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Optical Spin Hall Effect
Alexey Kavokin,
1,2
Guillaume Malpuech,
2
and Mikhail Glazov
2,3
1
Department of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom
2
LASMEA, Universite
´
Blaise Pascal, 24 Avenue des Landais, 63177 Aubiere, France
3
A. F. Ioffe Physico-Technical Institute, 26 Politechnicheskaya, 194021 St. Petersburg, Russia
(Received 10 May 2005; published 19 September 2005)
A remarkable analogy is established between the well-known spin Hall effect and the polarization
dependence of Rayleigh scattering of light in microcavities. This dependence results from the strong spin
effect in elastic scattering of exciton polaritons: if the initial polariton state has a zero spin and is
characterized by some linear polarization, the scattered polaritons become strongly spin polarized. The
polarization in the scattered state can be positive or negative dependent on the orientation of the linear
polarization of the initial state and on the direction of scattering. Very surprisingly, spin polarizations of
the polaritons scattered clockwise and anticlockwise have different signs. The optical spin Hall effect is
possible due to strong longitudinal-transverse splitting and finite lifetime of exciton polaritons in
microcavities.
DOI: 10.1103/PhysRevLett.95.136601 PACS numbers: 72.25.Fe, 71.36.+c, 72.25.Rb, 78.35.+c
The spin Hall effect (SHE) had been proposed by
Dyakonov and Perel’ in 1971 [1]. It was revisited and
has attracted a lot of attention in the last five years after
its rediscovery by Hirsch in 1999 [2] and the suggestion of
the so-called intrinsic SHE by two groups in 2003 and
2004 [3,4]. The SHE is now largely studied theoretically
and experimentally [5]. The spin Hall effect is the spin
current induced by the electric current. One can distinguish
between the extrinsic and intrinsic SHE, first one due to
elastic spin-flip scattering of carriers with impurities and
second one due to the spin-splitting of conduction band
induced by the spin-orbit interaction. The SHE has a huge
potentiality for spintronics and quantum informatics, as
many works show [6].
In this work, we demonstrate theoretically that the ex-
trinsic SHE has a remarkable analogy in semiconductor
optics, namely, in Rayleigh scattering of light in micro-
cavities. The spin polarization in the scattered state can be
positive or negative dependent on the orientation of the
linear polarization of the initial state and on the angle of
rotation of the polariton wave vector during the act of
scattering. Very surprisingly, spin polarizations of the po-
laritons scattered clockwise and anticlockwise have differ-
ent signs. Despite the visible similarity with the electronic
extrinsic SHE, the optical SHE has a different physical
mechanism which reminds to some extent that of the
intrinsic SHE. The optical SHE is possible due to strong
longitudinal-transverse splitting and finite lifetime of ex-
citon polaritons (polaritons) in microcavities. The effect
we propose should not be confused with the ‘‘Hall effect of
light’’ or ‘‘optical Hall effect’’ proposed in Refs. [7,8],
which consist in the drift of a wave packet of light in a
media having a gradient of the refractive index [7], or in the
presence of the magnetic field [8].
We consider a semiconductor microcavity in the strong-
coupling regime between the optical mode of the cavity
and the heavy hole excitons of the embedded quantum
wells (QWs) [9]. In this regime, new eigenmodes appear
called polaritons. They are characterized by two branches
of in-plane dispersion which are splitted by so-called
vacuum-field Rabi splitting. We shall refer to the familiar
experimental configuration of the resonant Rayleigh scat-
tering [10]. We shall suppose that one of the
~
k 0 states of
the lower polariton branch (LPB) is resonantly excited by a
linearly polarized light [
~
k k
x
;k
y
is the in-plane polar-
iton wave vector] [see the scheme in Fig. 1(a)]. The elas-
tically scattered signal comes from the quantum states
whose wave vector is rotated with respect to the initial
~
k
by some nonzero angle . We shall study polarization of
the scattered light as a function of and k.
We shall use the pseudospin model [11] that describes
the dynamics of polarization of polaritons in microcavities.
The pseudospin is a three-dimensional vector. Its in-plane
components describe both linear polarization orientations
in a given polariton state, while its normal-to-plane com-
ponent s
z
is proportional to the circular polarization of the
polariton state and hence the total average spin of polar-
itons in the given quantum state.
It has been shown theoretically [11] and experimentally
[12,13] that the main mechanism of spin relaxation of
polaritons in microcavities in the linear regime (i.e., if
polariton-polariton interactions are not important) is the
pseudospin precession induced by the longitudinal-
transverse splitting of polaritons. This splitting (
LT
)gives
rise to an effective magnetic field H
eff
oriented in plane of
the microcavity which rotates the polariton pseudospin if
the latter is not parallel to it [11]. Taking into account this
effect, propagation of the polaritons in microcavities can
be described by the following effective Hamiltonian:
^
H
@
2
k
2
2m
B
gH
eff
; (1)
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where m
is the polariton effective mass, is the Pauli
matrix vector, the effective magnetic field H
eff
@
B
g
k
,
and
k
has the following components:
x
k
2
k
2
x
k
2
y
;
y
2
k
2
k
x
k
y
; (2)
with
LT
@
. The orientation of the effective field with
respect to the in-plane polariton wave vector is shown in
Fig. 1(b).
Let us assume that light is incident in the x; z plane. In
the reciprocal space, it excites resonantly a polariton state
having an in-plane wave vector directed along the x axis, as
shown in green in Fig. 1(c). This polariton state is polarized
in the xz plane (TM polarization), which means that its
pseudospin s
0
is parallel to the x axis. As the pseudospin is
parallel to the effective field, it does not experience any
precession at the initial point. Consider now the scattering
act which brings our polariton into the state k
0
x
;k
0
y
, with
k
0
x
k cos and k
0
y
k sin. Following the classical the-
ory of Rayleigh scattering [14], we assume that the polar-
ization does not change during the scattering act, so that at
the beginning the pseudospin of the scattered state keeps
oriented in the x direction [Fig. 1(c)]. As the effective field
is no more parallel to the pseudospin, it starts precessing.
One can note that the precession takes place in opposite
directions for two opposite scattering angles. Figure 2
shows schematically the resulting dependence of the cir-
cular polarization degree of light scattered by the cavity in
different directions. Red (blue) corresponds to the right
(left) circularly polarized light. One can note the inequi-
valence of clockwise and anticlockwise scattering: if the
spin-up majority of polaritons (right-circular polarization)
dominates scattering at the angle , the signal at angle
is mostly emitted by spin-down polaritons (left-circular
polarization), and vice versa. In order to obtain the polar-
ization distribution in scattering of TE-polarized light (in-
cident electric field in the y direction), one should simply
interchange the blue and red in Fig. 2. This effect can be
detected experimentally by measuring the circular polar-
ization degree of the scattered light versus the scattering
angle. Note that in the four directions corresponding to the
boarders between the blue and red areas the linear polar-
ization of light is conserved. An experimental observation
of conservation of the linear polarization in the four direc-
tions and its modification in the areas between them have
been recently reported by Langbein [15], which represents
indirect evidence of the effect we propose.
This analysis can be made more quantitative by writing
the equation of motion for the pseudospin of a scattered
state which reads
@s
@t
s
k
0
ft
s
; (3)
where the first term describes precession and the second
term describes the flux of polaritons coming from the
initial state. It has only the x component (or the y compo-
σ
+
σ
+
σ
-
σ
-
x
y
z
FIG. 2 (color). The black arrows show the polariton pseudo-
spin orientation in the reciprocal space at a time t
2
after ar-
rival of the TM-polarized excitation pulse. In the first and third
quarter (emission angles [0
–90
] and [180
–270
], respec-
tively) polaritons have their pseudospins parallel to the z direc-
tion, which corresponds to the right-circularly polarized emis-
sion (
), shown in red. In the second and fourth quarter (emis-
sion angles [90
–180
] and [270
–360
], respectively) the po-
lariton pseudospins are antiparallel to the z axis, which corre-
sponds to the left-circularly polarized emission (
), shown in
blue.
a)
QW
k
x
k
y
k
x
k
y
b) c)
FIG. 1 (color). (a) Scheme of the considered experimental
configuration: linearly polarized light is incident on a semicon-
ductor microcavity under oblique angle. Polarization of the
scattered light is analyzed. (b) The red arrows show the distri-
bution of the effective magnetic field induced by the TE-TM
splitting on an elastic circle in reciprocal space. (c) The green
circle sketches the polariton state resonantly excited by a pulse
having an in-plane wave vector along the x axis. This initial
polariton state is TM polarized and has its pseudospin parallel to
the x direction (green arrow). The blue circles and blue arrows
show the short-time distribution of particles and their pseudospin
orientation. The red arrows show the distribution of the effective
magnetic field induced by the TE-TM splitting.
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nent), as the pseudospin in the initial state is always
parallel to the x axis (y axis) for TM-polarized (TE-
polarized) excitation, ft
s
0
1
e
t=
, with
1
being the
elastic scattering time-constant. The last term in Eq. (3)
accounts for the finite polariton lifetime . The polariton
population dynamics at the scattered state is described by a
simple rate equation:
@N
@t
2
s
0
1
e
t=
N
: (4)
Solutions of Eqs. (3) and (4) with the initial condition N
0 for the z component of the pseudospin yield
s
z
t; s
0
y
2
1
e
t=
1 cost;
N 2
s
0
t
1
e
t=
:
(5)
Here and further ‘‘’’ holds for TM excitation and ‘‘’’
holds for TE excitation.
The circular polarization degree of light emitted by the
cavity from the given polariton state is
c
2s
z
N
: (6)
For our scattered state, it writes
c
t;
y
1 cost
2
2
t
: (7)
The time-averaged value for the circular polarization is
c
2
R
1
0
s
z
t; dt
R
1
0
Ntdt
sin2
1
2
2
: (8)
Equation (8) shows that the maximum value of j
c
j is 1=2,
and it is achieved when 1 and for 45
, 135
for TM excitation and for and 45
, 135
for TE
excitation. The minimum value
c
1=2 is achieved at
the angles symmetric to the above ones (see Fig. 2).
For numerical modeling, we consider a realistic
GaAs based microcavity containing six 20 nm wide GaAs
QWs. The distributed Bragg reflectors are made of
AlAs=Al
0:1
Ga
0:9
As with 17 and 27 pairs in the upper and
lower mirror, respectively. The vacuum-field Rabi splitting
in such structures is about 8 meV. The LPB calculated for
the exciton-cavity mode detuning of 11 meV is shown in
Fig. 3(a). The wave-vector dependence of
LT
calculated
using the transfer matrix technique is shown in Fig. 3(b). It
shows a maximum of about 0.1 meV at k 2:5
10
6
m
1
. For such a structure the width of the bare cavity
mode is 0.16 meV, which corresponds to the cavity photon
lifetime
ph
4:1ps. The polariton lifetime is essentially
radiative in the photonic part of the polariton dispersion
and is given by
ph
C
, where C is the photon fraction of
the polariton mode. The Rayleigh scattering time constant
1
for the polariton mode is linked with the inhomogene-
ous broadening of the bare exciton line by
1
1C
@.
The typical values of in these kind of structures are of the
order of 50 eV [10], which yields
1
40 ps. This is
much longer than the polariton lifetime, which allows us to
safely neglect multiple-scattering processes. Figure 3(c)
shows the wave-vector dependence of the product .It
reaches one at k
0
2:6 10
6
m
1
, which corresponds to
the incidence angle of 20
. The state corresponding to this
incidence angle also has the advantage to be apart both
from the bare exciton resonance and from the ‘‘magic
angle’’ [16]. Therefore, the corresponding polariton states
are only very weakly affected by phonon scattering and
polariton-polariton scattering. In the low density limit for
these states, elastic scattering by disorder (Rayleigh scat-
tering) is the main scattering mechanism.
Figure 4(a) shows the time dependence of circular po-
larization degree
c
t for the signal scattered at 45
and
45
calculated for the incident wave vector k
0
at the TM-
polarized excitation. Both curves oscillate with a frequency
and decay with a characteristic time
1
. The only
difference between them is the sign. The inset shows the
time evolution of the population of these states. Figure 4(b)
shows the time-averaged circular polarization degree ver-
sus the scattering angle for two values of the incident light
wave vector (k
0
and k
0
=2). The curves are antisymmetric
1438
1440
1442
1444
1446
1448
1450
LPB energy (meV)
(a)
0,00
0,02
0,04
0,06
0,08
0,10
TE-TM splitting (meV)
(b)
0
1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
0,0
0,5
1,0
1,5
Ωτ
Wave vector (m
-1
)
Pumping wave vector
(c)
FIG. 3. (a) Dispersion of the LPB in the model microcavity.
The bare exciton energy is 1450 meV, and the bare photon
energy at k 0 is 1439 meV. (b) Wave-vector dependence of
the energy splitting between TE- and TM-polarized polariton
modes. (c) Wave-vector dependence of the product , where
is the polariton radiative lifetime.
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with respect to the zero-angle direction. Within the 90
to 90
range of angles, the circular polarization achieves
its maximum absolute value for the scattering angles of
45
and 45
. This maximum varies between 0.25 and
0.5, depending on the incident wave vector.
Let us discuss at this point the similarity of the optical
spin Hall effect with intrinsic and extrinsic electronic
SHEs. Our effective Hamiltonian (1) is formally equivalent
to the electronic Hamiltonian containing the spin-orbit
interaction (Rashba or Dresselhaus) term leading to the
intrinsic SHE [4], while the wave-vector dependence of our
effective magnetic field and the Rashba field is different.
The similarity of Hamiltonians allows us to suggest that the
optical spin Hall effect we describe is similar to the intrin-
sic rather than extrinsic electronic SHE. In the intrinsic
SHE for electrons, an external field is needed to change the
particle’s wave vector and thus to create a spin polarized
flux. In our case, the Rayleigh scattering of light plays the
role of the field changing the polariton wave vector. On the
other hand, the Rayleigh scattering in the optical SHE
seems analogous to the impurity scattering of electrons in
the extrinsic SHE, which allows us to suggest that our
effect has also common features with the extrinsic effect.
An essential difference between optical SHE and extrinsic
electronic SHE comes from the fact that the Rayleigh
scattering of light is isotropic and polarization indepen-
dent, while the impurity scattering of electrons is aniso-
tropic and spin sensitive [1,2]. In our case, spin polar-
ization of scattered polaritons is gained after the scattering
act, while in the extrinsic SHE the electron spin flips during
the scattering act.
Finally, the optical spin Hall effect we described invokes
the scattering of particles by a static disorder. However, a
similar effect can be expected for the acoustic phonon
assisted scattering which follows the same spin conserva-
tion rules as the Rayleigh scattering [9]. Therefore the
similar angle dependence of the circular polarization of
the scattered light might be observed at upper or lower
energies.
In conclusion, the optical spin Hall effect consists of the
polarized Rayleigh scattering of light having an antisym-
metric angular dependence. At linearly polarized excita-
tion, the scattered signal gets circularly polarized, which
means that photons gain a nonzero average spin. The
orientation of this spin is critically sensitive to the direction
of scattering and the orientation of the linear polarization
of the exciting light. We foresee various applications in
optical switches as well as for creation of polarization
entangled photon pairs.
This work has been supported by the Marie-Curie
research-training network ‘‘Clermont2,’’ Contract
No. MRTN-CT-2003-503677, and RFBR.
[1] M. I. Dyakonov and V. I. Perel’, Phys. Lett. 35A, 459
(1971).
[2] J. E. Hirsch et al., Phys. Rev. Lett. 83, 1834 (1999).
[3] S. Murakami et al., Science 301, 1348 (2003).
[4] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).
[5] See, e.g., C. Day, Phys. Today 58, No. 02, 17 (2005).
[6] Y. Kato et al., Science 306, 1910 (2004); J. Wunderlich
et al., Phys. Rev. Lett. 94, 047204 (2005).
[7] M. Onoda et al., Phys. Rev. Lett. 93, 083901 (2004).
[8] J. Singh et al., Phys. Rev. A 61, 025402 (2000).
[9] A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier,
Amsterdam, 2003).
[10] W. Langbein et al., Phys. Rev. Lett. 88, 47401 (2002).
[11] K. Kavokin et al., Phys. Rev. Lett. 92, 017401 (2004).
[12] M. D. Martin, Phys. Rev. Lett. 89, 77 402 (2002).
[13] I. Shelykh et al., Phys. Rev. B 70, 035320 (2004).
[14] Lord Rayleigh (J. W. Strutt), Philos. Mag. 47, 375 (1899).
[15] W. Langbein, in Proceedings of the 26th International
Conference on Physics of Semiconductors (Institute of
Physics, Bristol, 2003), p. 112.
[16] P. G. Savvidis et al., Phys. Rev. Lett. 84, 1547 (2000).
0 50 100 150 200
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
(a)
ρ
c
(t)
Time (ps)
-80-60-40-200 20406080
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
(b)
ρ
c
Scattering angle (degrees)
050100
0.1
0.2
0.3
0.4
0.5
0.6
Population
Time (ps)
FIG. 4. (a) Time dependence of the circular polarization de-
gree of light emitted by the states whose in-plane wave vector
makes an angle of 45
(solid line) and 45
(dashed line)
with respect to the wave vector of the incident light. The inset
shows the time dependence of the populations of the correspond-
ing states. (b) Integrated circular polarization degree versus the
scattering angle for two different in-plane wave vectors of the
incident light, namely, 2:6 10
6
m
1
(solid line) and 1:3
10
6
m
1
(dashed line).
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