representing the SO coupling. The coupling constant λ can be exper-
imentally determined, and can be controlled by the gate voltage [32].
Sinova et al. applied the Kubo formula to this Rashba Hamiltonian [7].
For this procedure, they defined the spin current Jz
product of the spin Szand the velocity vy =
der, the resulting spin Hall conductivity is e/(8π), which is independent of
the Rashba coupling λ. The Rashba term in (7) can be regarded as a k-
dependent effective Zeeman field Beff= λ(ˆ z×k). In an equilibrium the spins
are pointing either parallel or antiparallel to Beff for the lower and upper
yto be a symmetrized
∂ky. By assuming no disor-
∂H