Quantum transport in nonuniform magnetic fields: Aharonov-Bohm ring as a spin switch

VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17D ECEMBER 2001 Quantum Transport in Nonuniform Magnetic Fields: Aharonov-Bohm Ring as a Spin Switch Diego Frustaglia,1Martina Hentschel,1and Klaus Richter1,2 1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany 2Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany (Received 4 May 2001; published 30 November 2001) We study spin-dependent magnetoconductance in mesoscopic rings subject to an inhomogeneous in-plane magnetic field. We show that the polarization direction of transmitted spin-polarized electrons can be controlled via an additional magnetic flux such that spin flips are induced at half a flux quantum. This quantum interference effect is independent of the strength of the nonuniform field applied. We give an analytical explanation for one-dimensional rings and numerical results for corresponding ballistic microstructures. DOI: 10.1103/PhysRevLett.87.256602 PACS numbers: 72.25.–b, 03.65.Vf, 05.30.Fk, 73.21.–b Recent experimental progress [1] in creating spinpolarized charge carriers in semiconductors indicates the principle ability to perform spin electronics [2] based on nonmagnetic semiconductor devices. This widens the field of usual magnetoelectronics in metals and opens up the intriguing program of combining the rich physics of spinpolarized particles with all the advantages of semiconductor fabrication and technology, e.g., precise design of nanoelectronic devices with controllable charge carrier densities and optoelectronical applications. Besides, the spin relaxation times involved can be rather long; coherence of spin states can be maintained up to scales of more than 100 mm [3]. Hence coherent control and quantum transport of spin states in semiconductor heterojunctions or quantum dots are attracting increasing interest [4], also in view of proposed future applications including spin transistors [5], filters [6], and scalable devices for quantum information processing [7], to name only a few. In nonmagnetic semiconductors the coupling of the carrier spin to an applied magnetic field can be used to control the spin degree of freedom. In this respect, nonuniform magnetic fields whose direction varies on mesoscopic length scales (textured fields) are of particular interest. Besides the usual Zeeman spin splitting, they give rise to a variety of additional effects absent in conventional charge quantum transport. In the limit of a strong magnetic field the electron spin can adiabatically follow the spatially varying field direction, and the spin wave function acquires a geometrical or Berry phase [8]. In mesoscopic physics, Berry phases were first theoretically studied for one-dimensional (1D) rings [9,10]. They are also expected to give rise to clear signatures in the magnetoconductance of two-dimensional (2D) ballistic microstructures [11]. Nonuniform magnetic fields on mesoscopic scales have been realized in semiconductors, for instance, by placing micromagnets [12,13] or ferromagnetic stripes [14] above or into the plane of a 2D electron gas in high-mobility semiconductor heterostructures. However, although magnetic inhomogeneities of up to 1T have been reported [15], it is difficult to experimentally reach the truly adiabatic regime. The coupling of the carrier spins to more realistic moderate inhomogeneous fields generally leads to nonadiabatic, spin-flip processes counteracting geometrical phases. Hence, despite various experimental efforts [13,16] a clear-cut demonstration of Berry phases in mesoscopic transport remains an experimental challenge. In this Letter we study nonadiabatic, spin-dependent coherent transport through ballistic mesoscopic rings in the presence of textured fields. This enables us, on the one hand, to quantitatively investigate for unpolarized electrons the relevant conditions necessary to observe geometrical phases or their nonadiabatic generalizations, Aharonov-Anandan phases [17]. On the other hand, we show for spin-polarized charge carriers how to use inhomogeneous fields to induce spin flips in a controlled way. For Aharonov-Bohm (AB) ring geometries (with in-plane nonuniform field) coupled symmetrically to two leads we demonstrate that the spin direction of polarized particles transversing the rings can be tuned and even reversed by applying an additional small control field. This quantum effect exists irrespective of adiabaticity. We consider symmetric 1D and 2D ballistic rings with two attached leads as shown in Fig. 1. For the inhomogeneous magnetic field we assume a circular configuration ៬ Bi共៬ r兲苷Bi共r兲ˆw苷共a兾r兲ˆw(in polar coordinates) centered around the inner disk of the microstructure [18]. The Hamiltonian for noninteracting electrons with effective mass mⴱand spin given by the Pauli matrix vector ៬ sreads, in the presence of a magnetic field ៬ B苷៬ =3៬ A, H苷1 2mⴱ∑៬ p1e c ៬ A共៬ r兲∏2 1V共៬ r兲1m ៬ B?៬ s.(1) The potential V共៬ r兲defines the confinement of the ballistic conductor. In our case the vector potential has two contributions, ៬ A苷៬ A01៬ Ai. The term ៬ Ai共៬ r兲generates the inhomogeneous field ៬ Bi共៬ r兲and ៬ A0represents a (weak) perpendicular uniform field ៬ B0or an AB flux fto be used as an additional tunable parameter to study the magnetoconductance. In Eq. (1), m苷gⴱmB兾2苷gⴱe¯h兾共4m0c兲 where mBis the Bohr magneton, m0is the bare electron mass, gⴱis the effective gyromagnetic ratio, and e.0. 256602-1 0031-9007兾01兾87(25)兾256602(4)$15.00 © 2001 The American Physical Society 256602-1 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17D ECEMBER 2001 FIG. 1. Geometries of ballistic microstructures used in the quantum calculations of the spin-dependent conductance for a circular (in-plane) magnetic field texture plus a magnetic flux f. Spin directions are defined with respect to the yaxis. We compute the spin-dependent conductance G共E,Bi,f兲for two-terminal quantum transport through the microstructures using the Landauer formula. We focus on the case where the two leads of width wsupport only one open channel [19]. The spin-dependent conductance then reads, for zero temperature, G共E,Bi,f兲苷e2 h共jt""j21jt##j21jt#"j21jt"#j2兲.(2) We define the spin direction with respect to the yaxis in Fig. 1. The transmission coefficients T#" 苷jt#"j2共T"# 苷 jt"#j2兲describe transitions between an incoming state from the right with spin up (down) to an outgoing state to the left with spin down (up). They vanish for Bi苷0. In the opposite, adiabatic limit of a strong magnetic field, the magnetic moment associated with the electron spin traveling around the ring stays (anti)aligned with the local inhomogeneous field. Hence, for the field geometry in Fig. 1(a) an incoming spin-up state is then converted into a spin-down state upon transmission through the ring, and vice versa. In the strong-field limit, T"" 苷jt""j2苷0and T## 苷jt##j2苷0. For the experimentally relevant, intermediate case of moderate magnetic fields one must solve coupled equations for the spin states to account for spin flips. We calculate the four spin-dependent transmission amplitudes by projecting the corresponding Green function matrix of the system onto the transverse mode spinors (of incoming and outgoing states) in the leads. We obtain the Green functions for the Hamiltonian (1) numerically after generalizing the recursive Green function method for spinless particles to the case with spin. This requires one to replace the on-site and hopping energies in a tight-binding approach by 232spin matrices [20]. We first study how adiabaticity is approached in mesoscopic rings by considering the spin-dependent transmission of unpolarized electrons in the full crossover regime between Bi苷0and the adiabatic limit. The appearance of geometrical phases requires an adiabatic separation of time scales: For 1D rings of radius r0the Larmor frequency of spin precession, vs苷2mB兾¯h, must be large compared to the frequency v苷yF兾r0of orbital motion with Fermi velocity yFaround the ring [10]. In the adiabatic limit a geometric phase g"共#兲is acquired during a round-trip. For the in-plane field considered, g"共#兲苷pgiving rise to a geometric flux 2f0兾2with f0苷hc兾e. With an AB flux fit adds up to an effective flux f2f 0兾2. This causes a shift in the AB magnetooscillations of T"# and T#" such that the overall transmission T共f苷0兲苷0[11], since also T"" and T## tend to zero; see above. The condition for adiabaticity can be written as q⬅v vs 苷kFr0 gⴱ共mⴱ兾m0兲共pr2 0B兾f0兲ø1, (3) with kF苷mⴱyF兾¯h. For 2D rings of width dand mean radius r0the angular 共ˆw兲component of ៬ kFis relevant for adiabaticity. For the mth propagating mode in a 2D ring, qin Eq. (3) is then replaced by the rescaled parameter qw⬅qp12关m兾共kFd兾p兲兴2(provided that d兾r0ø1). To show how adiabaticity is approached we consider transport through a ring with one open channel, m苷1. The solid line in Fig. 2 depicts the numerically obtained average transmission 具T共E,f苷0兲典Eas a function of 1兾qwfor the quasi-1D ring of Fig. 1(b) 共d兾r0苷0.25兲at f苷0. The average is taken over an energy interval (between the first and the second open channel) at fixed qwto smooth out energy-dependent oscillations. With increasing 1兾qwthe transmission 具T共E,f苷0兲典Etends to zero which is a clear signature of the geometrical phase as discussed above. The overall decay is Lorentzian, ⬃共11q22 w兲21 (dotted line in Fig. 2). This curve and the dashed line, which agrees well with the numerical result, are obtained in an independent transfer matrix approach for a 1D ring [Fig. 1(a), qw⬅q] to be discussed below. In our calculations 具kFr0典⯝15. In a typical experimental setup, kFr0苷2pr0兾lF⯝60 for r0⯝500 nm and 1兾q⯝0.07B关T兴and 0.86B关T兴for GaAs and InAs. However, despite the relatively large fields necessary for satisfying Eq. (3) for q, the scaling factor entering into qw allows one to reach adiabaticity for considerably lower FIG. 2. Energy-averaged quantum transmission as a function of the adiabaticity parameter qw(see text) for unpolarized electrons through rings with circular in-plane magnetic field (as in Fig. 1) and zero flux. The solid line represents numerical results for the geometry in Fig. 1(b). The dashed curve shows results from a corresponding transfer-matrix approach for a 1D ring [Fig. 1(a); see text]. The dotted line shows an overall Lorentzian dependence 0.916兾共11q22 w兲. Inset: 1D approximate [Eq. (4), dashed line] and full (solid line) results for strongly coupled leads (e苷0.5). In all curves 具T共E,f苷0兲典E!0for qw!0 owing to geometrical phases. 256602-2 256602-2 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17D ECEMBER 2001 field strengths. This is achieved by reducing either the width of quasi-1D rings or kFvia the electron density [21]. In the following we study how the spin-dependent transmission changes as a function of an additional flux f苷 pr2 0B0with B0øBi. Our main results are summarized in Fig. 3 showing 具T共E,f兲典Efor three scaled strengths qw艐20, 1.4, 0.25 of the inhomogeneous field. We consider up-polarized, incoming spins; equivalent results are obtained for spin-down states. In the weak-field limit, Fig. 3(a), 具T#"典(dotted line) is close to zero, and the total transmission (solid line) shows usual AB oscillations predominantly given by 具T""典(dashed line). The behavior is reversed in the adiabatic limit, panel (c), where 具T#"典exhibits AB oscillations, shifted by f0兾2due to the geometrical phase as discussed above. Panel (b) shows the general case of an intermediate field. With increasing flux the polarization of transmitted electrons changes continuously. Most interestingly, 具T#"典苷0 at f苷0, while 具T""典苷0for f苷f0兾2. For zero flux an ensemble of spin-polarized charge carriers is transmitted always keeping the spin direction, while for f苷f0兾2 FIG. 3. Averaged transmission for up-polarized incoming electrons (see Fig. 1) through a quasi-1D ring as function of a flux f苷pr2 0B0in the presence of a circular in-plane field Bi¿B0 of increasing strength: (a) weak, (b) moderate, and (c) strong. The overall transmission (solid line) is split into its components 具T""典(dashed line) and 具T#"典(dotted line). Note the change in the polarization upon tuning the flux and the spin-switch mechanism at f苷f0兾2. the transmitted electrons just reverse their spin direction. In other words, by tuning the flux from 0to f0兾2, one can reverse the polarization of transmitted particles in a controlled way. Hence, the AB ring plus the rotationally symmetric magnetic field acts as a tunable spin switch, independent of the field strength Bi.0, which determines only the size of the spin-reversed current. Alternatively, for afixed flux 0,f,f 0兾2(vertical dotted line in Fig. 3) the spin polarization is reversed upon going from the nonadiabatic to the adiabatic regime, while the total transmission remains nearly constant. This mechanism for changing the spin direction neither relies on the spin coupling to the control field B0nor on the Zeeman splitting often exploited in spin filters. It is a pure quantum interference effect which exists also for the nonaveraged transmission at a given energy. In the following we give an analytical explanation for the numerically observed effects (Figs. 2 and 3). To this end we consider the model of a 1D AB ring coupled to 1D leads, Fig. 1(a), and extend the transfer matrix approach for spinless particles [22] to the case with spin. We follow the method outlined in [23] but consider fluxes instead of probabilities to work with unitary transfer matrices. The eigenstates of the Hamiltonian (1), which are analytically obtained for a ballistic 1D ring [10], are necessary for implementing the transfer matrix algorithm. They read Cn,s苷exp共inw兲≠cs n共w兲, where the first factor describes the motion along the ring and the second refers to the spin state s苷",#[with respect to the vertical 共z兲axis]. The Zeeman term causes a slight difference in the kinetic energy of spin-"and spin-#electrons traveling clockwise or counterclockwise around the ring so that we must distinguish four possible n:n" j,n# j共j苷1, 2兲. They are given by n0⬅n1f兾f0, where the n0are the solutions of the equation [20] ˜ EF苷n0412n031共122˜ EF兲n022 2共˜ EF1˜mBcosa兲n01˜ E2 F2˜mBcosa2共˜mB兲2. Here, ˜ EF苷共2mⴱr2 0兾¯h2兲EFis the scaled Fermi energy, ˜m苷 共2mⴱr2 0兾¯h2兲m, and ais the tilt angle of the textured magnetic field with respect to the zaxis. In the general, nonadiabatic case four angles g" j,g# j#atake the role of a and characterize the spin eigenstates which read c" nj 苷0 B @ cosg" j 2 6ieiwsing" j 2 1 C A,c# nj 苷0 B @ sing# j 2 7ieiw0cosg# j 2 1 C A, with cotgj苷6关cota1共2n0 j11兲兾共2˜mBsina兲兴. In the adiabatic limit g!a. The transfer matrices in the generalized basis enter into the transmission formulas which generally require a numerical evaluation. For an in-plane field 共a苷p兾2兲without additional flux 共n苷n0兲the equations above simplify considerably and we find n"共#兲 1苷2共n#共"兲 211兲. Though being involved, all expressions leading to the transmission can be handled analytically. The transmission depends strongly on the coupling at the junctions between the ring and the leads. It 256602-3 256602-3 VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17D ECEMBER 2001 is given by a parameter e[22] where e苷0共0.5兲describes zero (strongest) coupling. Adjusting ethe analytical model allows us to estimate the effective coupling to the leads in ballistic rings used in the numerical calculations above. The dashed line in Fig. 2 共e苷0.316兲fits well with the numerical result (solid line). For e苷0.5 and f苷0,an approximate analytical expression for 具T共E,0兲典Ecan be given in compact form, if we replace the energy averages over rapidly oscillating angular functions involved by their mean. We find, leaving the details to [20], 具T共E,f苷0兲典E⯝16 cos2¯g1sin2共Dnp兾2兲 41cos4¯g1关12cos共Dnp兲兴2. (4) Dn⬅n" 12n# 1and ¯g1⬅共1兾2兲共g" 11g # 1兲can be expressed through qwas Dn苷共11q22 w兲1兾2and cos¯g1苷 共11q22 w兲21兾2. The inset in Fig. 2 shows the result (4) (dotted line) compared to the exact 1D result (solid line) for e苷0.5. All the general features of 具T共E,0兲典Ein Fig. 2 are well described by Eq. (4). Owing to destructive interference, the transmission vanishes at points where Dnis an even integer corresponding to 1兾qw苷p3, p15, . . . . Equation (4) gives a complicated overall decay factor for 具T共E,0兲典Ewhich reduces to the Lorentzian cos2¯g1苷 1兾共11q22 w兲in the limit e!0. Already for e,0.4 this is a good approximation for the overall crossover from the diabatic to the adiabatic regime (dotted line in Fig. 2). Within the 1D model we further reproduce the flux dependence for spin-dependent transport, Fig. 3, and find an analytical proof [20] for the spin-switch effect discussed above. The transmission coefficient T"" vanishes completely at f苷f0兾2, if the magnetic field to which the spins couple has no component perpendicular to the plane of the ring. Furthermore, we find numerically that the spinswitch effect does not occur only in quasi-1D rings but also in doubly connected structures with more than one open mode and arbitrary shape, as long as reflection symmetry with respect to the horizontal axis is preserved [20]. However, the effect requires single-channel leads [19]. We further note that rings with Rashba (spin-orbit) interaction [24] (yielding an effective in-plane magnetic field in the presence of a vertical electric field) might lead to a similar spin-switch effect. To summarize, we have studied nonadiabatic spin transport through ring geometries in inhomogeneous magnetic fields. We obtain, numerically and analytically, the explicit dependence of the transmission on the scaled field strength qw, which acts as an adiabaticity parameter, elucidating the role of geometrical phases in quantum transport and possible experimental realizations. For in-plane fields and symmetric ballistic structures we demonstrate how an additional small flux fcan be used to control spin flips and to tune the polarization of transmitted electrons. This quantum mechanism does not require adiabaticity. In combination with a spin detector such a device may be used to control spin polarized current, similar to the spin field-effect transistor of Ref. [5]. Whether related effects may prevail in diffusive ferromagnetic conductors remains as a further interesting problem. We thank H. Schomerus for very helpful comments and J. Fabian and D. Weiss for useful discussions. [1] R. Fiederling et al., Nature (London) 402,787 (1999); Y. Ohno et al., Nature (London) 402,790 (1999); P. R. 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