**Spin-current generation and detection in the presence of an ac gate**

A. G. Mal’shukov,1C. S. Tang,2C. S. Chu,3and K. A. Chao4 1

*Institute of Spectroscopy, Russian Academy of Science, 142190, Troitsk, Moscow oblast, Russia*
2_{Physics Division, National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu 30013, Taiwan}

3* _{Department of Electrophysics, National Chiao-Tung University, Hsinchu 30010, Taiwan}*
4

_{Solid State Theory Division, Department of Physics, Lund University, S-22362 Lund, Sweden}共Received 15 August 2003; published 24 December 2003兲

We predict that in a narrow gap III-V semiconductor quantum well or quantum wire, an observable electron spin current can be generated with a time-dependent gate to modify the Rashba spin-orbit coupling constant. Methods to rectify the so generated ac current are discussed. An all-electric method of spin-current detection is suggested, which measures the voltage on the gate in the vicinity of a two-dimensional electron gas carrying a time-dependent spin current. Both the generation and detection do not involve any optical or magnetic media-tor.

DOI: 10.1103/PhysRevB.68.233307 PACS number共s兲: 73.63.⫺b, 71.70.Ej, 72.25.Dc

One key issue in spintronics based on semiconductor is the efficient control of the spin degrees of freedom. Datta and Das1 suggested the use of gate voltage to control the strength of Rashba spin-orbit interaction 共SOI兲2 which is strong in narrow gap semiconductor heterostructures. In InAs-based quantum wells a variation of 50% of the SOI coupling constant was observed experimentally.3,4 Conse-quently, much interest has been attracted to the realization of spin-polarized transistors and other devices based on using electric gate to control the spin-dependent transport.5

In addition to using a static gate to control the SOI strength and so control the stationary spin transport, new physical phenomena can be observed in time-dependent spin transport under the influence of a fast varying gate voltage. Along this line, in this article we will consider a mechanism of ac spin current generation using time-dependent gate. This mechanism employs a simple fact that the time variation of Rashba SOI creates a force which acts on opposite spin elec-trons in opposite directions. Inversely, when a gate is coupled to a nearby electron gas, the spin current in this electron gas also induces a variation of the gate voltage, and hence affects the electric current in the gate circuit. We will use a simple model to clarify the principle of such a new detection mechanism without any optical or magnetic media-tor. The systems to be studied will be 1D electron gas in a semiconductor quantum wire共QWR兲 and 2D electron gas in a semiconductor quantum well共QW兲.

We consider a model in which the Rashba SOI is
*de-scribed by the time-dependent Hamiltonian H*SO*(t)*⫽ប␣*(t)*

*⫻(k*ជ⫻*ˆ ) _{•s}*ជ

*, where k*ជ is the wave vector of an electron,

*បs*ជis the spin operator, and

*ˆ is the unit vector. For a QWR*

*ˆ is*perpendicular to the wire axis, and for a QW perpendicular to the interfaces. The time dependence of the coupling pa-rameter␣

*(t) is caused by a time-dependent gate.*6To explain clearly the physical mechanisms leading to the spin-current generation, we will first consider the 1D electron gas in a QWR, and assume ␣

*(t) to be a constant*␣

*for t*⬍0, and

␣*(t)⫽0 for t⬎0. For the 1D system we choose the x *

*direc-tion as the QWR axis and y axis parallel to* *ˆ , to write the*
*SOI coupling in the form H*SO*(t)*⫽ប␣*(t)kxsz. For t*⬍0 the

spin degeneracy of conduction electrons is lifted by SOI,
producing a splitting ⌬⫽ប␣*kx* *between sz⫽1/2 and sz*

⫽⫺1/2 bands, as shown in Fig. 1 by solid curves together

*with the Fermi energy EF*. The spin current in this state is

zero, as it should be under thermal equilibrium.

*Indeed, the spin current is defined as I _{s}(t)⫽I_{↑}(t)*

*⫺I↓(t), where I↑(t)* *关or I↓(t)] is the partial current *
*associ-ated with the spin projections s _{z}⫽1/2 共or s_{z}*⫽⫺1/2). Hence,

*I _{s}共t兲⫽* ប

*2L* *E(k*

### 兺

*)*

_{x}*⬍E*

_{F}*关v↑共kx兲⫺v↓共kx*兲兴, 共1兲

*where L is the length of the QWR. Taking the momentum*derivative of the Hamiltonian, we obtain the velocity as

*v _{↑,↓}共kx兲⫽បkx/m**⫾␣共t兲/2. 共2兲

The spin current is then readily obtained as

*Is共t兲⫽共បn/4m***兲共បk _{↑}⫺បk_{↓}*兲⫹ប␣共t兲n/4, 共3兲

*where n is the 1D electron density, and k _{↑}*

*共or k*) is the average momentum in the

_{↓}*↑-spin 共or ↓-spin兲 band.*

For a parabolic band *បk _{↑}⫽⫺m**␣/2 and

*បk**␣/2. Although

_{↓}⫽m*បk*

_{↑}⫺បk_{↓}gives a finite contribution to Is(t) in Eq.*共3兲, for t⬍0 where* ␣*(t)*⫽␣, this contribution is

compen-sated by the contributionប␣*n/4 due to the SOI. Hence, the*
*total spin current Is(t)⫽0 for t⬍0. However, when the SOI*

*is switched off at t*⫽0, ␣*(t)*⫽0 and so the spin current is
finite, because the average electron momenta retain the same
*as they were at t*⬍0. As time goes on, the electron momenta
relax with a relaxation time . Therefore, *Is(t)*

⫽⫺(ប␣*n/4)exp(⫺t/) for t*⬎0.

*It is instructive to make a Fourier transform of Is(t) to*

obtain a Drude-like expression

*I _{s}*共⍀兲⫽

## 冋

បn*2m***共i⍀⫺1兲*

## 册

### 冋

*m**

2 *i*⍀␣共⍀兲

### 册

. 共4兲Since the units of our spin current isប/2, the above
expres-sion is a complete analogy to the electric conductivity.
*In-stead of an electric driving force eE, here we have an*
**PHYSICAL REVIEW B 68, 233307** 共2003兲

*equivalent driving force (m**/2)*关d␣(t)/dt*兴, the Fourier
*component of which is (m***/2)i*⍀␣(⍀). Under this driving
force we have the classic equation of motion

*m***dv↑,↓*
*dt* ⫽⫾
*m**
2
*d*␣共t兲
*dt* . 共5兲

This force acts in opposite directions on electrons with op-posite spin projections. When such a force creates a spin current, it does not induce an electric current.

The above conclusion of spin-current generation can be
demonstrated with a rigorous linear response analysis, which
will be performed on a 2D electron gas共2DEG兲. The simple
Drude expression共4兲 will then appear as a general result. Let
*the 2DEG be in the xy plane with the unit vector**ˆ along the*
*z axis, which is the spin-quantization axis. We will use the*
equation of motion for the spin-density operator to
general-ize the 1D expressions 共1兲, 共2兲 for the spin current. For a
homogeneous system the spin-current density operators can
*be expressed in terms of the electron creation operator c _{kជ,}*†

_{␥}

*and destruction operator ckជ,*␥, where ␥ labels the spin

*pro-jection onto the z axis. This current is then derived as*
*Jj*
*i _{⫽J}*

*j*

*i*

_{⫹J}*j,*

_{SOI}

*i*, 共6兲

*where the superscript i⫽x,y,z specifies the direction of spin*
*polarization, and the subscript j⫽x,y refers to the direction*
of the spin-current flow. The partial current

*J _{j}i*⫽

### 兺

*kជ*

### 兺

␥ ប2

_{k}*j*

*m**

*ckជ,*␥ †

*s*

_{␥}

*i*

*ckជ,* 共7兲

is the ordinary kinematic term and
*J _{j,}*

SOI

*i* _{⫽}*i jz*_{ប}_{␣}_{n/4}_{共8兲}

is the contribution of SOI.7Here*i jz*denotes the Levy-Civita
symbol. The SOI induced current resembles the diamagnetic
current of electrons under the action of an external
electro-magnetic vector potential.

We note that the SOI Hamiltonian can be conveniently written in terms of the kinematic current as

*H*SO*共t兲⫽关m**␣共t兲/ប兴共J*y*

*x _{⫺J}*

*x*

*y*_{兲.} _{共9兲}

When an ac bias with frequency⍀ is applied to the front or
the back gate of a 2DEG,3,4 the Rashba coupling constant
contains two terms ␣*(t)*⫽␣0⫹␦␣*(t), where* ␣0 is constant
in time and␦␣*(t)*⫽␦␣*ei⍀t*. We assume that the only effect
of the ac bias is to add a time-dependent component to the
SOI coupling constant, although in practice it is not simple to
avoid the bias effect on the electron density.4 The SOI
Hamiltonian is separated correspondingly into two parts
*H*SO*(t)⫽H*SO

0 * _{⫹H}*
SO

### ⬘

*(t). The time-independent part H*

_{SO}0 does not produce a net spin current in the thermodynamically equilibrium state. However, as pointed out in the above

*analysis on the 1DEG system, the time-dependent H*SO

*(t)*can give rise to a spin current.

*We will incorporate H*SO
0

into our unperturbed
*Hamil-tonian and treat H*_{SO}

### ⬘

*(t) within the linear response regime.*The so-generated ac spin current

### 具J

*i*

_{j}(t)### 典

has the form### 具

*Jj*

*i*

_{共t兲}_{典}

_{⫽}

*i*ប

### 冕

⫺⬁*t*

*dt*

### ⬘具

*关H*

_{SO}

### ⬘

*共t*

### ⬘

*兲,J*

_{j}i共t兲兴### 典

⫹*i jz*ប␦␣共t兲n/4. 共10兲

In the above equation the first term can be written in the

form ␦␣*(t)Ri _{j}*(⍀). For zero temperature and with ⍀⬎0,

the response function*Ri _{j}*(⍀) can be represented as the
Fou-rier transform of the correlator

*Rj*
*i _{共t兲⫽⫺i}* ប
2

*m**

*kជ*⬘

### 兺

␣⬘⬘*k*

_{j}### ⬘

*s*

_{␣}⬘

_{⬘}

*i*⫻

### 兺

*kជ*␣

*h*ជ

*kជ•s*ជ␣具

*T*兵

*c*

_{kជ}_{⬘}

_{␣}

_{⬘}†

*共t兲ckជ*其

### 典具

*T*兵

*ckជ*⬘⬘

*共t兲c*

_{kជ}_{␣}† 其

### 典

, 共11兲*where h*ជ* _{kជ}⫽k*ជ⫻

*ˆ . In the above equation, the bar over the*product of two one-particle Green functions means an en-semble average over impurity positions.

We will use the standard perturbation theory9to calculate
this ensemble average, which is valid when the elastic
scat-tering time due to impurities is sufficiently long such that
*EF*Ⰷប. We will assume that the electron Fermi energy E*F*

is much larger than bothប⍀ and ប␣0*hkជ*. To the first-order

approximation, we neglect the weak localization corrections
to the correlator 共11兲, since these corrections simply
renor-malize the spin-diffusion constant.8 Consequently, the
con-figuration average of the pair product of Green functions is
expressed in the so-called ladder series.9We found that since
*h*ជ* _{kជ}⫽⫺h*ជ

*many of such ladder diagrams vanish after angu-lar integration in Eq.共11兲, similar to suppression of ladders in the electric current driven by the vector potential.9At the same time, some of nondiagonal on spin indice diagrams do not turn to 0 after the angular integration. Employing the analysis of similar diagrams done in it can be shown that they cancel each other.8 Hence, the configuration average in*

_{⫺kជ}Eq.共11兲 decouples into a product of average Green functions

and Eq.共11兲 becomes

FIG. 1. The dashed curve is the electron energy band without
SOI. The SOI splits the energy band into the*↑-spin and the ↓-spin*
bands, as shown by the solid curves, with corresponding average
*wave vectors k _{↑}and k_{↓}*.

BRIEF REPORTS **PHYSICAL REVIEW B 68, 233307** 共2003兲

*Rj*
*i _{共⍀兲⫽⫺i}* ប
2

*m**

### 兺

*l,n*

*lnz*

### 兺

*kជ*

*kjkn*⫻

### 冕

*d* 2Tr

*关s*

*l*ជ

_{G}_{共k}_{,}

_{}

*ជ*

_{兲s}i_{G}_{共k}_{,}

_{⫹⍀兲兴, 共12兲}

*where G(k*ជ,) is the average Green’s function which
*con-tains fully the effect of H*_{SO}0 . This function is represented by
the 2⫻2 matrix

*G共k*ជ,兲⫽关⫺E*kជ*/ប⫺␣0*h*ជ*kជ•s*ជ*⫹i⌫ sgn共*兲兴⫺1, 共13兲

where⌫⫽1/2*, and Ekជis defined with respect to EF*.

Sub-stituting Eq. 共13兲 into Eq. 共12兲, and then into Eq. 共10兲, we obtain the spin current

### 具J

*j*

*i*_{共⍀兲}

_{典}

_{⫽}

*i jz*ប

4␦␣*n*

⍀

*⍀⫹2i⌫*. 共14兲

It is important to point out that the spin density under the gate area is zero. This is the reason why even in a 2DEG the D’yakonov-Perel spin relaxation10 does not appear in Eq.

共14兲 for the generated spin current, although this spin current

is determined by the response function 共12兲 which involves spin degrees of freedom. Hence, in the homogeneous system with zero spin density, only electron momentum relaxation occurs in the process of spin-current generation by a time-dependent gate.

Unlike the spin current共4兲 in a 1D system, in a 2DEG the
current given by Eq.共14兲 has no specific direction. To clarify
the spatial distribution of the spin flux induced by an ac gate,
let us take the chiral component*J*_{chir}*(t) of the spin current*

*J*chir*共t兲⫽关*

### 具

*Jy*

*x _{共t兲}*

_{典}

_{⫺}

_{具}

_{J}*x*

*y _{共t兲}*

_{典}

_{兴/2.}

_{共15兲}

It is easily seen that this chiral projection has the same form
as the expression *共4兲 for a 1D system, if n represents the*
electron density of the 2DEG. In Fig. 2 we illustrate the
spin-current distribution for a circular gate which is marked
as the gray area. The spin polarization at any point under the
gate has two components parallel to the 2DEG. For any
*di-rection specified by the unit vector N*ជ, the two spin-polarized
fluxes with polarization directions parallel and antiparallel to
*N*ជ will oscillate out of phase by the amount of along the
*direction perpendicular to N*ជ. Such out of phase oscillation is
schematically plotted in Fig. 2. The amplitude of the spin
density flow in each of the opposite directions, as marked by
the dashed-line arrows, is just*J*chir*(t). In the 2DEG outside*
the gate area, the spin current can be supported only by spin
diffusion. Therefore the chiral ac spin polarization is
accu-mulated in the vicinity of the circumference of the gate, and
from where diffuses away from the gate area. It can also
diffuse under the gate. For small gates such back diffusion
can diminish the efficiency of the spin generation. On the
other hand, for large gates with the size larger than the
spin-diffusion length the spin-diffusion counterflow does not reduce
much the total spin current.

The so-generated current amplitude can be easily
esti-mated. With ␦␣⫽3⫻106 _{cm/s,}4 _{for} _{⍀⫽2⫻10}9 _{s}⫺1_{, n}

⫽1012_{cm}_{⫺2}_{, and} _{⫽1 ps, from 共14兲 we derive (2e/ប)}

⫻

### 具

*Jj*

*i*_{(}_{⍀)}

_{典}

_{⯝10}

_{⫺3}

_{Amp/cm. This ac spin current can be }

de-tected by various methods. For example, if holes can tunnel into the neighborhood of the gate edge, their recombination with spin-polarized electrons will produce the emission of circular-polarized light.11

However, here we will discuss a method of direct electric
detection of the dc or the ac spin current. This method is
based on a simple fact that the Rashba SOI couples the spin
current to the gate voltage. We have shown in our above
analysis that due to this coupling, spin current can be
in-duced by a time-dependent gate voltage. In this case the
voltage variation plays the role of a source which drives
electrons out of thermodynamic equilibrium, and the spin
current is the linear response to this perturbation. The reverse
process is to create a spin current in a 2DEG by some source,
and so inducing a voltage shift in a nearby gate. This is also
possible to realize. We thus consider a model where the SOI
constant ␣*(U) is a function of the gate voltage U(t)⫽U*0

*⫹V(t). U*0is the static equilibrium value in the absence of a

*spin current, while V(t) is a dynamic variable. The mean*
value

### 具

*V*

### 典

*of V(t) has to be calculated as a linear response to*the perturbation associated with the presence of the spin-polarization flow. The explicit form of this perturbation can be obtained by averaging the Hamiltonian of the system over an electronic state with the given time-dependent spin cur-rent.

Let

### 具

•••### 典

*J*be such type of average. To the lowest order

with respect to SOI, the coupling of the gate voltage to the
spin current is thus determined by the average of the Rashba
interaction in Eq.共9兲 with␣⫽␣*(U). The coupling between*
*the gate voltage U(t) and the spin current* *J _{j}i* is via the

*kinetic current J*

_{j}i. To derive the coupling Hamiltonian H_{int}, we use Eq.

*共6兲 to express Ji*in terms of

_{j}*Ji*, and expand FIG. 2. Distribution of spin currents induced by a time-dependent circular gate which is marked as the gray region. Under the gate, electrons with opposite spins 共solid arrows兲 move in op-posite directions indicated by the dashed-line arrows. Arrows out-side the gate area show the accumulated spin polarization during a half period of ac gate voltage oscillation.

_{j}BRIEF REPORTS **PHYSICAL REVIEW B 68, 233307** 共2003兲

␣*(U)*⫽␣*(U*_{0})⫹␣

### ⬘

*V(t)*

*for small V(t). The coupling*Hamiltonian is then derived from Eq.共9兲 as

*H*int⫽
*m**␣

### ⬘

ប*V*关

### 具

*Jy*

*x*

_{典}

*J*⫺

### 具J

*j*

*y*

_{典}

*J*兴. 共16兲

*The charging of the gate Q⫽CV is related to the gate *
ca-pacitance. Hence, Eq. 共16兲 can be expressed in the
*conve-nient form H*_{int}*⫽QE, where*

*E⫽m _{បC 关}**␣

### ⬘

### 具J

*y*

*x*

_{典}

*J*⫺

### 具J

*j*

*y*

_{典}

*J*兴 共17兲

is the effective electromotive force.

To illustrate our proposed method of direct electric
detec-tion, let us consider a circuit connected to the gate. The
prin-cipal scheme of the spin current detection is shown in Fig. 3.
In it, an additional back gate can be utilized to tune the
electron density 共not shown兲. The circuit is characterized by
*a frequency-dependent impedance Z(*⍀). The voltage
in-duced on the gate by the electromotive force 共17兲 is then
easily obtained as

### 具

*V*

### 典

*⫽E*

*i⍀CZ共⍀兲*

1*⫹i⍀CZ共⍀兲*. 共18兲

When the spin-current frequency is in resonance with the circuit eigenmode, the gate voltage becomes very large. In the limit of high impedance 共open circuit兲,

### 具

*V*

### 典

*⫽E. Using*

*the spin current (2e/*ប)

### 具

*Ji*(⍀)

_{j}### 典

⯝10⫺3 Amp/cm derived above, and the fact that### 具

*Ji*

_{j}### 典

*J⫽A*

### 具J

*j*

*i*

(⍀)

### 典

*, where A is the*area under the gate, let us estimate the electromotive force induced in a probe gate by this spin current generated by a nearby source gate. For the reasonable parameter values ␣

### ⬘

⫽3⫻107 _{cm/Vs,}4 _{m}_{*}_{⫽0.03 m}

*e, and C*⫽⑀0*A/l with*

*⫽10 and l⫽10*⫺5_{cm, from Eqs.}_{共14兲 and 共18兲 we obtain E}

⯝10⫺5 _{V.}

The generated ac spin current can be rectified with vari-ous methods. For example, one can use a shutter gate which is/2 phase shifted with respect to the generation gate. The shutter gate can be placed in the neighborhood of the gen-eration gate or between two such gates. The evaluation of the rectifying efficiency of such a setup requires a thorough analysis of spin relaxation and diffusion processes caused by the spin accumulation during the shutter cycle.

We would like to add one relevant piece of information
which we became aware of after we completed this paper.
*The preprint of Governale et al. on the quantum-spin *
pump-ing in a 1D wire is also based on the idea of creatpump-ing spin
current via a time-dependent gate.12 However, our results
involving dissipative transport in 2DEG and 1DEG cannot
be compared directly to those in Ref. 12.

This work was supported by the National Science Council of Taiwan under Grant Nos. 91-2119-M-007-004 共NCTS兲, 91-2112-M-009-044 共CSC兲, the Swedish Royal Academy of Science, and the Russian Academy of Sciences and the RFBR Grant No. 03-02-17452. A.G.M. acknowledges the hospitality of NCTS in Hsinchu where this work was initiated.

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FIG. 3. Schematic illustration of spin current detection. ac spin
current flows from the right to the left under the gate with spin
*polarized as shown by arrows. V denotes the voltmeter and Z is the*
outer circuit impedance.

BRIEF REPORTS **PHYSICAL REVIEW B 68, 233307** 共2003兲