Abstract
The calculation of the electronic energy levels of ntype deltadoped quantum wells in a GaAs matrix is presented. The effects of hydrostatic pressure on the band structure are taken into account specially when the host material becomes an indirect gap one. The results suggest that under the applied pressure regime the GaAs can support twodimensional conduction channels associated to the deltadoping, with carrier densities exceeding 10(13) cm2.
GaAs; Deltadoping; Hydrostatic pressure
ELECTRONIC AND MAGNETIC PROPERTIES OF NANOSCOPIC SYSTEMS
Electronic states in ntype GaAs deltadoped quantum wells under hydrostatic pressure
M. E. MoraRamos^{I}; C. A. Duque^{II}
^{I}Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, C.P. 62210, Cuernavaca, MOR. México
^{II}Instituto de Física, Universidad de Antioquia, AA 1226, Medellín, Colombia
ABSTRACT
The calculation of the electronic energy levels of ntype ddoped quantum wells in a GaAs matrix is presented. The effects of hydrostatic pressure on the band structure are taken into account specially when the host material becomes an indirect gap one. The results suggest that under the applied pressure regime the GaAs can support twodimensional conduction channels associated to the deltadoping, with carrier densities exceeding 10^{13} cm^{2}
Keywords: GaAs; Deltadoping; Hydrostatic pressure
I. INTRODUCTION
Ultrathin semiconducting layers with exceptional quality can be obtained with the use of modern crystal growth techniques. Impurity seeding is achieved up to the atomic layer scale (ddoping). The localization of ionized impurities in a very thin layer gives rise to a very intense electric field which in turns causes a bending of the energy bands and the occurrence of a particular Vshaped potential. Work on ddoped structures was primarily in ntype, on Si and IIIV semiconducting materials (see, for instance, [18]). However, there is also an early report on this kind of system grown on Al_{x}Ga_{1x}As alloy [9]. In all cases, the main application sought for these systems is the fabrication of high electron mobility transistors provided the formation of a high density twodimensional electron gas.
The GaAsbased deltadoped systems are among the most studied both experimentally and from the theoretical point of view. It is known that for the ntype GaAs delta wells, the upper limit for the twodimensional density of ionized impurities is of about 10^{13} cm^{2} (see for instance [10] and references therein). This is a saturation limit and above it no additional increase in the electron concentration, N_{2D}, is achieved.
The aim of the present work is to present the calculation of the electron energy levels of ntype deltadoped quantum wells in the conduction band of GaAs, including the effect of hydrostatic pressure. This is done with the use of the localdensity ThomasFermi approximation [10,11]. Such approach has proven to be a simple and accurate alternative to selfconsistent electronic calculation in the twodimensional electron gas (2DEG) of deltadoped structures. Details of the model can be found in Refs. [11,12]. Particular attention will be paid to the effect of the transition from direct to indirect gap as a result of the increasing applied pressure (see Table I). To illustrate the situation, the figure 1 shows, in schematic form, the relative position of the G and X minima in the GaAs conduction band and the corresponding deltadoped quantum wells for a doping density of 10^{13} cm^{2}. It can be observed that the ground electronic energy level in the system moves from the Brillouin zone center (at P = 0) to locate at the Xpoint (the transition from direct to indirect energy gap occurs when P goes above 36.69 kbar).
The inclusion of the effects of the hydrostatic pressure is done by introducing a pressure dependence for each of the basic input parameters. That is, the position of the G and X minima with respect to the top of the valence band, the corresponding electronic effective masses, and the dielectric constant [13].
Tables II to V show the results of the calculation for the ground and first excited energy levels in ndeltadoped GaAs quantum wells considering the effects of the applied hydrostatic pressure. In each case, the formation of the quantum well is assumed for both the G and X minima of the conduction band. The idea is to study the conditions for which the ground state in the system will move away from being located at the Brillouin zone center. In order to give a homogeneous picture, the levels are reported considering the zero of the energy scale located at the valence band top edge.
For twodimensional densities of 5×10^{12} cm^{2} (Table II) and 10^{13} cm^{2} (Table III), it is clearly seen that even for pressures above the GX crossover, the ground state stays at k = 0. More specifically, the ground energy level starts locating at X only for pressures around 60 kbar. The reason for this to happen is that even at P = 40 kbar, the changes in the G band parameters are not large enough as to cause a significant modification of the quantum well features (energy position of its edge, depth, and average width) in this point of the Brillouin zone.
The two mentioned are admissible values for the ionized impurity density in GaAs, according to the above referred studies. It should be noticed that in all the previous literature on the subject the formation of the delta quantum well at G for atmospheric pressure is taken for granted. Here, we go beyond and calculate the spectrum of the deltadoped quantum wells assuming the possibility of having densities of 5×10^{13} cm^{2} (Table IV) and 10^{14} cm^{2} (Table V). The results for the ground level at the G minimum are only reported in the cases where they arise from a physically meaningful situation within the model. In this sense, even with the increase with pressure of the effective mass, and the decrease of the dielectric constant, the electrical environment in the material reflected in the effective Bohr radius will not allow for the formation of a delta well with such characteristics [14] (obviously, situations where the quantum well bottom turns out to be below the valence band top can not be accepted). This is equivalent to say that the system saturates and that those values of N_{2D} for the given pressures become unrealistic for they do not reflects in higher 2DEG densities.
However, such inconvenient is not present in the case of the X minimum. The higher values of the electron effective mass conditions the shape of the quantum well potential profile to be wider and not too deep for concentrations well above 10^{13} cm^{2}. In addition, the ground level of the system always locates at this point. The effective Bohr radius is smaller (at very high pressures it approaches the lattice constant), and it has been already shown that in such a case carrier concentrations can reach the order of 10^{14} cm^{2} [15,16]. Therefore, there is the possibility of having highdensity electronic channels.
III. CONCLUSIONS
The output of the present calculation indicates that the application of hydrostatic pressure to GaAs makes possible to attain high density twodimensional conduction channels associated to the X minimum in the conduction band of that material. This is a desirable feature that can lead, for instance, to higher electron mobilities. The results of our work suggest that it is worth to perform some experimental study in this direction aiming to determine whether such Xassociated conduction channels can be present; if not at normal pressure, at least for the case of an applied one.
IV. ACKNOWLEDGEMENTS
The authors acknowledge support from CONACYT (México) and COLCIENCIAS (Colombia) through bilateral agreement J200.729/2004. This work has been partially supported by CODIUniversidad de Antioquia and by the Excellence Center for Novel Materials ECNM, under Colciencias contract No. 0432005. M.E.M.R. wishes to thank Spanish Ministry of Science and Education for support through grant SAB20040199.
[1] E. F. Schubert, A. Fischer, and K. Ploog, IEEE Trans. Electron Devices ED33 (1986) 625.
[2] H. P. Zeidl, T. Wegehaupt, I. Eisele, H. Resinger, G. Tempel, and F. Koch, Appl. Phys. Lett. 50 (1987) 1164.
[3] R. L. Headrick, B. E. Weir, A. F. J. Levi, D. J. Eaglesham, and L. C. Feldman, Appl. Phys. Lett. 57 (1990) 2779.
[4] W. X. Ni, G. V. Hansson, J. E. Sundgren, L. Hultman, L. R. Wallenberg, J. Y Yao, L. C. Markert, and J. E. Greene, Phys. Rev. B 46 (1992) 7551.
[5] H. J. Gossmann, C. S. Rafferty, A. M. Vredenberg, H. S. Luftman, F. C. Unterwald, D. J. Eaglesham, D. C. Jacobson, T. Boone, and J. M. Poate, Appl. Phys. Lett. 64 (1994) 312; H. J. Gossmann, and F. C. Unterwald, Phys. Rev. B 47 (1993) 12618.
[6] B. E. Weir, L. C. Feldman, D. Monroe, H.J. Gossmann, R. L. Headrick, and T. R. Hart, Appl. Phys. Lett. 65 (1994) 737.
[7] L. M. R. Scolfaro, D. Beliaev, P. Enderlein and J. R. Leite, Phys. Rev. B. 50 (1994) 8699.
[8] Y. Wang, R. J. Hamers and E. Kaxiras, Phys. Rev. Lett. 74 (1995) 403.
[9] E. F. Schubert, C. W. Tu, R. F. Kopf, J. M. Kuo, and L. M. Lunardi, Appl. Phys. Lett. 54 (1989) 2592.
[10] L. M. GaggeroSager and R. PérezAlvarez, J. Appl. Phys. 78 (1995) 4566.
[11] L. Ioriatti, Phys. Rev. B 41 (1990) 8340.
[12] L. M. GaggeroSager, M. E. MoraRamos and D. A. ContrerasSolorio, Phys. Rev. B 57 (1998) 6286.
[13] See C. A. Duque, N. PorrasMontenegro, Z. Barticevic, M. Pacheco, and L. E. Oliveira, Microelectronics Journal 36, 231 (2005), and references therein.
[14] L. M. GaggeroSager, M. E. MoraRamos, and M. A. OlivaresRobles, Microelect. J. 36 (2005) 413.
[15] A. Aleksov, A. Vescan, M. Kunze, P. Gluche, W. Ebert, E. Kohn, A. Bergmaier, and G. Dollinger, Diam. Relat. Mater. 8 (1999) 941.
[16] M. E. MoraRamos, Diam. Relat. Mat. 17 (2003) 33.
Received on 8 December, 2005
 [1] E. F. Schubert, A. Fischer, and K. Ploog, IEEE Trans. Electron Devices ED33 (1986) 625.
 [2] H. P. Zeidl, T. Wegehaupt, I. Eisele, H. Resinger, G. Tempel, and F. Koch, Appl. Phys. Lett. 50 (1987) 1164.
 [3] R. L. Headrick, B. E. Weir, A. F. J. Levi, D. J. Eaglesham, and L. C. Feldman, Appl. Phys. Lett. 57 (1990) 2779.
 [4] W. X. Ni, G. V. Hansson, J. E. Sundgren, L. Hultman, L. R. Wallenberg, J. Y Yao, L. C. Markert, and J. E. Greene, Phys. Rev. B 46 (1992) 7551.
 [5] H. J. Gossmann, C. S. Rafferty, A. M. Vredenberg, H. S. Luftman, F. C. Unterwald, D. J. Eaglesham, D. C. Jacobson, T. Boone, and J. M. Poate, Appl. Phys. Lett. 64 (1994) 312;
 H. J. Gossmann, and F. C. Unterwald, Phys. Rev. B 47 (1993) 12618.
 [6] B. E. Weir, L. C. Feldman, D. Monroe, H.J. Gossmann, R. L. Headrick, and T. R. Hart, Appl. Phys. Lett. 65 (1994) 737.
 [7] L. M. R. Scolfaro, D. Beliaev, P. Enderlein and J. R. Leite, Phys. Rev. B. 50 (1994) 8699.
 [8] Y. Wang, R. J. Hamers and E. Kaxiras, Phys. Rev. Lett. 74 (1995) 403.
 [9] E. F. Schubert, C. W. Tu, R. F. Kopf, J. M. Kuo, and L. M. Lunardi, Appl. Phys. Lett. 54 (1989) 2592.
 [10] L. M. GaggeroSager and R. PérezAlvarez, J. Appl. Phys. 78 (1995) 4566.
 [11] L. Ioriatti, Phys. Rev. B 41 (1990) 8340.
 [12] L. M. GaggeroSager, M. E. MoraRamos and D. A. ContrerasSolorio, Phys. Rev. B 57 (1998) 6286.
 [13] See C. A. Duque, N. PorrasMontenegro, Z. Barticevic, M. Pacheco, and L. E. Oliveira, Microelectronics Journal 36, 231 (2005), and references therein.
 [14] L. M. GaggeroSager, M. E. MoraRamos, and M. A. OlivaresRobles, Microelect. J. 36 (2005) 413.
 [15] A. Aleksov, A. Vescan, M. Kunze, P. Gluche, W. Ebert, E. Kohn, A. Bergmaier, and G. Dollinger, Diam. Relat. Mater. 8 (1999) 941.
 [16] M. E. MoraRamos, Diam. Relat. Mat. 17 (2003) 33.
Publication Dates

Publication in this collection
04 Dec 2006 
Date of issue
Sept 2006
History

Received
08 Dec 2005