Spatial spin splitter based on a hybrid ferromagnet, Schottky metal and semiconductor nanostructure

Journal of Magnetism and Magnetic Materials 401 (2016) 231–235

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Spatial spin splitter based on a hybrid ferromagnet, Schottky metal and semiconductor nanostructure

art ic l e i nf o Keywords: Hybrid magnetically modulated nanostructure Lateral displacement Spin polarization Spatial spin splitter

a b s t r a c t We theoretically investigate the lateral displacement of the spin electron across a hybrid magnetically modulated nanostructure. Experimentally, this nanostructure can be produced by depositing a ferromagnetic stripe with in-plane magnetization and a Schottky metal stripe on top and bottom of a semiconductor heterostructure, respectively. Theoretical analysis reveals that the inclusion of the Schottky metal stripe in single ferromagnetic-stripe nanostructure can break the intrinsic symmetry and a sizeable spin polarization in the lateral displacement will occur. Numerical calculations demonstrate that both magnitude and sign of the spin polarization can be controlled by adjusting the width and/or the position of the Schottky metal stripe in the device. Thus, based on such a hybrid magnetically modulated nanostructure, a spatial spin splitter can be proposed successfully. & 2015 Elsevier B.V. All rights reserved.

1. Introduction The use of electron spins to store and process digital information calls for the ability to inject, propagate and manipulate spin with high efficiency [1,2]. Motivated by the field, how to spinpolarize electrons from spatial domain into the conventional semiconductor materials in hybrid magnetically modulated nanostructures (HMMN) [3], where the HMMN is exploited as a spatial spin splitter [4], has attracted much interest in recent years [5–10]. In general, such a scheme spin polarizes electrons into semiconductors by means of significant difference of spatial positions (such as angle, shift and lateral displacement) between spin-up and spin-down electrons across a HMMN [11]. A simple, experimentally attractive proposal [12–14] for spintronic devices is to exploit a single nanosized ferromagnetic (FM) stripe on top of a semiconductor heterostructure. In such a device, the FM stripe with a horizontal magnetization produces an antisymmetric magnetic field, which acts perpendicularly on the twodimensional electron gas (2DEG) formed usually in a modulationdoped semiconductor heterostructure. Due to this intrinsic symmetry, there is no spin polarization in the single FM-stripe HMMN device [15]. However, in parallel configuration placing a Schottky metal (SM) in the vicinity of the FM stripe can break the intrinsic symmetry, and a spin filter was proposed successfully by Zhai et al. [16]. By adding another FM stripe with an in-plane magnetization on bottom of the semiconductor heterostructure, this intrinsic symmetry also can be broken [17–19]. Therefore, another kind of spin filters also was proposed correspondingly, which may be useful for spintronics applications [20]. Very recently, a tunable δpotential was introduced into the single FM-stripe device with the help of the atomic layer doping technique or the δ-doping technique [21], and its effect on spin-polarized transport was taken into account. It is found that the δ-doping can break the intrinsic symmetry. As the result of the broken symmetry, such a δ-doped single FM-stripe HMMN device can serve as a structurally http://dx.doi.org/10.1016/j.jmmm.2015.10.040 0304-8853/& 2015 Elsevier B.V. All rights reserved.

controllable spin filter [22] or spatial spin splitter [23]. Edified by the above these brief reports, in the present work, we deposit a SM stripe on the bottom of the semiconductor heterostructure in the single FM-stripe HMMN device, and focus our attention on the influence of the SM stripe on the lateral displacement of the spin electron. Theoretical analysis reveals that the SM stripe will break the intrinsic symmetry, and such a HMMN device can be exploited consequently as a spatial spin splitter for spintronics applications.

2. Model and theoretical method The HMMN device under consideration is shown in Fig. 1(a), → where a FM stripe with a horizontal magnetization ( M0 ) and a SM stripe under an applied negative voltage ( −Vg ) are deposited [24] on top and bottom of a semiconductor heterostructure, respectively. The model of this device is presented in Fig. 1(b). The magnetized FM stripe will produce a magnetic field, acting on the 2DEG in ( x, y ) plane, can be written by [25] ∧ → B = Bz (x) z ,

Bz (x) = B [δ (x + L/2) − δ (x − L/2)],

(1)

where B is the magnetic strength of the δ-function magnetic barrier and L stands for the width of the FM stripe. Correspondingly, the magnetic vector potential is given in Landau gauge, by [26,27]

→ A (x) = [0, Ay (x), 0] Ay (x) = BΘ (L/2 − |x|),

(2)

in which Θ (x ) is the Heaviside step function. The SM stripe applied by a negative voltage also will induce an electric potential U(x), which can be viewed as [12] a rectangular barrier for a small

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is referred to as the effective potential of an electron in the HMMN device. Apparently, this effective potential depends on not only the wave vector ky, magnetic configuration Bz(x) and electron spins σ , but also on the U(x) induced by the SM stripe. In fact, it is this dependence of the Ueff (x ) on the SM stripe that breaks the intrinsic symmetry in single FM-stripe device and makes such a device serve as a spatial spin splitter. Obviously, the reduced Schrödinger equation (5) can be analytically solved by linearly combining wave function with plane waves. No loss of generality, the wave function for the electron with the energy E projects onto the HMMN device in an incident angle α [see Fig. 1(c)] can written as Ψin = exp {i [kl (x + L/2) + k y y]}, x < − L/2, while the transmitted and reflected wave functions can be expressed and in Ψref = γ exp {i [ − kl (x + L/2) + k y y]}, x < − L/2 respectively, where Ψout = τ exp {ik r (x − L/2) + k y y}, x > L/2,

ky =

2E sin α , kl = k r =

2E − k y2 , and τ /γ is the transmission and

reflection amplitudes. By matching wave function at the ±L/2, we can obtain with the help of the transfer matrix method [28,29]

⎛ τ⎞ ⎛ 1 1 ⎞ ⎛ 1⎞ ⎟ ⎜ ⎟ = M ⎜ ⎟, ⎜ ⎝ 0⎠ ⎝ ikl − ikl ⎠ ⎝ γ ⎠

(6)

with

M=

⎛ m11 m12 ⎞ ⎜ ⎟ ⎝ m21 m22 ⎠

⎛ ⎜ cos k1 [x 0 + (L − d)/2] =⎜ ⎜ ⎝ k1 sin k1 [x 0 + (L − d)/2] ⎛ sin k2 d ⎞ ⎟ ⎜ cos k2 d − k2 ⎟ × ⎜ ⎟ ⎜ ⎝ k2 sin k2 d cos k2 d ⎠

Fig. 1. (a) Schematic illustration of the HMMN device, where a FM stripe with a → horizontal magnetization ( M0 ) and a SM stripe under an applied voltage ( −Vg ) are deposited on top and bottom of the semiconductor heterostructure, and L and d are width of the FM and SM stripes, respectively (b) the model of the device, where B is magnetic strength for magnetic profile produced by FM stripe and U(x) is the electric potential induced by SM stripe, and (c) the lateral displacement for the electron tunnelling through this HMMN device.

⎛ ⎜ cos k3 [(L − d)/2 − x 0 ] ⎜ m⁎g ⁎σB ⎜ − sin 2m0 k3 ⎜ ⎜ ⎜ k3 [(L − d)/2 − x 0 ] ⎜ ⎜ k3 sin k3 [(L − d)/2 − x 0 ] ⎜ m⁎g ⁎σB ⎜ + cos 2m 0 ⎜ ⎜ ⎝ k3 [(L − d)/2 − x 0 ]

vertical distance between the SM stripe and the 2EDG with a homogeneous resistivity, i.e.,

U (x) = UΘ (d/2 − |x − x 0 |).

(3)

Here, U is the electric barrier (EB), and the SM stripe with a width d is assumed to locate at x0. And then, the Hamiltonian describing such a 2DEG system, within the single particle, effective mass approximation, is

H=

px2 2m⁎

+

[py + eAy (x)]2 2m⁎

+

em⁎g ⁎σ ℏ Bz (x) + U (x), 4m 0

where m⁎, m0 and e are the effective mass, free mass and charge of → an electron, respectively, p = (px , py ) is the electronic momentum, g ⁎ is the effective Landé factor of electron, and σ =+ 1/ − 1 for spinup/spin-down electrons. Because of the translational invariance along the y-axis in the HMMN system, the solution of the stationary Schrödinger equation for the electron HΨ (x, y ) = EΨ (x, y ) can be written as Ψ (x, y ) = ψ (x ) exp (ik y y ), where ky is the electronic wave-vector component in y-direction, and the wave function ψ (x ) satisfies the following one-dimensional (1D) Schrödinger equation: {

d2 dx 2

+

⎤ ⎡ ⎤2 eg ⁎σ ℏ 2m⁎ ⎡ e ⎢E − Bz (x ) − U (x ) ⎥ − ⎢ k y + Ay (x ) ⎥ } ψ (x ) = 0, ⎦ ⎥⎦ ⎣ ℏ 4m0 ℏ2 ⎢⎣

where k1 = k3 =

2k l , P + iQ

where the quantity Ueff (x ) = [ℏk y + eAy (x )]2 /(2m⁎) + em⁎g ⁎σ ℏBz (x )/4m0 + U (x )

2E − k y2 and k2 =

(7)

2 (E − U ) − k y2 . And then, we

(8)

where P = kl m11 + k r m22 and Q = kl k r m12 − m21. Therefore, the phase shift for the electron across the HMMN device can be expressed as ϕ = tan−1(P /Q ). According to the stationary phase method [5,30,31], the lateral displacement of the spin electron can be calculated by

Sσ = −

dϕ . dk y

(9)

The electron-spin polarization can be defined by considering the relative difference of the lateral displacement [32]

PS = S↑ − S↓, (5)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ cos k3 [(L − d)/2 − x 0 ] ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

− sin k3 [(L − d)/2 − x 0 ] k3

readily write out

τ= (4)

sin k1 [x 0 + (L − d)/2] ⎞ ⎟ k1 ⎟× ⎟ cos k1 [x 0 + (L − d)/2] ⎠

−

(10)

where S↑ and S↓ are lateral displacements for spin-up and spin-down electrons, respectively.

/ Journal of Magnetism and Magnetic Materials 401 (2016) 231–235

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3. Results and discussion For convenience of calculation, we express the relevant quantities in the dimensionless form by introducing two basic quantities, the cyclotron frequency ωc = eB0/m⁎ and the magnetic length ℓB = ℏ/eB0 , e.g., x⟹xℓB , E⟹EE0 = ℏωc and Ay ⇒ Ay B0 ℓB . At the same time, we take the InAs as the 2DEG material with m⁎ = 0.024m0 and g ⁎ = 15 (resulting in E0 = ℏωc = 0.48 meV and ℓB = 81.3 nm for B0 = 0.1 T ), and choose some structural parameters as L¼ 2.0 and B ¼3.0. On the other hand, we merely consider the electron with an incident angle α = − 85° throughout whole work, as an example of demonstrating the principle of operation of the HMMN device. As we know, in single FM-stripe HMMN device the electronic transmission in independent of its spins due to the intrinsic symmetry [15]. However, the intrinsic symmetry will be consequently broken if we deposit one SM stripe on the bottom of the semiconductor heterostructure as shown in Fig. 1(a). Therefore, it can be expected that the lateral displacement will depend on the spins for the electron traversing such a HMMN device with the SM stripe. As the function of the incident energy, Fig. 2(a) shows the lateral displacement for the electron tunnelling through this HMMN device, where the solid and dashed curves correspond with spin-up and spin-down electrons, respectively, while the SM stripe is assumed to locate at x0 = 0.5 and its width is taken as d ¼0.6. Indeed, we can observe clearly from this figure that the lateral displacement depends strongly on the spins for the electron across the HMMN device, that is to say, the lateral displacement of spin-up electron is different distinctly from that for spin-down electron. In other words, the spin splitting in the lateral displacement occurs or the HMMN device shows up a sizeable spin polarization effect in the lateral displacement in involved energy range. In fact, such a spin polarization can be observed more obviously from Fig. 2(b), where the spin polarization PS as the function of the incident energy E is shown. This figure confirms an apparent spin polarization effect in the lateral displacement for the spin electron across the HMMN device shown in Fig. 1(a). Moreover, both magnitude and sign switch rapidly with the incident energy, especially in the region of 0.6 ≤ E ≤ 1.8. After a respectable spin polarization in the lateral displacement is observed when a SM stripe is introduced into the single FMstripe HMMN device, we wonder that the SM stripe has a what impact on its degree. Beyond all doubt, the SM stripe will produce a significant influence on the spin polarization, because the effective potential Ueff of the HMMN device is related closely to the SM stripe via its structural parameters [d and x0 , see Eq. (5)]. Next, we explore how the width (d) and the position (x0) of the SM affect the degree of the spin polarization (PS) in the lateral displacement for the electron traversing the HMMN device shown in Fig. 1(a). Firstly, we take the influence of the width of the SM stripe into account and present in Fig. 3 the spin polarization PS in the lateral displacement for the electron tunnelling through the HMMN device [see Fig. 1(a)], where the position of the SM stripe is fixed at x0 = 0.5. Fig. 3(a) gives the dependence of the spin polarization PS on the incident energy E for three concrete width of the SM stripe: d ¼0.2 (black curve), 0.5 (red curve) and 0.8 (blue curve), respectively. Significant change of the spin polarization PS with the width d of the SM stripe can be evidently observed. When the SM stripe becomes wide, in the considered incident energy range, the spin polarization of the lateral displacement shifts towards the highenergy region, its valley is widened and its magnitude slightly becomes small. In Fig. 3(b), the spin polarization PS is presented directly as the function of the width d of the SM stripe, where the incident energy is chosen as E ¼0.7 (blue line), 1.2 (black line) and 1.7 (red line). From this figure, we can observe clearly the

Fig. 2. (a) Spin-dependent lateral displacement versus the incident energy, where solid and dashed curves stand for spin-up and spin-down electrons, respectively, and (b) the spin polarization of the lateral displacement, where the SM stripe is set to be d¼ 0.6 and x0 = 0.5.

modulation of the width d of the SM to the spin polarization Ps (including its magnitude and its sign). In other words, the degree of the spin polarization in the lateral displacement is controllable expediently by properly choosing the width of the SM stripe deposited on the bottom of the semiconductor heterostructure in single FM-stripe HMMN device shown in Fig. 1(a). Meanwhile, such a modulation still shows up a great dependence on the incident energy, especially for the E ¼1.7 (cf. red curve). Finally, the effect of the position of the SM stripe on the spin polarization in the lateral displacement for the electron across the HMMN device is given in Fig. 4, where the width of the SM stripe remains unchanged d¼ 0.6. Here, Fig. 4(a) presents that the spin polarization PS switches with the incident energy E for three different positions of the SM stripe: x0 = − 0.5 (red curve), 0.0 (black curve) and þ0.5 (blue curve), respectively. We can observe apparently from the black curve that if the SM stripe locates at the centre of the single FM-stripe device, i.e., x0 = 0.0, there is no spin polarization ( PS = 0) in the HMMN device shown in Fig. 1(a), because the intrinsic symmetry still exists even if the SM stripe is included. However, when the SM stripe deviates from the centre of

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/ Journal of Magnetism and Magnetic Materials 401 (2016) 231–235

Fig. 3. (a) Dependence of spin polarization on the incident energy for three different widths of the SM stripe: d ¼ 0.2 (black curve), 0.5 (red curve) and 0.8 (blue curve), respectively, and (b) direct variation of spin polarization with the width of the SM stripe under three fixed incident energies: E¼ 0.7 (blue line), 1.2 (black line) and 1.7 (red line), where the position of the SM stripe is taken as x0 = 0.5. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the device, a sizeable spin polarization in the lateral displacement appears and its degree varies strongly with the incident energy, as indicated by red curve or blue curve in Fig. 4(a). Another observation from this figure is that the spin polarization complies with the relation, PS ( − x0 ) = − PS (x0 ). For the HMMN system [see Fig. 1(b)], its magnetic profile Bz(x) and the corresponding magnetic vector potential Ay(x) are antisymmetric and symmetric with respect to the x-axis, respectively. As a result of such a symmetry, the spin polarization is antisymmetric under the replacement x0 → − x0 , since the transmission is always equal for a particle traversing a potential barrier in opposite directions. The influence of the position of the SM stripe can be observed more clearly from Fig. 4(b), which directly shows the dependence of spin polarization PS on the position x0 for the incident energy E ¼1.2. The modulation of the position of the SM stripe on the spin polarization of the lateral displacement can be obviously seen from this figure. Both magnitude and sign of the spin polarization PS vary rapidly with

Fig. 4. (a) Spin polarization versus incident energy for three different positions of the SM stripe: x0 = − 0.5 (red curve), 0.0 (black curve) and þ 0.5 (blue curve), respectively, and (b) direct variation of spin polarization with the position of the SM stripe under a fixed Fermi energy E = 1.2, where the width of the SM stripe is chosen as d ¼ 0.6. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the position x0. Moreover, the antisymmetric behaviour, PG ( − x0 ) = − PG (x0 ), in change of the PS with the x0 appears again.

4. Conclusions In summary, with the help of modern nanofabrication technique, we deposit a SM stripe on the bottom of the semiconductor InAs heterostructure in single FM-stripe HMMN device, and theoretically investigate its influence on the lateral displacement for the electron across this device. Theoretical analysis indicates that the intrinsic symmetry in the single FM-stripe system is broken by the SM stripe, giving rise to a strong dependence of the lateral displacement of the electron on its spins. Numerical calculation confirms that both magnitude and sign of the spin polarization in the lateral displacement are sensitive to the width and/or the position of the SM stripe. These interesting properties may provide an alternative scheme to achieve a spin-polarized source, and this

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HMMN device can be used as a spatial spin splitter.

Acknowledgement

[17] [18] [19] [20] [21]

This work was supported by the Scientific Research Project (Grant no. 12C0873) in Department of Education of Hunan Province of China.

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Li-Hua Shenn, Wen-Yue Ma, Gui-Xiang Liu, Lin Yuan Department of Science, Shaoyang University, Shaoyang 422004, People's Republic of China E-mail address: [email protected] (L.-H. Shen) Received 9 August 2015 28 September 2015 10 October 2015

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Corresponding author. Fax: þ86 7396866173.