Abstract
The dependence of the electron Landé gfactor on carrier confinement in quantum wells recently gained both experimental and theoretical interest. The g factor of electrons in GaAs(Ga,Al)As quantum wells is of special interest, as it changes its sign at a certain value of the well width. In the present work, the effects of an inplane magnetic field on the cyclotron effective mass and on the Landé g<FONT FACE=Symbol>^</FONT>factor in single GaAs(Ga,Al)As quantum wells are studied. Theoretical calculations are performed in the framework of the effectivemass and nonparabolicband approximations. The OggMcCombe Hamiltonian is used for the conductionband electrons in the semiconductor heterostructure, and the Landé g<FONT FACE=Symbol>^</FONT>factor theoretically evaluated is found in good agrement with available experimental measurements.
Magnetic fields; Quantum wells; gfactor; Cyclotron effective mass
ELECTRONIC AND MAGNETIC PROPERTIES OF NANOSCOPIC SYSTEMS
Effects of inplane magnetic fields on the electronic cyclotron effective mass and Landé factor in GaAs(Ga,Al)As quantum wells
E. ReyesGómez^{I}; C. A. Perdomo Leiva^{II}; L. E. Oliveira^{III}; M. de DiosLeyva^{I}
^{I}Dept. of Theor. Physics, Univ. of Havana, San Lázaro y L, Vedado 10400, Havana, Cuba
^{II}Departamento de Física, ISPJAE, Calle 127 s/n, Marianao 19390, Havana, Cuba
^{III}Instituto de Física, Unicamp, CP 6165, Campinas, São Paulo, 13083970, Brazil
ABSTRACT
The dependence of the electron Landé gfactor on carrier confinement in quantum wells recently gained both experimental and theoretical interest. The g factor of electrons in GaAs(Ga,Al)As quantum wells is of special interest, as it changes its sign at a certain value of the well width. In the present work, the effects of an inplane magnetic field on the cyclotron effective mass and on the Landé g_{^}factor in single GaAs(Ga,Al)As quantum wells are studied. Theoretical calculations are performed in the framework of the effectivemass and nonparabolicband approximations. The OggMcCombe Hamiltonian is used for the conductionband electrons in the semiconductor heterostructure, and the Landé g_{^}factor theoretically evaluated is found in good agrement with available experimental measurements.
Keywords: Magnetic fields; Quantum wells; gfactor; Cyclotron effective mass
Semiconductor heterostructures, such as quantum wells (QWs), quantumwell wires (QWWs), quantum dots (QDs), and superlattices (SLs) have been widely studied in the past three decades. Such interest was motivated by possible electronic and optoelectronic applications and by the need to understand fundamental properties of matter at nanoscale dimensions. In that respect, the transport of spinpolarized electrons by using ferromagnetic probe tips in a lowtemperature scanning tunnelling microscope opened up the possibility of investigating magnetic systems at spatial resolutions in the angstrom scale [1]. Of course, the ability to manipulate single spins [2] is one of the important aspects in the development of quantum information processing and spintronics. In particular, the dependence of the electronic effective mass and electron Landé gfactor on carrier quantum confinement in QWs and QDs has recently gained the community attention both experimentally as well as theoretically [38]. The gfactor of electrons in GaAsGa_{1x}Al_{x}As QWs is of special interest, as it changes its sign at certain values of the well width. In this study we are particularly interested in the experimental work by Hannak et al [4] who determined the electron Landé factor as a function of the GaAsGa_{1x}Al_{x}As well width from 1 to 20 nm under inplane magnetic fields by the technique of spin quantum beats. We are also concerned with the experimental data by Le Jeune et al [5] who studied the anisotropy of the electron Landé factor in GaAs QWs, and by Malinowski and Harley [6] who investigated the influence of quantum confinement and builtin strain on conductionelectron g factors in GaAs/Al_{0.35}Ga_{0.65}As QWs and strainedlayer In_{0.11}Ga_{0.89}As/GaAs QWs, for QW widths between 3 and 20 nm. Here, we are concerned with the effects of inplane magnetic fields on the cyclotron effective mass and Landé g_{^}factor in GaAsGa_{1x}Al_{x}As semiconductor QWs, within the effectivemass and nonparabolicband approximations [9,10], with theoretical results compared with available experimental measurements. Details of the present work will be presented elsewhere [11].
The effective Hamiltonian for the conductionband electrons in a GaAsGa_{1x}Al_{x}As QW, grown along the y axis, under an inplane magnetic field is given by
where =  i Ñ; = (yB,0,0) is the magnetic vector potential, are the Pauli matrices, m(y) and g(y) are the growthdirection positiondependent (with bulk values of GaAs or Ga_{1x}Al_{x}As) conductionelectron effective mass and Landé gfactor, respectively [12], µ_{B} is the Bohr magneton, l_{B} is the Landau length, a_{1}, a_{2}, a_{3}, a_{4}, a_{5} and a_{6} are constants appropriate to bulk GaAs [13], {,} is the anticommutator between the and operators, and V(y) is the confining potential, taken as 60 % of the Ga_{1x}Al_{x}As and GaAs bandgap offset [14]. The term proportional to G in (1) is the cubic Dresselhaus spinorbit term [15].
The eigenfunction of (1) may be chosen as
where n is the Landau magneticsubband index, and y_{0} = is the cyclotron orbitcenter position. Due to the low population of the conduction states at low temperatures, one may consider only the k_{z} = 0 contribution to the energy spectra, and by neglecting the offdiagonal terms in (1), the spin up and spin down states become uncoupled. We have denoted as _{ms} the diagonal terms of (1) for a given m_{s} projection (or) of the electron spin along the magneticfield direction, and expanded the corresponding wave functions in terms of the m,y_{0}ñ harmonicoscillator wave functions [11], i. e.,
After some algebraic manipulations, one straightforwardly obtains
and, by diagonalizing _{ms}, the eigenvalues E_{n} (y_{0},m_{s}). We would like to stress that the terms of order superior to the parabolic (^{2}) in (1) are quite often taken into account via perturbation theory [16], and in the present work they are exactly considered within the diagonal approach.
Here, the results refer to GaAsGa_{0.65}Al_{0.35}As QWs under inplane magnetic fields, as the experimental data from Hannak et al [4], Le Jeune et al [5], and Malinowski et al [6] on the electronic Landé g_{^}factor are for GaAsGa_{1x}Al_{x}As QWs with Al proportion corresponding to x = 0.35. In Figure 1 we display the ten lowest Landau levels as functions of the orbitcenter position in GaAsGa_{0.65}Al_{0.35}As QWs under an inplane magnetic field of B = 4 T and B = 20 T, and for the QW width of L = 50 Å and L = 500 Å. The orbitcenter position is in units of the well width L, and energies in units of the cyclotron energy w_{c} = , where m_{w} is the conductionelectron effective mass in the well material. Solid and dashed lines are associated with the Landau electron subbands with spin projections in the parallel () and antiparallel () directions of the inplane applied magnetic field along the + z axis, respectively. Note that the and electron subbands are essentially undistinguishable in the scale used in Figure 1. Also, as one may notice from Fig. 1 (a), for B = 4 T in an L = 50 Å GaAsGa_{0.65}Al_{0.35}As QW, and in the range of y_{0} orbit center considered, the lowest Landau energy subbands are essentially flat as a function of the orbitcenter position. This behavior is to be expected for small values of the applied magnetic field, as the QW width of 50 Å is small compared with the l_{B} = 128 Å Landau length. Therefore, in this case (L << l_{B}), the effect of the magnetic field is weak and the electronic Landau energylevel structure is essentially dominated by the barrier confining potential. As the GaAsGa_{0.65}Al_{0.35}As QW width is increased beyond the l_{B} Landau length, the orbitcenter position dependence of the electron Landau subbands becomes dispersive [cf. Fig. 1 (b)], with a minimum at the center of the well, i.e., y_{0} = 0. At low temperatures, therefore, electrons would tend to populate energy levels around y_{0} = 0. Notice that, at B = 20 T, the orbitcenter dependence of the electron Landau levels is more dramatic than for B = 4 T [cf. Figs. 1 (c) and (d)].
As it is well known, the technique of cyclotron resonance is a powerful tool in the study of the effective mass and transport properties of electrons in semiconductor heterostructures. An inplane magnetic field may modify the cyclotron effective mass due to the distortion of the Fermi contour by the applied field. In that respect, the inclusion of the band nonparabolicity is crucial in order to obtain a proper quantitative agreement with experimental measurements. For a given projection m_{s} of the electron spin, the m_{c} cyclotron effective mass associated with the nth Landau magnetic subband is defined by E_{n}_{+1}(y_{0},m_{s}) E_{n}(y_{0},m_{s}) = . The cyclotron effective mass (for n = 0 in the above equation) is shown in Figure 2 as a function of the orbitcenter position in GaAsGa_{0.65}Al_{0.35}As QWs of width L = 50 Å, and L = 500 Å, under inplane magnetic fields of B = 4 T and 20 T. In Figs. 2 (a) and ^{(c)}, the orbitcenter position dependence of the cyclotron effective mass is flat, which is due to the fact that the Landau energy levels are essentially independent of the orbitcenter position [see Figs. 1 (a) and (c)]. Moreover, the cyclotron effective mass is much smaller [cf. Figs. 2 (a) and ^{(c)}] than the bulk GaAs electron effective mass. One may argue that this is due to the large difference between the energies corresponding to the ground and firstexcited Landau levels. For L l_{B}, the cyclotron effective mass increases and tends to the bulk GaAs electron effective mass for the orbitcenter position at the center of the well, as one may see from Figs. 2 (b) and ^{(d)}. As the magnetic field increases, the effect of the barrier confining potential becomes less important than the magneticfield confining effect, the cyclotron effective mass increases, and for very large magnetic fields (L l_{B}), the difference between the n = 1 and n = 0 Landau levels is essentially given by w_{c} = , and m_{c} ® m_{w}.
With respect to the g_{^}factor, one may define DE_{n} = E_{n} (y_{0},,B)  E_{n}(y_{0},,B) = µ_{B} B, where is the effective Landé factor in the inplane direction (perpendicular to the ygrowth axis) associated to the E_{n}(y_{0},m_{s},B) Landau level, and the explicit dependence of the Landau levels on the applied inplane magnetic field is stressed. Notice the above equation is an adequate way of defining the effective Landé factor associated with the nth Landau magnetic subband and to the two lowest Zeeman and electronic sublevels. Moreover, it is clear that the effective factor of the the nth Landau magnetic subband will, in principle, depend on both the orbitcenter position and on the applied inplane magnetic field. Figure 3 shows the g_{^}  factor associated to and spin states in GaAsGa_{0.65}Al_{0.35}As QWs. In Fig. 3 (a) we display the magneticfield dependence of the g_{^}  factor corresponding to n = 0 Landau magnetic levels for various values of the QW width, and for the orbitcenter position at the center of the QW. Results were obtained for magnetic fields from 4 T to 20 T. In these range of magneticfield values, it is apparent that the g_{^}  factor depends weakly on the magnetic field. The fielddependence on the g_{^}  factor is due both by the modification of the energy bandstructure as well as by the redistribution of the wave function by the magnetic field. The orbitcenter position dependence of the g_{^}  factor associated to the n = 0 Landau magnetic level is shown in Fig. 3 (b) for B = 4 T and for various values of the well width. As the electronLandau levels are essentially flat for B = 4 T, L = 50 Å and L = 100 Å, for these values of the QW width, the g_{^}  factor does not appreciably depend on the orbitcenter position. As the QW width increases beyond l_{B}, the orbitcenter position dependence of the g_{^}  factor becomes appreciable. Finally, we show in Fig. 3 (c) the n = 0, n = 1 and n = 2 Landé g_{^}  factors as functions of the QW width for B = 4 T and for the orbitcenter position at the center of the well (solid lines). For n = 0, the sign of the electrong factors in the GaAs well and in the Ga_{0.65}Al_{0.35}As barrier are opposite. For the orbitcenter position at the center of the QW and for short values of the QWwidth (L l_{B}), the electron wavefunctions easily penetrate the Ga_{0.65}Al_{0.35}As barriers, and the g_{^}  factors are positive. On the other hand, for large values of the QW widths (L l_{B}), the g_{^}  factors are negative due essentially to the localization of the electron wavefunctions in the well material. Therefore, there must be a well thickness for which the g_{^}  factor is zero, which is clearly observed in Fig. 3 (c). The experimental results from Hannak et al [4], Le Jeune et al [5], and Malinowski et al [6] for the n = 0 Landau magnetic levels are also represented by squares, circles and triangles, respectively. One may notice that the present theoretical calculations are in excellent agreement with the experimental measurements.
In conclusion, we have studied the effects of an inplane magnetic field on the cyclotron effective mass and Landé g_{^}factor in single GaAs(Ga,Al)As QWs. Present theoretical results where carried out in the framework of the effectivemass approximation, with the nonparabolic conductionband effects taken into account via the OggMcCombe effective Hamiltonian for the conduction electrons. The QWwidth dependence g_{^}  factor reveals, as expected, a change in its sign, a fact which may be understood in terms of the electron wave function localization. Present theoretical calculations for the Landé g_{^}factor in single GaAs(Ga,Al)As quantum wells were found in excellent agreement with the experimental measurements reported by Hannak et al [4], Le Jeune et al [5], and Malinowski et al [6].
Acknowledgments
The authors would like to thank the Brazilian Agencies CNPq, FAPESP, Rede Nacional de Materiais Nanoestruturados/CNPq, and MCT  Millenium Institute for Quantum Computing/MCT for partial financial support. MdDL and ERG wish to thank the warm hospitality of the Institute of Physics, State University of Campinas, Brazil, where part of this work was performed.
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Received on 8 December, 2005
 [1] S. Heinze, M. Bode, A. Kubetzka, O. Pietzsch, X. Nie, S. Blugel, and R. Wiesendanger, Science 288, 1805 (2000);
 Z. Nussinov, M. F. Crommie, and A. V. Balatsky, Phys. Rev. B 68, 085402 (2003).
 [2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004);
 HA. Engel and D. Loss, Science 309, 586 (2005),
 [3] R. J. Nicholas, M. A. Hopkins, D. J. Barnes, M. A. Brummell, H. Sigg, D. Heitmann, K. Ensslin, J. J. Harris, C. T. Foxon, and G. Weimann, Phys. Rev. B 39, 10955 (1989);
 S. Huant, A. Mandray, and B. Etienne, Phys. Rev. B 46, 2613 (1992);
 B. E. Cole, J. M. Chamberlain, M. Henini, T. Cheng, W. Batty, A. Wittlin, J. A. A. J. Perenboom, A. Ardavan, A. Polisski, and J. Singleton, Phys. Rev. B 55, 2503 (1997).
 [4] R. M. Hannak, M. Oestreich, A. P. Heberle, W. W. Ruhle, and K. Kohler, Sol. Stat. Comm. 93, 313 (1995).
 [5] P. Le Jeune, D. Robart, X. Marie, T. Amand, M. Brosseau, J. Barrau, V. Kalevich, and D. Rodichev, Sem. Sci. Tech. 12, 380 (1997).
 [6] A. Malinowski and R. T. Harley, Phys. Rev B 62, 2051 (2000).
 [7] G. MedeirosRibeiro, M. V. B. Pinheiro, V. L. Pimentel, and E. Marega, Appl. Phys. Lett. 80, 4229 (2002);
 R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Ellerman, and L. P. Kouwenhoven, Phys. Rev. Lett. 91, 196802 (2003);
 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004);
 A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware, J. G. Tischler, A. Shabaev, Al. L. Efros, D. Park, D. Gershoni, V. L. Korenev, and I. A. Merkulov, Phys. Rev. Lett. 94, 047402 (2005).
 [8] E. I. Rashba and A. L. Efros, Phys. Rev. Lett. 91, 126405 (2003);
 R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 155330 (2003);
 S. J. Prado, C. TralleroGiner, A. M. Alcalde, V. LopezRichard, and G. E. Marques, Phys. Rev. B 69, 201310(R) (2004);
 C. F. Destefani and S. E. Ulloa, Phys. Rev. B 71, 161303(R) (2005).
 [9] N. R. Ogg, Proc. Phys. Soc. 89, 431 (1966);
 B. O. McCombe, Phys. Rev. 181, 1206 (1969);
 M. Braun and U. Rössler, J. Phys. C: Solid State Phys. 18, 3365 (1985).
 [10] J. Sabín del Valle, J. LópezGondar, and M. de DiosLeyva, Phys. Stat. Sol. (b) 151, 127 (1989);
 A. BrunoAlfonso, L. DiagoCisneros, and M. de DiosLeyva, J. Appl. Phys. 77, 2837 (1995).
 [11] M. de DiosLeyva, E. ReyesGómez, C. A. PerdomoLeiva, and L. E. Oliveira, Phys. Rev. B (in press).
 [12] E. H. Li, Physica E 5, 215 (2000);
 C. Hermann and C. Weisbuch, Phys. Rev. B 15, 823 (1977).
 [13] V. G. Golubev, V. I. IvanovOmskii, I. G. Minervin, A. V. Osutin, and D. G. Polyakov, Sov. Phys. JETP 61, 1214 (1985).
 [14] R. C. Casey, Jr., J. Appl. Phys. 49, 3684 (1978).
 [15] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
 [16] G. Lommer, F. Malcher, and U. Rossler, Phys. Rev. B 32, 6965 (1985);
 F. Malcher, G. Lommer, and U. Rossler, Superl. and Microstruct. 2, 267 (1986);
 G. Lommer, F. Malcher, and U. Rossler, Superl. and Microstruct. 2, 273 (1986);
 N. Kim, G. C. La Rocca, and S. Rodriguez, Phys. Rev. 40, 3001 (1989);
 B. Das, S. Datta, and R. Reifenberger, Phys. Rev. 41, 8278 (1990).
Publication Dates

Publication in this collection
29 Nov 2006 
Date of issue
Sept 2006
History

Received
08 Dec 2005