Magneto-subbands in spin orbit coupled quantum wires with anisotropic 2-D harmonic potential

Transcription

1 phys. stat. sol. (a 4, No., (7 / DOI 1.1/pssa.6737 Magneto-subbands in spin orbit coupled quantum wires with anisotropic -D harmonic potential In-Taek Jeong 1, Min-Gyu Sung 1,, Gun Sang Jeon 1, Keon-Ho Yoo 3, and Jong-Chun Woo *, 1 1 School of Physics and Astronomy, Seoul National University, Seoul, , Korea Hynix Semiconductor, San 136-1, Ami-ri, Bubal-eub, Icheon-si, Kyeongki-do, Korea 3 Department of Physics, Kyung Hee University, Seoul, 13-71, Korea Received 3 July 6, revised 8 January 7, accepted 8 January 7 Published online 6 February 7 PACS 71.7.Ej, 73.1.Hb The spin states of electrons in a perpendicular magnetic field (B are studied with the introduction of anisotropic harmonic confinement and spin orbit coupling (SOC (both Dresselhaus and Rashba interactions. The effective g-factors, as well as B-dependence of subbands, are numerically calculated for GaAs and their dependence on confinement strength and degree of anisotropy is estimated. This model is in good agreement with previous reports [PRB 7, (5 and so on] in the isotropic limit and weak SOC, indicating its usefulness for slightly deformed quantum dots and short quantum wire. However, dispersion caused by mixing of spin states imposes limitations when SOC and confinement are comparable. 1 Introductive remarks In recent years, the physics associating with electronic spin in semiconductors has been widely studied in conjunction with the fundamentals involved in spin interference and spin Hall effects, and for the feasibility in applications such as spin devices. The spin states in low-dimensional (low-d semiconductors such as quantum dots (QD, quantum wires (QWR and quantum wells (QW attract particular attention due to the ability to manipulate relevant parameters by controlling the nanoscale structure as well as the dimensionality. The spin orbit couplings (SOC, both Dresselhaus [1] and Rashba [], are known to be the most dominant interactions related to the spin states in non-magnetic semiconductors and have been extensively studied in QDs and QWs, but to a lesser extent in QWRs. Of further interest regarding the spin states is the field dependence of their separation, namely, the g-factor, in conjunction with dimensionality change of the electron confinement as in the case of split-gate biased QWs. The g-factors in -D and -D have been reported in numerous papers [3 6]. The dependence of 1-D confinement and enhancement by the subband of the effective g-factor have been modelled with the introduction of Rashba SOC in the theoretical B dependence of the subband [7, 8]. However, a preliminary result from a quasi-1-d GaAs/AlGaAs QWR array [9] indicates anisotropic confinement needs to be considered in the case of finite length QWRs. It also indicates the contribution of Dresselhaus SOC is significant in the observed effective g-factors. In this paper, we study the subband states and spin separation with the introduction of an anisotropic -D parabolic potential, and reveal its relevancy and limit. * Corresponding author: jcwoo@snu.ac.kr, Phone: , Fax:

2 Original Paper phys. stat. sol. (a 4, No. (7 57 The model of anisotropic harmonic potential with spin orbit coupling The Hamiltonian of the electron confined by a -D anisotropic harmonic potential V( x, y = m* ( ωx x + ω y y / and in the magnetic field B = Bzˆ (and thus A = B( - yxˆ+ xyˆ/ is represented as with H = H + HZ + HSO, (1 1 eb e B m* H = ( p + p + ( xp - yp + ( x + y + ( ω x + ω y, ( m* x y m* y x 8m* x y H = g* µ Bσ, Z 1 B z and λr eb eb λd eb eb HSO = È σ x py x σ y px y σ y py x σ x px y, ħ Í Ê + ˆ - Ê - ˆ + È Ê + ˆ - Ê - ˆ Î Ë Ë ħ ÍÎ Ë Ë (4 where m * is the effective mass, e is the absolute charge, g * is the effective g-factor of bulk and µ B is Bohr magneton. Here λ R is the Rashba SOC constant and λ D is the Dresselhaus SOC constant in the linear term, neglecting the cubic term [1, 11]. Introducing the unitary transformation with U = exp [ i( αxy+ β p p ] and the ansatz of x y n1 n g( x, y nn x y ± N 1 1 Ckl n1 n n1 k F n l H ± - - k= l= Ψ (, χ = ( ω, ω e ÂÂ (, Η ( Η ( χ, (5 the B-dependent subband eigenvalues ε ± + - n1, n and the Zeeman separation n1, n = εn1, n - εn1, n are computed by diagonalizing a matrix, where Η n( ξ is the Hermite polynomials and ± represents the up/down states. The detailed derivation of the spin independent terms and the notations used in Eq. (5 can be found in Ref. [1]. (3 Fig. 1 (online colour at: Magneto-subbands of (n 1, n states (a for ω x = ω y, (b for ω x =.5ω y, i.e., l y = 1.5l x, and (c for ω x = 5ω y, i.e., l y =.4l x, where n 1, n =, 1,... The corresponding (= /µ B B is presented in units of g * in the insets, where we used the same line styles as in Fig..

3 58 I.-T. Jeong et al.: Magneto-subbands in spin orbit coupled quantum wires 1. Fig. (online colour at: Anisotropy dependence of spin separations in vs. l y /l x for l SO = 37 nm and l x = at B =.1 T, respectively. Each line corresponds to ( n, n as indicated in the figure (1, (1, (, (, 1 (, log (ω x / ω y 3 Cases of numerical analysis The simulation is performed using the parameters, m * =.67m e and g * =.44 which are the bulk values of GaAs. The SOC constants, λ R = ev cm and λ D = ev cm are taken from Ref. [11], which give the characteristic length [8, 13] of l SO = ħ /m* λ + λ 37 nm. Taking lx = ħ/ m* ωx = nm or ω x ª s 1, the B-dependent eigenvalues for ly/ l x = 1, 1.5 and.4 are shown in Fig. 1(a, (b and (c respectively. In Fig. 1(a, the subbands of the isotropic potential, in the QD case are presented, which show the same result as Ref. [11] computed with cylindrical symmetry. It can be easily seen that, as B ( ωc ωx > ωy, the n 1 subbands converge to the Landau levels, while the splittings by n gradually decrease tending to zero. The mixing of spin states becomes dominant in 1 1 the vicinity of * - ωso = ħ/ mlso ª s, which is clearly recognizable in the B-dependent spin separations summarized as geff = µ / BB in the insets. R D (1, (1, (, (, 1 (, (a log(ω so / ω x (1, (1, (, (, 1 (, (b log(ω so / ω x Fig. 3 (online colour at: Influence of SOC and low-d confinement (weak to strong SOC for (a ωx = ωy and (b ωx = 5ωy. The same line styles are used as in Fig..

4 Original Paper phys. stat. sol. (a 4, No. (7 59 In Fig., the dependence of on the anisotropy of the spatial confinement is simulated for weak SOC at weak fields. For given n 1 the change of by the anisotropy is very sensitive to n. As ωx ωy, converges to QWR limits where only the n 1 separation appears. In the vicinity of ωy ª ωso, the computation becomes invalid because of significant mixing of the spin states. In Fig. 3, the contribution of the SOC strength is simulated for isotropic and anisotropic ( ωx = 5ωy confinements. It should be noted that in the ωso < ωx case n 1 contributes positively, while n contributes negatively when λ D > λ R as in the case of most materials. It also shows that the anisotropy adds more fine manipulation capability of the spin states, as seen from the comparison of Fig. 3(a and (b. If ωso > ωx, the -D or 1-D confinement is smaller than the SOC and rapidly restores -D limit behavior. 4 Discussion This model provides a useful tool for the calculation of electronic subbands and spin separation of laterally anisotropically confined structures, such as deformed QDs and finite-length QWRs. It also shows good agreement with other theoretical works on QDs [1, 11] and QWRs [7, 8]. However, it presents contrasting results with reference [14], which uses the unitary transformation U1 = exp { im* [ λr( yσ x - xσ y + λd( xσ x - yσ y]/ ħ } on the same Hamiltonian as ours. Polarization dependent magneto-photoluminescence (MPL data, obtained from MBE-grown GaAs/Al.5 Ga.5 As QWR arrays (QWR 6 nm width is compared with this model. Here, QWR arrays were grown by the Petroff method [15] on (1GaAs substrate slightly tilted by toward the [11] direction. The QWRs are arrayed in the [11] direction. The lengths of most QWRs are much larger than their widths. It is presented with the average and the difference of the spin-up and down states in Fig. 4(a and (b, respectively. Various levels of excitation power are used to get different levels of electronic density. When we identify the magneto-subbands from MPLs with the lowest three subbands indexed as (n 1, n = (,, (, 1 and (,, the B-dependence of the average of the subbands are in reasonable agreement with the model for ω x = s 1, ω y = s 1, Photon Energy (ev peak 3 peak peak 1 (a Magnetic Field (T Spin Energy Separation (mev peak 3 peak peak Magnetic Field (T Fig. 4 (online colour at: (a Magneto-subbands of GaAs/AlGaAs QWRs of 6 nm width, averaged over spin up and down states. Three curves are obtained from MPL measurements for different electron densities indicating the ground, the first and the second excited states. The solid lines are subbands of (n 1, n = (,, (, 1 and (, from the model with ω x = s 1, ω y = s 1, λ R = ev cm and λ D = ev cm (corresponding to l x = 7.1 nm, l y = 7. nm, l SO = 14.1 nm. (b Corresponding spin state separations. (b

5 53 I.-T. Jeong et al.: Magneto-subbands in spin orbit coupled quantum wires λ R = ev cm, and λ D = ev cm (other parameters are the same as in Section 3, as shown in Fig. 4(a. However, our model does not provide satisfactory explanation on the observed Zeeman splittings. The cause of the discrepancy is mainly due to the mixing of spin states. The cubic contribution of the bulk inversion asymmetry may also be worth revealing for the resolution of the discrepancy. In this work, with introduction of anisotropic -D confinement model and SOC, we provide a useful tool to understand the spin states of elliptically deformed QDs and finite-length QWRs. We demonstrate the feasibility of manipulating the effective g-factors by the anisotropy of these materials. Acknowledgements This work is supported in part by the Korea Research Foundation Grant (KRF 3-15C185. Magneto-photoluminescence measurements in Fig. 4 were performed at NHMFL, Florida State University, Tallahassee, FL, U.S.A. which is supported by NSF Cooperative Agreement No. DMR-84173, the State of Florida and the DOE. References [1] G. Dresselhaus, Phys. Rev. 1, 58 (1955. [] Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 639 (1984. [3] P. Le Jeune, D. Robart, X. Marie, T. Amand, M. Brousseau, J. Barrau, V. Kalevich, and D. Rodichev, Semicond. Sci. Technol. 1, 38 (1997. [4] E. Tutuc, S. Melinte, and M. Shayegan, Phys. Rev. Lett. 88, 3685 (. [5] T. Nakaoka, T. Saito, J. Tatebayashi, and Y. Arakawa, Phys. Rev. B 7, (4. [6] M. T. Björk, A. Fuhrer, A. E. Hansen, M. W. Larsson, L. E. Fröberg, and L. Samuelson, Phys. Rev. B 7, 137 (5. [7] J. Knobbe and Th. Schäpers, Phys. Rev. B 71, (5. [8] S. Debald and B. Kramer, Phys. Rev. B 71, 1153 (5. [9] I. T. Jeong, K. H. Yoo, X. Wei, Y. D. Park, H. S. Jeon, D. S. Kim, S. Hong, and J. C. Woo, to be published. [1] C. F. Destefani and S. E. Ulloa, Phys. Rev. B 7, (5 [also Phys. Rev. B 71, (5]. [11] P. Stano and J. Fabian, Phys. Rev. B 7, (5. [1] O. Dippel, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 49, 4415 (1994. [13] J. Carlos Egues, G. Burkard, D. S. Saraga, J. Schliemann, and D. Loss, Phys. Rev. B 7, 3536 (5. [14] M. Valin-Rodrogiez, A. Puente, and L. Serra, Phys. Rev. B 69, 8536 (4. [15] P. M. Petroff, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 45, 6 (