Spin dynamics through homogeneous magnetic superlattices

Transcription

1 See discussions, stats, and author profiles for this publication at: Spin dynamics through homogeneous magnetic superlattices Article in physica status solidi (c) February 27 DOI: 1.12/pssc CITATION 1 READS 1 2 authors: J. L. Cardoso Metropolitan Autonomous University 27 PUBLICATIONS 75 CITATIONS Pedro Pereyra Metropolitan Autonomous University 77 PUBLICATIONS 939 CITATIONS SEE PROFILE SEE PROFILE Available from: J. L. Cardoso Retrieved on: 1 May 216

2 Available at: off IC/25/61 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SPIN DYNAMICS THROUGH HOMOGENEOUS MAGNETIC SUPERLATTICES J.L. Cardoso Física Teórica y Materia Condensada, UAM-Azcapotzalco, México D.F., México and P. Pereyra Física Teórica y Materia Condensada, UAM-Azcapotzalco, México D.F., México and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract We develop a theory to study the spin dynamics of a 2DEG moving across a magnetic superlattice, where the external, tilted and sectionally homogeneous magnetic fields, induce spin transitions between the two spin components, along the superlattice. Using the transfer matrix approach, we provide a joint description of the spatial evolution of the spin-1/2 wave functions: amplitude and phase, in terms of the tilting angle, the Fermi energy and the magnetic field strengths. Clear signatures of coherent spin mixing and coherent spin flipping processes are also obtained. MIRAMARE TRIESTE August 25 Senior Associate of ICTP. ppereyra@correo.azc.uam.mx

3 1. INTRODUCTION The experimental observation of partially-polarized spin currents 1 3 represents a crucial property towards the understanding of spin-dependent transport and the possibility of designing real spintronic devices 4 6. To these purposes it is also important knowing the influence of the phase coherence phenomena on the spin-dependent tunneling processes. Optical experiments have shown that coherently precessing electronic spin reveals a polarization memory, which survives for nanoseconds 7, while a transport polarized current looses its features over large distances (approximately 1µm) 8. Giant magnetoresistance and the full spin polarized transmission coefficients arise when spin-1/2 electrons move through 2D diluted magnetic semiconductor systems. In this work, we study a system which bears some similarities with this kind of systems. We study a 2DEG moving through a homogeneous magnetic superlattice (HMS), within the framework of the transfer-matrix approach 9,1. In this system, electrons move along a quasi two-dimensional homogeneous semiconductor subject to external tilted magnetic fields B W and B S acting on alternating stripes (see figure 1, where for simplicity we choose B W = and B S = B). We show that in this 2D HMS, the dephasing and loosing of polarization memory depend strongly on the transmission probability and on the strength of the spin-components interaction, tuned through the magnetic-field tilting angle. We solve analytically the Pauli equation for spin-up and spin-down coupled channels and calculate the scattering amplitudes. We find solutions in terms of matrix confluent hypergeometric functions, which upon diagonalization determine the Landau energy levels for each spin projection. This scheme allows us to calculate only the longitudinal magnetoresistance at zero temperature but not the Hall resistance. We are interested on transport properties and spin-precession effects induced by the spin-field coupling interaction. We analyze the band structure behavior and the spatial evolution of the electron wave function in the superlattice. 2. THEORETICAL BACKGROUND To study the transport properties of a 2DEG in a HMS, we shall consider, in general, a two dimensional spin 1/2 charged-particle gas (2DCG) moving through a semiconductor guide of transversal widths w y and w z and sectionally constant external magnetic fields acting on alternating stripes W SW SW SW... along the x axis. In this superlattice, we denote by W the stripes where the strength of the magnetic is weaker than the magnetic B S in the stripes denoted by S. For the sake of simplicity, without any loos of generality, we will consider that B W = and B S = B. The lengths of these regions are l W and l S respectively. The magnetic field B has two components, perpendicular and parallel to the EG. As shown in figure 1, the angle θ H is measured from the positive z axis. A single-cell, with length l c = l W + l S, can be chosen in different ways, for example W 1/2 SW 1/2, where W 1/2 means one half of the stripe W. 2

4 θ H l C z y w z w y B x l S l W FIG. 1: The quasi-2d homogeneous magnetic superlattice is formed by a set of n cells, whose length l c = l W + l S and w y >> w z,. In each cell there are three regions, one is subject to an external magnetic field tilted with respect to z, which is in between of two regions were we assume that the magnetic field is zero, in general with different magnitude. This will be our choice. Whereas in each region W, the electron state vectors (for Fermi energies below the threshold of the second propagating mode 11 ), evolve according to the transfer matrix (here I is the unit matrix) M F (l F ) = Ieikl F, (1) Ie ikl F in the field regions S, the evolution of the spin-1/2 state vectors is governed by the two-channel 4 4 transfer matrix M H (l H ) = α β (2) β α where α = ( A + B i ( A /k Bk )) /2 and β = ( A B i ( A /k + Bk )) /2. Here, A and B are the hypergeometric matrices and A = 1 F 1 ( b 2 ; I ) 2 ; l2 H lb 2 cos θ H e (l2 H /l2 B ) cos θ H ( I b B = l H 1 F 1 ; 3 ) 2 2 I; l2 H lb 2 cos θ H e (l2 H /l2 B ) cos θ H, 3

5 which depend on ( ( ) k b = I 2 lb 2 2 cos θ H sin2 θ H wy 2 ( 1 2 cos θ H l B 2π 12) ) 1 (σ 2 z + tan θ H σ x ) /2. A and B are just the derivatives of A and B with respect to x 12. In order to calculate the single-cell transmission amplitude t 1, we need to obtain the singlecell transfer matrix M 1 which is given by M 1 = M F (l F /2) M H (l H ) M F (l F /2) = α β β α (3) Given M 1 we are ready to calculate the whole n-cells superlattice transmission amplitude 9,1 ( ) 1 t n = α n = t, t,, (4) where t i,j (here i and j label the spin-up ( ) or spin-down ( ) projections) is the transmission amplitude from channel j on the left to channel i on the right, and α n is the (1,1) block of the whole superlattice transfer matrix M n. The off-diagonal terms t, and t, are the transmission amplitudes for processes where an odd number of spin flips have taken place inside the 2D HMS. From now on, we omit the subindex n. The corresponding transmission coefficients are defined by T i,j = t i,j 2. Following the theory presented in Ref. magnetic superlattice. t, t, 1, we can calculate the wave function along the The right- and the left-side propagating state spinors at any point x inside the HMS, φ (x) and φ (x) respectively, are connected with the initial states at the beginning of the system (when x = x ) by a transfer matrix φ (x) φ = M (x ) (x x ) = α φ (x) φ (x ) β β φ (x ) α r. (5) φ (x ) Here r is the reflexion coefficient. The wave function φ (x) + φ (x) is related to the incoming wave function φ (x ) by φ (x) + φ (x) = a φ (x ) = a, a, a, a, where a i,j (with i, j = or ) are the amplitude wavefunctions at x. φ (x ) 3. TRANSMISSION COEFFICIENTS AND WAVE FUNCTIONS as We find convenient to keep the Zeeman energy constant, hence we write the magnetic field B = B z (tan (θ H ),, 1). (6) 4

6 2.5 T a) b) T T T Transmission Coefficient Energy [ev] Energy [ev] 1.7 c) Transmission Coefficient T T Energy [ev] FIG. 2: Transmission coefficients as functions of the energy for the angles θ H = o and θ H = 4 o. In a) we plot T, and T,. In b) we have T, and T,. The arrows indicate the resonant energies used to plot the wave functions along the superlattice. c) is an amplification of the dotted box drawn in b), a micro band appears in T, because the process of transition T, is supplied only by the spin up state. In this representation, it is clear that varying θ H the spin precession and the strength of the spincomponents interaction vary also. We can then use the tilting angle θ H as a tuning parameter. In this report, we will keep the geometrical parameters fixed. The relevance of varying the geometrical parameters and the magnetic field on spin filter devices, have been studied in Ref. 11. We are interested here on dephasing and memory loos, i.e. on physical processes where the two spin states interact. Before discussing the spatial evolution of phase and spin states, we shall start considering the effect of the tilting angle on the transmission coefficients. In the lower part of all panels in figure 2, we plot the transmission coefficients in the uncoupled channels limit 5

7 7, θ H = o a) Amplitude [u.a.] ,, 1π Phase [rad] 8π 6π 4π θ, θ, θ, b) 2π.1 l C.3 2l C.5 3l C.7 4l C.9 5l C x [µm] FIG. 3: Probability amplitudes a,, a, a, and phases along a superlattice with five cells and θ H o. Notice that a, 2 is localized at the surface of the system, while θ, has abrupt jumps because of the low transmission probability. θ, and θ, have similar behavior because of strong spin coupling. where θ H =. In the upper part, spin states are coupled and θ H = 4 o. For clarity reasons, the curves for T, and T, (in the coupled case) are vertically offset by 1.5 transmission units. The transmission coefficients T, and T, are plotted below the coupled T, and T,, respectively. Fig. 2c) is an amplification of the dotted box drawn in 2b). It is easy to see in these figures the influence of the parallel field component B x, not only on the well known band structure shifting, but also on the spin transition processes. The transmission coefficients T, and T,, exhibit the resonances of the two uncoupled-spin bandstructures. These resonances reflects certainly the passage of flux from one spin state to another. The origin of these transitions is the precession term, which stimulates a mixing between the propagating physical channels, in the HMS regions where the spin-field interaction is significant. Although the forbidden bands in T, and T, are well-defined within certain energy intervals, small resonances appear also in these curves when θ H. Therefore, we can assert conclusively that in the presence of channel mixing interactions, incoming electrons in a given propagating mode, will come out from the magnetic scattering system in the spin-state mode with opposite spin projection. It is easy to 6

8 6 5,, θ H = 4 o a) Amplitude [u.a.] 4 3 2, 1 1π 8π θ, b) Phase [rad] 6π 4π 2π θ, θ,.1 l C.3 2l C.5 3l C.7 4l C.9 5l C x [µm] FIG. 4: Probability amplitudes and phases as in figure 3 but with θ H = 4 o. Notice that a, 2 is now an extended state with phases strongly correlated when x 3µm. see that the third allowed band of T, was supplied by the second band of T,. We shall analyze the principal characteristics of the spin wave functions for a particular Fermi energy, specifically for the fourth resonance energy in the second allowed band of T,. This resonance energy is pointed out with an arrow in figure 2. At this energy and θ H =, the spin particles are fully transmitted while those with spin are completely reflected. The purpose is to see how these probabilities change as θ H differs from zero. In figures 3 and 4, the amplitudes a,, a, and a, and phases θ,, θ, and θ,, as functions of the position along the x-axis, are plotted for θ H o and θ H = 4 o, respectively. In both figures, the probability a, 2 is different from zero all along the magnetic superlattice. In these typical case of extended state, spin-up electrons are found almost everywhere inside the system. Since for this energy we are inside a spin forbidden band, the amplitude a, 2 for θ H = (i.e. in the absence of channel mixing interaction), shows a localized state behavior, different from zero only at the superlattice surface. However, when the channel mixing is turned up (see figure 4), the spin flux is supplied from the spin one. This passage of flux is maintained as far as the transitions are present. In this case, the wave behavior pattern of a, 2 and 7

9 a, 2 follow that of a, 2, except at the surface of the system. With the phases θ,, θ, and θ,, plotted in figures 3 b) and 4 b), one can obtain more information about the electron spin dynamics. The phases θ, and θ, in this specific case, have similar behavior along the superlattice, they grow gradually with jumps of π. The phase of a classical free particle at the left of potential barrier, vary by steps of π-value, due to backscattering. At variance with the classical case, in the quantum system we have the tunneling effect; the little curvatures drawn on the phase steps are precisely related to tunneling. Notice that when θ H, the phase θ, has abrupt jumps of π. Phase jumps occur where the probabilities a, 2, a, 2 and a, 2 approach a minima. When the minima is closer to zero, the corresponding jump is more pronounced and discontinuous. In the presence of spin-states coupling, such as for θ H = 4 o, the phase θ, decreases at the surface within the first cell, showing in this case a strong dephasing process, and rapidly couples with the behavior of θ,, with a phase delay of 2π. We can conclude that the incoming electrons with spin are totally reflected and are localized at the surface of the system, these electrons enter with their own phase, which they progressively lose. The projection spin state is gradually supplied from the spin flux. This implies that all the spin electrons, from the central cells to the end of the system, acquire a similar dynamic as the spin electrons, since all of them have the same origin. Notice that, the population in the magnetic superlattice is some how divided in two parts: one part containing localized spin electrons and the other, containing basically spin particles moving coherently. Finally, we can notice that the phase of the extended states grows monotonously with x. In contrast, phases of localized states decrease with x. 4. CONCLUSIONS We presented a theory to deal with spin 1/2 electrons moving through a 2D semiconductor subject to a sectionally homogeneous tilted magnetic field. We obtained the band structure of transmission coefficient as well as the wave function amplitude and phase for spin and spin electrons. For tilted magnetic fields the interaction between the spin and spin states lead not only to the well-known spin precession but also to clear spin coupling phenomena, neatly reflected in the wave function (amplitude and phase) behavior as function of x. Between the various effects that our results here exhibit, we have the passage of flux from one spin state to another and the dephasing and phase coupling phenomena. This approach allows also to study other process qualitatively different when the Fermi energy lies, for example in the allow bands of both spin components. In these case, the phase memory should remain longer in space and time. Acknowledgments. This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 8

10 References 1 M. Johnson, Science 26, 32 (1993). 2 H. Ohno, Science 281, 951 (1998). 3 D. J. Monsma, R. Vlutters, and J. C. Lodder, Science 281, 951 (1998). 4 G. A. Prinz, Science 282, 166 (1998). 5 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (21). 6 P. A. Ball, Nature 44, 918 (2). 7 J. M. Kikkawa, I. P. Smorchkova, N. Samarth, and D. D. Awschalom, Science 277, 1284 (1997). 8 J. M. Kikkawa and D. D. Awschalom, Nature 397, 139 (1999). 9 P. Pereyra, Phys. Rev. Lett. 8, 2677 (1998). 1 P. Pereyra and E. Castillo, Phys. Rev. B 65, 2512 (22). 11 J. L. Cardoso, P. Pereyra, and A. Anzaldo-Meneses, Phys. Rev. B 63, (21). 12 Details of our calculations will appear elsewhere. 9