Dependence of Spin-orbit Parameters in Al x Ga 1 x N/GaN Quantum Wells on the Al Composition of the Barrier

Transcription

1 Commun. Theor. Phys. 60 (2013) Vol. 60, No. 1, July 15, 2013 Dependence of Spin-orbit Parameters in Al x Ga 1 x N/GaN Quantum Wells on the Al Composition of the Barrier Li Ming ( ) College of Electrical and Information Engineering, Xuchang University, Xuchang , China (Received March 18, 2013; revised manuscript received May 28, 2013) Abstract In this paper, we obtain considerable spin-orbit (SO) parameters in Al xga 1 xn/gan quantum wells (QWs) with sheet carrier concentration N s = /cm 2. With increasing Al content (x) of the barrier, the SO parameters increase as a whole, and the two major contributions are found to be the decrease of the expansion region of the envelope functions and the increase of the polarized electric field in the well. Compared with the Rashba parameters for the first two subbands, the intersubband SO parameter is a bit smaller and varies more slowly with x. The results indicate the SO parameters, especially the Rashba parameters can be engineered by the Al composition of the barrier, which may be helpful to the spin manipulation of III-nitride low-dimensional heterostructures. PACS numbers: Ej, Fg, Fs Key words: spin-orbit coupling effect, Rashba spin splitting, intersubband spin-orbit coupling, 2DEG 1 Introduction Research in nanoscience is of great importance for both technological implications and for the fundamental exploration of the properties of nanostructures. Of particular interest is the study of spin dynamics, which hopes to explore a number of interesting physical phenomena and potential applications in spintronic devices. [1 5] In this context, the spin-orbit coupling effect in low dimensional semiconductor heterostructures has attracted a considerable theoretical and experimental attention. [2,6 16] There are mainly two types of SO interactions in semiconductor heterostructures. One is the Dresselhaus SO interaction induced by the bulk inversion asymmetry of the lattice structure (BIA), [17] the other is the Rashba SO interaction induced by the structure inversion asymmetry (SIA) of the confining potential. [18] Recently, Esmerindo Bernardes, et al. found a new SO interaction in GaInAs/GaInAs QWs, which is nonzero even in symmetric heterostructures. [19 21] Since the SO parameters decrease as the band gap increases, one can expect very small SO parameters in wide band-gap materials, such as Ga(Al)N. However, in systems with asymmetric potential, both the Rashba and the intersubband SO parameters are related to the electric field. [4,18,22 23] Moreover, a high internal electric field generates at the heterointerface of Al x Ga 1 x N/GaN heterostructures, as a result of the large conduction band discontinuity and the strong piezoelectric and spontaneous polarization effect. Therefore the SO parameters in IIInitride semiconductor heterostructures grown along the c-axis is considerable and could be greatly engineered by the polarized field. [9,24 26] mingli245@163.com c 2013 Chinese Physical Society and IOP Publishing Ltd In previous work, the characteristic equation is projected into the subspace of the conduction band, and a general expression for the SO parameters are obtained, including the Rashba SO parameter for the first two subbands (α 1, α 2 ) and the intersubband SO parameter (η 12 ). [27 28] It is found that the SO parameters in the Al 0.5 Ga 0.5 N/GaN QW can be modulated by the well thickness [27] and gate voltage. [28] What s more, compared with the Rashba SO parameters, the intersubband SO parameter is of the same order of magnitude. However, because of the Piezoelectric and Spontaneous polarization, the Al mole fraction of the barrier has more considerable impact on the polarized electric field in the well, the confined energy, the band gap, and the spatial distribution of electrons. Therefore, we expect the SO parameters can be greatly modulated by the Al content of the barrier. In this paper, we explore the factors contributing to the SO parameters for QWs and how they vary as a function of x. 2 Theory and Model By eliminating the three valence band components of the envelope functions, the eight-band Kane Hamiltonian is projected into the 2 2 conduction band space [23 24] and the SO parameter in QWs with two subbands (v and v ) can be written as [9,19,21,27 28] η vv = P 1 P 2 [Φ v (0)Φ v (0)(β W β L ) Φ v (L)Φ v (L)(β W β R ) + v B L F L v L + v B W F W v W + v B R F R v R ], (1) 3 β = (E v ε)(e v ε ) 2 2, 3

2 120 Communications in Theoretical Physics Vol. 60 B = 3 [2E g (ε V )] {(E v ε)(e v ε ) 2 2. (2) 3 }2 E c and E v are the conduction edge and the valence edge at Γ point after the strain, respectively. [25] The confined energy level ε v and the envelope function Φ v are determined by solving the Schrödinger and Poisson equations with the finite difference method. [29] V is the self-consistent potential, and F = (1/q)( V / z) is the space-dependent electric field. Φ v (0) and Φ v (L) denote the amplitude of the envelope function for the v subband at the left and right heterointerface, respectively. 1 [(22 80x) mev] is the spin-orbit split-off energy, [30] and 2,3 (6.0 mev) are the crystal-field split energy. [31] P 1 and P 2 are interband momentum matrix elements, and P 1 = P 2 = E/2m 0 (E = 20 ev). [9] 3 Results and Discussion In this section, the SO parameters in Al x Ga 1 x N/GaN QWs with sheet carrier concentration N s = /cm 2 are calculated, and the contributions from the well, the heterointerface, and the barrier of the QW are discussed quantitatively. We adopt the material parameters listed in Tabel 1 in the calculation. Table 1 Parameters of Al xga 1 xn/gan QWs. Effective mass along c axis (m 0 ) m(z) = x Static dielectric constant (ε 0 ) ε r = x Conduction band offset (mev) E c = 1900x Elastic constants (GPa) C 13 = x, C 33 = x Piezoelectric coefficients (C/m 2 ) e 13 = x, e 33 = x [32] Fig. 1 Conduction band profile of Al xga 1 xn/gan QWs, the first two confined energy levels and the corresponding envelope functions for the Al 0.4Ga 0.6N/GaN QW. Figure 1 illustrates the conduction band profile of the Al x Ga 1 x N/GaN QWs, the energy levels and the corresponding envelope functions for the first two subbands in the Al 0.4 Ga 0.6 N/GaN QW. We can see that both the confining potential and the spatial distribution of electrons in the QW are asymmetric. These are induced by the polarized electric field and will certainly influence the SO parameters. The envelope function for the first subband has one peak near the left heterointerface, while that of the second subband has a peak near the left heterointerface and a valley approximately in the middle of the well. Moreover, the magnitude of the electric field around the left heterointerface increases with the Al content of the barrier. Fig. 2 The envelope functions ((a) and (b)) and the average positions of electrons in the first two subbands in Al xga 1 xn/gan QWs ((c) and (d)).

3 No. 1 Communications in Theoretical Physics 121 As can be seen from Eq. (1), five terms contribute to the SO parameters. The first two terms represent the contributions of the left and right heterointerface (Γ Inter ) and they depend on the amplitude of the first two envelope functions at the heterointerface. The third and the fifth terms denote contributions of the left and right barrier (Γ B ), and the fourth term represents the well s contribution (Γ W ). Γ B and Γ W are related to the electric field of the corresponding region and the envelope functions of the first two subbands. Figures 2(a) and 2(c) show the envelope functions and the average positions of electrons taking up the first subband of Al x Ga 1 x N/GaN QWs. As x increases, the ability of the QW to confine electrons becomes stronger with the elevation of the right barrier. Then the peak of the envelope function moves towards the left heterointerface, and the envelope function becomes less expanded. So the average position of electrons occupying the first subband moves gradually to the left of the QW. potential barrier becomes too low to confine electrons occupying the second subband. Therefore, for a given sheet carrier concentration, there are at least two confined states when the Al content of the barrier exceeds a critical value x min. So we assume x > 0.3 when discussing quantities about the second subband. As x increases, the envelope function for the second subband becomes less expanded, and the peak, especially the valley of it moves towards the left heterointerface. Therefore, when x > 0.4, the average position of electrons occupying the second subband also decreases with x. Fig. 4 Al-content-dependent intersubband SO parameter. Fig. 3 Dependence of the Rashba SO parameters for the first two subbands on the Al content of the barrier. Figures 2(b) and 2(d) show the envelope function and the average position of electrons occupying the second subband in Al x Ga 1 x N/GaN QWs. Because of the strong built-in electric field, the higher excited states may be squeezed outside of the well region. We can see clearly in Figs. 2(b) and 2(d) that when x = 0.2, z 2 is smaller than 30 Å, and the second confined state in the QW does not exist. That is to say, when x decreases to a certain value, the Figure 3(a) shows the increase of the Rashba SO parameter for the first subband (α 1 = η 11 ) as a function of x. The well s contribution is the largest, and next comes the contribution from the heterointerface, while the barrier s contribution is relatively small and varies slowly. With increasing x, the expansion region of the envelope function decreases, the magnitude of the envelope function at the left heterointerface increases, while that of the right heterointerface decreases. So the contribution from the heterinterface increases with x. What is more, the electric field in the well increases with x, especially the electric field around the left heterointerface is much stronger than that of the right heterointerface. The decrease of the expansion region of the envelope functions further leads to the increase of the average electric field in the well. So the well s contribution increases with x. Since the probability of electrons located in the barrier is relatively small, the barrier s contribution is small and varies slowly with x. Figure 3(b) illustrates the Rashba parameter for the second subband (α 2 = η 22 ). When x is larger than 0.3, just as α 1, the magnitude of α 2 also increases with x, and the contribution of the well is the largest, the lesser contribution is from the heterointerface, and the contribution of the barrier is the smallest and varies slowly. However, compared with α 1, α 2 is smaller, because the envelope functions of the higher subbands are more expanded and the corresponding average electric field is weaker.

4 122 Communications in Theoretical Physics Vol. 60 Figure 4 shows the intersubband SO parameter (η 12 ). When x > 0.3, the magnitude of η 12 increases slowly, and the main contribution is from the heterointerface, the lesser contribution with the opposite sign is from the well, and the smallest contribution is from the barrier, which varies slowly with x. According to Eq. (1), the envelope functions of the first two subbands and their overlap are crucial to η 12. Just as stated above, as x increases, the expansion region of the envelope function for the second subband decreases, the amplitude of the envelope function at the left heterointerface increases, and the average electric field in the well increases. Therefore, contributions to η 12 from the well and the heterointerface increase simultaneously with x, but the heterointerface contribution is canceled to a certain degree by the well contribution for opposite signs. However, the heterointerface contribution increase a bit more rapidly. So η 12 increase slowly with x. We can see from Eq. (2) that β and B i are related to the confined energy level, the band gap, and V, both of which depend on the Al composition of the barrier. However, their effects on the variation of the SO parameters as a function of x are not obvious, which indicates the decrease of the expansion region of the envelope functions and the increase of the electric field in the well are the two main factors contributing to the increase of the SO parameters. In all, the Rashba SO parameters for the first two subbands and the intersubband SO parameter increase with x as a whole. However, compared with the Rashba SO parameters, the intersubband SO parameter is basically of the same order, but relatively small and increases much more slowly with x. Compared with the effect of the well thickness and the gate voltage on the SO parameters in Refs. [27] and [28], respectively, the Rashba SO parameters can be more greatly modulated by the Al content of the barrier, while the intersubband SO parameter is also relatively small and varies slowly with x. Moreover, the SO parameters in the AlGaN/GaN QW are at least one order smaller than that of AlInAs/GaInAs Qws calculated in Ref. [19], since the band gap of GaN is much larger. 4 Conclusions In summary, we obtain considerable SO parameters in Al x Ga 1 x N/GaN QWs with sheet carrier concentration N s = /cm 2. The Rashba parameters for the first two subbands and the intersubband SO parameter in Al x Ga 1 x N/GaN QWs increase with x as a whole. We find that the decrease of the expansion region of the envelope functions and the increase of the polarized electric field in the well are the two major contributions. However, compared with the Rashba parameters, the intersubband SO parameter is basically of the same order, but a bit smaller and increases more slowly with x. These results show the SO parameters, especially the Rashba parameters can be modulated by the Al mole fraction of the barrier, which may be helpful to the spin manipulation of III-nitride lowdimensional heterostructures. References [1] D.D. Awshalom, D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation, Springer, Berlin (2002). [2] I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76 (2004) 323. [3] S. Datta and B. Das, Appl. Phys. Lett. 56 (1989) 665. [4] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnár, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger, Science 294 (2001) [5] D. Loss and D.P. Divincenzo, Phys. Rev. A 57 (1998) 120. [6] Ikai Lo, M.H. Gau, J.K. Tsai, Y.L. Chen, Z.J. Chang, W.T. Wang, J.C. Chiang, T. Aggerstam, and S. Lourdudoss, Phys. Rev. B 75 (2007) [7] X.W. He, B. Shen, Y.Q. Tang, N. Tang, C.M. Yin, F.J. Xu, Z.J. Yang, G.Y. Zhang, Y.H. Chen, C.G. Tang, and Z.G. Wang, Appl. Phys. Lett. 91 (2007) [8] W. Weber, S.D. Ganichev, S.N. Danilov, D. Weiss, W. Prettl, Z.D. Kvon, V.V. Bel kov, L.E. Golub, Hyun-Ick Cho, and Jung-Hee Lee, Appl. Phys. Lett. 87 (2005) [9] V.I. Litvinov, Phys. Rev. B 68 (2003) [10] S.D. Ganichev, V.V. Bel kov, L.E. Golub, E.L. Ivchenko, Petra Schneider, S. Giglberger, J. Eroms, J. De Boeck, G. Borghs, W. Wegscheider, D. Weiss, and W. Prettl, Phys. Rev. Lett. 92 (2004) [11] W. Yang and K. Chang, Phys. Rev. B 73 (2006) , and Phys. Rev. B 74 (2006) [12] Z.J. Chao, Y.S. Gui, X.Z. Shu, N. Dai, S.L. Guo, and J.H. Chu, Acta Phys. Sin. 53 (2004) [13] Y.S. Gui, G.Z. Zheng, S.L. Guo, J.H. Chu, and D.Y. Tang, Acta Phys. Sin. 48 (1999) 121. [14] L.Y. Shang, G.L. Yu, T. Lin, W.Z. Zhou, S.L. Guo, N. Dai, and J.H. Chu, Chin. Phys. Lett. 25 (2008) [15] Y.F. Hao, Y.H. Chen, G.D. Hao, and Z.G. Wang, Chin. Phys. Lett. 26 (2009) ; Chin. Phys. Lett. 26 (2009) [16] H.Z. Song, P. Zhang, S.Q. Duan, and X.G. Zhao, Chin. Phys. 15 (2006) [17] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [18] Y.A. Bychkov and E.I. Rashba, J. Phys. C 17 (1984) [19] Rafael S. Calsaverini, Esmerindo Bernardes, J. Carlos Egues, and Daniel Loss, Phys. Rev. B 78 (2008)

5 No. 1 Communications in Theoretical Physics 123 [20] Esmerindo Bernardes, John Schliemann, Minchul Lee, J. Carlos Egues, and Daniel Loss, Phys. Rev. Lett. 99 (2007) [21] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Physica Slovaca 57 (2007). [22] E.A. de Andrada e Silva, G.C. La Rocca, and F. Bassani, Phys. Rev. B 50 (1994) 8523; Phys. Rev. B 55 (1997) [23] R. Winkler, Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems, Springer, Berlin (2003). [24] V.I. Litvinov, Appl. Phys. Lett. 89 (2006) [25] M. Li, R. Zhang, Z. Zhang, W.S. Yan, B. Liu, Deyi Fu, C.Z. Zhao, Z.L. Xie, X.Q. Xiu, and Y.D. Zheng, Superlattice and Microstructure 47 (2011) 522. [26] M. Li, G. Sun, and L.B. Fan, Chin. Phys. Lett. 29 (2012) [27] M. Li, Y.H. Lv, B.H. Yang, Z.Y. Zhao, G. Sun, D.D. Miao, and C.Z. Zhao, Solid State Communications 151 (2011) [28] M. Li, R. Zhang, B. Liu, D.Y. Fu, C.Z. Zhao, Z.L. Xie, X.Q. Xiu, and Y.D. Zheng, Acta Phys. Sin. 61 (2012) [29] I.H. Tan, G.L. Snider, L.D. Chang, and E.L. Hu, J. Appl. Phys. 68 (1990) [30] M. Kumagai, S.L. Chuang, and H. Ando, Phys. Rev. B 57 (1998) [31] M. Suzuki, T. Uenoyama, and A. Yanase, Phys. Rev. B 52 (1995) [32] F. Bernardini, V. Fiorentini, and D. Vanderbilt, Phys. Rev. B 56 (1997) R10024.