Tunable spin Hall effect by Stern-Gerlach diffraction

Tunable spin Hall effect by Stern-Gerlach diffraction Jun-Qiang Lu and X.-G. Zhang Center for Nanophase Materials Sciences, and Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Sokrates T. Pantelides Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Received 6 November 2006; published 18 December 2006 We propose an effect based on simultaneous real- and reciprocal-space magnetic inhomogeneities, combining features of the Stern-Gerlach and both the intrinsic and extrinsic spin Hall effects. The known difficulties of directly observing the spin Hall effect are circumvented as spin currents generated from the effect are well-defined, dissipative, and detectable. Simulations of a specific system are used to illustrate wide tunability, allowing formation of a periodic spin lattice distinct from the charge lattice, selective polarization flipping, or spin current detection. Similar effects can be produced by photons and neutrons. DOI: 10.1103/PhysRevB.74.245319 PACS number s : 72.25.Dc, 71.70.Ej, 85.75. d I. INTRODUCTION The pursuit of spintronics ultimately depends on our ability to steer spin currents and detect or flip their polarization. The 1922 Stern-Gerlach SG experiment 1 first demonstrated that electron currents can be steered by the Zeeman force, F= B, caused by a spatially inhomogeneous magnetic field B r, where is the electron magnetic moment. The effect was recently observed for electrons in a semiconductor nanostructure, where the spatially inhomogeneous magnetic field was generated by a ferromagnetic layer. 2 In a similar vein, the SG effect has been demonstrated for a beam of light passing through a medium where the photons morph into polaritons and experience a spatially inhomogeneous effective magnetic field. 3 In the last few years, there has been intense interest in the manipulation of spins without the need for ferromagnetic materials or the application of external magnetic fields, relying solely on spin-orbit SO coupling and external electric fields. 4 10 An SG filter was recently proposed 5 in which an electron current gets split by a Zeeman force from an effective magnetic field produced by SO interactions through the Rashba Hamiltonian 11 and made inhomogeneous by the application of suitable external electric fields. The spin Hall effect SHE is another altogether different way to manipulate spins without relying on the Zeeman force. It exploits the momentum dependence of the SO effective magnetic field. 12 In the extrinsic SHE 6 one relies on impurities with strong SO coupling to produce skew scattering of electron currents. In the intrinsic SHE, 7 10 one enhances SO coupling by an external electric field applied to a two-dimensional electron system 2DES. Thus unlike the SG effect that relies on a magnetic field that is inhomogeneous in real space, the SHE is caused by an effective magnetic field that is inhomogeneous in reciprocal space. The experimental observation of the SHE has proved to be quite challenging. 13 15 Kato et al. 13 reported an unambiguous signature of SHE, namely the accumulation of opposite spins in directions transverse to the electron current, in a semiconductor two-dimensional electron system 2DES, but the origins of the effect are not completely clear. The authors attributed the observations to the extrinsic SHE, though alternative opinions have since been expressed. 10 Observations of the intrinsic SHE remain an open issue. 16 A key difficulty is the fact that in a 2DES with strong SO coupling, spin is not a good quantum number so that spin currents are not well defined. In this paper, we propose to combine the principles that underlie the SG experiment and the SHE to arrive at an interesting phenomenon that exploits a designed real-space inhomogeneity with the intrinsic reciprocal-space dependence of the effective SO magnetic field in a 2DES. The effect avoids both the randomness of impurity distribution that is inherent in the extrinsic SHE and the intrinsic-she difficulties of defining spin currents in a SO-coupled region: the SO-coupled scatterers are confined in a nanoscale diffraction grating in a 2DES, offering wide tunability of end results. The diffraction grating can be implemented either by an array of SO-coupled quantum dots and a uniform external electric field perpendicular to the 2DES or by a uniform layer of SO-coupled material and a patterned gate to produce an electric-field-induced grating. Unlike the intrinsic SHE, the spin currents generated by such a setup are well defined outside the SO region. The transverse charge currents and spin currents are real, dissipative, and detectable. The grating acts as the SG filter of Ref. 5, but we can now control the number of emerging beams. In addition, the diffraction design takes advantage of the SHE-like spin-dependent velocities arising from the momentum-dependent effective magnetic field. The proposed tunable SHE TSHE can be used to produce a variety of interesting effects that allow the generation, manipulation, and/or detection of spin currents and spin-polarized electrical currents, including polarization flipping. The underlying physics is similar to what one would get by a periodic array of SO-coupled impurities, i.e., in some sense the effect is akin to the extrinsic SHE, but their spatial arrangement in a grating brings into action the SG force. In addition, by doing away with both SO impurities and quantum dots QDs, the effect can be generated entirely 1098-0121/2006/74 24 /245319 5 245319-1 2006 The American Physical Society

LU, ZHANG, AND PANTELIDES by a patterned external electric field, which imposes the SG force on the intrinsic SHE. We demonstrate the TSHE with numerical simulations for a model system. We also show that the effect is quite general and can be produced by a general class of Hamiltonians that have odd powers both of spin and momentum. It can in principle be realized in diffraction phenomena of photons, neutrons, and even ion beams. II. TUNABLE SPIN HALL EFFECT IN 2DES We start with the theory for a semiconductor 2DES. In the presence of SO coupling, the electrons or holes can be described by the Rashba Hamiltonian 11 H = p 2 /2m + p ê = p 2 /2m + p ê, 1 where m is the effective mass, p= i, is the Pauli spin matrix vector, is a coupling constant, and ê is a unit vector in the direction of the external electric field, which controls the value of. The effective magnetic field is B eff = / B p ê, 12 where B is the Bohr magneton. In Ref. 5, the authors explored the effect of an inhomogeneous external electric field, which makes a function of r, giving rise to a Zeeman force that splits electron currents SG filter. The effective magnetic field, however, is intrinsically also a function of p, which produces a spin-dependent transverse velocity given by v = p H = p m + B p B eff = p + ê. m 2 To see the effects of both a real- and reciprocal-space dependence of B eff r,p, consider a spin-polarized current in the x direction, being injected into a 2DES in an x-z plane, at a particular energy, typically the Fermi energy E F, with crystal momentum p= k F and velocity = k F /m, as shown in Fig. 1. In the presence of an array of SO-coupled QDs shaded region in the z direction with width a and spacing b, there will be diffracted beams at angles given by sin n =2 n/k F b, with transverse momenta p n z =2n /b. The distribution of diffraction angles are symmetric between left +z and right z as they are resulting from the periodic z. The diffracted beams, however, acquire additional spindependent transverse velocities: v n z = H/ p z = p n z /m x. 3 The two terms arise from the real-space and reciprocal-space inhomogeneities of the effective magnetic field, respectively. The second term, arising from SO coupling, breaks the spin symmetry in the diffracted beams. Equation 3 displays the main features of the TSHE and reveals the options for control: the periodicity of the SO grating either an array of SO-coupled dots, or simply a patterned external electric field applied to a SO-coupled region controls the number of diffracted beams and the strength of the electric field controls the magnitude of the spin-dependent transverse velocities. One might even envision a superposed gradient on the external electric field to produce additional variety of outcomes. In addition, by patterning a periodic gate with a very small period, one can apply electric fields in a way that the period FIG. 1. The schematics of the basic concept of the tunable spin Hall effect in a two-dimensional electron system: a spin-polarized electron beam is asymmetrically diffracted by an array of quantum dots with Rashba spin-orbit coupling. The shaded region is the quantum dot array, which has been modeled as a simple oscillatory function in the simulations. of the effective grating can be a multiple of the physical period of the patterned gate. By a suitable choice of the grating parameters, several distinct objectives can be pursued. At one extreme, one can rely on the interference of diffracted beams to produce a two-dimensional periodic lattice of maxima and minima. The charge lattice, defined by the maxima or minima of the total electron density and the spin lattice, defined by the maxima or minima of the spin polarization, depend on the polarization of the incoming beam and are different from each other. They offer interesting opportunities for applications. At the other extreme, one can choose parameters so that there is only one left-diffracted beam and one right-diffracted beam, in addition to the principal beam that goes through unhindered. The asymmetry produces a transverse charge or spin current. The transverse currents can have spinpolarization opposite to the injected one, accomplishing controllable spin flipping. More importantly, by measuring the net transverse charge current from the diffracted beam, one can deduce the polarization of the principal beam, opening the possibility of a spin-polarization detector. We have applied the above theory to a potentially realizable device shown schematically in Fig. 1. A periodic array of SO-coupled QDs produce a periodic x,z whose overall amplitude is controlled by an external electric field along the y direction. The complete effective-mass Hamiltonian of the 2DES has the form H = p2 2m + x,z p ê y i 2 x,z ê y + V c x,z + V s x,z L, 4 245319-2

TUNABLE SPIN HALL EFFECT BY STERN-GERLACH FIG. 2. Color online Spatial distributions of the wave densities x,z in the region near the quantum dot array, for an injected beam spin polarized in the a x or b z directions. where the second term contains the Rashba SO coupling and the third term is needed for current conservation. V c is the spin-independent part of the crystal potential, which includes the confinement potential of QDs and the disordered impurity scattering potential. The term V s is the SO coupling potential from V c, where L is the orbital angular momentum of electron. 6 We have modeled the effect of the diffraction grating on by an analytical form x,z that is oscillatory in the z direction with period b: = x,z k 0/m 1 + cos k b z if a x 0; 0 if x a or x 0, 5 where k 0 sets the SO coupling strength, and k b =2 /b. In pursuing numerical solutions, we checked the effect of the potentials V c and V s. We found that the confinement potential plays a secondary role, causing a superposed diffraction pattern. Disordered impurity scattering, on the other hand, if strong enough, can destroy the diffraction patterns altogether, producing only skew scattering as in Ref. 6. In order to avoid the confinement potential and the disordered scattering, the diffraction grating can be produced by a patterned external electric field, using a SO-coupled layer that, to the extent that is possible, contains no impurities. In order to focus on the TSHE, we do not include any impurity scattering and set the potentials V c and V s to zero. In order to vividly view the TSHE in the 2DES, we first present the results for InGaAs/InAlAs 2DES, 5,17,18 with a choice of parameters that produces only two diffracted beams in addition to the principal beam that goes through unhindered. The numerical solutions are obtained by expanding the electron wave functions in a plane-wave basis and solving the diffraction problem using the Lippmann- Schwinger equation. The spatial wave density distribution x,z = x,z 2 in the region near the QD array is shown in Fig. 2. We take the effective mass of the electrons m =0.05m e, the Fermi wave vector of the injected electrons k F =0.353 nm 1. 17,18 The QDs, with the SO coupling strength k 0 =0.05k F, 17 width a =15 nm, and period b =20 nm, are drawn schematically in the middle of Fig. 2 a. If the injected beam is spin polarized in the x direction, the diffraction pattern, shown in Fig. 2 a, is clearly spatially asymmetric. In contrast, if the injected beam is spin polarized in the z direction, as shown in Fig. 2 b, the diffraction pattern is symmetric because of the absence of asymmetric transverse term in the velocity, Eq. 3. The spin-dependent transverse charge current j= j L j R is plotted in Fig. 3 a. For 0 a 20 nm, if the injected beam is polarized in the x direction, then j is positive, which means that the left-diffracted beam has a larger flux than the right-diffracted beam. j is always 0 if the injected beam is polarized in the z direction. The flux of the principal beam, j 0, and its spin polarization, P 0, are shown in Fig. 3 b. Here, P 0 = j 0 j 0 / j 0, and are relative to the spin direction of the injected beam. j 0 is always greater than 0.95. The polar- III. APPLICATION TO INGAAS/INALAS 2DES FIG. 3. Color online a The transverse charge current j, b the principle beam j 0 and its polarization P 0 for an injected beam spin polarized in the x, y, orz directions, as a function of the width of the quantum dot array a. Inset: the angle of the spin precession after passing through the array for an injected beam spin polarized in the x or y direction. 245319-3

LU, ZHANG, AND PANTELIDES FIG. 4. Color online Spatial distributions of the x component of spin polarization, P x x,z, in the region near the quantum dot array, for an injected beam spin polarized in the a x or b z directions. ization P 0 is exactly 1.0 if the injected beam is spin-polarized in the z direction. If the injected beam is spin polarized in the x direction, P 0 decreases with a, but it is still above 0.85 at a=15 nm, the width corresponding to the maximum j in Fig. 3 a. Exploiting the transverse charge current in combination with the preservation of the spin polarization of the principal beam, this effect can be the basis for a spinpolarization detector design. The nonzero j in Fig. 3 a for the injected beam polarized in the y direction is not anticipated by Eq. 3. However, both this effect and the changes of P 0 in Fig. 3 b can be explained by spin precession due to the effective magnetic field, when electrons travel through the QD array. For the principal beam, the electron momentum is always in the x direction, yielding an effective magnetic field along the z direction, B z eff = p x /. There is no spin precession if the injected beam is spin polarized in the z direction. Thus P 0 remains exactly 1.0 in this case, as shown in Fig. 3 b. When the injected beam is spin polarized in either the x or the y direction, the spin precesses around the z axis, thus away from its original direction. This precession explains the nonzero j even if the injected beam is spin polarized in the y direction. The spin precession also explains the decrease in P 0 shown in Fig. 3 b for the injected beam spin polarized in either the x or the y direction. The angle of spin precession, can be directly calculated from P 0 as, =cos 1 P 0. The angle scales linearly with the distance a, as plotted in the inset of Fig. 3 b. On the other hand, the angular rate of the spin precession under the effective magnetic field is =2B z eff /. Thus the angle of spin precession when the electron emerges from the QD array can be calculated. The result is = t =2B z eff t/ =2k 0 p x t/m =2k 0 a. For our system, this expression give =0.0353a, which agrees perfectly with the slope in the inset of Fig. 3 b. The agreement confirms the presence of the spin precession and the effective magnetic field due to the SO coupling inside the QD array. Spin precession also leads to the different spin orientations of the different beams after diffraction, resulting in the separation of the spin lattice and charge lattice. Figure 4 shows the spatial distribution of the x component of the spin polarization, P x x,z, of an injected beam spin polarized in the x direction Fig. 4 a or z direction Fig. 4 b. Here, 6 P x x,z = x,z x,z / x,z, and denote the +x and x directions. Clearly, in both Figs. 4 a and 4 b, the spin polarization after the diffraction is different from that of the injected beam, which confirms the presence of spin precession. If the injected beam is spin polarized in the z direction, although the principal beam spin does not precess, the diffracted beam spin does, as shown in Fig. 4 b. The spin precession causes the spin lattices in Fig. 4 to be totally different from the charge lattices in Fig. 2. A more interesting feature in Fig. 4 b is the antisymmetric spin pattern. If the injected beam is spin polarized in the z direction, although the diffracted fluxes are symmetric between left and right Fig. 2 b, their spin polarizations are not Fig. 4 b. This effect causes a transverse spin current, somewhat similar to the SHE, but with an important difference that the spin current is outside of the SO coupling region, which makes it dissipative and detectable. Because the asymmetric diffraction depends on the spinpolarization of the injected beam, the TSHE effect can be easily used to design spintronics applications. Generally, applications based on the TSHE effect can benefit from two features: first, the applications can be controlled electrically as the SO coupling strength is proportional to the electric field; second, different functions can be realized through adjusting the structure parameters of the QD array, as indicated earlier. IV. POTENTIAL APPLICATION IN NEUTRON AND PHOTON SCATTERING The TSHE is more general than the Rashba Hamiltonian of Eq. 1, which applies to semiconductor 2DES. It can be produced by corresponding Hamiltonians that describe photon or neutron beams, which have odd powers both of spin and momentum. For example, the SO coupling Hamiltonian due to the motion of neutrons through an external electric field E, 19 H SO = s p E, is linear in both neutron spin s and momentum p. Here, = e /m 2 c 2 is a constant, and = 1.91 is the gyromagnetic ratio of neutron. For photons, a SO-coupling Hamiltonian, 7 245319-4

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