dc spin current generation in a Rashba-type quantum channel

dc spin current generation in a Rashba-type quantum channel

L. Y. Wang,1C. S. Tang,2 and C. S. Chu1

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan} 2_{Physics Division, National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu 30013, Taiwan}

共Received 15 August 2005; revised manuscript received 2 November 2005; published 2 February 2006兲 We propose and demonstrate theoretically that resonant inelastic scattering共RIS兲 can play an important role in dc spin current generation. The RIS makes it possible to generate dc spin current via a simple gate configuration: a single finger gate that locates atop and orients transversely to a quantum channel in the presence of Rashba spin-orbit interaction. The ac-biased finger gate gives rise to a time variation in the Rashba coupling parameter, which causes spin-resolved RIS and, subsequently, contributes to the dc spin current. The spin current depends on both the static and the dynamic parts in the Rashba coupling parameter,␣₀and␣₁, respectively, and is proportional to␣0␣1

2

. The proposed gate configuration has the added advantage that no dc charge current is generated. Our study also shows that the spin current generation can be enhanced significantly in a double finger-gate configuration.

DOI:10.1103/PhysRevB.73.085304 PACS number共s兲: 73.23.⫺b, 72.25.⫺b, 72.30.⫹q

I. INTRODUCTION

Spintronics is important in both application and funda-mental arenas.1,2 _{A recent key issue of great interest is the}

generation of dc spin current 共SC兲 without charge current. Various dc SC generation schemes have been proposed, in-volving static magnetic field,3–5 _{ferromagnetic material,}6 _or

ac magnetic field.7_{More recently, Rashba-type sporbit}

in-teraction 共SOI兲 in two dimension electron gas 共2DEG兲8–10 has inspired attractive proposals for nonmagnetic dc SC generation.11–13Of these recent proposals, including a time-modulated quantum dot with a static spin-orbit coupling,11

and time modulations of a barrier and the spin-orbit coupling parameter in two spatially separated regions,12 _{the working}

principle is basically adiabatic quantum pumping. Hence, si-multaneous generation of both dc spin and charge current is the norm. The condition of zero dc charge current, however, is met only for some judicious choices for the values of the system parameters.

It is known, on the other hand, that quantum transport in a narrow channel exhibits resonant inelastic scattering 共RIS兲 features when it is acted upon by a spatially localized time-modulated potential.14,15_{This RIS is coherent inelastic}

scat-tering, but with resonance at work, when the traversing elec-trons can make transitions to their subband threshold by emitting mប⍀.14,15_{Should this RIS become spin resolved in}

a Rashba-type quantum channel共RQC兲, of which its Rashba coupling parameter is time modulated locally, we will have a simpler route to the nonmagnetic generation of dc SC. Thus, we opt to study, in this work, the RIS features in a RQC. This requires us to go beyond the adiabatic regime and into the regime when either ␮ or ␮n⬃ប⍀. We solve the

time-dependent spin-orbit scattering 共SOS兲 for all possible inci-dent electron energies and obtain large RIS contribution. In the adiabatic regime, however, with␮,␮nⰇប⍀, we find that

the dc spin-pumping effect from a sole SOI time-modulated region is small.12

II. SYSTEM CONFIGURATION

The system configuration considered is based on a RQC that forms out of a 2DEG in an asymmetric quantum well

by the split-gate technique. As is depicted in Fig. 1共a兲, a finger gate 共FG兲 is positioned above while separated from the RQC by an insulating layer. A local time variation in the Rashba coupling parameter ␣共r,t兲 can be induced by ac biasing the FG.12,13 _{The Hamiltonian is given by} H=p2_{/ 2m +H}

so共r,t兲+Vc共y兲 where the Rashba term Hso共r,t兲 = M ·

1

2关␣共r,t兲p + p␣共r,t兲兴. 共1兲

Here, M = zˆ⫻␴, zˆ is normal to the 2DEG,␴is the vector of Pauli spin matrices, and Vc共y兲 is the confinement potential.

The unperturbed Rashba coupling parameter ␣共r,t兲 is ␣0

throughout the RQC, but becomes␣0+␣1cos⍀t in the

re-gion underneath the ac-biased FG. The Dresselhaus term is neglected for the case of a narrow-gap semiconductor system.16

To demonstrate the pumping mechanism, we consider a narrow RQC in which its subband energy spacing is much greater than the Rashba-induced subband mixing. As such, the unperturbed Hamiltonian, in its dimensionless form, is

H0= −ⵜ2+␣0␴y共i⳵/⳵x兲+Vc共y兲. Appropriate units have been

used such that all physical quantities presented here, and henceforth, are dimensionless.15_{In particular,␣is in unit of} v_F*/ 2, and spin in unit ofប/2. The right-going 共R兲 eigenstate of H0, in the nth subband, is ␾n共y兲␺n␴共x兲, where

␺n␴共x兲=exp关ikn,R␴ x兴␹␴. The wave vector kn,R␴ =

冑

␮n+␩_␴␣0/ 2,

FIG. 1.共Color online兲 共a兲 Top-view schematic illustration of the RQC. The ac-biased FG, of width l, is indicated by the gray area; 共b兲 the electron dispersion relation of an unperturbed RQC.

while ␩_␴= ± 1 denotes the eigenvalue of␹_␴ to the operator

␴y. ␮n is the energy measured from the nth subband

threshold such that the energy of the eigentstate is

E =␮n+␧n−␣20/ 4, for␧n=共n␲/ d兲2. This dispersion relation is

shown in Fig. 1共b兲. The subband with␮n⬃ប⍀ is found to

contribute most to the RIS-enhanced spin pumping. It is of import to note that right-going electrons have兩kR↑兩 ⬎兩kR↓兩 and

that, at the subband threshold, k_R↑共↓兲= k_L↑共↓兲.

III. PHYSICAL PICTURE

In this section, we show that the physical origin of the dc SC generation can be understood from two perspectives. A weak pumping regime result is then obtained for an explicit confirmation of our physical reasoning.

The first perspective is associated with the vector poten-tial. In the ac-biased region,H=Hx+Hy, the transverse part Hy= −⳵2/⳵y2+ Vc共y兲, and the longitudinal part

Hx共t兲 =

冉

− i ⳵ ⳵x+ ␣共x,t兲 2 M · xˆ

冊

2 −1 4␣共x,t兲 2_{. 共2兲}

The form of Eq.共2兲 suggests an effective vector potential, A共t兲=₂1␣共x,t兲M·xˆ, which depends on the spin and gives rise to a spin-resolved driving electric field E = −⳵A /⳵t. However,

inHx, the A2term does not depend on␴, while for the term

linear in A, A␹_␴= −1₂␩_␴␣共x,t兲␹_␴ gives rise only to a trivial spin dependence, which can be easily removed by a shift in the origin of time for the case of an oscillatory␣共x,t兲. Yet it turns out that the full term linear in A, given by −i共⳵/⳵x兲xˆ·A,

manages to give rise to nontrivial spin-resolved transmis-sions. In a perturbative sense, this term becomes k_R↑共↓兲Ax, for

the case of a right-going electron incident upon a spatially uniform␣共t兲. This renders the effective longitudinal driving field to become spin dependent, through the factor k_R↑共↓兲. The difference in the current transmissions, for up and spin-down cases, is proportional to the difference in k_R↑共↓兲, or␣0,

and is found to be amplified by RIS. This breaking of the longitudinal symmetry in the effective driving field by ␣₀ leads to the generation of dc spin current in a FG-RQC struc-ture that has but an apparent longitudinal configuration sym-metry, and with zero source-drain bias. No dc charge current will be generated, however, in such a structure.

An alternate perspective for the understanding of the ori-gin of the spin-resolved current transmission is associated with a unitary transformation. By introducing a unitary trans-formation ⌿_␴共x,t兲=exp关共i␩_␴/ 2兲兰_−l/2x ␣共x

⬘

, t兲dx

⬘

兴␺_␴共x,t兲, the Schrödinger equation关Eq. 共2兲兴 becomes

冋

− ⳵ 2 ⳵x2+ U1共t兲 + U2 ␴_共t兲

册

_␺ ␴共x,t兲 = i_⳵⳵ t␺␴共x,t兲, 共3兲

of which the two time-dependent potentials are U₁共t兲 = −␣共x,t兲2/ 4 and U₂␴共t兲=共⍀␣1/ 2兲共x+l/2兲cos共⍀t+␩␴␲/ 2兲.

Even though only U₂␴depends on spin, both the term in U1共t兲

that oscillates with frequency⍀ and U₂␴together constitute a pair of quantum pumping potential that pump SC. This is our major finding in this work: that spin pumping nature is built-in even in a single FG configuration.

The SC expression for a state ⌿_␴ is given by the SC density operator Jˆx y = i

冋

⳵⌿␴ † ⳵x ␴y⌿␴− H.c.

册

+ ␣ 2⌿␴ †_兵␴ y,M其x⌿␴. 共4兲

The SC conservation is maintained by the supression of sub-band mixing and the associated spin-flipping in a RQC. For a scattering state⌿_␴, the SC can be expressed in terms of the transmission coefficients. More specifically, the ratio be-tween the time-averaged transmitted and the incident SC gives the spin-resolved current transmission T_␤␣␴ , where␣,␤, are, respectively, the incident and the transmitting lead. Sum-ming over contributions from all states in reservoirs R and L, the SC is

Is= I↑− I↓, where

I␴=

冕

dEf共E兲关TRL␴ − TLR␴ 兴, 共5兲

and I␴ is the number current due to electrons with spin ␩_␴ from both reservoirs that are under zero source-drain bias condition. Here TRL␴ =兺n兺m共␮

n m_⬎0兲T

n,RL

m,␴ _{and f共E兲 is the}

Fermi-Dirac distribution. The transmission coefficient T_n,RLm,␴ =兩t_n,RLm,␴兩2

冑

_␮

n m

/␮n denotes the current transmission that an

electron incident from terminal L in the spin channel ␴, subband n, energy E, is scattered into terminal R, sideband

m, with kinetic energy ␮n

m

=␮n+ m⍀. The net charge

current is given by Iq_{= I}↑_{+ I}↓_{. In a symmetric FG}

configura-tion, we have TLR␴ = TRL−␴, so that the net spin current is Is_{= 2兰dEf共E兲共T}

RL ↑ _{− T}

RL

↓ _{兲 and the net charge current is}

identi-cally zero.

In the weak pumping共WP兲 regime, when␣1is small, we

can demonstrate analytically, and most unequivocally, that spin-dependent reflection arises merely from the aforemen-tioned linear A term in Hx共t兲. We outline the derivation here

while leaving the detail in Appendix A. Tracing up to the first order in␣1, our derivation retains the reflection

ampli-tudes to m = ± 1 sideband and drops that to the m = 0 side-band. Contribution to the total reflection includes thus reflec-tion at either the left or the right edges of the time-modulated region. For an electron incident from terminal L with wave vector k_n,R␴ 共E兲, the reflection at the left edge is obtained from the wave-function continuous condition and the boundary condition − ⳵ ⳵x兩⌿␴兩x=共− l/2兲++ ⳵ ⳵x兩⌿␴兩x=共− l/2兲− + i 2␩␴␣1cos⍀t兩⌿␴兩x=−l/2= 0. 共6兲 In the time-modulated region, the wave function ⌿_␴ consists of one-sideband terms, given by the form

eik_n,R␴ 共E±⍀兲x_e−i共E±⍀兲t_{, and the m = 0 term, given by the form} eik_n,R␴ 共E兲x_e−iEt_关1+␩

␴/共2⍀兲␣1kn,R␴ 共E兲共ei⍀t− e−i⍀t兲兴. The extra ⍀t

dependence in the m = 0 term is resulted from the time-dependent driving effect of A, which is obviated by the

weighting factor that involves␣₁k_n,R␴ . The reflection ampli-tude rLm,␴ at the left edge is obtained to be

rL m,␴_{= sgn共m兲}␩␴ 2

冋

␣1 ⍀kn,R␴ 共kn,R␴ − kn,Rm,␴兲 + ␣1m 2

册

k_n,Rm,␴− k_n,Lm,␴ ⫻ e−i共kn,R␴ −kn,L m,␴_兲l/2 共7兲 for m = ± 1. The first term in the numerator of Eq. 共7兲 is clearly due to A, because of the factor␣1kn,R␴ , and the second

term is due to the scattering at the edge. Here the wave vector k_n,Rm,␴_共L兲= ±共␮n

m_兲1/2₊␩

␴␣0/ 2, with upper 共lower兲 sign

corresponds to the right-共left-兲 moving electron in the nth subband, mth sideband, and with kinetic energy ␮n

m

. It is clear then that wave-vector differences in both the numerator and the denominator of rL

m,␴_{are spin independent. Hence, the}

spin dependence arises solely from the␣₁k_n,R␴ factor in the first term of the numerator in Eq.共7兲, or from A. This con-firms our understanding of the physical origin of the dc SC generation.

Including the reflection at the right edge, we obtain the total reflection amplitude

r_n,LLm,␴ =关1 − ei共kn,R␴ −kn,L m,␴_兲l

兴rLm,␴. 共8兲

We note that the spin dependence of this total reflection amplitude is associated with ␣0. In fact, it turns out that

the SC is proportional to ␣₀. The SC is related to the current transmission, which, within the aforementioned approximation, is given by TRL␴ ⬇1−兺n关Rn,LL1,␴ + Rn,LL−1,␴兴, where R_n,LLm,␴=兩r_n,LLm,␴兩2

冑

_␮

n m

/

冑

␮n. From Eq.共5兲, the energy derivative

of the zero-temperature SC is given by ⳵Is/⳵E = 2⌬TRL

= 2共TRL↑ − TRL↓ 兲 from which its explicit expression is given by

⳵Is ⳵E= 1 2␣0␣1 2

兺

n m=±1

兺

共␮n m_⬎0兲 兵1 − cos关共

冑

␮n+

冑

␮n m_兲l兴其

冋

冉

1 4

冊

2 −

冉

1 ⍀共␮n−

冑

␮n␮n m_{兲 +}m 4

冊

2

册

␮n

冑

␮n m . 共9兲

That this expression diverges when␮n m

= 0, for m⬍0, exhib-its the RIS feature unambiguously and also demonstrates the need to go beyond the one-sideband approximation near the RIS condition.

IV. NUMERICAL RESULTS AND DISCUSSIONS

In the following, we present results obtained from solving the time-dependent SOS exactly, in the numerical sense. An outline of the method is presented in Appendix B. Physical parameters are chosen to be consistent with the InGaAs-InAlAs–based narrow-gap heterostructures such that the electron density ne= 1⫻1012cm−2, effective mass m*_{= 0.04m}

0, and ␣0= 0.13 共ប␣0= 3⫻10−11eV m兲.9

Accord-ingly, the length unit l*_{= 4.0 nm and the energy unit} E*_{= 59 meV.}

For the case of one FG共N=1兲, the energy dependence of the spin-resolved transmission TRL␴ is plotted in Figs.

2共a兲–2共c兲, and that of the corresponding dc SC is plotted in Fig. 2共d兲. The FG width l=20 共80 nm兲, driving frequency ⍀=0.002 共␯=⍀/2␲⬇28 GHz兲, and energy ␮= E −␧₁. Dip features in TRL␴ at ␮/⍀=1 are the quasi-bound state 共QBS兲

features, where electrons undergo coherent inelastic scatter-ing to a QBS just beneath its subband bottom.14_Higher-order

QBS features at ␮/⍀=2 are barely shown by the small peaks. Of particular interest is the change in sign in the trans-mission difference⌬TRL= TRL↑ − TRL↓ across the dip structures,

namely, ⌬TRL共␮=⍀−兲⬎0 while ⌬TRL共␮=⍀+兲⬍0. This

leads to a nonzero dc SC, peaked at␮/⍀=1, and is exhibited in Fig. 2共d兲.

It is also shown that the dc SC increases with the oscil-lating amplitude␣1 of the ac-biased gate voltage. More

im-portantly, all the above dc SC characteristics, including even their order of magnitudes, are already captured by Eq. 共9兲. This lends strong support to our finding that RIS has played a pivotal role in the generation of dc SC. Similar RIS-induced peak in Is _{is found if we vary d instead of␮.}

The nonlinear enhancement in the dc SC by two FGs 共N=2兲 is presented in Figs. 3共a兲–3共c兲. The driving frequency

FIG. 2. 共Color online兲 Spin-resolved current transmissions T_RL↑ 共solid兲 and TRL↓ 共dashed兲 versus the incident energy␮/⍀.

Param-eters N = 1,␣₀= 0.13, ⍀=0.002, l=20, and␣₁=共a兲 0.03, 共b兲 0.04, and共c兲 0.05. The corresponding dc SC is plotted in 共d兲.

is chosen to be ⍀=0.001 共␯⬇14 GHz兲, and with l=22 共⯝88 nm兲. For comparison, the N=1 FG transmissions are plotted along with that of the N = 2 FG case, in Figs. 3共a兲 and 3共b兲, respectively. The corresponding dc SC, expressed in terms of pumped spins per cycle NP

s

=共2␲/⍀兲兩Is_{兩, is shown in}

Fig. 3共c兲. The pumping is optimized by a choice of the FG separation, with the edge-to-edge separation ⌬l=22. The QBS dip structures are significant up to the fourth sideband in Fig. 3共b兲. As indicated by arrows, the pumped spin-per-cycle peaks at␮/⍀⯝1.57 共1.92兲, with peak value 0.8 共0.1兲 for the N = 2共N=1兲 FG case. The enhancement in N_Ps is far greater than doubling the NP

s

of N = 1 FG. Finally, we discuss the effectiveness of tuning␣. Grundler showed that a static FG bias change ⌬VFG⬃0.075 V can tune ⌬␣⬃0.25␣0.10

This tuning ability should remain valid in ac FG bias if the wave function in the asymmetric quantum well responses adiabatically. We estimate the quantum-well energy-level spacing ⌬E⬃0.08 eVⰇប⍀⬃0.06 meV, for ⍀/共2␲兲=14 GHz. Thus, the adiabatic response of the wave function in the quantum well is established. Furthermore, the ac FG biases, with amplitude⌬VFG⬃0.075 V, is estimated

to be within reach of coaxiable cable technology.17

V. CONCLUSION

In conclusion, a nonmagnetic way of generating dc SC has been established. The proposed Rashba-type quantum channel driven by an ac-biased finger gate is a simple struc-ture and should be within reach of recent fabrication capa-bility. The spin pumping is studied, in detail, in both its nature and its pumping mechanism. A resonant inelastic pro-cess is the major factor that contributes to the robustness of the spin pumping. The coherent nature of the pumping sup-ports further enhancement of the spin pumping by invoking configuration consisting of more than one finger gate.

ACKNOWLEDGMENTS

The authors acknowledge valuable discussions with A. G. Mal’shukov. This work was funded by the National Science Council of ROC under Grants No. NSC92-2112-M-009-035,

No. NSC92-2120-M-009-010, No. NSC93-2112-M-009-036, and No. NSC93-2119-M-007-002共NCTS兲.

APPENDIX A: WEAK-PUMPING (WP) REGIME

In the WP regime, we can obtain analytic results. The WP regime refers to the case when the Rashba coupling param-eter oscillates with a small amplitude␣1. Keeping only up to

the lowest nonvanishing contribution of ␣₁, it is simpler to calculate the reflection amplitudes than the transmission co-efficients. The reflection amplitudes to m = ± 1 sidebands are first order in␣1and are the major objects of our calculation

here. The reflection amplitudes to m = 0 sideband, however, is second order in␣₁ and is neglected. When the Rashba cou-pling parameter oscillates in time within a spatial region −l / 2⬍x⬍l/2, the longitudinal Hamiltonian is given by

Hx= − ⳵2 ⳵x2+␣0␴y

冉

i ⳵ ⳵x

冊

+ ␣1cos⍀t 2 ␴y ⫻

冋

␪共l/2 − 兩x兩兲

冉

i ⳵ ⳵x

冊

+

冉

i ⳵ ⳵x

冊

␪共l/2 − 兩x兩兲

册

, 共A1兲

where␪共x兲 is the step function. For an electron incident from terminal L with wave vector k_n,R␴ 共E兲, the reflection coeffi-cients consist of contributions from reflections at the left and the right edges of the time-modulated region. We first calcu-late the reflection amplitudes due to reflection at x = −l / 2. The wave function is given by

␺n␴共x ⬍ − l/2兲 = e ik_n,R␴ _共E兲x e−iEt+

兺

m=±1 共m⫽0兲 rLm,␴e ik_n,Lm,␴x e−i共E+m⍀兲t, 共A2兲 ␺n␴共− l/2 ⬍ x ⬍ l/2兲 =

兺

m=±1 共m⫽0兲 tL m,␴ eikn,R m,␴_x e−i共E+m⍀兲t + tL0,␴e ik_n,R␴ 共E兲x e−iEt

冋

1 + ␩␴ 2⍀␣1kn,R␴ 共E兲共e i⍀t − e−i⍀t兲

册

. 共A3兲 Here, t_Lm,␴and r_Lm,␴denote, respectively, the transmission and reflection coefficients at the left edge of the time-modulated region. We have not included, in Eq.共A3兲, corrections to the wave functions associated with the coefficients t_Lm,␴, for

m = ± 1, that arise from the time-modulation of the Rashba

spin-orbit intraction共SOI兲. It is because the coefficients tLm,␴

are already first order in ␣₁. These coefficients are solved from the wave-function continuous condition and the bound-ary condition in Eq. 共6兲. The reflection coefficients are cal-culated, and the expression is presented in Eq.共7兲. It is worth noting that tL0,␴= 1, up to first order in␣1.

Following a similar procedure, the reflection at the right edge of the time-modulated region can be obtained from the following wave function:

FIG. 3. 共Color online兲 Current transmission versus ␮/⍀ for N =共a兲 1, and 共b兲 2. Pumped spins per cycle are plotted in 共c兲 for N = 1共thick curve兲 and N=2 共thin curve兲 with␣1= 0.065, and

␺n␴共− l/2 ⬍ x ⬍ l/2兲 = eikn,R␴ 共E兲x_e−iEt

冋

_{1 +}␩␴␣1 2⍀ kn,R␴ 共E兲共e i⍀t − e−i⍀t兲

册

+

兺

m=0,±1 rR m,␴ eikn,L m,␴_x e−i共E+m⍀兲t ⫻

冋

1 +␩␴␣1 2⍀ kn,L

m,␴_共ei⍀t_{− e}−i⍀t_兲

册

_{, 共A4兲}

␺n␴共x ⬎ l/2兲 =

兺

m=0,±1

tRm,␴e ik_n,Rm,␴x

e−i共E+m⍀兲t. 共A5兲 Again, t_Rm,␴ and r_Rm,␴ denote the transmission and the reflec-tion coefficients, respectively, at the right edge of the time-modulated region. It is noted that in Eq.共A4兲, only one in-cident wave needs to be considered. The inin-cident waves associated with coefficients tL

m,␴

in Eq.共A3兲, for m= ±1, is neglected because these coefficients are, themselves, first or-der in␣1. Invoking the wave-function continuous condition

and the boundary condition

⳵ ⳵x兩␺n ␴_兩 x=共l/2兲−− ⳵ ⳵x兩␺n ␴_兩 x=共l/2兲+− i 2␩␴␣1cos⍀t兩␺n ␴_兩 x=l/2= 0, 共A6兲 we obtain rRm,␴= − ei共kn,R ␴ _−k n,L m,␴_兲l rLm,␴, 共A7兲

and the total reflection coefficient, up to the first order in␣1

is given by r_n,LLm,␴= rRm,␴+ rLm,␴, for m = ± 1, which explicit

ex-pression is presented in Eq.共8兲.

APPENDIX B: FORMULATION FOR NUMERICAL CALCULATION

The Hamiltonian in the ac-biased region, given by Eq. 共2兲, can have its time-modulated term transformed

away by the use of a transformation: ⌿n,␴共x,t兲

= exp关␩_␴共␣₁/⍀兲sin ⍀t共⳵/⳵x兲兴␺_n␴共x兲, where ␺_n␴ satisfies Eq. 共2兲 with␣1sets to zero. Thus, the general form of the wave

function in the time-modulated region is given by

⌿n,␴共x,t兲 = exp

冉

␩␴ ␣1 ⍀ sin⍀t ⳵ ⳵x

冊

冕

d␧关A˜n ␴_共␧兲eik_n,R␴ 共␧兲x + B˜n␴共␧兲e ik_n,L␴ _{共␧兲x兴e}−i␧t_{. 共B1兲}

The matching of the wave functions at all times requires the form F˜n␴共␧兲=兺m⬘Fn␴共m

⬘

兲␦共␧−␮n− m

⬘

⍀兲.15 Hence, the nth-subband scattering wave function is of the form

⌿n,␴共x,t兲 = eikn,R␴ 共E兲x_e−iEt₊

兺

m r_n,LLm,␴eikn,L m,␴_x e−i共E+m⍀兲t, if x⬍ − l 2 共B2兲 ⌿n,␴共x,t兲 =

兺

m,m⬘ 共−␩␴兲m−m⬘

冋

An␴共m

⬘

兲e ik_n,Rm⬘,␴x Jm−m⬘

冉

␣1 ⍀kn,R m⬘,␴

冊

+ Bn␴共m

⬘

兲eikn,L m⬘,␴_x J_m−m⬘

冉

␣1 ⍀kn,L m⬘,␴

冊册

e−i共E+m⍀兲t, if兩x兩 ⬍l 2 共B3兲 ⌿n,␴共x,t兲 =

兺

m t_n,RLm,␴eikn,R m,␴_x e−i共E+m⍀兲t, if x⬎ l 2. 共B4兲 By invoking the boundary conditions in Appendix A, the reflection and the transmission coefficients can all be solved.

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