Manipulation of Persistent Spin Helix States by Weak External Magnetic Fields

Transcription

1 Commun. Theor. Phys. 61 (014) Vol. 61, No. 3, March 1, 014 Manipulation of Persistent Spin Helix States by Weak External Magnetic Fields HU Mao-Jin ( ), CHAI Zheng (Æ ), and HU Liang-Bin ( Í) Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou , China (Received July 31, 013; revised manuscript received October 9, 013) Abstract We study theoretically the effect of weak external magnetic fields on persistent spin helix states in semiconductor two-dimensional electron gases with both Rashba and linear-in-momentum Dresselhaus spin-orbit coupling. We show that in the presence of weak external magnetic fields, some basic properties of a persistent spin helix state, including the dispersion relation between the decay time and the magnitude of the wavevector, the maximum decay time and the value of the characteristic magnitude of the wavevector at which the maximum decay time occurs, will all depend sensitively on the directions of applied external magnetic fields. PACS numbers: d, 7.0.-i, Jt Key words: spin-orbit coupling, persistent spin helix state, effect of weak external magnetic field 1 Introduction In the recent years semiconductor spintronics has been an active research field in the community of science and technology, with the goal of using electron spins in semiconductors as an effective source for information storage and processing. [1 8] Within this context, the understanding of the physics related to intrinsic spin-orbit interaction (SOI) in semiconductors is crucially important. On one hand, SOI in semiconductors can lead to a variety of interesting phenomena, which may find some practical applications in spintronics. For example, the presence of SOI may provide a natural way of manipulating electron spins via purely electron means. [5 7] On the other hand, SOI also has the undesired effect of causing decay (i.e., relaxation) of nonequilibrium spin polarizations of electrons. A main obstacle for the practical realization of many semiconductor spintronic devices is just the rapid decay of non-equilibrium spin polarizations of electrons owing to SOI. Hence, the suppression and manipulation of spin relaxation in the presence of strong SOI is a major challenge in the field of semiconductor spintronics. In the study of spin-orbit coupled semiconductor electronic systems, semiconductor two-dimensional electron gases (DEGs) formed in III-V type semiconductor quantum wells or heterostructures have attracted a great deal of attention. [9 ] In such DEGs, there exist two types of SOI: Rashba SOI originating from structural inversion asymmetry (SIA) and Dresselhaus SOI due to bulk inversion asymmetry (BIA). [3 4] The band spin splitting caused by Rashba SOI is linear in the momentum k of electrons and the coupling strength can be controlled by an electric gate. Dresselhaus SOI can contain both k- linear and k-cubic contributions. The k-cubic contribution is usually much smaller than the k-linear contribution and can be neglected in many cases. For a semiconductor DEG with both Rashba and k-linear Dresselhaus SOI, the physics may exhibit some novelties when the strengths of the two types of SOI are equal. [9 16] For example, a new type of exact SU() symmetry may emerge if the strengths of the two types of SOI are equal, which may lead to the existence of long-lived spin helical states in such systems. [10] Such long-lived spin helical states are termed persistent spin helix (PSH) in the literature. [10 16] Experimentally, the existence of such long-lived PSH states has been demonstrated by means of transient spin-gating spectroscopy in GaAs/AlGaAs quantum wells. [11 13] An interesting question about PSH states is that whether they are stable in the presence of weak external magnetic fields or whether some of their basic properties can be manipulated by using of weak external magnetic fields. For the practical application of such long-lived spin helical states, it is necessary to understand how they will be affected in the presence of weak external magnetic fields. In the present paper, based on the spin diffusion model for spin-polarized transport, we present a theoretical investigation on the effect of weak external magnetic fields on PSH states in DEGs with both Rashba and k- linear Dresselhaus SOI. The theoretical results indicate that weak external magnetic fields may have some significant influences on such PSH states. On one hand, some basic properties of a PSH state may be manipulated effectively by using of weak external magnetic fields. On the other hand, the stability of a PSH state may also be destroyed by weak external magnetic fields. The sensitive response of a PSH state to weak external magnetic fields may provide a useful way for the manipulation of some basic properties of such states. Supported by the National Natural Science Foundation of China under Grant No c 013 Chinese Physical Society and IOP Publishing Ltd

2 386 Communications in Theoretical Physics Vol. 61 The paper is organized as follows: In Sec. we give a brief description on the theoretical model and the theoretical formalism used in the present paper. The results obtained will be presented and discussed in some detail in Sec. 3. Model and Theoretical Formalism We consider a disordered semiconductor DEG system formed in III-V type semiconductor quantum wells grown along the [001] direction. We set the [100] direction as the x axis, the [010] direction as the y axis, and the [001] direction as the z axis. In the presence of both Rashba and k-linear Dresselhaus SOI, the system can be described by the following single-particle effective-mass Hamiltonian, Ĥ = k m + α R(k yˆσ x k xˆσ y ) + α D (k xˆσ x k yˆσ y ) gµ Bˆσ B + V i (r). (1) Here m is the effective mass of electrons, k = (k x, k y ) is the two-dimensional electron wave vector, σ α (α = x, y, z) is the Pauli matrix, α R and α D are the Rashba and k- linear Dresselhaus spin-orbit coupling constants, B is the external magnetic field, g is the effective Lande factor, µ B is the Bohr Magneton, and V i (r) is a random spinindependent impurity potential. For the sake of convenience, the Rashba Dresselhaus spin-orbit coupling terms can also be written in a unified form as the following, H SO = k A σ k j A jαˆσ α, () j=x,y α=x,y where A is a second rank tensor, with the elements defined by A xx = α D, A xy = α R, (3) A yx = α R, A yy = α D. (4) In the semiclassical approximation, the quantum states of an electron with spin can be described by the Wigner distribution function, which is a matrix in spin space. In the Boltzmann limit, the following equation of motion can be obtained for the Wigner distribution function by using of standard nonequilibrium Green s function approach, [17 1,5] St[f k ] = f k t + m k rf k + 1 { A jα σ α, f } k x j=x,y α=x,y j + i k j A jα [σ α, f k ] j=x,y α=x,y 1 gµ B α=x,y,z B α [σ α, f k ], (5) where f k f(r, k, t) is the Wigner distribution function and St[f k ] is the scattering integral due to random impurity potentials. The effect of SOI is introduced by the third and fourth terms on the right-hand side of Eq. (5), where [σ,...] and {σ,...} denote commutator and anticommutator with the Pauli spin matrixes, respectively. Assuming that the random impurity potentials are short-ranged and spin-independent, the scattering integral St[f k ] can be expressed as St[f k ] = πn i v0 k δ(ǫ k ǫ k )(f k f k ), (6) where n i is the density of impurities, v 0 is the scattering amplitude, and f k f(r, k, t). By using of the Wigner distribution function, the time- and position-dependent spin polarization densities S α (r, t) and spin current densities Jα j (r, t) (flowing to the j-th direction with spin parallel to the α-th direction) can be computed as the following, S α (r, t) = 1 Tr {ˆσ α, f(r, k, t)}, (7) k Jα(r, j t) = 1 Tr {ˆν j α, f(r, k, t)}, (8) k where ˆν α j is the spin current operator and Tr denotes the trace over spin indices. By definition, the spin current operator is given by ˆν α j = (1/4){ˆσ α, ˆv j }, where ˆv j 1 Ĥ/ k j = k j /m + 1 α=x,y A jασ α is the electron velocity operator, which contains a spin-dependent part in the presence of SOI. The equation of motion for the Wigner distribution function can be solved by usual linear-response approximation. Within the linear-response approximation, from Eq. (8) one can show that the spin current density flowing to the j-th direction with spin parallel to the α-th direction can be expressed as Jα j = D S α + D ǫ αβγ à jβ S γ, (9) x j β=x,y γ=x,y,z where Jα j Jj α (r, t), S α S α (r, t), à jβ (m / )A jβ, ǫ αβγ denotes the usual three-dimensional Levi-Civita tensor, D = (1/)v τ is the diffusion constant of electrons, F and τ = 1/n i v0m is the momentum scattering time of electrons. The first term in Eq. (9) represents the diffusion contribution to the spin current, while the effect of SOI appears in the second term as a rotational term. From Eqs. (7) (8) one can further show that the timeand position-dependent spin polarization densities and spin current densities shall satisfy the following continuity equation, S α + Jα j ǫ αβγ à jβ Jγ j t x j j=x,y j=x,y β=x,y γ=x,y,z + Γ α S α gµ B (B S) α = 0, (10)

3 No. 3 Communications in Theoretical Physics 387 where Γ α = 1/τ s describes the Dyakanov Perel spin relaxation rate, [] with the relaxation time τ s defined by τ τ s = 8(mD/ ) (α + β ). (11) Substituting Eq. (9) into Eq. (10), we can get a set of coupled spin diffusion equations, which can be written as S x S x S x / x S y Ω S y + Γ S y / x t S z S z S z / x S x / y S x + S y / y Λ S y = 0, (1) S z / y S z where Ω, Γ,, and Λ are all 3 3 matrices, which are defined, respectively, by D 0 0 Ω = 0 D 0, (13) 0 0 D 0 0 DÃxy Γ = 0 0 DÃxx, (14) DÃxy DÃxx DÃyy = 0 0 DÃyx, (15) DÃyy DÃyx 0 D j=x,y à jy 1/τ s D j=x,y ÃjxÃjy gµ B B z gµ B B y Λ = D j=x,y ÃjxÃjy + gµ B B z D j=x,y à jx 1/τ s gµ B B x gµ B B y gµ B B x D j=x,y α=x,y à jα 1/τ s. (16) Based on the spin diffusion equations (1), the decay of a helical spin density wave can be described as the following. For a helical spin density wave with a given wavevector q, the time development of the spin polarization density can be expressed as S(r,t) = s(q, t)e iq r, where s(q, t) is the time-dependent factor. According to Eqs. (1), s(q, t) shall satisfy the following equation of motion, s(q, t) = t ˆL(q)s(q, t), (17) where ˆL(q) is a 3 3 matrix, whose elements can be written down directly from Eqs. (1) (16). In general, from Eq. (17) we can get three eigenmodes. We denote the eigenvalues and normalized eigenvectors of the three eigenmodes as λ i (q) and n i (q) (i = 1,, 3), respectively. For a helical spin density wave with a given initial condition S(r,0) = s 0 e iq r, we can expand s 0 in terms of the set of eigenvectors {n i (q)}. The solution of the time-dependent factor s(q, t) is then of the form s(q, t) = a i n i (q)e λi(q)t, (18) i=1,,3 where a i (i = 1,, 3) are the expansion coefficients. From Eq. (18) one can see that the decay rate of a helical spin density wave with a given wavevector q will be determined by the set of eigenvalues { λ i (q)} of the matrix ˆL(q), and the decay time of the i-th eigenmode can be defined by the inverse of λ i (q). 3 Results and Discussions In this section we will use the formalism introduced in Sec. to study the decay of a PSH state in the presence of weak external magnetic fields. For DEGs formed in III-V type semiconductor quantum wells grown along the [001] direction, a PSH state is a helical spin density wave with the wavevector q along the [110] direction. According to the SU() symmetry analysis of Ref. [10], if the strengths of Rashba and Dresselhaus SOI are equal (i.e., α R = α D ), a new type of exact SU() symmetry shall emerge when the magnitude q of the wavevector is equal to a certain characteristic value q max. This SU() symmetry shall render the decay time of a helical spin density wave with this characteristic wavevector being enhanced significantly. This conclusion can be reproduced by solving the spin diffusion equations established in Sec.. To illustrate this point, we first neglect the effect of external magnetic fields. In all results presented below, the direction of the wavevector q will be fixed along the [110] direction and the strengths of Rashba and Dresselhaus SOI will be set to be equal. In Fig. 1 we show the dependences of the decay times of the three eigenmodes of the spin diffusion equations on the magnitude q of the wavevector. From Fig. 1 we can see that the decay times of two eigenmodes decrease monotonously as q is increased. These two modes are two ordinary spin diffusion modes. In addition to the two ordinary modes, there is an anomalous mode whose features contrast significantly with ordinary spin diffusion. For this anomalous mode, the decay time first increases rapidly as q is increased towards a characteristic value q max and then decreases rapidly as q is increased further. The decay time of this anomalous mode is enhanced significantly at q = q max, hence the dispersion curve in q-space shows a sharp peak at q = q max. The value of q max is determined principally by the equal Rashba Dresselhaus spin-orbit coupling constants, with q max 4mα R / = 4mα D /. If disorder scattering is neglected, the decay time of this anomalous mode shall tend to be infinite at q q max, and due to this fact, such long-lived anomalous modes are termed persistent spin helix (PSH) states in the literature. [10 16] The ex-

4 388 Communications in Theoretical Physics Vol. 61 istence of such long-lived PSH mode had been verified experimentally by means of transient spin-gating spectroscopy in GaAs/AlGaAs quantum wells, and the dispersion curves shown in Fig. 1 are in good qualitative agreement with the corresponding experimental results. [11 1] Since at q = q max the decay time of the PSH mode is significantly larger than that of the two ordinary modes, from Eq. (18) one can see that, even if a local spin excitation is initially not in the form of a PSH mode, it will evolve diffusively into a PSH mode. Recently a direct mapping of the diffusive evolution of a local spin excitation into a PSH mode has also been demonstrated experimentally by a time- and spatially resolved magneto-optical Kerr rotation technique. [13] Fig. 1 The dependences of the decay times of the three eigenmodes of the spin diffusion equations on the magnitude of the wavevector in the absence of external magnetic fields. The input parameters used in the plot are: the electron effective mass m = 0.067m e, the electron density n = cm, the electron momentum scattering time τ = s, the Rashba and Dresselhaus spin-orbit coupling constants α R = α D = ev cm. Now we investigate what influences weak external magnetic fields may have on the properties of the anomalous PSH mode. The external magnetic field will be expressed as B = (B sin θ cosφ, B sin θ sinφ, B cosθ), where B is the strength of the magnetic field, θ is the polarization angle of the magnetic field with respect to the [001] axis, φ is the in-plane azimuthal angle of the magnetic field with respect to the wavevector q. By using of the formalism introduced in Sec., we find that the anomalous PSH mode shown in Fig. 1 is unstable and shall degenerate into an ordinary mode in the presence of strong external magnetic fields, i.e., the decay times of all eigenmodes of the spin diffusion equations shall decrease monotonously as the magnitude of the wavevector increases. In contrast, if the strength of the external magnetic field is weak enough, from the spin diffusion equations we can still get an anomalous mode in general, whose features are similar to that of the PSH mode as shown in Fig. 1. For this anomalous mode, these exists a characteristic magnitude q max of the wavevector at which the decay time will have a maximum value. This fact is illustrated clearly in Figs. (a) (d), where we show the dependences of the decay times of the three eigenmodes of the spin diffusion equations on the magnitude of the wavevector in the presence of an in-plane weak external magnetic field. For the four cases shown in Figs. (a) (d), the strength of the external magnetic field is set to be the same (B = Tesla), but the direction of the external magnetic field is set to be different, with the in-plane azimuthal angle φ = π/, 11π/16, 13π/16, and 15π/16, respectively. For each case shown in Figs. (a) (d), there are two ordinary spin diffusion modes whose decay times decrease monotonously as q increases, and there is also an anomalous mode whose decay time has a maximum value when q is equal to a certain characteristic value q max. In the region of q < q max, the decay time of this anomalous mode increases monotonously as q increases, and in the region of q > q max, the decay time decreases monotonously as q increases. These features are similar to that of the PSH mode as shown in Fig. 1. But from Figs. (a) (d) we can see that, in the presence of weak external magnetic fields, some basic properties of a PSH mode, including the dispersion curve in q-space, the maximum decay time at q = q max, and the value of the characteristic magnitude q max of the wavevector, will all depend sensitively on the direction of the magnetic field. For example, as can be seen from Figs. (a) (d), if we keep the strength of the magnetic field unchanged but its direction is turned away gradually from the direction perpendicular to the wavevector (φ = π/) to the direction parallel to the wavevector (φ = π), the value of the characteristic magnitude q max of the wavevector will become smaller and smaller, and the value of the maximum decay time of the PSH mode at q = q max will also be decreased gradually. When the direction of the magnetic field is turned to the direction parallel to the wavevector (i.e., φ = π), the PSH mode shall degenerate completely into an ordinary mode. These facts are more clearly illustrated in Figs. 3(a) 3(b), where we show the in-plane azimuthal angle dependences of the maximum decay time of the PSH mode and the characteristic magnitude q max of the wavevector when the direction of the external magnetic field is turned away gradually from the direction with φ = π/ to the direction with φ = π, while keeping the strength of the magnetic field unchanged. The results shown in Figs. (a) (c) are obtained in the presence of in-plane weak external magnetic fields. We find that if the magnetic field is out-of-plane, similar results can also be obtained. In general, in addition to two ordinary spin diffusion modes, an anomalous PSH mode

5 No. 3 Communications in Theoretical Physics 389 can still be obtained from the spin diffusion equations in the presence of out-of-plane weak external magnetic fields, but the dispersion curve of the PSH mode in q-space, the maximum decay time of the PSH mode, and the value of the characteristic magnitude q max of the wavevector at which the maximum decay time occurs, shall all depend sensitively on the direction of the magnetic field, similar to the results shown in Fig.. Fig. The dependences of the decay times of the three eigenmodes of the spin diffusion equations on the magnitude of the wavevector in the presence of in-plane weak external magnetic fields. For the four cases shown in (a) (d), the strength of the magnetic field is set to be the same (B = Tesla), but the direction of the magnetic field is set to be different, with the in-plane azimuthal angle φ = π/, 11π/16, 13π/16, and 15π/16, respectively. Other input parameters used in the plot are: m = 0.067m e, n = cm, τ = s, α R = α D = ev cm. Fig. 3 (a) The in-plane azimuthal angle dependence of the maximum decay time of the PSH mode when the direction of the external magnetic field is turned away gradually from the direction perpendicular to the wavevector (φ = π/) to the direction parallel to the wavevector (φ = π). (b) The corresponding in-plane azimuthal angle dependence of the characteristic magnitude q max of the wavevector. Other input parameters used in the plot are: m = 0.067m e, n = cm, τ = s, α R = α D = ev cm, B = Tesla.

6 390 Communications in Theoretical Physics Vol. 61 In summary, we have presented a detailed theoretical investigation on the effect of weak external magnetic fields on PSH states in DEGs with both Rashba and linearin-momentum Dresselhaus SOI. The results indicate that weak external magnetic fields may have some significant influence on some basic properties of a PSH state. In the presence of weak external magnetic fields, the dispersion curve between the decay time of a PSH state and the magnitude of the wavevector, the maximum decay time and the value of the characteristic magnitude of the wavevector at which the maximum decay time occurs, shall all depend sensitively on the directions of applied external magnetic fields. Such sensitive dependences suggest that some basic properties of a PSH state may be manipulated effectively by using of weak external magnetic fields. It should be noted that, since the emergency of a PSH state requires that the strengths of Rashba and Dresselhaus SOI be equal, if we want to manipulate these basic properties of a PSH state (e.g., to manipulate the value of the characteristic magnitude q max of the wavevector) through the tuning of the strengths of the two types of SOI, we need to change the strengths of the two types of SOI simultaneously. Though the strength of Rashba SOI can be tuned via usual electric means, the strength of Dresselhaus SOI is determined by the intrinsic bulk inversion asymmetry and cannot be tuned via usual electric means. The sensitive response of a PSH state to weak external magnetic fields may provide a useful alternative way for manipulating some basic properties of a PSH state. References [1] D.D. Awschalom, D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation, Springer, Berlin (00). [] R. Winkler, Spin-Orbit Coupling Effects in Two Dimensional Electron and Hole Systems, Springer, Berlin (003). [3] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76 (004) 33. [4] D.D. Awschalom and M.E. Flatte, Nature Physics 3 (007) 153. [5] S. Murakami, N. Naogaosa, and S.C. Zhang, Science 301 (003) [6] Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Science 306 (004) [7] B.A. Bernevig, T.L. Hughes, and S.C. Zhang, Science 314 (006) [8] M.M. Wu, J.H. Jiang, and M.Q. Weng, Phys. Rep. 493 (010) 61. [9] J. Schliemann, J.C. Egues, and D. Loss, Phys. Rev. Lett. 90 (003) [10] B.A. Bernevig, J. Orenstein, and S.C. Zhang, Phys. Rev. Lett. 97 (006) [11] C.P. Weber, J. Orenstein, B.A. Bernevig, S.C. Zhang, J. Stephens, and D.D. Awschalom, Phys. Rev. Lett. 98 (007) [1] J.D. Koralek, C.P. Weber, J. Orenstein, B.A. Bernevig, S.C. Zhang, S. Mack, and D. Awschalom, Nature (London) 458 (009) 610. [13] M.P. Walser, C. Reichl, W. Wegscheider, and G. Salis, Nature Physics 8 (01) 757. [14] M.C. Lüffe, J. Kailasvuori, and T.S. Nunner, Phys. Rev. B 84 (011) [15] Y. Kunihashi, M. Kohda, and J. Nitta, Phys. Rev. B 85 (01) [16] M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P. Olbrich, C. Schohuber, I. Caspers, V.V. Belkov, L.E. Golub, D. Weiss, K. Richter, J. Nitta, and S.D. Ganichev, Phys. Rev. B 86 (01) (R). [17] A.A. Burkov, A.S. Nunez, and A.H. MacDonald, Phys. Rev. B 70 (004) [18] A.G. Malshokov, L.Y. Yang, C.S. Chu, and K.A. Chao, Phys. Rev. Lett. 95 (005) [19] M. Hruška, Š. Kos, S.A. Crooker, A. Saxena, and D.L. Smith, Phys. Rev. B 73 (006) [0] A.V. Shytov, E.G. Mishchenko, H.A. Engel, and B.I. Halperin, Phys. Rev. B 73 (006) [1] C.H. Chang, J. Tsai, H.F. Lo, and A.G. Malshukov, Phys. Rev. B 79 (009) [] Z.F. Jiang, R.D. Li, S.C. Zhang, and W.M. Liu, Phys. Rev. B 7 (005) [3] Y. Bychkov and E.I. Rashba, J. Phys. C 17 (1984) [4] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [5] G.D. Mahan, Many-Particle Physics, Plenum, New York (1990).