The spin of electrons and holes in semiconductor heterostructures ...
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The spin of electrons and holes in semiconductor heterostructures ...

Spin-polarized reflection of electronsin a two-dimensional electron systemHong Chen, J. J. Heremans*, J. A. Peters, J. P. Dulka, andA. O. Govorov,Department of Physics and Astronomy, and The Nanoscale and Quantum PhenomenaInstitute, Ohio University, Athens OH 45701N. Goel, S. J. Chung, and M. B. SantosDepartment of Physics and Astronomy, and Center for Semiconductor Physics inNanostructures, The University of Oklahoma, Norman OK 73019AbstractWe present a method to create spin-polarized beams of ballistic electrons in atwo-dimensional electron system in the presence of spin-orbit interaction. Scattering of aspin-unpolarized injected beam from a lithographic barrier leads to the creation of twofully spin-polarized side beams, in addition to an unpolarized specularly reflected beam.Experimental magnetotransport data on InSb/InAlSb heterostructures demonstrate thespin-polarized reflection in a mesoscopic geometry, and confirm our theoreticalpredictions.PACS: 72.25.Dc ; 73.23.Ad ; 73.63.Hs*Corresponding author: Department of Physics and Astronomy, ClippingerLaboratories, Ohio University, Athens OH 45701 ( Chen et al. Page 1 of 14 9/12/2003

The spin of electrons and holes in semiconductor heterostructures has attractedmuch interest, as a factor to realize new spin-based electronic device concepts [1], and forits potential in realizing quantum computational schemes [2]. In heterostructures, spincan manifest itself through strong and tunable spin-orbit interaction terms [3]. Recentstudies have often regarded spin-orbit interaction (SOI) as deleterious, since it can lead toshort spin-coherence times. However, semiconductor heterostructures can be fabricatedwith a long carrier mean free path, longer than lateral dimensions within reach of presentlithographic techniques. If the mean free path is longer than the lateral dimensions,charge transport in the geometry occurs ballistically, i.e. the preponderant scatteringevents involve the device boundaries [4]. In such mesoscopic devices, the decoherencedue to SOI is minimized, and SOI, together with the device geometry, can be exploitedfor spin manipulation, and for the preparation of spin-polarized carrier states. Theoreticalstudies have explored the effect of SOI on one-dimensional mesoscopic transport, and onvertical transport through heterostructures [5]. Here we present a method to create spinpolarizedbeams of ballistic electrons by utilizing elastic scattering off a barrier in astraightforward open geometry, and present experimental results verifying the realizationof the method. As illustrated in the upper panels of Fig. 1, a beam of two-dimensionalelectrons in a heterostructure is injected towards a barrier. Both energy and themomentum parallel to the barrier are conserved during the scattering event off the barrier.However, in the presence of SOI, scattering off the barrier leads to spin-flip events, andresults in different reflection angles for different spin polarizations. The spin-polarizedreflected beams can then be captured through suitably positioned apertures (upper leftpanel in Fig. 1). The multi-beam reflection process can be utilized to create spin-Hong Chen et al. Page 2 of 14 9/12/2003

from the stepwise increase in resistance as B is varied, added to the negativemagnetoresistance background. We also note here that the wet-etching process results inuncertainty in the structure’s dimensions, and that therefore a non-zeroBmay have to beapplied to center the three beams on the exit aperture. Hence, the 6 minima need not becentered around = 0 . Sample S underwent a deeper wet-etch, resulting in narrowerB2apertures, as betrayed by the higher resistance values. Hence, the range ofBwhere threebeams fit into the exit aperture of is reduced as compared to . Two steps inS 2resistance occur in such a narrow range of B that they are observed as one, resulting in 5observable minima. Assuming that the Bychkov-Rashba mechanism leads to theobserved minima, the data can be used to estimate the magnitude of the spin splitting.S 1SOI can be evaluated by the spin-splitting∆SOat the Fermi level E F, given by∆ = 2α k , where k denotes the Fermi wave vector andSOSOFFαSOdepends on materialand heterostructure parameters. Estimatingα SOfrom the experiments, we havecalculated the values ofBwhere cutoffs occur, using the equations derived below for theangular deviations from specular reflection,∆θ+→−and−→+∆θ (Fig. 1). The followingparameters are consistent with our experimental observations:α SO≈ 1× 10−6meV cm11and ∆ SO≈ 2.5 meV, at NS= 2.6 × 10 cm-2 and EF= 35 meV (the effective massm e= 0.014m0). This value for∆SOapproaches that obtained from the opticalmeasurements on similar InSb/InAlSb heterostructures [9]. The literature does not yetcontain experimental values for the Dresselhaus SOI parameters in InSb-basedheterostructures. Returning to the negative magnetoresistance background, we haveconsistently observed only a weak-localization peak at B ≈ 0 in mesoscopic geometriesHong Chen et al. Page 5 of 14 9/12/2003

fabricated in the InSb/InAlSb heterostructure, in contrast to the antilocalization signatureobserved in GaAs or InAs based 2DESs [11]. Another example of a weak-localizationpeak in a mesoscopic geometry is shown in the inset in Fig. 2, namely the resistance vs.applied perpendicularB , measured over an anti-dot lattice fabricated on the sameheterostructure [12]. The absence of antilocalization is not surprising in InSb.Antilocalization requires the Dyakonov-Perel’ spin scattering mechanism to dominate,leading to a randomization of the spin precession process due to a weak SOI [11]. Yet,due to large spin splitting in InSb, the impurity broadening of the electron energy is lessthan the spin-splitting, invalidating the conditions for Dyakonov-Perel’ scattering andantilocalization ( h / τ ≈ 0.5 meV

ˆx ywhere σ( )represent the Pauli matrixes. The first term in Eq. 1 originates from theperpendicular electric fieldF z, the second from the in-plane electric field. The materialparameterγ describes the strength of the SOI, andα = −eγF. The operator (1)SOzassumes averaging in the z-direction over the wave function in the well.Since the potential in the Hamiltonian depends only on the x-coordinate, thegeneral solution of the Schrödinger equation takes a form Ψ = Φ(x)e , where k is they-component of the momentum. Outside of the interaction zone with the barrier the wavefunction and energy of a single incident electron, have a form:ik y yyΨ 1 ⎛ 1 ⎞= ⎜ ⎟e±2 ⎝±e / i⎠2 2h kk , (2)2meinikrk, , Ekiϕ ( k )± = ±α sowhere k k x, k ) andtan ( ) = k / k . For the above spin states (+ or -), the spins are= (yϕ ky xperpendicular to the momentum due to the SOI. During the reflection process, thek yelectron conserves both energy and , leading to: Ψ = A(+ ) Ψ + A(−)Ψ ,outq+, + q−where the momenta of the reflected waves, q , k ) and q , k ) , are+= ( qx + ydetermined by kinematics equations. If the incident electron is in the state Ψ, −−= ( qx− yink,+, themomentumq, whileq k δ q is determined by conservation of energy:x+ = −k xx− = −x+x−inE+ ( kx, ky) = E −( −kx+ δ qx−, ky) . In the case of incoming state Ψ k,−, the momenta of thereflected waves will beqx=−− k x, andx+= −k x+ qx+q δ . From the above consideration,we deduce that the reflected wave for each incident state is composed of two beamsHong Chen et al. Page 7 of 14 9/12/2003

inpropagating at different angles: for Ψk,+at θ and θ+, and forinΨ k,−, atθ and θ−( θ+> η , the0− EF> ∆SO. If = ( ∆ / ) 1 / 2soEF θcthe solution ofSchrödinger equation in the regionx > 0contains an exponentially decayingβxcontribution, e − . In Fig. 3, T −+at θ > θcrepresents a squared amplitude of thisexponentially decaying contribution.We now assume that the incident beam of electrons is not spin-polarized andcontains electrons in both and , as is the case in our experiment. Due to theinΨ k,+inΨ ,k −k ySOI and conservation of the momentum, the scattered wave will consist of a triplebeam (Fig. 1). The side beams are fully spin-polarized whereas the middle beam does notHong Chen et al. Page 8 of 14 9/12/2003

carry spin polarization. Charge conservation arguments can be used to calculate thecurrent carried by the unpolarized middle beam () and by the polarized side beams( I and I ), assuming unit current in the incident beam. For example, if θ = 45°and+−η

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FIGURE CAPTIONSFIG. 1: Upper left: schematic of the geometry. Electrons are injected at the upperaperture, scatter from the left barrier and are collected at the lower aperture. Aperpendicular magnetic fieldTrajectories are indicated forimage of sampleB allows the trajectories to sweep the lower exit aperture.B = 0 (dotted line) and B = 7 mT (solid line). Lower left:. Upper right: the scattering geometry and the nomenclature for thespin-polarized beams. The spin states are denoted + and -. The deviations from specularreflection,S 1∆θ+→−and ∆θ−→+, lead to spin-polarized reflected beams. Lower right:geometrical interpretation of the spin-polarized scattering event, with incident andreflected wave vectors at the Fermi surface (for clarity only scattering of incident + spinstates is depicted). Energy and the momentum parallel to the barrier are conserved.FIG. 2: The four-contact resistance of the triangular structures and S , versus theS12perpendicular applied magnetic field B . The arrows indicate the values of B wherebeam cutoffs occur. Insert: magnetoresistance of an anti-dot lattice fabricated on thesame heterostructure (anti-dot diameter 0.4 µ m , periodicity 0.8 µm ), showing, forcomparison, a featureless negative magnetoresistance background. Geometricalresonances appear at higher B (not shown).FIG. 3: Calculated spin-dependent angular deviations ∆θ −→+and ∆θ +→−(top panel),and transmission coefficients (lower panels), as a function of the incident angle. Insertsillustrate the kinematics of the spin-dependent reflection. Parametersα SO≈1×10 −6meVcm and γ ≈10 −14cm2 were used.Hong Chen et al. Page 11 of 14 9/12/2003

1.50.0µmθθ+−±yx∆θ+→−∆θ−→+-1.5–+k yk x5 µmFIGURE 1(Hong Chen et al.)Hong Chen et al. Page 12 of 14 9/12/2003

280186T = 0.5 KResistance (Ohm)185184S 1545S 2278276183540274-10 10-10 -5 0 5 10B (mT)FIGURE 2(Hong Chen et al.)Hong Chen et al. Page 13 of 14 9/12/2003

FIGURE 3(Hong Chen et al.)Hong Chen et al. Page 14 of 14 9/12/2003

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