Photon-assisted electron transmission resonance through a quantum well with spin-orbit coupling

Cun-Xi Zhang,1 Y.-H. Nie,1,2,* and J.-Q. Liang1

1Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China2Department of Physics, Yanbei Normal Institute, Datong, Shanxi 037000, China

�Received 5 November 2005; published 8 February 2006�

Using the effective-mass approximation and Floquet theory, we study the electron transmission over aquantum well in semiconductor heterostructures with Dresselhaus spin-orbit coupling and an applied oscilla-tion field. It is demonstrated by the numerical evaluations that Dresselhaus spin-orbit coupling eliminates thespin degeneracy and leads to the splitting of asymmetric Fano-type resonance peaks in the conductivity. Inturn, the splitting of Fano-type resonance induces the spin-polarization-dependent electron current. The loca-tion and line shape of Fano-type resonance can be controlled by adjusting the oscillation frequency and theamplitude of the external field as well. These interesting features may be a very useful basis for devisingtunable spin filters.

DOI: 10.1103/PhysRevB.73.085307 PACS number�s�: 73.23.�b, 72.25.Dc, 73.63.Hs

The Fano-type resonance, which arises from interferencebetween a localized state and the continuum band,1 has beenwidely known across many different branches of physics andobserved in a great variety of experiments including atomicphotoionization,2 electron and neutron scattering,3 Ramanscattering,4 and photoabsorption in quantum wellstructures.5,6 In recent years, Fano-type resonances havebeen reported to appear also in experiments on electrontransport through mesoscopic systems. Fano-type resonancesin electronic transport through a single-electron transistor al-low one to alter the interference between the two paths bychanging the voltages on various gates.7 Reference 8 re-ported the first tunable Fano experiment in which a well-defined Fano system is realized in an Aharonov-Bohm �AB�ring with a quantum dot �QD� embedded in one of its arms,which is the first convincing demonstration of this effect inmesoscopic systems. The Fano effect in a quantum wire witha side-coupled quantum dot occurs in a way different fromthat in the QD-AB-ring system, because only reflected elec-trons at the QD are involved for its emergence.9–11 However,except Ref. 12 which reported the first Fano-type resonancesdue to the interaction of electron states with opposite spinorientation, so far the Fano effects have been studied theo-retically and experimentally for electron transport throughmesoscopic systems but not considering, to our knowledge,the electron spin degree of freedom which plays an importantrole in spintronics. Recently, a large number of spin-dependent phenomena—for instance, Datta-Das spin field-effect transistor13 spin transport,14 spin Hall effect,15 spin-dependent tunneling phenomena in semiconductorheterostructures,16 and so on—have attracted a great atten-tion because of the prospect of technological applications.

In general, the condition for the Fano-type resonance tooccur is the presence of two scattering channels at least: thediscrete level and continuum band. In the mesoscopic sys-tems mentioned above, the nature of two scattering channelsis dependent on the geometry of the device under consider-ation. In this paper we adopt the interferometer geometrywhich is realized by a time-periodic quantum well in semi-

conductor heterostructures and study the spin-dependentFano resonance and electron transport over through thequantum well with Dresselhaus spin-orbit coupling. Floquetscattering through a time-periodic potential induces mixingof the continuum and bound states and leads to the appear-ance of asymmetric Fano-type resonances in the conductiv-ity. The effect of Dresselhaus spin-orbit coupling on the mo-tion of electrons makes the effective band mass of electronsdependent on the spin orientation; consequently, the asym-metric Fano-type resonance splits into two different peakscorresponding to electrons with opposite spin polarization,which leads to the spin-polarization-dependent transmissionelectron current.

We consider the transmission of a single electron withincident wave vector k= �k� ,kz� through a one-dimensionaltime-periodic potential well which extends from 0 to a—thatis,

V�z,t� = �0, z � 0 and z � a ,

− V0 + V1 cos��t� , 0 � z � a .� �1�

Here the well depth V0 grows along z � �001� and can beadjusted by an applied field, k� is the wave vector parallel tothe plane of the well, and kz is the wave vector normal to thewell plane along the direction of tunneling. The electronpasses three regions from the left to the right with I, III, andII denoting the right, left, and central regions, respectively.We assume a low enough temperature such that the electron-phonon interactions considered in Ref. 17 can be neglected.Thus the electron motion in semiconductor heterostructuresmay be described by a time-dependent Schrödinger equation

i��

�t��z,s,t� = H��z,s,t� , �2�

with Hamiltonian

H = −�2

2�

�2

�z2 +�2k�

2

2�+ V�z,t� + HD, �3�

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HD = �0, z � 0 and z � a ,

�xkx�ky2 − kz

2� + yky�kz2 − kx

2� + zkz�kx2 − ky

2�� , 0 � z � a ,� �4�

where � is the effective mass of the electron and HD denotesthe spin-dependent Dresselhaus term in zinc-blende-structuresemiconductors.18 � are the Pauli matrices, and a materialconstant describing the strength of the spin-orbit coupling.We have assumed that �0 outside the well. In the quantumwell with finite depth the confinement of the electron wavefunction in the growth direction forces quantization of thecorresponding component of the wave vector; thus, oneshould consider kz in the Hamiltonian as an operator −i� /�z.Assuming that the kinetic energy of the incidence electron ismuch smaller than the well depth V0 the Dresselhaus term

HD may be simplified to16

HD = 0, z � 0 and z � a ,

�xkx − yky��2

�z2 , 0 � z � a . �5�

Using Pauli matrices the Hamiltonian �3� inside the potentialwell can be written as

H = �−�2

2�

�2

�z2 +�2k�

2

2�+ V�z,t� k+ �2

�z2

k− �2

�z2 −�2

2�

�2

�z2 +�2k�

2

2�+ V�z,t� � ,

�6�

with k±=kx± iky. Inserting ��z ,s , t�=��z , t� s with s thespin wave function and Eq. �6� into Schrödinger equation �2�we obtain

± =1 2

� 1

�e−i� � , �7�

which describes the electron spin states of the opposite spinpolarizations. Here � is the polar angle of the wave vector k�

in the xy plane, k�� = �k� cos � ,k� sin ��, The orientation ofelectron spins s�±�k���= ±

+ ±= ��cos � , ±sin � ,0� corre-sponding to the eigenstates ± depends on the direction ofthe wave vector k�� in the xy plane. In the spin subspace the

Hamiltonian H may be reduced as

H± = ±+H ± = −

�2

2�±

�2

�z2 +�2k�

2

2�+ V�z,t� , �8�

where the modified effective mass of the electron dependsnot only on the Dresselhauss coupling constant and in-plane electron wave vector k��, but also on the orientation ofthe electron spin, and is given by �±=��1±2�k� /�2�−1.Although the modification of the electron effective mass isvery small, it plays an important role in the generation of theasymmetric Fano resonance peak splitting. Thus theSchrödinger equation can be rewritten as

i��

�t�±�z,t� = H±�±�z,t� , �9�

with

�±�z,t� = ±�±�z,t�exp�ik�� · ��� , �10�

where �� = �x ,y� is a vector in the well plane. Inserting Eqs.�8� and �10� into Eq. �9� we obtain the equation for �±�z , t�:

i��

�t�±�z,t� = −

�2

2�±

�2

�z2�±�z,t� +�2k�

2

2��±�z,t� + V�z,t��±�z,t� .

�11�

Inside the potential well, the potential V�z , t� in Eq. �11� is aperiodic function of time; thus, according to the Floquettheorem,19–21 Eq. �11� has a solution of the form

�±F�z,t� = �±�z,t�exp�− iEF

±t/�� , �12�

where EF± is the Floquet energy eigenvalue and �±�z , t� is a

periodic function of time: �±�z , t�=�±�z , t+T� with periodT=2� /�. Substituding Eq. �12� into Eq. �11� for �±�z , t�=g±�z�f±�t�, we have two separate equations with an intro-duced constant E± �Refs. 22 and 23�:

−�2

2�±

d2

dz2g±�z� = �E± + V0 −�2k�

2

2��g±�z� , �13�

i�d

dtf±�t� − V1 cos��t�f±�t� = �E± − EF

±�f±�t� . �14�

The solution of Eq. �14� is found as

f±�z,t� = exp�− i�E± − EF±�t/�� �

n=−�

+�

Jn� V1

���exp�− in�t� ,

�15�

where we have taken the initial condition f±�0�=1 and Jn�x�is the nth-order Bessel function of the first kind. Sincef±�t�= f±�t+T�, Eq. �14� requires that E±=Em

± =EF± +m�� with

m being an integer.The incoming and outgoing waves �channels� of Floquet

scattering form the sidebands �or Floquet channels� with en-ergy spacing �� according to Em

± =EF± +m�� �m is the side-

band index�. The mode of Em�0 is an evanescent mode, andthe corresponding sideband is called an evanescent sidebandbecause such a mode with imaginary km cannotpropagate.23,24 The equation for g±�z� has a solution

g±�z� = �m=−�

+�

�am± eiqm

± z + bm± e−iqm

± z� , �16�

where am± and bm

± are constant coefficients and

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qm± = �2�±

�2 �EF± + m�� −

�2k�2

2�+ V0��1/2

.

Thus the wave function inside the oscillation-potential wellcan be expressed as

�±II�z,t� = ± �

n=−�

+�

�m=−�

+�

�am± eiqm

± z + bm± e−iqm

± z�

�Jn−m� V1

���e−iEzn

± t/� exp�ik�� · �� − iE�±t/�� .

�17�

Since electrons incident to the oscillating region will be scat-tered inelastically into an infinite number of Floquet side-bands, so the wave function outside the well can be writtenas the superposition of waves with all values of energy:

�±I �z,t� = ±�eikz0

± z−iEz0± t/� + �

n=−�

+�

rn0± e−ikzn

± z−iEzn± t/��

�exp�ik�� · �� − iE�±t/�� , �18�

�±III�z,t� = ± �

n=−�

+�

tn0± eikzn

± z−iEzn± t/� exp�ik�� · �� − iE�

±t/�� ,

�19�

where Ez0± +E�

±=E0± ,E�

±=�2k�2 /2�1 and kzn

±

= �2�1 /�2��Ez0± +n���. For the sake of simplicity we con-

sider the case that E0±��� corresponding to the propagating

mode of the lowest energy. rn0± and tn0

± are the probabilityamplitudes of the reflecting waves and outgoing waves fromthe sideband 0 to sideband n, respectively. The continuity of�± and �1/���� /�z��± at the interfaces z=0 and z=a re-quires

�m=−�

+�

Jn−m� V1

����am

± + bm± � = �n0 + rn0

± , �20�

�1

�±�

m=−�

+�

Jn−m� V1

���qm

± �am± − bm

± � = kzn± ��n0 − rn0

± � , �21�

�m=−�

+�

Jn−m� V1

����eiqm

± aam± + e−iqm

± abm± � = eikzn

± atn0± , �22�

�1

�±�

m=−�

+�

Jn−m� V1

���qm

± �eiqm± aam

± − e−iqm± abm

± � = kzn± eikzn

± atn0± .

�23�

The continuity conditions of the wave functions �20�–�23�can be expressed as the matrix forms

J�A + B� = � + R ,

�1

�±JQ�A − B� = K�� − R� ,

J�LA + L−1B� = ST ,

�1

�±JQ�LA − L−1B� = KST , �24�

with the help of the square matrixes defined by the matrixelements Jnm=Jn−m�V1 /���, Qnm=qn

±�nm, Knm=kn±�nm, Snm

=eikzn± a�nm, and Lnm=eiqm

± a�nm and the column matrixes by thematrix elements An=an

±, Bn=bn±, �n=�n0, Rn=rn0

± , and Tn= tn0

± , where R and T denote the matrices of reflecting andtransmission amplitudes. From the matrix equation �24� onecan obtain the matrix of the transmission amplitude:

T = S−1�M2−1L−1M1 − M1

−1LM2��M2−1M1 − M1

−1M2�� ,

�25�

where M1= �J−1+ ��± /�1�Q−1J−1K� and M2= �J−1

− ��± /�1�Q−1J−1K�. The total electron-transmission prob-abilities of spin-up and spin-down components are given by

T± = �m=0

+�

�tm0± �2, �26�

from which the conductance of the electrons through semi-conductor heterostructures can be obtained by the Landauer-Buttiker formula25,26

G± =2e2

hT± =

2e2

h�m=0

+�

�tm0± �2. �27�

We now study numerically the scattering of a incidentwave by an oscillating quantum well of InP-GaSb-InP semi-conductor hereostructures and calculate the conductivitywith the help of Eqs. �25�–�27�. The minimum number ofsidebands needed to be included in the sum of Eq. �26� de-pends on the oscillation amplitude of the potential well. Ingeneral, it is enough to take N�V1 /�� �Ref. 27�.

The mechanism of resonance considered here is differentfrom Ref. 17 where the resonance originates from an accu-mulation of electrons in bound states of the well �this accu-mulation of electrons conversely produces strong feedbackon the transmission in the incident channel�. In our model,the interaction of electrons with the oscillating field leads tophoton-mediated transmission resonances. The incident elec-trons can emit photons and drop to the bound states of thepotential well. Similarly, the electrons in bound states canalso jump to incident channels or other Floquet channels byabsorbing photons. This forms the discrete channel of scat-tering required by Fano-type resonance. Once the energy dif-ference between the incident electrons and bound states ofthe well is equal to the integer times the energy of one pho-ton, transmission resonance occurs. In Fig. 1, we plot theconductivity G± as a function of the incident electron energyEz0 for V0=300 meV, V1=10 meV, k� =106 cm−1, and ��=10 meV. The parameters of the semiconductor heterostruc-ture are chosen as =187 eV Å3, �=0.041me �me is the massof the free electron� for GaSb, 1=8 eV Å3, �1=0.081me forInP according to Ref. 16. In actual calculation we take 1�0 because �1. Sidebands of n=0, ±1,… , ±5 are takeninto account so that T±=�n=0

5 �tn0± �2. The conductivity pattern

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in Fig. 1 shows two obvious asymmetric resonance peaks, atEz

+=5.2 meV for the spin-up state “�” and Ez−=2.44 meV

for the spin-down state “�.” The corresponding bound- stateenergies are Eb

+=−4.796 meV, Eb−=−7.559 meV for V1=0,

and resonance energies satisfy the relation Ez±=��+Eb

±.When Ez increases continuously and enters the second inci-dent channel—i.e., ���Ez�2��—the second set of asym-metric resonance peaks appears �see the inset of Fig. 1�, butthey are very small because the probability of a two-photonprocess is much less than that of one-photon process. For anincident electron of given energy there exist similar reso-nance peaks in the conductivity when the oscillation fre-quency increases.

The amplitude of oscillating field indicates actually thestrength of electron coupling with the external field. The cor-respondence between the resonance location and energylevel of the potential well makes sense only when, strictlyspeaking, V1→0. The electron coupled with the applied fieldleads to the broadening of the level and thus the asymmetricresonance can take place within certain range of energyrather than at a single level. Figure 2 shows the variance of

spectrum in which the asymmetric resonance peaks are get-ting “fat” with increasing oscillation amplitude for spinstates “�” and “�,” which provides a means to adjust theline shape of Fano-type resonance by external parameters.When the amplitude of the oscillating field is very small, theeffect of energy level broadening is inconspicuous and theFano-type resonance displays sharp peaks. Thus the bound-state energy can be determined by the location of the sharpresonance peaks according to Ez

±=��+Eb±, which can be

used to measure the structure parameters of semiconductorherostructures such as thickness and band gaps and even thespin-orbit coupling constant.

The locations of the asymmetric resonance peaks in en-ergy parameter space depend not only on the width and depthof the well, but also on the oscillation frequency of the ex-ternal field. In Fig. 3, we plot the frequency dependence ofthe conductivity for spin states “�” and “�.” The asymmet-ric resonance peaks move toward the direction of high en-ergy as the frequency increases. It is apparent that the loca-

FIG. 1. Splitting of Fano-type resonance in conductivity G± forV0=300 meV, V1=10 meV, a=60 Å, k� =106 cm−1, ��=10 meV,�=0.041me, �1=0.081me, =187 eV Å3, and 1=0. The inset is adetail of the tiny resonance peaks resulting from electrons exchang-ing two photons with an applied field.

FIG. 2. Dependence of the width of Fano-type resonance peaksin conductivity G± on amplitude of applied field V1 for spin states“�” and “�” with the parameters in Fig. 1.

FIG. 3. Conductivity G± as a function of Ez for different ��,indicating that resonance peaks move toward the direction of highenergy as �� increases, with V1=15 meV and the other parametersin Fig. 1.

FIG. 4. Dependence of the polarization efficiency of the trans-mission electrons on the energy of incident electron and spin orien-tation. The solid curve denotes polarization along spin “�” and thedashed curve denotes spin “�” with V1=30 meV and the otherparameters in Fig. 1.

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tion control of resonance peaks is easien by adjusting thefield frequency than the parameters of the semiconductorheterostructure in practical experiments.

The spin-polarization-dependent splitting of Fano-typeresonance peaks is advantageous to the realization of spincurrent. Figure 4 shows the spin-polarization-dependenttransmission of electrons as a function of the electron energyaccording to P= �T+−T−� / �T++T−�. Figure 4 indicates thatthe transmission current with the spin “�” polarization �solidline� is dominate �over 80% comparing with spin “�” polar-ization current �dashed line�� for the energy range Ez from1 meV to 3.5 meV. The characteristics of the energy Ez de-pendence of the spin-polarization current suggests that onecan exploit the Fano-type resonances as the basis of a spinfilter and control the external parameters—e.g., the oscilla-tion frequency � and the amplitude V1 of the external field—to tune the energy to align with specific resonances.

In summary, we have investigated theoretically the prop-

erty of the electron transport through a quantum well insemiconductor heterostructures with spin-orbit coupling andan applied field. The numerical results demonstrate thatDresselhaus spin-orbit coupling eliminates the spin degen-eracy and leads to a splitting of the asymmetric Fano-typeresonance peaks in the conductivity. The spin polarizationarising from the splitting of Fano-type resonance has an ad-vantage to realize the spin current. The location and lineshape of Fano-type resonance can be controlled by adjustingthe oscillation frequency � and amplitude V1 of the externalfield. These interesting features not only deepen our funda-mental understanding of the role of spin-orbit coupling insolids, but also may be a very useful basis for devising tun-able spin filters.

This work was supported by the National Nature ScienceFoundation of China �Grant No. 10475053� and ShanxiNature Science Foundation �Grant No. 20051002�.

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