All-electric spin pumping in quantum channels with a single finger-gate capacitor

All-electric spin pumping in quantum channels with a single finger-gate capacitor

L. Y. Wang

Department of Physics, Fu Jen Catholic University, New Taipei City 24205, Taiwan C. S. Chu

Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan

(Received 18 October 2016; revised manuscript received 17 January 2017; published 7 February 2017) In this paper, we show that a single finger-gate capacitor (FGC) can generate pure spin pumping in a quantum channel (QC). Two dynamic fields, ac spin-orbit interaction and ac potential energy, both induced by the FGC onto the QC, are the agents driving the spin pumping. Smooth spatial profiles of the two ac fields are taken into account both perturbatively and full numerically for the nonadiabatic spin pumping. Our perturbative approach reveals that the spin-pumping mechanism is resonant sideband processes associated with simultaneous coupling of the two ac fields with traversing carriers. Full sideband-process treatment is carried out numerically by a time-dependent scattering matrix method. The same spin-pumping mechanism holds also for the case of a single finger-gated QC, albeit with smaller pumping amplitudes.

DOI:10.1103/PhysRevB.95.075406 I. INTRODUCTION

Spintronics aim to generate, manipulate, and detect spin current for both application and fundamental arenas [1,2]. Spin current injection in metallic heterostructures involving ferromagnetic components is well established, and more recent developments include, among others, spin pumping at a ferromagnetic/paramagnetic interface, where a precessing magnetization on the ferromagnetic side induces spin currents on the paramagnetic side [3–5]. Spin current injection at a ferromagnetic/semiconducting interface, however, remains a challenge [6,7] subjecting to further advancement in interface physics and technology. In the presence of Tesla-scale mag-netic fields, spin current injection in all-semiconductor nanos-tructures was demonstrated [8–10], where quantum point con-tacts (QPC) transmit spin-polarized currents in the presence of the magnetic field. External magnetic field was also invoked for spin pumping driven by electric dipole spin resonance in all-semiconductor devices [11,12], when gate-induced oscillating electric fields in parallel with a static magnetic field are both in plane [11] or out of plane [12] to the same device region where spin-orbit interaction (SOI) [13–17] is at work.

At the heart of spintronics, however, is all-electric spin cur-rent generation in all-semiconductor devices [18–32], where no ferromagnetic materials and magnetic fields are involved. Configurations with two spatially separated time-modulated regions for spin pumping were proposed. These were adiabatic quantum (charge) pumping [33–36] configurations extended, for instance, to include SOI to a mesoscopic quantum dot (QD) acted upon by two dot-confinement-modulating gates [18], to replace one of the two potential-modulating gates in a quantum channel (QC) by a SOI-modulating gate [19], or to replace, also in a QC, the pair of potential-modulating gates by two SOI-modulating finger-gate capacitors (FGC) oriented orthogonally to each other [26]. The proposals were prompted by the recently demonstrated electric modulation of the Rashba SOI in structure inversion asymmetric quantum wells [37,38]. Other adiabatic spin-pumping proposals involve time-modulating SOI quantum dots in both their energy levels and the coupling between the QD and the leads [27,29].

Aside from pumping, spin-polarized current schemes for QC of high intrinsic SOI had been studied. It is by imposing a highly asymmetric lateral QC confinement [25], a potential barrier located at and along the QC central line [30], or a transverse lead, as in a mesoscopic spin Hall effect configura-tion, for spin current extraction when a charge current passes through the source and drain leads [31]. These spin-polarized current schemes, however, operate with a finite source-drain bias. We thus opt to consider spin pumping, which has the added advantage of zero source-drain bias. Our focus is upon the simplest gated QC, namely, a single finger-gated QC.

Consensus has not been reached on whether a single finger-gated QC generates spin pumping. On the one hand, nonadiabatic spin pumping for one ac gate-induced field case had been demonstrated [21,23]. The ac field is a SOI field, which is gate induced and was assumed to have a stepwise spatial profile. Besides, the QC has uniform and static SOI, which is Rashba SOI in Ref. [21], and Rashba and Dresselhaus SOIs in Ref. [23]. The nonadiabatic spin-pumping results have left concerns about the simplified profile for the ac SOI field [12], and the possible effects of the ac gate-induced potential field, which was not considered in [21,23]. On the other hand, spin pumping was not found in two ac gate-induced fields case [12]. The ac fields are the gate-induced SOI and potential fields; both are of smooth spatial profiles. Besides, the QC has static gate-induced SOI and potential fields (both having smooth spatial profiles), but no uniform static SOI. Including only up to±1 sidebands in their calculations, no spin pumping was found [12]. Thus, a comprehensive study is needed to fully explore and to understand the spin-pumping nature in a single finger-gated QC. Issues including spatial profiles of the ac fields, the case for two ac fields, and many sideband effects need to be studied in detail.

On the spatial profile issue, we show that a single finger-gated QC can generate spin pumping, regardless of the spatial profiles of the gate-induced ac fields. To obtain a transparent physical picture, we develop a time-dependent perturbation theory that has incorporated the smooth induced-field-profile feature. Expressions for spin-dependent transmissions are obtained, up to the lowest (second) order in the gate bias.

FIG. 1. Schematic illustration of the finger-gate-capacitor quan-tum channel. The quanquan-tum channel, depicted by the dashed lines (channel width d), is formed in the two-dimensional electron gas (2DEG) layer due to split-gate (dark gray colored cross section traced by the dashed lines) biases. The single finger-gate capacitor consists of an aligned gate pair (orange) of longitudinal length L at z= zBand

z= zA. Indicated also are the ac biases of the finger-gate capacitor

for the spin pumping.

Higher-order treatment is found to become important when states near the subband bottom are involved due to resonant inelastic processes. We thus perform also a full high-order numerical calculation by extending a time-dependent scatter-ing matrix method [39] to the present problem. The QCs that we consider have gate-induced ac SOI and potential energy fields but no static SOI fields. Nonzero difference in the spin-dependent transmissions leads to spin pumping.

The perturbation expression that we obtain for the trans-mission difference has the induced-field profiles entered in the form of spatial integrals. This in turn shows that variations in the spatial profiles should only have quantitative rather than qualitative effects on the spin-pumping characteristics. We note that, in this work, a single finger-gate-capacitor configuration is considered instead of a single finger-gate con-figuration. The finger-gate-capacitor configuration provides an additional knob for the tuning of the relative strengths between the induced SOI and the potential energy fields. It is by way of symmetric or asymmetric sandwiching of the QC by the finger-gate capacitor, as is shown in Fig.1.

Our perturbation results also exhibit clearly the mechanisms for the spin pumping. In the absence of a static and uniform SOI, the spin pumping is resulted from simultaneous coupling

of the ac gate-induced SOI and potential energy fields to the traversing electrons. In the presence of a static and uniform SOI, additional spin-pumping mechanism arises from simultaneous coupling of the ac gate-induced SOI and the static SOI with the electrons.

We have shown that these spin-pumping mechanisms and the resonant inelastic nature in a QC together contribute appreciably to spin pumping. The inelastic processes are sideband processes where the traversing electrons have their energy μ changes to μ± n¯h, where  is the frequency of the ac gate bias, n is an integral sideband index, and μ is measured from the subband bottom. The sideband processes become resonant when states around the subband bottom are involved, and the resonant condition is μ= n¯h [21,23,39–41]. It is due to the singular density of states at a QC subband bottom. The resonance typically features a dip structure in the energy dependence of the transmission [21,23,39–41] which, in a way, exhibits the temporal trapping characteristics of the electron by the time-modulated agent [40].

This paper is organized as follows. In Sec.II, we present our time-dependent perturbation method. This serves to illustrate the physical mechanisms of the spin pumping. Time-dependent scattering matrix method is presented in Sec.III. This is for the treatment of general pumping parameters. Numerical results and discussions are presented in Sec.IV, and, finally, presented in Sec.Vis our conclusion.

II. PERTURBATION METHOD

This section presents our time-dependent perturbation method that has incorporated the smooth induced-field-profile feature of the ac fields. We have included, for the sake of completeness, a static and uniform Rashba SOI in the QC. The perturbation will treat the static Rashba SOI, the ac SOI, and the ac potential energy fields all on the same footing. The perturbation scheme here provides a framework to treat systematically terms that involve both spatial and time dependencies simultaneously, to extract from the framework useful quantities such as scattering coefficients, and to see explicitly the effects of the density of states from the derived expressions. In particular, the need for higher-order perturbation, as is prompted by the density of states relevant to sideband processes, becomes transparent in our expressions in, for instance, Eq. (24). For simplicity, we consider QC with sufficiently narrow widths such that only the lowest subband is needed for our considerations [12,21–23,26]. The purpose of this section is to show that spin-dependent transmission difference [Eq. (24)] is nonzero for any given incident energy. This finding leads to nonzero net spin pumping, where all incident states with energies below the Fermi energy contribute [Eq. (40)], and is shown numerically in Sec.IV.

The Hamiltonian of the QC is given by

H = H₀+ H_SO0 + U(x,t) + H_SO(t), (1) where H0= p2/(2m)+ Uc(y) with Uc(y) the confinement

potential; H_SO0 = α0(ˆz× σ) · p is the static uniform Rashba

SOI [16,17]; andσ is the vector of Pauli matrices (σx,σy,σz)

for the electron spin. Both electric field E(x,t) and potential energy U (x,t) are induced from the same ac-biased gate structures (see Fig.1). The ac SOI term HSO(t)= λ₂{(σ × p) ·

E(x,t)+ [E(x,t) × σ] · p} [13]. The SOI coupling constants are α0and λ. By focusing on one-dimensional QCs where only

the lowest subband is involved, SOI terms in H that contain py,

the transverse momentum, can be neglected. Subsequently, the only spin component that appears in H is σy. We then choose

the wave function |η(x,t) χη to be spin eigenstates where

σyχη = η χηand η= ±. The Schr¨odinger equation becomes

Hη|η(x,t) = i¯h ∂ ∂t|η(x,t), (2) where Hη=  ε₀−1 2mα 2 0  + H0 η + λη 2 {Ez(x,t),px} + U(x,t) (3) and H_η0= 1 2m[px+ mηα0] 2_{. (4)}

Here, ε0is the lowest subband energy, η= −η, and the third

term in Eq. (3) is an anticommutator. It is convenient that we absorb the constant energy term in Eq. (3) by considering Hη

instead in the following, where

H_η = Hη−



ε0−1₂mα20



. (5)

The finger-gate capacitor and the bias, as shown in Fig. 1, generates electric potential V (x,z,t) [12,42], given by

V(x,z,t)= V1cos(t)[ξ (x,z− zA)− ξ(x,z − zB)], (6) where ξ(x,z)= 1 π  tan−1  L/2+ x |z|  + tan−1L/2− x |z|  . (7) The potential in the vicinity of the QC is uniform in y as long as the length of the finger gate along ˆy is much greater than zA, |zB|, and d. This potential form, given by Eq. (7),

is valid in the quasielectrostatic regime, when the relevant wavelength of the ac field, given by = 2πc/, is much larger than all length scales in the system. For f = /(2π) = 1 THz, the = 300 μm justifies the choice of the potential form. Here, c is the speed of light. We point out also that

ξ(x,z)= (φL+ φR)/π where φL (φR) is the azimuthal angle

between the (x,z) point and the line (parallel to ˆz) passing through the left (right) edge of the gate. The sign convention is that φL = φR = π/2 at the origin of the x-z plane, and the

satisfaction of ξ (x,z) to the Laplace equation is self-evident. The induced electric field acting upon the QC, located at

z= 0, is

Ez(x,t)= V1cos(t) [ξ(x,zA)− ξ(x,zB)], (8)

where ξ(x,z)≡ ∂ξ/∂z. The induced potential energy acting on the QC is U (x,t)= −eV (x,0,t) where −e is an electron charge.

For the following time-dependent perturbation treatment, it is convenient to cast Hη = Hη0+ H

so

η(t)+ U(x,t) and to

introduce dimensionless field profile functions f (x) and g(x) for, respectively, the ac SOI and potential energy terms such that H_η = H_η0+α1η 2 cos(t){f (x),px} + U1cos(t)g(x), (9) with f(x)= h0[ξ(x,zA)− ξ(x,zB)], (10) g(x)= ξ(x,zA)− ξ(x,zB),

where h0= zA+ |zB| is the vertical gate separation of the

finger-gate capacitor, α1= λV1/ h0, and U1= −eV1. The

form of Hη as defined in Eqs. (9) and (5) has the time

dependence shown explicitly while the SOI coupling constants

α₁and α0are in the same dimension.

Starting from Eq. (9) and going to the interaction picture, where|η = e−iH 0 ηt/¯h|ψ_η, we have i¯h∂ ∂t|ψη = α1η 2 cos(t) e ζ t_{f I(x,t),px} |ψη + U1cos(t) eζ t gI(x,t)|ψη, (11) where fI(x,t)= eiH 0

ηt/¯h_f(x,t)e−iHη0t/¯h_{, and g}

I(x,t) is defined

similarly. Adiabatic factors eζ t are introduced in Eq. (11) to facilitate our perturbation in the following.

A right-going incident wave |ψ0

η = |kηR of energy μ satisfies H0 η|kηR = μ|kηR, where kηR = K − mα0η/¯h, μ= ¯h2K2/(2m), and x|kηR = eikηRx/ √ Lx. Here, Lx is a

repre-sentative system size along x which we will take Lx = 1 in

the following for presentation sake.

The first-order correction, in V1, to the wave function

|ψη = |kηR + |ψη(1) has taken the expansion form

ψ_η(1) =

k 

C_k(1) (t)|k, (12)

where the primed summation has excluded the case k= kηR.

Substitution of Eq. (12) into Eq. (11) gives

dC_k(1) dt = − i ¯hcos(t) e ζ t_eiωη_{K K}t_k|Fη KK(x)|kηR. (13) Here, K= k+ mα0η/¯h, ωη_KK = ¯h(K2− K 2_{)/(2m), (14)} and F_KηK(x)= α1η¯h 2  K+ K −2mα0η ¯h  f(x)+ U1g(x). (15)

It is noted that α0appears in Eq. (15) through the wave vectors kηR and k, whereas the corresponding wave vectors K and

K, respectively, directly associate to their group velocities and energies, in the conventional way. Integrating Eq. (13) gives C_k(1) (t)= − F_KηK 2¯h  ei(ωηK K+−iζ )t ωη_KK +  − iζ + ei(ω η K K−−iζ )t ωη_KK−  − iζ  (16) for k= kηRand F η KK = K|F η KK(x)|K.

We substitute Eq. (16) into Eq. (12) to obtain ψη(1)(x,t).

More importantly, our interest is to extract from the integral form of ψ(1)

η (x,t) the reflection coefficients rηn for n= ±1.

On the left-hand side of the finger-gate capacitor, and far from the gate-induced fields, we impose the expected form

ψ_η(1)(x,t)=

n=±1

rηneik

n

where knηL= −Kn− mα0η/¯h is the wave vector for

left-going state at energy μn= μ + n¯h = ¯h2Kn2/(2m). In short,

ψ_η(1)(x,t) in Eq. (17) is for the reflection waves. The fact that the reflection waves do not depend on t, even though Ck(1) (t) does,

should not cause any alarm. It is because the appropriate time dependencies will be recovered in the Schr¨odinger picture, and are in accordance to the sideband-process picture.

Conversion of ψ(1)

η (x,t) in Eq. (12) into the form given by

Eq. (17) is facilitated by writing in C_k(1) (t) the factor FKηK in

its integral form, and keeping x to locate on the left-hand side of all locations involved in the spatial integral. The reflection waves expression is given by

ψ_η(1) = − n_=±1 e−imα0¯h ηx 4π ¯h _∞ −∞ dxeiKx _∞ −∞ dK × Fη KK(x) e−iK|x−x| ei(ωK K−n)t ωKK− n − iζ , (18)

where superscript η is dropped from ωKK when Kbecomes

the integration variable, and the relevant range of x is determined by the field’s range even though the integration is over the formal full range of x.

Performing the Kintegral gives

ψ_η(1) = − n_=±1 mi 2¯h2 e −imα0 ¯h ηx _∞ −∞ dxeiKx ×eiKn|x−x _| Kn F_−Kη n,K(x _{), (19)}

where K±1=K2± 2m/¯h. For the case when K_−becomes

imaginary, the convention K−= i2m/¯h− K2_{is used. It is}

straightforward to see that Eq. (19) is of the form in Eq. (17). The reflection coefficients are then obtained to give

rηn = − mi 2¯h2Kn _∞ −∞dx _Fη −Kn,K(x _{) e}i(K+Kn)x_{. (20)}

Substituting Eq. (15) into Eq. (20) gives

rηn= imη 4¯h  α₁  1− K Kn+ 2mα0 ¯hKn η  FnK− 2U1η ¯hKn G nK  , (21) where FnK = _∞ −∞ dxf(x) ei(K+Kn)x_, (22) GnK = _∞ −∞dx _g(x_{) e}i(K+Kn)x

are the integrals involving the spatial profiles of, respectively, the ac SOI and potential energy fields. Terms inside the square brackets of Eq. (21) that contain the η factor contribute to genuine spin-dependent transmissions. In addition, terms inside the parentheses contain the breaking of spin symmetry by α0during reflection in the sideband processes. This is due

to the anticommutator in Eq. (9), which will be proportional to the sum of the incident wave vector (K− mα0η/¯h) and

the reflection wave vector (−Kn− mα0η/¯h) in a n-sideband

process. The magnitude of the wave-vector sum, given by

(K− Kn)− 2mα0η/¯h, becomes spin dependent as long as n= 0. Moreover, that the resonant sideband nature weighs

more on n= −1 process guarantees its breaking of the spin symmetry by a nonzero α0.

The total dc spin-dependent transmission T_RLη , from the left to right electrodes, is given by T_RLη =_nT_RLη (n). Here,

T_RLη (n)= |tηn|2



μn

μ, the primed summation sums over n

where μn 0, and the factor

√

μn/μ is the group velocity

ratio [40,43]. The calculation of TRLη is more easily done than

its direct calculation, through using the current conservation relation T_RLη + Rη_RL= 1, (23) where R_RLη =_n|rηn|2  μn μ.

Furthermore, our interest is on the difference in the total spin-dependent transmission TRL↑ − TRL↓ . For presentation

sake, we have replaced η= 1 (−1) by η = ↑ (↓). Substituting the first-order rηn expressions (21) into Eq. (23), we obtain,

after some algebra, the lowest- (second-) order results for the dc spin-dependent transmission difference, as is presented below: T_RLs = T_RL↑ − T_RL↓ = n=±1 1 2 _m ¯h 2_mα1 ¯hK  1− K Kn  ×  α0α1|FnK|2− U1 mRe(F ∗ nKGnK)  . (24) Equation (24) is our key result. First of all, it shows that

Ts

RLis nonzero. It remains nonzero even when α0= 0. The K−1 factor in mα1/(¯hK) comes from the incident density

of states, and in (1− K/Kn) the factor Kn−1 is reckoned to

represent the corresponding density of states for the n= ±1 sidebands.

In a spin-pumping situation, when incident states come from both electrodes, the total charge current, which is related to (TRL↑ + TRL↓ )− (TLR↑ + TLR↓ ), is zero. This is because of the

time-reversal-symmetry result, given by

T_RL± = T_LR∓. (25)

On the other hand, the total spin current, which is related to (TRL↑ − TRL↓ )− (TLR↑ − TLR↓ ), becomes related to 2TRLs .

Two mechanisms that contribute to TRLs are shown in

Eq. (24). The term that involves α1 and U1 corresponds to

the simultaneous coupling of the ac SOI and potential energy fields with the electrons. The term that involves α1 and α0

corresponds to the simultaneous coupling of the static SOI and ac SOI fields with the electrons.

Resonant nature of the mechanisms is shown also in Eq. (24). It is through the factor Kn−1, or the corresponding

sideband density of states, for n= −1. When K₋₁ → 0+(or

μ− ¯h → 0+), the n= −1 sideband process will hit the subband bottom. Contribution from this sideband process is large due to the singular density of states. In the vicinity of this resonant sideband regime, high-order processes are important and a full sideband numerical treatment is provided by our time-dependent scattering matrix method (Sec.II). It is worth

pointing out that our numerical results are found (not shown) to match those given by Eq. (24) in small-U1cases.

The induced-field profiles entering Eq. (24) throughFnK

andGnK, in Eq. (22), show that our nonzero TRLs finding

should hold regardless of the profiles. The induced-field profiles could only affect our finding quantitatively and not qualitatively. We drive this finding home by showing, in the following, that Eq. (24) has covered the perturbation results in [21] as a special case. Setting U1 = 0 while keeping

nonzero α1, which can be achieved for a symmetric finger-gate

capacitor (zA= |zB|), and choosing f (x) = (L/2 − |x|) and

α0finite, we get from Eq. (20) r_ηna =α1mnηi 2¯hKn sin  (K+ Kn) L 2  n+¯hkηR m(K− Kn)  . (26) Here, n= ±1, (x) is a step function, and superscript “a” denotes the case for a steplike profile for the ac SOI. The expression in Eq. (26) equals that in [21]. The essential spin dependence in ra

ηnenters through kηRin Eq. (26).

III. SCATTERING MATRIX METHOD

The time-dependent scattering matrix method is to dis-cretize the induced-field region into small slices of ac

potentials, each has a width  along x. The reflection and transmission coefficients of each such slice is then calculated. The scattering properties of neighboring pieces will be convoluted to obtain the scattering matrix of a finite width of such an ac potential. This convolution procedure will stop when the entire range of the induced fields is covered. The scattering matrix method aims to provide a decent numerical scheme for the handling of the evanescent waves. The present method has included the SOI to an earlier work, in Ref. [39], of one of the author.

The Hamiltonian describing an ac potential slice, located within |x − xj|  /2, is taken from Eq. (9) except that a

slicelike envelope (/2− |x − xj|) is imposed upon the

field profile functions f (x) and g(x), to give

H_ηj = H_η0+ U₁cos(t)g(x) (/2− |x − xj|)

+α1η

2 cos(t){f (x) (/2 − |x − xj|),px}. (27) We further approximate g(x) by g(xj) and f (x) by f (xj) after

the evaluation of the anticommutator. Consider a right-going state |kni

ηR incident upon the slice potential (Hηj − Hη0) in

Eq. (27), the total scattering state, represented by reflection coefficients rηj(n,ni) and transmission coefficients tηj(n,ni),

are given by the form

ψηj(x,t)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  n  eikηRnix_δ nni+ rηj(n,ni)e ikn ηLx_e−iμnt/¯h _{for x < x} j−, e−isin(t)¯h [(α1η2 )[2f (xj)px+¯hif(xj)]+U1g(xj)] n  Aηj(n,ni)eik n ηRx+ B_ηj(n_,n i)eik n ηLx_e−iμnt/¯h _for |x − x_j| <  2,  n  tηj(n,ni)eik n ηRx_e−iμnt/¯h _{for x > x} j+, (28)

where xj_±= xj± . The wave-function form in the |x −

xj| < ₂ region of Eq. (28) can be shown, by direct

substitu-tion, to satisfy the time-dependent Schr¨odinger equation for the Hamiltonian in Eq. (27). The slicelike envelope in the anticommutator contributes to the wave-function matching conditions, given by ∂ψηj ∂x  xj±+0 + xj±−0+ ± mα1η i¯h cos(t) f (x) ψηj   x_j± = 0. (29) By invoking the relation eizsin(t)₌

nJn(−z)e−int, we

separate different time dependencies to obtain, after some algebra, the equations for coefficients Aηj(n,ni) and Bηj(n,ni)

inside the potential slice. Here, Jn(z) is Bessel function

of the first kind. The equations for the coefficients are given by  M11 M12 M21 M22  Aηj(ni) Bηj(ni)  =  Cηj(ni) 0  . (30)

Here, Aηj(ni), Bηj(ni), and Cηj(ni) are column vectors

of (2NSB+ 1) × 1 dimensions, and 2NSB+ 1 is the number

of sidebands used. Typically, in this work, we have NSB∼

30. At x= xj−, the matrices and their matrix elements

are (M11)nn = (−1)n−n  Jn−n  Z_{j R}n eikηRnxj− ×  kn_ηL− k_ηRn −mα1η(n− n ₎ ¯h Zn j R f(xj₋)  ,  M12  nn = (−1) n_−n_J n−n  Z_{j L}n eiknηLxj− ×  kn_ηL− k_ηLn −mα1η(n− n ₎ ¯h Zj Ln f(xj−)  . (31) At x= xxj+, the matrix elements are

(M21)nn = (−1)n−n  Jn−n  Z_{j R}n eikηRnxj+ ×  kn_ηR− k_ηRn −mα1η(n− n ₎ ¯h Zj Rn f(xj+)  , (32) (M22)nn = (−1)n−n  Jn_−n(Zn  j L) e ikn ηLxj+ ×  k_ηRn − kn_ηL −mα1η(n− n ₎ ¯h Zj Ln f(xj+)  . (33)

The column vector (Cηj)_n= δnni[k

n ηL− k n ηR]e ikn ηRxj− _and Zn j R= 1 ¯h[α1ηf(xj)¯hk n ηR+ α1η¯h 2i f(xj)− U1g(xj)].

Solving Eq. (30) for Aηj(ni) and Bηj(ni), the reflection and transmission coefficients can then be calculated from the following equations: rηj(n,ni)= n (−1)n−nJn−n  Z_{j R}n ei(kηRn−kηmLn )xj−_A ηj(n,ni)+ Jn−n  Zn_{j L}ei(kηLn−knηL)xj−_B ηj(n,ni)  − δnnie i(kn ηR−knηL)xj− ₍₃₄₎ and tηj(n,ni)= n (−1)n−n_J n−n  Z_{j R}nei(knηR−kηRn )xj+_Aηj(n_,n i)+ Jn−n  Z_{j L}nei(knηL−knηR)xj+_Bηj(n_,n i)  .

We consider also left-going incident state |kni

ηL and obtain

the reflection ˜rηj(n,ni) and transmission ˜tηj(n,ni) coefficients

with the same procedure above. All these coefficients are used in our transmission calculation through the entire ac induced fields with smooth spatial profiles.

In the total transmission calculation, the total waves between the j th and (j+ 1)th potential slices are denoted by coefficients aηj(n) and bηj(n) for, respectively, right- and

left-going waves in the nth sideband. The scattering matrix Sη(j−

1,j ), of dimensions (4NSB+ 2) × (4NSB+ 2), connecting the

waves on the two sides of the j th potential slice is given by

Sη(j− 1,j) =  tηj ˜rηj rηj ˜tηj  , (35) where  aηj bη,j−1  = Sη(j − 1,j)  aη,j−1 bηj  . (36)

For the convolution of the scattering matrices to obtain

Sη(0,N ), when N is the total number of potential slices, we

need to use an inverse transfer matrix Iη(j ), given by

Iη(j )=  t−1_ηj −t−1_ηj ˜rηj rηjt−1ηj ˜tηj− rηjt−1ηj ˜rηj  , (37) where  aη,j−1 bη,j−1  = Iη(j )  aηj bηj  . (38)

Finally, the scattering matrix Sη(0,j ) can be expressed in terms

of Iη(j ) and Sη(0,j− 1) [39], given by Sη(0,j )11 = [Iη(j )11− Sη(0,j− 1)12 Iη(j )21]−1 × Sη(0,j− 1)11, Sη(0,j )12 = [Iη(j )11− Sη(0,j− 1)12 Iη(j )21]−1 × [Sη(0,j− 1)12 Iη(j )22− Iη(j )12], Sη(0,j )21 = Sη(0,j− 1)21 + Sη(0,j− 1)22Iη(j )21Sη(0,j )11, Sη(0,j )22 = Sη(0,j− 1)22Iη(j )22 + Sη(0,j− 1)22Iη(j )21Sη(0,j )12. (39)

Iterating Eq. (39) leads us eventually to Sη(0,N ).

Now, for the case of right-going state|kηR incident upon the

smooth-profiled ac induced fields bηN = 0 and aη0(n)= δn0.

Then, the transmission and reflection coefficients are given, respectively, by tη = Sη(0,N )11aη0 and rη = Sη(0,N )21aη0.

On the same token, we have ˜tη = Sη(0,N )12bηN and ˜rη =

Sη(0,N )22bηNfor the left-going incident case. Here, bηN(n)=

δn0and aη0= 0.

The net spin current Is

RL, in units of ¯h/2, due to right-going

incident carriers is given by

I_RLs = η η h μ 0 T_RLη (E) dE, (40) and the net pumped spin current

I_ps= I_RLs − I_LRs = 2I_RLs . (41) Subsequently, the net pumped spin per cycle Ns

p =2π I

s p. IV. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we present numerical results for the spin-pumping characteristics in a quantum channel due to a single finger-gate capacitor (FGC). Modulated by ac biases, the FGC induces both ac potential and SOI in the quantum channel, which has no static SOI (α0= 0). The general spin-pumping

characteristics of our interest are exhibited in their dependence on both μ and .

Material parameters for the numerical calculations are the same as for InSb, with effective mass m= 0.0139 meand SOI

coupling constant ¯hλ= 5.23 e · nm2_{[13]. Here, m}

eand e are,

respectively, the electron mass and charge (magnitude). We define physical quantity units out of typical electron density

ne= 1012cm−2, with wave vector unit k∗=

√

2π ne= 2.5 ×

108m−1, length unit l∗ = 1/k∗= 4 nm, and frequency unit

f∗= E∗/ h= 41 THz, where E∗= ¯h2k∗2/(2m)= 171 meV. For the numerical examples below, the FGC parameters are chosen to be L= 12 l∗, zA= 6 l∗, and zB= −3 l∗. The ac bias

parameters, in Eq. (6), are chosen to be V1= 0.15 V for the

bias amplitude, and  ranges between 0.01∗∼ 0.03 ∗ for the bias angular frequency. Here, ∗= 2πf∗.

Presented in Fig.2is TRLs and its μ and  characteristics.

The most important result is that the Ts

RLare quite significant

in their values. For instance, peak values Ts

RL≈ 0.12, near

μ= 0.0065 for  = 0.013, and T_RLs ≈ −0.08, near μ =

0.055 for = 0.02. Equally important is that for the patches of positive, or negative, TRLs in the μ- plane, the positive

Ts

RLpatches appear to dominate over their counterparts in

both their Ts

RL magnitudes and their areal sizes. This is

crucial for the net spin pumping, as will be presented in Fig.3. The patches of positive, or negative, TRLs also exhibit

two other features. First, smaller-sized (larger-sized) patches occur in the smaller (larger)  region. On the one hand, the resonant sideband processes introduce a natural energy scale ¯h, which tends to bring forth smaller μ structures in the smaller  region. We indicate, as a guide, the sideband-process

µ (¯hΩ

₎

Ω

(Ω

)

FIG. 2. Colored contours for T_RL↑ − T_RL↓ as a function of incident energy μ and pumping frequency . The finger-gate parameters are

L= 12l∗, zA= 6l∗, zB = −3l∗, with l∗= 4 nm, and pumping bias

amplitude V1= 0.15 V. Indicated by dashed lines are cases for μ = n¯h. Frequencies depicted by black arrows are selected for further analysis in Figs.4–6.

conditions μ= n¯h by black dashed lines. On the other hand, as is shown in Figs.4and5, smaller (larger)  has a larger (smaller) sideband effect which, in turn, causes modifications to the patch structures. Second, Ts

RL shows nonmonotonic

variations in either increasing μ, for a given , or increasing

, for a given μ. We will look into the nonmonotonic variation along μ in Fig.5for three  values depicted by black arrows in Fig.2. The nonmonotonic variation along the vicinity of

μ=  will be shown in Fig.6. The net spin pumped per cycle Ns

p and its characteristics

in the μ- plane are presented in Fig. 3. This is a direct integration of Fig.2, over energy and up to μ, for a given , as is given by Eqs. (40) and (41). That Ns

p>0 in the entire μ-

plane, shown in Fig.3, reflects the dominance of the positive

Ts

RLpatches over its counterparts in Fig.2. The optimal Nps

does not necessarily occur at where TRLs is peaked, such as at

µ (¯hΩ

₎

Ω

(Ω

)

FIG. 3. Colored contours for net spin Ns

ppumped per cycle as a function of μ and . Positive Ns

pcorresponds to N s

pnet spin-up (along ˆ

y) electrons being pumped from left to right electrodes. Pumping bias amplitude V1= 0.15 V. Finger-gate parameters are the same as in Fig.2.

FIG. 4. μ characteristics of T_RLη , the right-going spin-dependent transmission, for some ac potential field strength γ U1. Pumping frequencies fixed at (a) = 0.011 ∗, (b) = 0.013 ∗, and (c)

= 0.025 ∗are indicated in Fig.2by arrows. The finger-gate and pumping parameters are as in Fig.2except that the ac potential field strength has γ= 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Vertical dashed lines indicate μ= n¯h.

= 0.013. Rather, whether Ts

RLcould manage to maintain

the same sign over a large-μ region is another important factor. It turns out that the optimal spin pumping occurs at = 0.011, or f = 0.45 THz, for μ = 0.06. The optimal pumped spin per cycle is Ns

p = 0.45. This would be equivalent to an electric

current of Ns

pef = 32 nA, had each pumped spin been given

a charge e. The spin-pumping effect is thus significant. Resonant sideband origin for the spin pumping shown in Fig.2is identified through our analysis in Figs.4and5. The

μcharacteristics of TRLη and TRLs are presented for various

values of the ac potential field strength γ U1 in, respectively,

Figs. 4 and 5. For analysis purposes, the parameter γ is introduced artificially to help tune only the coupling strength of the ac potential U1 while keeping that of the ac SOI α1

constant. Cases for γ = 1 correspond to those shown in Fig.2. Presented in Fig. 4 (Fig. 5) is T_RLη (Ts

RL) in incremental

order of γ between [0, 1]. For the clarity of presentation, a relative upward shift of 0.3 between curves of consecutive γ is

FIG. 5. μ characteristics of Ts RL= T ↑ RL− T ↓ RL for some ac

potential field strength γ U1. Pumping frequencies are fixed at (a) = 0.011 ∗, (b) = 0.013 ∗, and (c) = 0.025 ∗ (indicated in Fig. 2). The finger-gate and pumping parameters are as in Fig. 2 except that the ac potential field strength has

γ = 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Vertical dashed lines indicate μ= n¯h.

adopted in Fig.5. Selected pumping frequencies , indicated in Fig.2and presented in Figs.4and5, are of values (a) 0.011, (b) 0.013, and (c) 0.025.

Typical resonant sideband features, the dip structures in TRLη

at μ= ¯h, are shown in Fig.4for γ = 0 and γ = 0.2 cases. It is associated with the temporal trapping, or the forming of a quasibound state just beneath the subband bottom, of the electron in the vicinity of the ac field [40]. The dip structures are small for the γ = 0 case, showing weak time-modulation effects of the ac SOI, and TRL↑ = TRL↓ shows that there is no

spin pumping, as is expected according to Eq. (24) when

α₀= 0. Deeper and broader dip structures for the γ = 0.2

case show an increase in the time-modulation effects. Of the three frequencies (for γ = 0.2) shown, the dip structures in Fig.4(c)(= 0.025∗) bear the resonant characteristics closest to typical low sideband processes [21], namely, with the dip occurring at μ= ¯h while T_RLη recovers to high values, less than but close to unity, on both sides of the μ= ¯h

FIG. 6. μ characteristics of T_RL↑ − T_RL↓ along ¯h= μ+for fixed ac potential field strength γ U1. The finger-gate and pumping parameters are as in Fig.2except that the ac potential field strength has γ = 0, 0.2, 0.4, 0.6, 0.8, and 1 denoted by, respectively, the brown (thin solid), red (dotted-dashed), orange (dashed), green (dotted), gray (medium solid), and black (thick solid) curves. μ+= μ + 10−6.

position. As  decreases, from = 0.013∗ in Fig. 4(b)

to = 0.011∗ in Fig. 4(a), the progressively lowering in

T_RLη in the μ < ¯h region shows that more sideband processes are involved in the establishment of the resonance. This more sideband-processes interpretation is further confirmed by the fact that the dip structures are more leftward shifted in Fig.4(a)

than in 4(b) away from the μ= ¯h position for γ = 0.2. This is corroborated with the lowering of the quasibound state by the increasing in the time-modulation field strengths [40]. That lower frequency tends to bring forth more sideband processes is also consistent with the −1 dependence in the dimensionless strengths Zn

j R and Z n

j Lin Eq. (31). For higher

γ values in Fig.4, even more sideband processes come into play, the resonant sideband features thus evolve from simple dip structures to diplike or peaklike or kinklike structures at

μ= n¯h. In our numerical calculation, we have included up

to 2NSB+ 1 sidebands, for NSB= 30.

The Ts

RL curves obtained from Fig. 4 are presented in

Fig.5. Two key features are observed. The energy structures in TRLs are essentially given by ¯h, which is a manifestation

of the resonant sideband features, the μ= n¯h structures we obtained in Fig.4. As γ increases toward unity, the negative

Ts

RL regions are either being suppressed to much smaller

magnitudes, such as the trend shown in Fig.5(a), or being pushed towards larger μ, such as the trend shown in Figs.5(b)

or5(c). Both trends are important for spin pumping. Thus, we have established that resonant sideband processes contribute significantly to the spin pumping. On the other hand, one should be cautioned that too large an ac potential field might not be a favorable choice for spin pumping. Towards this end, we consider the case of a single-finger gate, when the zBgate of

the single-gate capacitor is removed. The ac potential field in the QC is increased while the ac SOI is decreased. Indeed, our calculation shows that Ns

p, though remaining within a

frequency of = 0.011∗ (or f = 0.45 THz.), zA= 7 l∗,

L= 14 l∗, V1= 0.09 V, and NSB= 36, we obtain Nps= 0.02.

Finally, we present in Fig.6 the characteristics of TRLs

along the μ=  line for various ac potential field strength

γ U₁. To focus on the μ < ¯h regime, we have chosen a slightly larger ¯h= μ+. Our results show that as γ increases, or more sidebands are involved, the μ dependence of TRLs

evolves from flat-out zero (γ = 0) to monotonic decreasing (γ = 0.2) to oscillatorylike (or nonmonotonic). Furthermore, the oscillatorylike characteristics (for γ  0.4 in Fig.6) have the number of Ts

RLundulations increasing with γ . This shows

unequivocally that the nonmonotonic behavior in TRLs , along

in Fig.2, arises from large sideband processes.

In our calculations, we have included only the lowest subband for the QC. The subband energy spacing ε, when estimated by a hard wall of width d= 5 l∗ (200 ˚A), gives

ε= 3(π/5)2¯h∗≈ 200 meV. On the other hand, the ac SOI term that we have neglected is λV1[ξ(x,zA)− ξ(x,zB)]¯hky

where ky is taken to be zero for intrasubband processes.

For intersubband processes, we replaceky ≈ π/(5 l∗) and,

using the same pumping parameters in this work, we estimate the intersubband transition amplitude from the above ac SOI term (setting x= 0) to be of order 4 meV. This is much smaller than ε and thus validates our one-subband treatment. Here, we have |ξ(0,zA)− ξ(0,zB)| ≈ 0.138/l∗. We would

also like to comment on the possible effect of Dresselhaus SOI (DSOI) on our results. The form of DSOI to be considered is H_D0= −βk2zσxkx [13], where for kz2 ≈ [π/(5 l∗)]2 and

β = 761 eV ˚A3gives us H_D0 ≈ −(0.187 eV ˚A)σxkx. This could

cause the rotation of the spin onto ˆz in the presence of the ac SOI term in Eq. (9). However, from our other calculations (not shown) of including α0, or HSO0 , into our spin pumping,

we find its fractional contribution to be quite small, of order 5% of our spin-pumping results. As H_SO0 = ¯hα0σykx ≈

(0.523 eV ˚A)σykx, when an interface normal electric field of

order Ez≈ 105 V/cm is assumed [37], it is larger than H_D0

magnitudewise. Thus, we reckon that the contribution from the H_D0, though differing in the spin directions, should be at best only of comparable order of magnitudes as that from H_SO0 . As a result, the correction from DSOI is expected to be small.

V. CONCLUSION

Through this work, we come to realize that adiabatic spin pumping cannot be invoked by only one gate. It is obvious if the gate gives rise to only one ac field to the system. It cannot be invoked even when the gate gives rise to two ac fields, if the two fields differ only by a phase of either 0 or π [see Eq. (9)]. On the other hand, this work has firmly established

that nonadiabatic spin pumping can be invoked by only one gate. This includes cases of one ac field, when a uniform static SOI is needed, and two ac fields, when a uniform static SOI is not needed. The key is the coherent sideband-processes nature of the nonadiabatic spin pumping.

It is worth putting our finite spin-pumping results in the perspectives of symmetry in our system. We focus on the two ac fields case (α0 = 0). The time-reversal symmetry (TRS)

in Eq. (9) is clearly seen when we restore σy in place of

η. Furthermore, the x→ −x symmetry in the dimensionless field profile functions f (x) and g(x) [see Eqs. (10), (6), and (7)] leads to TRLη (n)= T

η

LR(n), and to Eq. (25). A consequence

of Eq. (25) is that dc charge pumping is zero, as is expected for nonadiabatic pumping when both TRS and spatial symmetry are conserved [44]. On the other hand, for a given η, the SOI term (due to α1) in Eq. (9) causes the breaking of the x→ −x

symmetry in Hη. Since the other time-dependent term, namely,

the U1g(x)cos(t) term, has the x→ −x symmetry, thus we

must have TRLη (n)= T η

LR(n) [44], or, summing over sidebands,

to give

T_RLη = T_LRη . (42)

The net pumped spin current Is

p, given by I_ps= 2 h μ 0 dE[T_RL↑ − T_LR↑ ], (43) is hence nonzero.

In conclusion, we have shown that significant spin pumping can be achieved in quantum channels with a single finger-gate capacitor. Resonant sideband processes are the major contributors to the spin pumping. Simultaneous couplings of the traversing electrons to both the ac SOI and ac potential fields provide the mechanisms to the spin pumping. Our perturbation theory has demonstrated clearly the physical mechanisms and the nonadiabatic nature for the spin pumping in this work. Our time-dependent scattering matrix approach provides an efficient numerical scheme for the calculations. The same spin-pumping mechanism holds also for a single finger gate, giving rise to smaller yet discernible spin-pumping results. These results should be of interest to the all-electric spin-pumping research, in particular, and to the spintronics research, in general.

ACKNOWLEDGMENT

This work was supported by Taiwan MOST (Contracts No. NSC 009-005-MY3, No. NSC 102-2112-M-030-001-MY3, No. MOST 105-2112-M-009-008, No. MOST 105-2112-M-030-005), FJU Grant No. 3101311043504, and a MOE-ATU grant.

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