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Spin Photovoltaic Effect in Quantum Wires with Rashba Spin Photovoltaic Effect in Quantum Wires with Rashba
Interaction Interaction
Yuriy V. Pershin Dr
University of South Carolina - Columbia
, pershin@physics.sc.edu
Carlo Piermarocchi
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Publication Info Publication Info
Published in
Applied Physics Letters
, ed. Nghi Q. Lam, Volume 86, Issue 21, 2005, pages
212107-1-212107-3.
Pershin, Y. V., & Piermarocchi, C. (2005). Spin photovoltaic effect in quantum wires with Rashba
interaction.
Applied Physics Letters, 86
(21), 212107-1 - 212107-3. DOI: 10.1063/1.1935747
© Applied Physics Letters, 2005, American Institute of Physics
http://apl.aip.org/resource/1/applab/v86/i21/p212107_s1
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Spin photovoltaic effect in quantum wires with Rashba interaction
Yuriy V. Pershin
a兲
and Carlo Piermarocchi
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320
共Received 4 February 2005; accepted 13 April 2005; published online 19 May 2005兲
We propose a mechanism for spin-polarized photocurrent generation in quantum wires. The effect
is due to the combined effect of Rashba spin-orbit interaction, external magnetic field, and
microwave radiation. The time-independent interactions in the wire give rise to a spectrum
asymmetry in k space. The microwave radiation induces transitions between spin-splitted subbands,
and, due to the peculiar energy dispersion relation, charge and spin currents are generated at
zero-bias voltage. We demonstrate that the generation of pure spin currents is possible under an
appropriate choice of external control parameters. © 2005 American Institute of Physics.
关DOI: 10.1063/1.1935747兴
The Rashba spin-orbit interaction 共SOI兲共Ref. 1兲 in trans-
port and equilibrium phenomena
2–8
plays a central role in the
fast growing fields of spintronics and quantum computation.
9
In particular, it was recently discovered that the joint action
of the Rashba SOI and in-plane magnetic field on electrons
confined in one-dimensional quantum wires 共QW兲 results in
unique properties.
2–4
Several useful applications based on
these properties were proposed, including a scheme for mea-
suring nuclear spin polarization
2
and a spin filter.
3
In this letter, we theoretically investigate the effect of a
microwave radiation in a QW with Rashba SOI and an in-
plane magnetic field. This setup was stimulated by recent
experiments on the modifications of the Hall effect in the
presence of a microwave field.
10
We will show below that
spin and charge photocurrents can be generated in micro-
wave irradiated QWs 共spin photovoltaic effect兲. The effect
originates in the broken symmetry of QW subbands caused
by the interplay of SOI and constant magnetic field. We em-
phasize that this mechanism is based primarily on spin de-
grees of freedom in contrast to other mechanisms of the pho-
tovoltaic effect considered before 共see, e.g., Ref. 11兲 and
differs from the optical spin current generation.
12
In our model, a ballistic QW of length L is connected to
two electron reservoirs having equal chemical potentials
.
This geometry is realized, for instance, in a QW created by a
split gate technique in a two-dimensional electron gas. For
the sake of simplicity, we assume that only the QW region is
irradiated by the microwave field. Without microwave radia-
tion, the currents in QW from the left to the right reservoir
and from the right to the left reservoir balance each other so
that the total current through the QW is zero. The microwave
radiation induces transitions between spin-splitted subbands.
The electron wave vector k is conserved in such transitions,
however, in the presence of SOI, the electron velocity is not
simply proportional to k. The direction of the electron veloc-
ity, in specific intervals of k, can be reversed after the tran-
sition. The intersubband transition rate, due to the asymmetry
of QW subbands, is different for left- right-moving electrons,
and produces a net charge current. The spin current is also
influenced by the microwave field, since spin flips occur in
these transitions.
We consider a QW in the x direction created via a lateral
confinement 共in the y direction兲 of a 2DEG in the 共x,y兲
plane. The Hamiltonian for the conduction electrons in the
QW in the presence of the microwave radiation can be writ-
ten in the form,
2–4
H =
p
2
2m
*
+ V共y兲 − i
␣
y
x
+
g
*
B
2
· B共t兲 + U共z,t兲. 共1兲
We assume the microwave field propagating in the 共x,y兲
plane and we fix the electric-field component in the z direc-
tion. In Eq. 共1兲, p is the momentum of the electron, m
*
is the
electron effective mass, V共y兲 is the lateral confinement po-
tential due to the gates,
␣
is the SOI constant,
is the vector
of Pauli matrices,
B
and g
*
are the Bohr magneton and
effective g factor, and U共z,t兲 is the potential due to the elec-
tric field component of the radiation. B=B
0
+B
1
cos共
t兲,
where B
0
is the in-plane constant magnetic field and B
1
is the
magnetic-field component of the microwave radiation, and
is the radiation frequency. U共z,t兲 can be neglected because it
does not couple spin-splitted subbands. The third term in Eq.
共1兲 represents the Rashba SOI for an electron moving in the
x direction. We assume that the effects of the Dresselhaus
SOI can be neglected.
13
At B
1
=0, the solutions of the Schrödinger equation can
be written in the form
⌿
l,±
共k兲 =
e
ikx
冑
2
冉
±e
i
1
冊
l
共y兲, 共2兲
where
=arctan关−B
0,y
/B
0,x
−2
␣
k/共g
*
B
B
0,x
兲兴 and
l
共y兲 is
the wave function of the transverse modes 关due to the con-
finement potential V共y兲兴. The eigenvalue problem can be
solved to obtain
E
ᐉ,±
共k兲 =
E
Z
2
E
␣
k
˜
2
+ E
l
tr
± E
Z
冑
1+
2k
˜
B
0,y
B
0
+ k
˜
2
. 共3兲
Here, k
˜
=k
␣
/E
Z
, E
␣
=2m
*
␣
2
/ប
2
, E
Z
=g
*
B
B
0
/2, and E
l
tr
is the
lth eigenvalue of V共y兲. Assuming the parabolic confinement
potential in the y direction, we have E
l
tr
=ប
0
共l+1/2兲.
The energy spectrum corresponding to Eq. 共3兲 is illus-
trated in Fig. 1, for the two spin-split subbands characterized
by l=0 and B
0
directed at
/4 with respect to the x axis in
the 共x,y兲 plane. Notice that the down-splitted subband has a
a兲
Electronic mail: pershin@pa.msu.edu
APPLIED PHYSICS LETTERS 86, 212107 共2005兲
0003-6951/2005/86共21兲/212107/3/$22.50 © 2005 American Institute of Physics86, 212107-1
clearly defined asymmetry. This subband features several lo-
cal extrema, namely, two minima and one maximum. The
energy branches avoid the crossing and form a local gap. The
expectation values of spin polarization in the states 共2兲 are
具±兩
x
兩±典= ±cos关
共k兲兴 and 具±兩
y
兩±典= ⫿sin关
共k兲兴. While the
external magnetic field realigns the electron spins in the gap
region, far from this region the spins are polarized in y di-
rection by the Rashba SOI 共Fig. 1兲. The velocity of an elec-
tron is determined by
v
±
共k兲=
E
±
/ប
k. Denoting by k
+
min
,
k
−
min,1
, k
−
max
, k
−
min,2
the positions of the local extrema of E
l,+
and E
l,−
, we find from Eq. 共3兲 that
v
+
⬍0 for k⬍k
+
min
,
v
+
⬎0 for k⬎k
+
min
,
v
−
⬍0 for k⬍k
−
min,1
and k
−
max
⬍k⬍ k
−
min,2
,
v
−
⬎0 for k
−
min,1
⬍k⬍ k
−
max
and k⬎ k
−
min,2
. Generally, k
+
min
⫽k
−
max
. Thus, there are two intervals of k when the directions
of
v
−
共k兲 and
v
+
共k兲 are opposite.
Next, we consider transitions generated by the time-
dependent magnetic field B
1
cos共
t兲. It can be shown that
the transition rate differs from zero only for transitions be-
tween subbands characterized by the same number l with
conservation of k 共examples of such vertical transitions are
presented in Fig. 1兲. We calculate the transition rate using the
expression
W =
2
ប
冏
冓
+
冏
g
*
B
2
B
1
冏
−
冔
冏
2
␦
共E
l,+
− E
l,−
− ប
兲. 共4兲
It follows from Eq. 共4兲 that the transition rate depends on the
relative orientation of the spin polarization in a state k and
the direction of B
1
. This property allows one to use the di-
rection of the oscillating magnetic field as an additional pa-
rameter that controls spin and charge currents in the QW.
In order to take into account the effect of the time-
dependent magnetic field on transport properties of the QW
we solve the Boltzmann equations
v
⫿
共k兲
f
⫿
x
= Wf
±
共1−f
⫿
兲 − Wf
⫿
共1−f
±
兲, 共5兲
for the distribution functions f
±
共k,x兲. Assuming that the
chemical potential is below the minimum of E
+
共k兲 and a low
temperature, we supplement Eq. 共5兲 by the following bound-
ary conditions: For
v
−
共k兲⬎0, f
−
共x=0兲= f共
兲; for
v
−
共k兲⬍0,
f
−
共x=L兲= f共
兲; for
v
+
共k兲⬎0, f
+
共x=0兲=0; for
v
+
共k兲⬍0,
f
+
共x=L兲=0, where f共
兲 is the Fermi function. The solution
of linearized Eq. 共5兲 with the specified above boundary con-
ditions was found analytically. We calculate the current as:
14
I=
e
h
兺
=±
兰
−⬁
⬁
v
共k兲f
共k兲dk. We found that the transitions con-
serving the velocity direction do not change the charge cur-
rent. The current as a function of the excitation frequency is
shown in Fig. 2 and has a two-peak structure. The first peak
in this structure corresponds to transitions between the states
with k near k
−
max
, the second peak corresponds to transitions
between states with k near k
−
min,2
. The direction and ampli-
tude of the charge photocurrent shows a significant depen-
dence on the direction of B
1
, especially in the first peak
region, because the spin structure in this region is strongly
affected by the magnetic field direction.
As electrons carry spin as well as charge, the time-
dependent magnetic field also influences the spin current
through the wire. The spin current can be defined as the
transport of electron spins in real space. When the electron
transport is confined to one dimension, the spin current is a
vector. Its components can be calculated using I
␥
s
=兺
=±
兰
−⬁
⬁
具
兩
␥
兩
典
v
共k兲f
共k兲dk where
␥
=共x,y,z兲. Figure 3
shows the x and y components of the spin current 共I
z
s
=0兲
calculated for two different values of the chemical potential
. The main features of the spin current are: 共i兲 The spin
current is coordinate dependent; 共ii兲 transitions conserving
the direction of the electron velocity also contribute to the
spin current; 共iii兲 generation of a pure spin current 共without a
charge current兲 occurs for transitions with
v
−
共k兲
v
+
共k兲⬎0.
The spin current components calculated at
=0 have a com-
plex dependence on
.At
=−0.3E
␣
, the role of different
transitions can be more easily understood. With increase of
, we first observe excitations with
v
−
共k兲
v
+
共k兲⬍0, and, then,
after passing the second minimum of E
−
共k兲共see Fig. 1兲, with
v
−
共k兲
v
+
共k兲⬎0. The insets in Fig. 3 show that the first type of
transitions leads to changes in spin current at x=0 and x=L,
while the second type of transitions changes the spin current
at x=L only. The asymmetry of the spin current components
at x=0 and x=L is a signature of pure spin currents in the
system.
We now discuss the conditions for an experimental ob-
servation of this spin photovoltaic effect. First, a QW should
be fabricated from a structure with large Rashba SOI. A
promising candidate are InAs-based semiconductor hetero-
structures, which have a relatively large
␣
.
15
The character-
istic energy of the SOI for these structures 共
␣
=4.5
FIG. 1. 共Color online兲 Energy dispersion E
0,±
共k兲共with respect to E
0
tr
兲 for
E
Z
=0.1E
␣
and B
0
=共B
0
/
冑
2,B
0
/
冑
2,0兲. Spin orientation is illustrated by ar-
rows 关具
x
典 is plotted along k
˜
, 具
y
典 is plotted along E, and 具
z
典=0兴.
FIG. 2. 共Color online兲 Current through QW as a function of the excitation
frequency
.
is the in-plane angle between B
1
and x axis,
=0, T=0.
212107-2 Y. V. Pershin and C. Piermarocchi Appl. Phys. Lett. 86, 212107 共2005兲
⫻10
−11
eV m, m
*
=0.036m
e
兲
16
is E
␣
=1.9 meV. Assuming
E
Z
=0.1E
␣
and taking g
*
=6,
17
we obtain B
0
=1.1 T. Second,
we note that extremely low temperatures are not required for
experimental observation of the spin photovoltaic effect.
From the condition k
B
TⱗE
␣
, E
Z
, we estimate Tⱗ 2 K. Fi-
nally, the condition that B
1
⫽0 only in the QW region was
used only for convenience. In a real experiment, the whole
system 共QW and leads兲 is subjected to a finite B
1
. The effect
of the leads depends on the particular system studied, but
should not considerably affect the scheme proposed here,
especially if there is no appreciable SOI in the leads. The
spin and charge currents can be measured by any appropriate
experimental technique. We believe that the charge current
can be measured in a standard way, for example, using a
sensitive amperemeter. The most convenient technique for
spin current measurement is the scanning Kerr rotation spec-
troscopy, which was recently employed in the detection of
the spin Hall effect in semiconductors.
18
This research was supported by the National Science
Foundation, Grant NSF DMR-0312491.
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FIG. 3. 共Color online兲 Spin current components as a function of the excita-
tion frequency
,
=0, T=0. The region of pure spin currents is to the right
of the vertical dashed line. Insets: Spin current components at
=−0.3.
212107-3 Y. V. Pershin and C. Piermarocchi Appl. Phys. Lett. 86, 212107 共2005兲
