Tunable spin spatial splitter based on a δ -doped realistic magnetic-barrier nanostructure

Accepted Manuscript Tunable spin spatial splitter based on a -doped realistic magnetic-barrier nanostructure Mao-Wang Lu, Xue-Li Cao, Xin-Hong Huang, Ya-Qing Jiang, Shuai Li, ShiPeng Yang PII: DOI: Reference:

S0749-6036(14)00445-5 http://dx.doi.org/10.1016/j.spmi.2014.11.019 YSPMI 3494

To appear in:

Superlattices and Microstructures

Received Date: Revised Date: Accepted Date:

15 July 2014 9 November 2014 11 November 2014

Please cite this article as: M-W. Lu, X-L. Cao, X-H. Huang, Y-Q. Jiang, S. Li, S-P. Yang, Tunable spin spatial splitter based on a -doped realistic magnetic-barrier nanostructure, Superlattices and Microstructures (2014), doi: http://dx.doi.org/10.1016/j.spmi.2014.11.019

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Tunable spin spatial splitter based on a δ-doped realistic magnetic-barrier nanostructure Mao-Wang Lu,∗ Xue-Li Cao, Xin-Hong Huang, Ya-Qing Jiang, Shuai Li, Shi-Peng Yang College of Science, Guilin University of Technology, Guilin 541004, People’s Republic of China November 27, 2014 Abstract We theoretically investigate a tunable δ-potential effect on a spin spatial splitter, which can be realized experimentally by depositing a ferromagnetic stripe with a vertical magnetization on top of semiconductor heterostructure and using atomic-layer doping technique. With the help of stationary phase method, the δ-doping dependent spin-polarized lateral displacement is numerically calculated by means of the improved transfermatrix technique. Both amplitude and sign of the spin polarized lateral displacement are found to vary dramatically with the weight and/or location of the δ-doping. Thus, a structurally tunable spin spatial splitter can be achieved for spintronic applications. Keywords: Magnetic nanostructure; δ-doping; Lateral displacement; Spin polarization; Spin spatial splitter * Corresponding author. Tel.: +86 773 5896179; fax: +86 773 5896179. E-mail address: [email protected] (M.W. Lu).

1

1

Introduction

In semiconductor spintronics1 the digital information is carried by the spins of an electron rather than its charge. Consequentially, a newfashioned spin-based device operates by virtue of the spin-polarized electrons in semiconductors. Thus, how to spin polarize electrons into a conventional semiconductor is one of basic scientific questions, which needs to be imminently resolved for spintronic applications.2 Many alternative methods have been proposed both theoretically and experimentally, one of which is to spin polarize electrons by separating the spins form the spatial domain. In principle, this method mainly is to utilize the optics-like behaviors of electrons in a two-dimensional electron gas (2DEG) nanostructure for realizing the electron-spin polarization.3 For example, after an electron with a nonzero incidence angle is refracted at an interface between two regions in the presence of different spin-orbital coupling (SOC), the transmitted electron will be spin split by different refraction angles.4,5 Magnetically modulated semiconductor nanostructure (MMSN)6 is a significant kind of 2DEG nanostructures, which can be experimentally fabricated by confining the motion of a 2DEG in a modulation-doped semiconductor heterostructure with an inhomogeneous magnetic field on nanometer scale, e.g., depositing a nanosized ferromagnetic (FM) stripe on the top of a conventional semiconductor heterostructure can form the magnetic-barrier (MB) MMSN.7 Due to the small size, the low dimensionality, and the magnetic restriction on nanometer scale, a MMSN possesses plentiful properties, such as the wave vector filtering,8 the spin filtering,9 and the giant magnetoresistance (GMR) effect.10 Recently, stimulated by the rapidly growing spintronics, there are increasing works devoted to the spin polarization realized by utilizing the spin-dependent lateral displacements (LD) of electrons across MMSNs. Chen et al.11 and Yuan et al.12 studied the LD of electrons in a double δ-MB nanostructure, and found that, for two δ-MBs pointing at the same direction, the electron spins can be spatially separated completely by significantly different LDs between spin-up and spin-down electrons. Subsequently, the spin-dependent LD of electrons was studied in the other MMSNs, and some spin spatial splitters were proposed.13−15 In particular, the LD of spin electrons in realistic MB MMSNs was studied recently, and it has been found that, only for MMSNs with the symmetric magnetic profiles, the LD’s of electrons are dependent their spins or show up a considerable spin polarization.16,17 Investigations on spin-dependent LDs of electrons in MMSNs still are going along, how to effectively manipulate the spin-polarized LDs of electrons in MMSNs is, however, very lacking. In fact, a controllable spin-polarized source is desired for spintronic applications.18 With the advancement of the modern material growth techniques such as molecular beam epitaxy (MBE) and metal-organic chemical-vapor deposition (MOCVD), the dimensional control close to interatomic spacing has become possible in experiments, literally referred to as the atomic layer doping technique or the δ-doping technique.19 By means of such a technique, a tunable δ-potential with the controllable weight and position can be intentionally introduced into a semiconductor nanostructure, e.g., resonant tunneling devices with 2

a controllable δ-doping in barriers or wells,20 have been realized experimentally. Very recently, the influence of the δ-doping on the spin filtering in realistic MB MMSNs was studied, and an alternative method to structurally manipulate a spin filter was theoretically proposed by the δ-doping technique.21,22 Motivated by these works, in this paper we focus on the modulation of the δ-doping to the LDs of spin-polarized electrons in a realistic MB MMSN. Theoretical analysis together with numerical calculation shows that both magnitude and sign of the spin-polarized LDs vary strongly with the weight and/or the position of the δ-doping. Thus, an alternative scheme to structurally manipulate a spin spatial splitter via the δ-doping can be proposed, and the considered MMSN could be employed as a δ-doping controllable spin-polarized source for spintronics applications.

2

Model and Theoretical method

The MMSN under consideration is a 2DEG formed usually in a modulationdoped semiconductor heterostructure, subject to the modulation by a FM stripe with vertical magnetization,23 as schematically depicted in Fig. 1(a). Here, the  0 , d and h are the magnetization, width and thickness of the FM stripe, M respectively, z0 stands for the distance between the FM stripe and the 2DEG plane, and the left and right ends of the MMSN are assigned at x = x− and x+ , respectively. Experimentally, a δ-potential with the tunable weight V and convertible position x0 can be introduced into the MMSN device by the δdoping technique [as presented in Fig. 1(b)], where the solid curve represents the magnetic profile produced by the magnetized FM stripe as7  = Bz (x)∧ B z,  Bz (x) = B0

2d (x + d/2) 2

(x + d/2) + z02

−

2d (x − d/2) 2

(x − d/2) + z02

(1)

 .

Here B0 = M0 h/d, h << d and z0 << h. The Hamiltonian describing such a 2DEG system in the (x, y) plane, within the single particle and effective mass approximation, can be written as H=

p2x h [py + eAy (x)]2 eg ∗ σ¯ + + Bz (x) + V δ(x − x0 ), 2m∗ 2m∗ 4m0

(2)

where m∗ , m0 , g ∗ , and p = (px , py ) are the effective mass, the free mass in vacuum, the effective Land´e factor and the momentum of an electron, respectively, σ = +1/ − 1 corresponds to spin-up/spin-down electrons, and Ay (x) is the y-component of the magnetic vector potential given, in Landau gauge, by

3

 = [0, Ay (x), 0], A   (x + d/2)2 + z02 Ay (x) = B0 d ln , (x − d/2)2 + z02

(3)

which complies with Bz (x) = dAy (x)/dx. All the relevant quantities can be expressed as dimensionless form by introducing the cyclotron frequency ω c =  eB0 /m∗ and the magnetic length B = ¯ h/eB0 with a typical magnetic field B0 , e.g., Bz (x) → B0 Bz (x), Ay (x) → B0 B Ay (x), x → B x, and E → E¯ hω c . The electronic transmission in MMSN’s is invariant along the y-axis, the solution of the stationary Schr¨ odinger equation, HΨ(x, y) = EΨ(x, y), can be, therefore, written as a product, Ψ(x, y) = ψ(x) exp(iky y), where ky is the longitudinal wave vector. Thereby, the wave function ψ(x) satisfies the following one-dimensional (1D) Schr¨ odinger equation: {

d2 + 2[E − Uef f (x, ky , V, x0 , σ)]}ψ(x) = 0, dx2

(4)

where Uef f (x, ky , V, x0 , σ) = [ky +Ay (x)]2 /2+m∗ g ∗ σBz (x)/4m0 +V δ(x−x0 ) is referred to as the effective potential of the MMSN device. Clearly, this effective potential Uef f depends not only on the magnetic configuration Bz (x), the longitudinal wave-vector ky and the spin σ, but also on the δ-doping, V δ(x − x0 ). It is the dependence of the Uef f of the MMSN device on the δ-doping that gives rise to the possibility to manipulate the spin spatial splitter via the δ-doping. On the other hand, It is impossible to analytically solve the Eq. (4) due to the complicated Uef f . In the following we will employ our previously improved transfer-matrix method (ITMM)7 to numerically settle it. No loss of generality, one can assume that the incident wave function of a 2D electron from the left of the MMSM device in an angle of incidence α, as is indicated inFig. 1(c), is Ψin (x, y) = exp{i[kxl (x − x− ) + ky y]}, x > x− , √ where kxl = 2E − ky2 cos α and ky = 2E sin α with the incident energy E of electron. The wave functions of the transmitted and reflected electrons can be expressed as Ψout (x, y) = τ exp{i[kxr (x − x+ ) + ky y]}, x > x+ , and Ψref (x, y) = γ exp{i[kxl (x− − x] + ky y}, x < x− , where τ /γ is transmission/reflection amplitudes, and kxr = kxl , respectively. With the help of the ITMM7 we can obtain by matching the wave function at the left and right boundaries x± ,     1 1 1 m11 m12 τ = , (5) l l m21 m22 γ 0 ikx − ikx where mij (i, j = 1, 2) are the total transfer matrix elements. Thus, the transmission amplitude can be expressed as τ=

2kxl , A + iB 4

(6)

with A = kxl m11 + kxr m22 and B = kxl kxr m12 − m21 . The phase shift

 the of transmitted electron with respect to the incident one is ϕ = tan−1 B A . Consequently, the LD of the transmitted electron [as depicted in Fig. 1(c)] can be calculated, according to the stationary phase method,10,24,25 by Sσ =

dϕ . dky

(7)

The electron-spin polarization can be characterized by considering the relative difference of the LD,12 PS = S↑ − S↓ ,

(8)

where S↑ and S↓ are LD’s for spin-up and spin-down electrons, respectively. Here, we would like to point out that, our theoretical method presented above merely considers the single-electron effect in calculating the lateral shift for the electron across the realistic magnetic-barrier nanostructure under a δdoping. However, the influence of the electron beam (such as its width) on the lateral shift is interesting in practical applications for a spin spatial splitter. For a finite-sized electron beam, the incident wave function will be26 1 A(ky − ky0 ) exp{i[kxl (x − x− ) + ky y]}dky , Ψin (x, y) = √ (9) 2π while the transmitted wave functions are then expressed by 1 Ψout (x, y) = √ 2π



τ A(ky − ky0 ) exp{i[kxr (x − x+ ) + ky y]}dky .

(10)

where the A(ky − ky0 ) is angular spectrum distribution around the central wave vector ky0 , Nevertheless, it is impossible to calculate the lateral shift of the electron beam for the realistic magnetic-barrier nanostructure as shown in Fig. 1(a). Fortunately, Chen’s works11 have demonstrated the validation of the stationary phase method in calculating the lateral shift for the electron through a MMSN.

3

Results and Discussion

In the following calculation, the InAs system is taken as the 2DEG material (m∗ = 0.024m0 and g ∗ = 15, which gives rise to B = 81.3nm and ¯hω c = 0.48meV for B0 = 0.1T ), and the structural parameters are set to be d = 1, z0 = 0.1 and x± = ±1.5 for convenience. In addition, in order to demonstrate the principle of operation of the device, we merely consider that an electron projects onto the MMSN device in a specific incidence angle α = 350 . For the MMSN device shown in Fig. 1(a), a recent theoretical investigation16 demonstrated that there exists a sizeable spin polarization effect owing to a significant discrepancy of the LDs between spin-up and spin-down electrons. A 5

δ-potential is included into this device by the δ-doping technique, whether does the obvious spin polarization effect still exist? Figure 2 shows the LD (Sσ ) as the function of the incident energy (E) for electrons across the MMSN device [see Fig. 1(a)], where the dashed and dotted curves correspond to spin-up and spin-down electrons, respectively, while the δ-doping is assumed to locates at the centre (x0 = 0.0) of the MMSN device and its weight is set to be V = 1.0. Contrasting the dashed and dotted curves, one can clearly observe from this figure that the LD of the spin-up electron differs obviously from that of the spin-down electron, especially at the resonance. Namely, for the MMSN device with a δ-doping, the spin splitting or the spin polarization in the LD spectrum of electrons still remains, just as is confirmed in the inset of Fig. 2 that gives the dependence of the spin polarization (PS ) in the LD of electrons on the incident energy. From the polarization curve, one also can observe that the polarization reaches its maximal value and switches its sign at the resonance. This spin polarization attributes to the Zeeman interaction between the electron-spins and the magnetic field of the MMSN device, and the existence of such an interaction in the MMSN device is independent of the δdoping. As a result of the independence, the considered MMSN device possesses a significant spin polarization in the LD of the electron, regardless of whether a δ-doping exists or not. After a considerable spin polarization effect in LD of the electron is seen when a δ-doping is included in the MMSN device, one wonders how the δ-doping impacts the degree of the spin polarization? Next, we explore in detail the effects of the weight (V ) and the position (x0 ) of the δ-doping on the electronspin polarization of the LD in the MMSN device presented in Fig. 1(a). To begin with, the position of the δ-doping is fixed, such as x0 = 0, i.e., locating at the center of the MMSN device, to reveal the influence of its weight (V ) on the degree of spin polarization (PS ) in the LD of electrons. Figure 3 gives the numerical results, where Fig. 3(a) shows the spin polarization PS as the function of the incident energy E for three fixed weights: V = 0.0 (solid curve), 1.0 (dashed curve) and 2.0 (dotted curve). Without the δ-doping (i.e., V = 0.0), the solid curve tells us that an appreciable spin polarization in the LD of electrons exists in the considered MMSN device, as demonstrated in the previous investigation.16 When a δ-doping is comprised in the MMSN device (V = 0.0; see the dashed and dotted curves), the degree of the spin polarization becomes weak. At the same time, the spin polarization curve PS shifts towards high-energy region if the weight of the δ-doping becomes large. In other words, the weight of the δ-doping can greatly alter the degree of the spin polarization in LD of the electron in the MMSN device. In order to more clearly see the modulation of the weight of the δ-doping to the spin polarization, figure 3(b) shows that the degree of spin polarization PS varies directly with the weight V of the δ-doping, where the incident energy of electrons is set to be E = 6.0. From this figure, one can clearly observe that both magnitude and sign of the electron-spin polarization PS of the MMSN device switches dramatically with the weight of the δ-doping, especially for the small weight region. The variation of the PS with the V implies that the degree of spin polarization of the MMSN 6

device can be handily adjusted by the weight of the δ-doping. This modulation to the spin polarization stems from the dependence of the effective potential Uef f of the MMSN device on the weight of the δ-doping. Besides the weight V , the effective potential Uef f still depends in fact on the position x0 of the δ-doping. Consequently, the spin polarization of the LD for the electron across the MMSN device also should be modulated by changing the position of the δ-doping. In Fig. 4 we present the spin polarization of the LD by taking the position of the δ-doping into account. Here, Fig. 4(a) shows that the spin polarization Ps versus the incident energy E for three concrete positions: x0 = 0.0 (solid curve), 0.5 (dashed curve) and 1.0 (dotted curve), where the weight of δ-doping is fixed at V = 1.0. We can observe obviously a strong dependence of the spin polarization PS on the position of the δ-doping. When the δ-doping is away from the centre of the MMSN device, the degree of the spin polarization becomes large and the spin polarization curve shifts the low-energy end. In order to observe better the influence of the position of the δ-doping, Fig. 4(b) plots the variation of the spin polarization PS with the position x0 of the δ-doping, where the incident energy of electrons is taken as E = 6.0. One can observe that both magnitude and sign of spin polarization PS are terribly sensitive to the location x0 of the δ-doping, which attributes to the dependence of the effective potential Uef f [cf. Eq. (4)] on the position of the δ-doping, especially for a small x0 . Another observation is that the variation of the spin polarization with the position of the δ-doping shows up a symmetrical behavior with respect to the centre x0 = 0, i.e., PS (−x0 ) = PS (x0 ). The transmission characteristics are the same for a particle tunnelling through a potential barrier in opposite directions, therefore, for the MMSN device [see Fig. 1(a)] with a symmetric magnetic profile Bz (−x) = Bz (x) and an antisymmetric magnetic vector potential Ay (−x) = −Ay (x), its spin polarization PS will possess a centre symmetry. Thus, the electron-spin polarization in the LD also is tunable by means of changing the position of the δ-doping in MMSN device.

4

Conclusion

In summary, we have investigated theoretically the δ-doping with tunable weight and position in a spin spatial splitter, which can be fabricated by depositing a ferromagnetic stripe with a vertical magnetization on the top of a semiconductor heterostructure and using the δ-doping technique in experiments. By theoretical analysis together with numerical calculation, we find that both magnitude and sign of electron-spin polarization in the LD depend strongly on the weight and/or position of the δ-doping. Namely, behavior of spin polarized electrons can be manipulated expediently by adjusting the weight and/or the position of the δ-doping. Thus, it is possible to open a new door for effectively manipulating spin polarized source by the δ-doping, and the considered MMSN can serve as a structurally-controllable spin spatial splitter. Recently, researchers suspected that, in semiconductor 2DEG nanostructure in the presence of a magnetic field, the Goos-H¨anchen shift of the electron perhaps is closely related to the quantum

7

Hall effect.27−29 Therefore, the Goos-H¨anchen effect for the electron in MMSNs deserves further investigations. Moreover, we believe that the theoretical results obtained here will be confirmed by the relevant experimental works in the near future. Acknowledgement 1 This work was supported by the Natural Science Foundation of China Under Grant No. 11164006 and 61464004.

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[19] E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, R. Kucharczyk, M. Ste´slicka, Phys. Rev. B 56 (1997) 9603. [20] H. Z. Xu, Z. Shi, Phys. Rev. B 69, 237201 (2004). [21] M.W. Lu, Z.Y. Wang, Y.L. Liang, Y.B. An, L.Q. Li, Appl. Phys. Lett. 102 (2013) 022410. [22] M.W. Lu, Z.Y. Wang, Y.L. Liang, Y.B. An, L.Q. Li, Europhys. Lett. 101 (2013) 47001. [23] V. Kubrak, F. Rahman, B.L. Gallagher, P.C. Main, M. Henini, C.H. Marrows, M.A. Howson, Appl. Phys. Lett. 74 (1999) 2507. [24] D. W. Wilson, E. N. Glytsis, T. K. Gaylord, IEEE J. Quantum Electron. 29 (1993) 1364. [25] D. Bohm, Quantum Theory (Prentice-Hall, New York, 1951), pp257-261. [26] X. Chen, X.J. Lu, Y. Wang, C.F. Li, Phys. Rev. B 83 (2011) 195409. [27] N.A. Sinitsyn, Q. Niu, J. Sinova, K. Nomura, Phys. Rev. B 72 (2005) 045346. [28] M. Onoda, S. Murakami, N. Nagaosa, Phys. Rev. Lett. 93 (2004) 083901. [29] K.Yu. Bliokh, V.D. Freilikher, Phys. Rev. B 74 (2006) 174302.

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Figure Captions Fig. 1. (a) Schematic illustration of the MMSN device, where a FM stripe with a vertical magnetization is deposited on top of the semiconductor heterostructure, (b) the magnetic profile and the δ-doping, and (c) the LD of spin electrons. Fig. 2. The LD as the function of the incident energy, where the δ-doping is taken as V = 1.0 and x0 = 0.0, dashed and dotted curves correspond to spin-up and spin-down electrons, respectively, and the inset shows the corresponding spin polarization in the LD. Fig. 3. (a) Energy dependence of the spin polarization in the LD for fixed weight of the δ-doping V = 0.0 (solid curve), 1.0 (dashed curve) and 2.0 (dotted curve) and (b) its spin polarization as the function of the weight of the δ-doping for incident energy EF = 6.0, where the position of the δ-doping is x0 = 0.0. Fig. 4. (a) Variation of spin polarization in the LD with incident energy for fixed positions of the δ-doping: x0 = 0.0 (solid curve), 0.5 (dashed curve) and 1.0 (dotted curve), and (b) the spin polarization versus the position of the δ-doping for incident energy E = 6.0, where the weight of the δ-doping is set to be V = 1.0.

10

FM

(a)

d

0

h y

z

z0

2DEG

x

o

Bz(x) Vδ(x-x0) x+

x-

x

Spin-up electron

(c) SĖ MMSN

α

SĘ

x Spin-down electron

incident electron X-

X+

Fig. 1 M.W. Lu et al.

11

12

3

9 PS 0

Sσ 6 -3

3

6

E

9

12

σ = +1 σ = -1

3

V = 1.0,x 0 =0.0 0

0

3

6

E

Fig. 2 M.W. Lu et al.

12

9

12

6

V=0.0 V=1.0 V=2.0 x0=0.0

(a)

3

PS 0 -3 -6 3

6

6

E

9

12

(b)

3

PS 0 -3

E=6.0, x0=0.0

-6 -9 -3

-2

-1

0

1

V

Fig.3 M.W. Lu et al.

13

2

3

6

x0=0.0 x0=0.5 x0=1.0 V=1.0

(a)

3

PS 0 -3 -6 3

6

E

9

12

3

(b) 0

PS

-3

E=6.0,V=1.0

-6 -9 -1.5

-1.0

-0.5

0.0

0.5

x0

Fig.4 M.W. Lu et al.

14

1.0

1.5

Highlights 1) Effect of a tunable į-doping on a spin spatial splitter is studied. 2) Spin-polarized lateral displacement of the electron depends greatly on weight and/or position of the į-doping. 3) A tunable spin spatial splitter is achieved for spintronics applications.