Size- and voltage-dependent spin polarization in hybrid ferromagnetic-Schottky-metal and semiconductor nanostructure

Solid State Communications 150 (2010) 1409–1412

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Size- and voltage-dependent spin polarization in hybrid ferromagnetic-Schottky-metal and semiconductor nanostructure M.W. Lu ∗ , S.Y. Chen, J.R. Xiao College of Science, Guilin University of Technology, Guilin 541004, Guangxi, People’s Republic of China

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Article history: Received 4 January 2010 Received in revised form 31 March 2010 Accepted 23 April 2010 by F. Peeters Available online 5 May 2010 Keywords: A. Hybrid magnetic–electric barrier structure D. Quantum size effect D. Spin polarization

abstract Recently, an electron-spin filter was proposed by depositing two nanosized ferromagnetic metal stripe and Schottky normal metal stripe on the top of the semiconductor heterostructure [F. Zhai, H.Q. Xu, Y.Guo, Phys. Rev. B 70 (2004) 085308]. In this paper, we theoretically investigate the effect of device parameters on electron-spin polarization in the spin filter. It is shown that the electron-spin polarization is dependent greatly on the sizes and the position of the stripes. Thus, a quantum size effect exists in this device and the optimal spin polarization can be achieved by felicitously fabricating the stripes. It also is shown that the spin polarization can be altered by adjusting the electric-barrier height induced by an applied voltage to the Schottky metal stripe, which can result in a voltage-tunable electron-spin filter. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Spintronics is a multidisciplinary field whose central theme is the active manipulation of spin degree of freedom in solid-state systems [1]. Its goal is to understand the interaction between the particle spin and its solid-state environments and to fabricate efficient spintronic devices using the acquired knowledge. The realization of spintronic devices depends on the ability to inject spinpolarized current into a semiconductor and this has attracted great current interest [2–4]. The conventional method to inject spinpolarized current into a semiconductor is from either a ferromagnetic metal or magnetic semiconductor [5,6]. However, in almost all of the cases, the measured spin-polarization ratio is very small, and such low efficiency arises from the conductance mismatch between ferromagnetic and semiconductor [7]. Experiments have demonstrated that we can use a ferromagnetic semiconductor or dilute-magnetic semiconductor as a spin aligner, [8] but the Curie temperature is low when magnetic semiconductors are used as the spin injector [5,6]. The use of spin filters is, therefore, an alternative approach, which can significantly enhance spin injection efficiencies [9]. Recently, it is demonstrated that [10–16] spin polarization of current also can be achieved by passing it across a two-dimensional

electron gas (2DEG) plane. Such a type of magnetically modulated nanostructure is a combination of semiconductor and magnetic material, where the semiconductor is usually a heterostructure which comprises a 2DEG, while the latter provides a magnetic field which can influence locally the motion of the electrons in the semiconductor heterostructure. More recently, an electron-spin filter, based on a ferromagnetic metal (FM) stripe and a Schottky metal (SM) stripe on the 2DEG, was proposed by Zhai et al. [15]. Experimentally, this device can be realized by placing a SM stripe parallel to the FM stripe on top of the InAs heterostructure [17]. It is found that this device possesses a considerable electron-spin polarization effect. A tunable spin-polarized source is desirable for spintronic applications [18]. In the present work, we study the influence of the structural parameters on spin filtering in this device. Numerical calculations show that, not only the amplitude of the spin polarization, but also its sign varies with the sizes and the position of the FM stripe and SM stripe. Thus, a quantum size effect of electron-spin filter exists in the device, and the optimal electronspin filtering can be achieved by felicitously fabricating the FM stripe and SM stripe. It is also shown that in such a device, the degree of spin polarization depends drastically upon the electricbarrier (EB) height induced by an applied voltage to the SM metal stripe, which can result in a voltage-tunable spin filter whose spin filtering can be altered by adjusting this applied voltage. 2. Model and method

∗

Corresponding author. Tel.: +86 7735896179; fax: +86 7735896179. E-mail address: [email protected] (M.W. Lu).

0038-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2010.04.025

The electron-spin filter we consider here is a 2DEG in the (x, y) plane subject to modulations by a FM stripe and a SM stripe as

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sketched in Fig. 1(a). Here, dM and dE are the width of the FM stripe and SM stripe, respectively, D is the spacing between the FM stripe and SM stripe, dz is the thickness of the FM stripe, and the parameter z0 stands for the upright distance that the 2DEG is located below the stripes. For simplicity, the magnetic field Bz (x) provided by the FM stripe and the electrical potential U (x) induced by an applied voltage to the SM stripe are approximated [19] as a delta function and the rectangular electrical barrier, respectively, as shown in Fig. 1(b). We assume that the magnetic field Bz (x) and the electrical potential U (x) are homogeneous in the y direction and vary only along the x axis. The motion of an electron in such a modulated 2DEG system can be described by the single-particle Hamiltonian, *

[PE + e A ]2

H =

2m∗

1

+ U (x) + g ∗ µB σz Bz (x),

d2

 −

dx2

−

e h¯

Ay (x) + ky

em∗ g ∗ σz Bz (x) 2m0 h¯

2 +

2m∗ h¯ 2

[E − U (x)]

d2

) ψ(x) = 0.

dx2

+ 2 E − U x, ky , σz 






(2)

ψ(x) = 0, 2

(3) m∗ g ∗ σ B (x)

z z where U (x, ky , σz ) = 12 Ay (x) + ky + U (x) + is the 4m0 effective potential of the corresponding structure, which depends not only on the magnetic configuration Bz (x) via the vector potential Ay (x), the transverse wave vector ky , and the electrical potential U (x), but also on the electron spin σz . The reduced one-dimensional Schrödinger equation (3) can be exactly solved by using the transfer-matrix method [20]. In the left and right regions of the structure, the wave functions are ψ` (x) = exp(ik` x) + γ exp(−ik` x), qx < x− and ψr (x) =



τ exp(ikr x), x > x+ , where k` = kr =

Fig. 1. (a) Schematic illustration of the device. (b) Magnetic field and electric potential profiles exploited in this work.

averaging the electron flow over half the Fermi surface from the well-known Landauer–Buttiker formula, and is given as follow [21] Gσz (EF ) = G0 (EF )

2E − k2y and γ /τ is the

reflection/transmission amplitude. Matching wave function at interface, we can obtain the transmission probability for electrons with incident energy E, wave vector ky and spin orientation σz by T (E , ky , σz ) = |τ |2 . Once the transmission probability T (E , ky , σz ) is calculated, the ballistic conductance at zero temperature can be calculated by

Z

π /2 −π /2



T EF ,

p



2EF sin θ , σz cos θ dθ ,

(4)

where θ is the incident angle relative to the x direction, G0 (EF ) = e2 m∗ υF Ly /h2 with Ly being the length of the system in the y direction, and υF being the Fermi velocity. The spin-dependent conductance components Gσz are presented in units of G0 (EF ). The spin polarization can be defined by the relative difference between the spin-up and spin-down conductances at the Fermi energy, PG =

For convenience, we express quantities in dimensionless units by introducing two characteristic parameters, the cyclotron √ frequency ωc = eB0 /m∗ and the magnetic length `B0 = h¯ /eB0 with a typical magnetic field B0 = 0.1 T. For example, the magnetic field Bz (x) → B0 Bz (x), the magnetic vector potential A(x) → B0 `B0 A(x), the coordinate x → `B0 x, the wave vector k → k/`B0 and the energy E → h¯ ωc E = E0 E. So the Schrödinger equation becomes



b

(1)

2

E are the effective mass and momentum, e is the where m∗ and p absolute value of the electron’s charge, µB = eh¯ /2m0 is the Bohr magneton (m0 is the free electron mass), g ∗ is the effective Landè factor of the electron in the 2DEG, σz represents the z-component of the electron spin (σz = +1/ − 1 for spin up/down), and the E = [0, Ay (x), 0] in magnetic vector potential can be written as A the Landau gauge. Since the system is translation invariant along the y direction, the solution of the stationary Schrödinger equation H Φ (x) = E Φ (x) can be written as a product Φ (x, y) = eiky y ψ(x), where ky is the wave vector component in the y direction. Thus, the wave function ψ(x) satisfies the following one-dimensional Schrödinger equation: (

a

G↑ − G↓ G↑ + G↓

(5)

where G↑ and G↓ are the conductance for spin-up and spin-down electrons, respectively. 3. Results and discussion In this section we present and discuss numerical results of spin polarization of the conductance for electrons tunneling through the nanostructures described by Fig. 1. In our numerical calculation, the InAs system is taken as the 2DEG material (g ∗ = 15 and m∗InAs = 0.024m0 ) and the reduced units are `B0 = 81.3 nm and E0 = 0.48 meV. In the following, we explore in detail the effects of structural parameters in this device on electron-spin filtering properties. First of all, in Fig. 2 we plot the spin polarization of the conductance PG defined by Eq. (5) as a function of the width of the FM stripe dM at three fixed Fermi energies EF = 7.0 (solid curve), 9.0 (dashed curve), and 11.0 (dotted curve), where other structural parameters are chosen as B = 6.0, U = 8.0, dE = 1.0 and D = 1.0. We can see clearly from this figure that the spin polarization depends strongly on the width of the FM stripe dM . With dM increasing, the spin polarization shows obvious oscillations. At the same time, the spin polarization reduces when the Fermi energy EF becomes large and the value of the spin polarization becomes a negative one when the Fermi energy reaches a certain value. When the thickness of the FM stripe dz increases or the upright distance z0 between the 2DEG and the FM stripe reduces, the magnetic field B becomes large [22]. In Fig. 3, we give the spin polarization of the conductance PG versus the magnetic field B, where the parameter U is taken to be U = 8.0, the width of the FM stripe and SM stripe are dM = 1.0 and dE = 1.0 and the spacing between the FM stripe and SM stripe is D = 1.0. The solid, dashed and dotted curves correspond to three fixed Fermi energies EF =

M.W. Lu et al. / Solid State Communications 150 (2010) 1409–1412

Fig. 2. Spin polarization of conductance PG as a function of the width dM of FM stripe for three different Fermi energies: EF = 7.0 (solid curve), 9.0 (dashed curve), and 11.0 (dotted curve). The other structural parameters are chosen as B = 6.0, U = 8.0, dE = 1.0 and D = 1.0.

Fig. 3. The spin polarization of conductance PG versus the magnetic field B. The solid, dashed and dotted curves correspond to EF = 7.0, 9.0 and 11.0. Other device parameters are assumed as U = 8.0, dE = dM = D = 1.0.

7.0, 9.0 and 11.0, respectively. It is evident that the degree and sign of the spin polarization PG greatly changes when the magnetic field B increasing. For low magnetic field the spin polarization is small, the electron-spin polarization changes its sign quickly with the magnitude of magnetic field becomes large, and further increasing the magnetic field the spin polarization begins to become smooth. Moreover, it is apparent from this figure that the spin polarization is relative to the Fermi energy. The curve shifts rightwards and its peaks become sharp with the increment of the Fermi energy. Fig. 4 shows the calculated spin polarization of the conductance PG as a function of the width of the SM stripe dE for three fixed Fermi energies EF = 7.0 (solid curve), 9.0 (dashed curve), and 11.0 (dotted curve). Here, the magnetic field, the electric-barrier height, the width of the FM stripe and the spacing between the FM stripe and SM stripe are set to be B = 6.0, U = 8.0, dM = 1.0 and D = 1.0. One can see clearly that the spin filtering effect depends upon the width of the SM stripe. The degree of spin conductance polarization PG changes obviously when the width of the SM stripe dE increases. Moreover, for different Fermi energies the degree of spin polarization is also altered. The spin polarization of conductance PG as a function of the spacing D between the FM stripe and SM stripe for three fixed Fermi energies: EF = 7.0 (solid line), 9.0 (dashed line), and 11.0 (dotted line) are calculated as Fig. 5, where dE = dM = 1.0 and the magnetic-field strength and electrical-barrier height are the same as Fig. 4. Apparently, the spin polarization of conductance PG shows dramatic oscillations including its amplitude and sign with the increment of the spacing between the FM stripe and SM stripe. Besides, the spin polarization of conductance PG changes with increasing Fermi energy.

1411

Fig. 4. The spin polarization of conductance PG versus the width dE of SM stripe. The structural parameters are taken as B = 6.0, U = 8.0 and dM = D = 1.0, where the Fermi energies are chosen as EF = 7.0 (solid curve), 9.0 (dashed curve), and 11.0 (dotted curve).

Fig. 5. Spin polarization of conductance PG versus the spacing D of the FM stripe and SM stripe for three fixed Fermi energies EF = 7.0 (solid line), 9.0 (dashed line), and 11.0 (dotted line). Other device parameters are chosen as B = 6.0, U = 8.0, dE = dM = 1.0.

All these features of the effects of the structural parameters on the electron-spin polarization attribute to the change of the effective potential U (x, ky , σz ) when the structural parameters alter. Nevertheless, the dependence of spin polarization PG on the Fermi energy is due to energy-dependent spin electron transmission. Thus, a quantum size effect exists in this electronspin filter. Also, these interesting properties hint that a much larger spin polarization or much better spin-filtering properties in the device can be obtained by properly fabricating FM and SM stripes. From the point of view of applications, a controlled spin polarization is desirable for the spintronic devices. Now, we explore the tunability of the spin polarization by changing the electric-barrier height. We have calculated the spin polarization of the conductance PG as a function of the electric-barrier height U for three fixed Fermi energies: EF = 7.0 (solid line), 9.0 (dashed line) and 11.0 (dotted line), as presented in Fig. 6. The structural parameters are chosen as B = 6.0 and dE = dM = D = 1.0. We can see clearly from this figure that the spin polarization curve coincides very well with the Fig. 4 in Ref. [15] when the magnetic-field density and Fermi energy are set to be B = 6.0 and EF = 9.0, respectively. From these three polarization curves one can clearly see that the spin polarization exhibits an obvious variation with increasing EB height in positive EB range, while for all negative values of electric-barrier height U, the spin polarization is small. Another observation from this figure is that only for proper Fermi energies (EF = 9.0) the spin polarization may exhibit a drastic variation with the increment of the electric-barrier height U. Moreover, the curves of the spin

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M.W. Lu et al. / Solid State Communications 150 (2010) 1409–1412

tum size effect exists in the electron-spin filtering of the device, and one can achieve the optimal spin filtering effect by felicitously fabricating the FM and SM stripes in the device. It is also shown that the spin polarization depends drastically upon the electric-barrier height induced by an applied voltage to the SM stripe, and such a device can serve as a voltage-tunable electron-spin filter. References [1] [2] [3] [4] [5] Fig. 6. Spin polarization of conductance PG calculated as a function of the EB height at a given magnetic-field strength B = 6.0 for three fixed Fermi energies EF = 7.0, 9.0, 11.0, where other structural parameters are assumed as dE = dM = D = 1.0.

[6] [7] [8]

polarization shift rightwards with increasing Fermi energy EF . This character stems from the strong dependence of the effective potential U (x, ky , σz ) for our considered device on electric-barrier height U. As the electric potential barrier is induced by an applied voltage to the SM stripe, such a device can serve as a voltage-tunable electron-spin filter whose spin polarization can be changed by adjusting this applied voltage. 4. Conclusions In summary, we theoretically investigate the structural parameters on the spin filtering properties in an electron-spin filter device. This device is based on a hybrid ferromagnetic-Schottkystripe and semiconductor nanosystem and can be realized by placing a SM stripe parallel to the FM stripe on top of the semiconductor heterostructure. We have revealed that, in such an electron-spin filter device, the degree of spin polarization depends strongly upon the sizes and the position of the FM and SM stripes. Thus, a quan-

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