Physics Letters A 375 (2011) 4198–4202

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Physics Letters A

www.elsevier.com/locate/pla

Voltage-tunable spin electron beam splitter based on antiparallel doubleδ-magnetic-barrier nanostructure

Lin Yuan, Mao-Wang Lu ∗, Yun-Hui Zhao, Li-Hua Shen

Department of Physics, Shaoyang University, Hunan 422004, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 August 2011Received in revised form 22 September2011Accepted 10 October 2011Available online 14 October 2011Communicated by R. Wu

Keywords:Goos–Hänchen effectMagnetic barrier nanostructureSpin polarizationSpin beam splitter

We report on a theoretical study of spin-dependent Goos–Hänchen (GH) shift of electrons in antiparalleldouble δ-magnetic-barrier (MB) nanostructure under an applied voltage, which can be experimentallyrealized by depositing two metallic ferromagnetic (FM) stripes on top and bottom of the semiconductorheterostructure. GH shifts for spin electron beams across this device, is exactly calculated, with the helpof the stationary phase method. It is shown that a considerable spin polarization of GH shifts can beachieved in this device for two δ-MBs with unidentical magnetic strengths. It also is shown that bothmagnitude and sign of spin polarization of GH shifts can be controlled by adjusting the electric potentialinduced by the applied voltage. These interesting properties may provide an effective approach of spininjection for spintronics application, and this device can be used as a voltage-tunable spin beam splitter.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Recently, optics-like effects of electrons in two-dimensionalelectron gas (2DEG) nanosystems, which originates from thequantum-mechanical wave feature of electrons, have attractedgrowing attentions [1,2], due to their potential application in de-veloping electronic devices. In particular, some of optics-like effectsare now used to realize spin-polarized injection into the con-ventional semiconductors for spintronics [3], in which the spindegree of freedom of electrons is employed for device operation.Basic principle of the spin injection realized by optics-like behav-iors of electron beams, is mainly to separate spatially spin-up andspin-down electron beams, when the spatial position (such as an-gle, shift and displacement) of spin-up electron beam is differentgreatly from that of spin-down one across the 2DEG nanosystem.Namely, such a 2DEG nanosystem serves as a spatially separatingspin filter or a spin beam splitter, which operates in optics-likeeffects of spin electron beams. For example, Khodas et al. [4]have studied the refraction of a spin electron beam at the lat-eral interface between two regions with different strengths ofRashba spin–orbit coupling (SOC). They found that for a beamwith nonzero incident angle the transmitted electrons will splittwo spin polarization components propagating at different angles.Subsequently, Zhang [5] further investigated the refraction of spinelectron beam at the interface by considering SOCs of both Rashba

* Corresponding author. Tel.: +86 773 5886179, fax: +86 773 5886179.E-mail address: M_W_Lu@126.com (M.W. Lu).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2011.10.012

and Dresselhaus types, and found that an electron beam with someangles of incidence after passing through such an interface splitsinto two beams with different spin polarizations propagating innegative and positive directions, respectively. Therefore, such aninterface can serve as a spin beam splitter, which operates in spin-dependent refraction of electron beams. Frustaglia et al. [6] studiedspin injection by using pure quantum interference effect of spinelectron beams in 2DEG ballistic microstructures. Dragoman [7]presented an alternative spin beam splitter in 2DEG nanostruc-tures, in terms of magnetic depopulation of subbands in magneticfields.

Very recently, Chen et al. [8] investigated the Goos–Hänchen(GH) effect [9] of transmitted spin electron beams in a paral-lel double δ-function magnetic-barrier (MB) nanostructure [10].They found that, GH shift of transmitted electron beam is stronglydependent on the electron-spins, and can be controlled by themagnetic strength of ferromagnetic (FM) stripes and the appliedvoltage to the metallic FM stripe. Thus, based on such a MB nanos-tructure, they proposed a spin beam splitter, which completelyseparates spin-up and spin-down electron beams by their corre-sponding spatial GH shifts.

In this Letter, we explore in detail the GH effect of spin elec-tron beams in antiparallel double δ-MBs nanostructure under anapplied voltage. GH shifts for spin electron beams through this MBnanostructure are analytically calculated by using the stationaryphase method. By numerical calculations for realistic InAs materialsystem, we demonstrate that the device possesses a considerablespin polarization effect due to the significant discrepancy of GHshifts between spin-up and spin-down electron beams, when two

L. Yuan et al. / Physics Letters A 375 (2011) 4198–4202 4199

Fig. 1. (a) Schematic illustration of the MB nanostructure with two FM stripesare deposited on top and bottom of the semiconductor heterostructure, (b) themagnetic–electric barrier model exploited here, and (c) the (1) positive and (2) neg-ative GH shifts of ballistic electron beam in this nanostructure.

δ-MBs have unidentical magnetic strengths. Moreover, not only themagnitude of spin polarization but also its sign varies stronglywith the electric potential produced by the applied voltage. Thus,the considered nanostructure can be used as the spin beam splitterwith the spin polarity controllable by the applied voltage.

2. Model and formulas

The nanosystem under consideration is a magnetically modu-lated 2DEG formed usually in a modulation-doped semiconductorheterostructure, which can be experimentally realized [11] by de-positing two FM stripes on top and bottom of the semiconductorheterostructure [12], as schematically depicted in Fig. 1(a). The in-plane magnetization of FM layers produces an out-of-plane fringemagnetic fields at two ends, forming nonhomogeneous MBs withinthe 2DEG. The distances of two FM layers relative to the 2DEG areassumed to be different, which will result in MBs with unidenti-cal strengths created by two FM layers. Making use of the modernnanotechnology, such a system can be deliberately designed to fallshort of the left-hand edges of two FM layers, so that effects offringe fields there can be ignored. For the small distances be-tween 2DEG and FM stripes, the magnetic field provided by twoFM stripes is approximated [13] as

�B = Bz(x)z,

Bz(x) = B1δ(x + d/2) − B2δ(x − d/2), (1)

where B1 and B2 are the magnetic strengths of two δ-functionbarriers, and d is their separation. The electric potential inducedby the negative voltage applied to the metallic FM stripes, U (x) =UΘ(d/2 − |x|), is homogenous in the y direction and varies onlyalong the x axis [14], which is shown in Fig. 1(b). The Hamilto-nian describing such a device in the (x, y) plane, within the singleparticle effective mass approximation, is

H = p2x

2m∗ + [p y + e A y(x)]2

2m∗ + eg∗σ h

4m0Bz(x) + U (x), (2)

where m∗ is the electron effective mass and m0 is the free elec-tron mass, (px, p y) are the components of the electron momen-tum, g∗ is the effective Landé factor of electron, σ = +1/ − 1for spin-up/spin-down electrons, and A y(x) is the y-componentof the magnetic vector potential given, in Landau gauge, by �A(x) =[0, A y(x),0], i.e.,

A y(x) =⎧⎨⎩

0, x < −d/2,

B1, −d/2 < x < d/2,

(B1 − B2), x > d/2,

(3)

which results in Bz(x) = dA y(x)/dx. For convenience, we expressall the relevant quantities in dimensionless form: (1) the magneticfield Bz(x) → B0 Bz(x), (2) the vector potential A y(x) → B0lB A y(x),(3) the coordinate x → lB x, and (4) the energy E → Ehωc , whereωc = eB0/m∗ is the cyclotron frequency and lB = √

h/eB0 is themagnetic length with B0 as some typical magnetic field. In our nu-merical calculation, the InAs system (g∗ = 15, and m∗ = 0.024m0)is taken as the 2DEG material, thus for a typical B0 = 0.1 T, wehave lB = 81.3 nm, hωc = 0.48 meV.

A two-dimensional electron beam of incident energy E comesfrom the left with an incident angle θ0 in (x, y) plane, as isdepicted in Fig. 1(c). The plane wave component of the inci-dent beam is Ψi(�x) = C(ky)exp{i[kl

x(x + d/2) + ky y]} with kl =√2E , ky = kl sin θ , kl

x = kl cos θ , where θ stands for the incidentangle of the contributed plane wave, and C(ky) represents theangular-spectrum distribution. Because the system is translation-ally invariant along the y direction, the solution of the station-ary Schrödinger equation HΨ (x, y) = EΨ (x, y) can be written asΨ (x, y) = ψ(x)exp(iky y). The wave function ψ(x) satisfies the fol-lowing one-dimensional (1D) Schrödinger equation:

{d2

dx2+ 2

[E − Ueff (x,ky,σ )

]}ψ(x) = 0, (4)

where Ueff (x,ky, σ ) = [ky + A y(x)]2/2 + m∗ g∗σ B(x)/4m0 + U canbe called as the effective potential of the corresponding nanos-tructure. Clearly, it depends not only on the magnetic config-uration Bz(x) of the system, the electric potential U (x), andthe transverse wave-vector ky of the electron, but also on theelectron-spin σ . The reflected and transmitted wave functionscan be written as Ψr(�x) = r(ky)C(ky)exp{i[kl

x(x + d/2) + ky y]}and Ψt(�x) = t(ky)C(ky)exp{i[kr

x(x − d/2) + ky y]}, where krx =√

2E − (ky + B1 − B2)2, and r and t are the reflection and trans-

mission amplitudes, respectively. In the barrier region, −d/2 <

x < d/2, the wave function is Ψ (�x) = C1(ky)C(ky)exp{i[kix(x +

d/2) + ky y]} + C2(ky)C(ky)exp{i[−kix(x + d/2) + ky y]} with ki

x =√2(E − U ) − (ky + B1)2, where C1 and C2 are determined by

boundary conditions, that is, the continuity of the wave functionsand their derivations at x = −d/2 and d/2,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 + r = C1 + C2,

(1 − r)klx − i

m∗g∗σ B1

2m0(1 + r) = ki

x(C1 − C2),

C1 exp(iki

x d) + C2 exp

(−ikix d

) = t,(iki

x − m∗g∗σ2m0

B2

)C1 exp

(iki

x d)

−(

ikix + m∗g∗σ

2m0B2

)C2 exp

(−ikix d

) = ikrxt.

(5)

So, the amplitude transmission coefficient t(ky) = 2klx/Z is ob-

tained by the complex number Z = M + iN with

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

M = (kl

x + krx

)cos

(ki

x d) + m∗g∗σ

2m0kix

(−klx B2 + kr

x B1)

sin(ki

x d),

N = m∗g∗σ2m0

(B1 − B2) cos(ki

x d)

+[

kix + kl

xkrx

i+

(m∗g∗σ

2m

)2 B1 B2i

]sin

(ki

x d).

(6)

kx 0 kx

4200 L. Yuan et al. / Physics Letters A 375 (2011) 4198–4202

Fig. 2. Energy dependence of GH shifts of spin-up (solid curve) and spin-down(dashed curve) electron beams with incident angle θ0 = 00 for three fixed EBheights (a) U = 0, (b) U = 2, and (c) U = 4, where structural parameters are takenas B1 = 3, B2 = 6, and d = 1.

The total phase shift of the transmitted electron beams at x = d/2with respect to the incident one at x = −d/2 is ϕ = −arctg( B

A ).It should be noticed that the ki

x is either real or imaginary, butthe kl

x and krx must be real. And that, for an imaginary ki

x , cos(kixa)

and sin(kixa) become cosh(ki

xa) and sinh(kixa), respectively. Conse-

quently, the GH shifts of the transmitted electron beams [as indi-cated in Fig. 1(c)] is calculated, according to the stationary-phasemethod [8,15,16], by

Sσ = − dϕ

dky0, (7)

where the subscript 0 in the Letter denotes the values taken atky ≡ ky0, i.e., θ = θ0.

3. Results and discussion

In Fig. 2, we plot GH shifts Sσ calculated as a function ofthe incident energy E for spin-up (solid curve) and spin-down(dashed curve) electron beams through the device [as shown inFig. 1(b)] with different EB heights (a) U = 0, (b) U = 2, and U = 4,where the transverse wave vector is set to be ky = 0 (i.e., theincident angle θ0 = 00), and the structural parameters are takenas B1 = 3, B2 = 6, and d = 1. Clearly, there exists a remarkablediscrepancy in GH shift for electron beams with opposite spin ori-entations, regardless of the EB height U (x). That is to say, for sucha device the evident spin splitting appears in the calculated GHshifts, especially in low-energy region. Due to intrinsic symmetry

Fig. 3. GH shifts of spin-up (dashed curve) and spin-down (dotted curve) electronbeams versus the incident energy E for different magnetic field differences B ≡B2 − B1 = 0, 1, and 2, where other parameters are B1 = 3, U = 0, d = 1, and θ0 = 00.

[17] there exists no spin splitting in the magnetic–electric barriernanostructure if two δ-MBs possess the same magnetic strengths.However, this symmetry is broken if two δ-MBs have not the iden-tical strengths, B1 �= B2, which will lead to the spin splitting ofGH shifts for electron beams across this device. Moreover, The GHshift and its spin splitting show a strong dependence on the EBU (x), which originates from the variation of the effective poten-tial of spin electron beams Ueff [see Eq. (4)] with the EB heightU . For different EB height U , the GH shifts of spin electron beamsSσ can be positive as well as negative; c.f. Fig. 2(a)–(c). The de-gree of difference in GH shifts between spin-up and spin-downelectron beams is strongly changed by the U (x). In particular, itis striking that, at U = 0 and in low energy [see Fig. 2(a)], si-multaneously large and opposite GH shifts between spin-up andspin-down electron beams can be achieved. This allows the nanos-tructure to realize the spin beam splitter [5,8], i.e., to completelyseparate spin-up and spin-down electron beams by their differentspatial GH shifts.

Clearly, the magnetic field difference between two δ-MBs willinfluence the GH shifts of spin electron beam through our consid-ered system as shown in Fig. 1(a). Indeed, our calculated resultsalso prove this point. In Fig. 3, we plot GH shifts Sσ of spin-up(dashed curve) and spin-down electron beams with the incidentangle θ0 = 00 calculated as a function of the incident energy Efor different magnetic field discrepancy B ≡ B2 − B1 = 0, 1, and2, where B1 = 3, U = 0, and d = 1. When B = 0, i.e., B1 = B2,one can evidently see that the GH shift of spin-up electron beamis the same as that of spin-down electron beam, which is dueto an intrinsic symmetry [17] in our device. However, this intrin-sic symmetry can be broken if two δ-MBs have not the identicalmagnetic field strength, i.e., B1 �= B2, will lead to spin splittingof GH shifts for electron beam through this device. From othertwo group curves of B = 1 and 2 in Fig. 3, we can apparentlyobserve that the evident spin splitting appears in calculated GHshifts of electron beams. Moreover, the spin splitting of GH shiftsshows a strong dependence on the magnetic field difference intwo δ-MBs. The degree of the spin splitting in GH shifts of elec-tron beams is significantly enhanced as the discrepancy B in-creases. In addition, one also observes from Fig. 3 that GH shiftcurves move rightward with increasing B . Because wave vector

krx =

√2E − (ky + B1 − B2)2 must be real, the incident energy E

increases if B = B1 − B2 becomes large, and GH shift curves movethus toward the high-energy direction.

In order to characterize the spin splitting of GH shifts, we in-troduce the spin polarization of GH shift, Sσ ≡ S↑ − S↓ , whereS↑ and S↓ stand for GH shifts for spin-up and spin-down electronbeams, respectively. Fig. 4 gives the spin polarization Sσ of GHshifts versus the incident energy E for three wave vectors of elec-

L. Yuan et al. / Physics Letters A 375 (2011) 4198–4202 4201

Fig. 4. Spin polarization of GH shifts for electron beams with (a) ky = 0, (b) ky =−0.5 and (c) ky = 0.5 through the magnetic–electric barrier nanostructure as shownin Fig. 1(b), where the structural parameters are the same as in Fig. 2.

Fig. 5. GH shift Sσ calculated as a function of the EB height U for spin-up (solidline) and spin-down (dashed line) electron beams with the incident angle θ0 = 00

and the incident energy E = 5, where the device and the structural parameters arethe same as in Fig. 3, and the corresponding spin polarization Sσ is shown in theinset.

tron beam: (a) ky = 0, (b) ky = −0.5, and (c) ky = 0.5, where thestructural parameters are the same as in Fig. 2, and the EB heightis taken as U = 0 (solid curve), 2 (dashed curve), and 4 (dottedcurve). A considerable spin polarization effect of GH shifts for elec-tron beams through the device shown in Fig. 1(a) can be evidentlyseen, especially in low-energy region. The spin polarization Sσ

changes its value and sign when the incident energy varies. Fur-

thermore, the spin polarization Sσ presents a great dependenceon the EB U (x), that is, its degree can be significantly altered fordifferent EB height U .

Finally, to see more clearly the influence of the EB U (x), we cal-culate the GH shift Sσ and its spin polarization Sσ as the func-tion of the EB height U . In Fig. 5, we give the variation of the GHshifts Sσ with the EB height U for spin-up (solid line) and spin-down (dashed line) electron beams through the magnetic–electricbarrier nanostructure as shown in Fig. 1(b), where the energy andincident angle are chosen as E = 5 and θ0 = 00, respectively, andthe structural parameters are B1 = 3, B2 = 6, and d = 1. From thisfigure, we can observe apparently the modulation of GH shifts bythe EB U (x). When the EB height U changes from negative to pos-itive, both magnitudes of GH shifts for spin-up and spin-downelectron beams vary dramatically, and GH shifts can switch frompositive to negative. Another observation from Fig. 5 is that thediscrepancy of GH shifts between spin-up and spin-down electronbeams can be altered greatly by the EB height, i.e., the spin split-ting of GH shifts show a strong dependence on the EB U (x). Thisdependence of spin splitting in GH shift on the EB height, that is,the spin polarization of GH shifts of spin electron beams, Sσ , ispresented correspondingly in the inset of Fig. 5. Once again, onecan see clearly that the spin polarization Sσ switches greatlywith the EB height U , including its magnitude and sign. All thesefeatures result from the dependence of the effective potential Ueffon the EB U (x), which results in the modulation of GH shifts andits spin polarization by the EB height U . Due to the EB U (x) pro-duced by an applied voltage under the metallic FM stripes in thedevice [cf. Fig. 1(a)], one can expect that spin polarization in GHshifts of spin electron beams is tuned by adjusting this appliedvoltage, which may give rise to a voltage-controllable spin beamsplitter [18].

4. Conclusions

In summary, we have investigated theoretically spin-dependentGoos–Hänchen effect in an antiparallel double δ-barrier magneticnanostructure under an applied voltage, which can be experimen-tally realized by depositing two metallic FM stripes on the top andbottom of heterostructure. By using the stationary phase method,we obtain exactly the GH shifts of spin electron beams, which areused to study the spin GH effect in the device. It is shown thatthis magnetic–electric barrier nanostructure possesses a consider-able spin polarization effect in GH shifts of electron beams dueto the different magnetic strengths in two δ-MBs. It is also shownthat the spin polarization of GH shifts depends strongly on the EBinduced by the applied voltage. These interesting properties mayprovide an effective approach of spin injection in semiconductor,and this device can be used as a voltage-tunable spin beam split-ter.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China (11164006).

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