Spin filter due to spin Hall effect with axially asymmetric potential

ARTICLE IN PRESS Physica E 42 (2010) 956–959

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Spin filter due to spin Hall effect with axially asymmetric potential Tomohiro Yokoyama , Mikio Eto Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

a r t i c l e in f o

a b s t r a c t

Available online 10 November 2009

We examine a three-terminal spin filter including an artificial potential created by antidot, scanning tunnel microscope (STM) tip, etc., fabricated on semiconductor heterostructures with strong spin–orbit interaction. When the potential is attractive and its strength is properly tuned, the resonant scattering takes place, which enhances the extrinsic spin Hall effect. As a result, the efficiency of the spin filter can be more than 50% when the potential is axially symmetric. The efficiency becomes smaller when the symmetry is broken, but we still expect an efficient spin filter unless the degree of asymmetry is too large. & 2009 Elsevier B.V. All rights reserved.

Keywords: Spin–orbit interaction Spin Hall effect Spin filter Antidot Resonant scattering

1. Introduction The injection and manipulation of electron spins in semiconductors are important issues for spin-based electronics, ‘‘spintronics’’ [1]. The spin–orbit (SO) interaction has attracted much attention for the spin manipulation without a magnetic field or ferromagnets. The SO interaction is written as HSO ¼

l r  ½p  =VðrÞ; ‘

ð1Þ

where VðrÞ is an external potential and r indicates the electron spin s ¼ r=2. The coupling constant l is largely enhanced in narrow-gap semiconductors such as InAs, compared with the value in the vacuum [2]. The SO interaction may be also useful for the spin injection in the spintronic devices [3–11]. The spin Hall effect (SHE) is one of the phenomena to create spin current due to the SO interaction. There are two types of SHE. One is an intrinsic SHE which creates a dissipationless spin current in the perfect crystal [12,13]. The other is an extrinsic SHE caused by the spin-dependent scattering of electrons by impurities [14]. In our previous papers [15,16], we formulated the extrinsic SHE in semiconductor heterostructures with an artificial potential created by antidot, scanning tunnel microscope (STM) tip, etc., and proposed three- or four-terminal spin filters including the artificial potential. For an axially symmetric potential VðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðr ¼ x2 þ y2 Þ for two-dimensional electron gas (2DEG) in the xy plane, Eq. (1) yields HSO ¼  l

1 dV lz sz ; r dr

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E-mail address: [email protected] (T. Yokoyama). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.10.061

ð2Þ

where lz and sz are the z component of angular momentum ½l ¼ ðr  pÞ=‘  and spin operators, respectively. The artificial potential is electrically tunable and may be attractive as well as repulsive. We found that (i) the SHE is significantly enhanced by the resonant scattering when the attractive potential is properly tuned and (ii) the efficiency of the multi-terminal devices can be more than 50% by tuning the potential strength to the resonance. Note that the enhanced SHE by the resonant scattering was first pointed out by Kiselev and Kim, considering a three-terminal device without the antidot potential [4,6]. A three-terminal spin filter including an antidot was proposed by Yamamoto and Kramer [10] in the case of repulsive potential. In actual devices, the artificial potential cannot be precisely axially symmetric. The asymmetric effect should be carefully evaluated since the resonant condition is sensitive to the symmetry. In the present paper, we perform numerical studies for the three-terminal spin filter with an elongated potential well. For the potential Vðx; yÞ in general, the SO interaction is given by   l @V @V  py : ð3Þ HSO ¼ sz px ‘ @y @x We show that the axial asymmetry weakens the spin Hall effect since the degeneracy is removed for the energy levels with lz ¼ 7m ðm 40Þ of virtual bound states. Even in this case, the spin-filtering effect may be large enough for the device application unless the degree of asymmetry is too large. We also discuss the displacement of the potential from the center of the junction.

2. Model We consider a three-terminal device with an antidot, or STM tip, as shown in Fig. 1. Three leads (quantum wires) of width W are joined to one another at a junction, which is a square area

ARTICLE IN PRESS T. Yokoyama, M. Eto / Physica E 42 (2010) 956–959

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the spin polarization in the z direction is given by Pz qp ¼

qp Gqp þ  G : qp Gqp þ G þ 

ð8Þ

We assume that the Fermi wave number is kF ¼ 2=R0 . The Fermi energy is EF =t ¼ 2  2cosðkF a0 Þ. There are two conduction channels in the leads. The strength of SO interaction is l~ ¼ 0:1, 2 which corresponds to the value for InAs, l ¼ 117:1 A˚ [2], with a0 ¼ W=30 and W  50 nm. The temperature is T ¼ 0. We focus on the transport from reservoir 1 to 2 and omit the superscript 21 of 21 G21 7 and Pz , unless otherwise stated. Fig. 1. Model for a three-terminal spin filter fabricated on semiconductor heterostructures with strong spin–orbit interaction. The device includes an antidot, or STM tip, which creates a tunable potential on 2DEG in the xy plane. Three ideal leads connect the junction (square area surrounded by broken lines) to reservoirs. Reservoir 1 is a source from which spin-unpolarized electrons are injected into the junction. The voltages are equal in the drains, reservoirs 2 and 3.

surrounded by broken lines. The leads are formed by the hardwall potential and connected to the reservoirs. Reservoir 1 is a source from which unpolarized electrons are injected. Reservoirs 2 and 3 are drains, the voltages of which are equal. We study smooth potential wells for the attractive potential. In the axially symmetric case, the potential is given by 8   DR0 > > ; r  R o  V > 0 0 > > 2 > >       > < V0 r  R0 DR0 jr  R0 jr 1  sin p ; VðrÞ ¼ ð4Þ 2 D R 2 > 0 > >   > > DR0 > > r  R0 4 > :0 2 with V0 o 0, where r is the distance from the potential center. The radius of the potential well is R0 ¼ W=4 and DR0 ¼ 0:7R0 . In the asymmetric case, we examine potential wells of elliptical shape, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vðr0 Þ with r 0 ¼ ðx=aÞ2 þðy=bÞ2 ðab ¼ 1Þ, where VðrÞ is given by Eq. (4). The gradient of V gives rise to the SO interaction in Eq. (3). Although the SO interaction is also created by the hard-wall potential at the edges of the leads, it is negligible because of the small amplitude of the wave function there [16]. Hence we only consider the SO interaction induced by the antidot potential, Eq. (4). For numerical study, we use a tight-binding model [17,18], discretizing the real space in two dimensions. The width of the leads is W ¼ 30a0 , with lattice constant a0 . The Hamiltonian in the tight-binding model is H¼t

3. Axially symmetric potential First, we discuss the spin-filtering effect in the case of axially symmetric potential. The potential well is located at the center of the junction. Figs. 2(a) and (b) show the conductance G 7 for sz ¼ 7 12 and spin polarization Pz , respectively, when the potential depth jV0 j is gradually changed. As seen in Fig. 2(a), G 7 shows three minima. At the first minimum at jV0 j=EF  0:6, the difference in the conductance for sz ¼ 7 12 is small. At the second and third minima at jV0 j=EF  2 and 5, respectively, the difference is remarkable, which results in a large spin polarization in the z direction, as shown in Fig. 2(b). Pz is enhanced to 25% around the second minimum of G 7 and 61% around the third minimum. The behavior of G 7 can be attributed to the resonant scattering by the potential well. The resonant scattering takes place through virtual bound states in the potential well, which enhances the electron scattering. This makes the minima of G 7 in our situation [16]. The virtual bound states correspond to lz ¼ 7 2 (D-wave) and 7 3 (F-wave) at the second and third minimum of G 7 , respectively, as discussed in our previous paper [16]. The SO interaction effectively enhances the scattering potential for sz ¼ 12 and suppresses the scattering potential for sz ¼  12 in the current from reservoir 1 to 2. Around the sharp resonance, the influence of SO interaction becomes remarkable, and as a result, we observe a

X X y y y V~ i;j ci;j; ðTi;j;i þ 1;j;s ci;j; s ci;j;s  t s ci þ 1;j;s þ Ti;j;i;j þ 1;s ci;j;s ci;j þ 1;s þ h:c:Þ; i;j;s

i;j;s

ð5Þ y where ci;j; s and ci;j;s are creation and annihilation operators of an 2 electron at site ði; jÞ with spin s. t ¼ ‘ =ð2m a20 Þ, where m is the ~ effective mass of electrons. V i;j is the potential energy at site ði; jÞ, in units of t. The transfer term in the x direction is given by Ti;j;i þ 1;j; 7 ¼ 1 7il~ ðV~ i þ 1=2;j þ 1  V~ i þ 1=2;j1 Þ; ð6Þ

whereas that in the y direction is Ti;j;i;j þ 1; 7 ¼ 1 8il~ ðV~ i þ 1;j þ 1=2  V~ i1;j þ 1=2 Þ;

ð7Þ

with l~ ¼ l=ð4a20 Þ. V~ i þ 1=2;j is the potential energy at the middle point between the sites ði; jÞ and ði þ 1; jÞ, and V~ i;j þ 1=2 is that of ði; jÞ and ði; j þ 1Þ. The z component of spin is conserved with the SO interaction in Eq. (3). Therefore, we can evaluate the conductance for sz ¼ 7 12 ¨ separately. Using Green’s function and Landauer–Buttiker formula, we calculate the conductance Gqp 7 from reservoir p to reservoir q, for qp spin sz ¼ 7 12 [19]. The total conductance is Gqp ¼ Gqp þ þ G , whereas

Fig. 2. Numerical results in the case of axially symmetric potential well. The potential is located at the center of the junction. kF R0 ¼ 2, where R0 is the radius of potential well. (a) Conductance G 7 from reservoir 1 to 2 in Fig. 1 for sz ¼ 7 12 and (b) spin polarization Pz of the output current in reservoir 2, as functions of the potential depth jV0 j. In (a), solid and broken lines indicate G þ and G , respectively. A dotted line shows the conductance per spin in the absence of SO interaction.

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Fig. 3. Numerical results for spin polarization Pz as a function of potential depth jV0 j, in the q case of elongated potential well. The potential is expressed as V ðr 0 Þ, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where r 0 ¼ ðx=aÞ2 þ ðy=bÞ2 ðab ¼ 1Þ and VðrÞ is given by Eq. (4), kF R0 ¼ 2. In (a), solid and broken lines indicate the cases of a=b ¼ 54 and 32, respectively. In (b), solid and broken lines indicate the cases of b=a ¼ 54 and 32, respectively. Dotted lines show the axially symmetric case with a ¼ b ¼ 1.

Fig. 4. Numerical results of spin polarization Pz as a function of potential depth jV0 j, when the axially symmetric potential well is located apart from the center of the junction, kF R0 ¼ 2. In (a), the position of the well is shifted in the x direction by W=10 (solid line) and W=10 (broken line). In (b), the position of the well is shifted in the y direction by W=10 (solid line) and W=10 (broken line). Dotted lines indicate the case that the well is located at the center of the junction.

5. Displacement of potential well large spin polarization. It should be mentioned that the effect of SO interaction for sz ¼ 7 12 is opposite in the current from reservoir 1 to 3. Hence the relation of Pz31 ¼  Pz21 holds when the symmetric potential is located at the center of the junction. Around the third minimum of conductance at jV0 j=EF  5, the minimum of G þ is sufficiently separated from that of G . The former is located at a smaller value of jV0 j than the latter. In consequence, Pz shows a pair of negative dip ðG þ oG Þ and a positive peak ðG þ 4G Þ.

4. Elongated potential well Now we examine axially asymmetric potential wells. The shape qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the wells is an ellipse; Vðr 0 Þ with r 0 ¼ ðx=aÞ2 þ ðy=bÞ2 ðab ¼ 1Þ, where VðrÞ is given by Eq. (4). The potential center is located at the center of the junction. Figs. 3(a) and (b) show the spin polarization Pz , as a function of the potential depth jV0 j, when the potential well is elongated in the x and y directions, respectively: (a) a=b ¼ 1, 5 3 5 3 4, and 2, while (b) b=a ¼ 1, 4, and 2. In both cases, Pz around the resonance is the largest in the axially symmetric case and gradually decreases with an increase in the asymmetry. The decrease in Pz should be ascribable to the energy splitting of the virtual bound states in the potential well. In the axially symmetric potential well, the states with lz ¼ 7m ðm4 0Þ are degenerate. They are mixed with each other by the SO interaction even when the SO interaction is smaller than the average of level spacings in the well. The asymmetry lifts off the degeneracy and consequently weakens the effect of SO interaction. Even in the most asymmetric case in Fig. 3, we still observe so large a spin-filtering effect as about 20% when the potential is tuned to the resonant condition. We expect a sufficiently large efficiency unless the degree of asymmetry is too large.

We discuss the influence of displaced position of artificial potential in the three-terminal junction. In Fig. 4, we present the spin polarization Pz as a function of potential depth jV0 j, when the axially symmetric potential well is located apart from the center of the junction. In Fig. 4(a), the position of the well is shifted in the x direction by W=10 (solid line) or W=10 (broken line). The influence of these displacements seems very small. We can observe a large spin polarization of 60% around the resonance. In Fig. 4(b), the position of the well is shifted in the y direction. For the shift by W=10 (solid line), the influence is small. The shift by W=10 (broken line), on the other hand, results in a serious decrease in the spin polarization. Pz  20% at the F-wave resonance at jV0 j=EF  5. Around the resonance, the scattering amplitude significantly depends on the scattering angle. When the diameter of the potential well is comparable to the width of wire (2R0 ¼ W=2 in our case), the position of the well could seriously influence the efficiency of the spin filter. In Fig. 4(b), we still observe a large value of Pz  20% at the resonance.

6. Conclusions We have examined the three-terminal spin filter including a tunable potential created by antidot, STM tip, etc. In the case of attractive potential, the resonant scattering takes place by tuning the potential strength, which remarkably enhances the spin Hall effect. We obtain the efficiency of more than 50% by tuning the resonant condition. The spin Hall effect is reduced when the axial asymmetry increases for the shape of the potential well. This is because the degeneracy of energy levels is removed in the virtual bound states for the resonant scattering. The position of the potential may also influence the spin-filtering effect.

ARTICLE IN PRESS T. Yokoyama, M. Eto / Physica E 42 (2010) 956–959

Acknowledgments This work was partly supported by the Strategic Information and Communications R&D Promotion Program (SCOPE) from the Ministry of Internal Affairs and Communications of Japan, and by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science. References ˇ ´ , J. Fabian, S. Das Sarma, Rev. Modern Phys. 76 (2004) 323. [1] I. Zutic [2] R. Winkler, Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer, Berlin, Heidelberg, 2003. [3] E.N. Bulgakov, K.N. Pichugin, A.F. Sadreev, P. Streda, P. Seba, Phys. Rev. Lett. 83 (1999) 376. [4] A.A. Kiselev, K.W. Kim, Appl. Phys. Lett. 78 (2001) 775.

959

[5] T. Koga, J. Nitta, H. Takayanagi, S. Datta, Phys. Rev. Lett. 88 (2002) 126601. [6] A.A. Kiselev, K.W. Kim, J. Appl. Phys. 94 (2003) 4001. [7] T.P. Pareek, Phys. Rev. Lett. 92 (2004) 076601. [8] M. Eto, T. Hayashi, Y. Kurotani, J. Phys. Soc. Japan 74 (2005) 1934. [9] J.I. Ohe, M. Yamamoto, T. Ohtsuki, J. Nitta, Phys. Rev. B 72 (2005) 041308(R). [10] M. Yamamoto, B. Kramer, J. Appl. Phys. 103 (2008) 123703. [11] J.J. Krich, B.I. Halperin, Phys. Rev. B 78 (2008) 035338. [12] S. Murakami, N. Nagaosa, S.-C. Zhang, Science 301 (2003) 1348. [13] J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth, A.H. MacDonald, Phys. Rev. Lett. 92 (2004) 126603. [14] H.A. Engel, B.I. Halperin, E.I. Rashba, Phys. Rev. Lett. 95 (2005) 166605. [15] M. Eto, T. Yokoyama, J. Phys. Soc. Japan 78 (2009) 073710. [16] T. Yokoyama, M. Eto, Phys. Rev. B 80 (2009) 125311. [17] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [18] T. Ando, Phys. Rev. B 40 (1989) 5325. [19] M. Yamamoto, T. Ohtsuki, B. Kramer, Phys. Rev. B 72 (2005) 115321.