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Progress In Electromagnetics Research Symposium, Beijing, China, March 2327, 2009 1361

Total Density of States in Rectangular Quantum Wells

S. Jelev-Vlaev1, Romeo de Coss2,A. Del Ro de Santiago1, and J. C. Martnez-Orozco1

1Unidad Academica de Fsica, Universidad Autonoma de Zacatecas

Calzada Solidaridad Esquina con Paseo La Bufa s/n, C. P. 98060, Zacatecas, ZAC, Mexico2Departamento de Fsica Aplicada, CINVESTAV-MeridaMerida 97310, Yucatan, Mexico

Abstract The total density of states in rectangular AlGaAs/GaAs quantum wells are cal-culated numerically summing in the two-dimensional Brillouin zone. The quasi-two-dimensionalenergy bands are observed for typical wells and the integrated spectral strengths are presented.The conditions for the formation of quantum wells for all electron K-vectors are discussed.

1. INTRODUCTION

The applications of the quantum wells are based mainly on the optical transitions around the highsymmetry points in the two-dimensional Brillouin zone [1]. By this reason the electronic structure

of the quantum wells has been studied extensively only in these points [2]. But all macroscopicproperties of any condensed matter system depend on the integrated (total) density of states [3].The experimental methods photoemission and inverse photoemission measure directly the totaldensity of states of the occupied and non-occupied electronic states respectively in atoms, molecules,crystals and nanostructures [4]. There is few experimental and theoretical information about thetotal density of states in quantum wells [1, 2]. In the present work we have conducted numericalcalculations of this quantity in typical rectangular quantum wells. The method of the special pointswas applied summing in 10, 36, 136, 528 and 2080 points of the two-dimensional Brillouin zone [5].We have studied the specific properties of the electronic spectrum in these quasi-two-dimensionalsystems within the framework of the semi-empirical tight-binding model and the Green functionformalism [6].

2. MODEL AND METHOD

The semi-empirical tight-binding model was applied within the framework of the Surface GreenFunction Matching (SGFM) method [6]. Nearest neighbor interactions and sp3s atomic baseincluding spin were taken into account. The (001) crystal growth direction was considered and theintegration over the two-dimensional Brillouin zone was made in 10, 36, 136, 528 and 2080 specialpoints. The bulk material Al(x)Ga(1 x)As was treated in the virtual crystal approximationand the Al barrier concentration was x = 0.35. The well width has been varied between 10 and50 monolayers. The small imaginary energy part was taken to be 0.01 eV or 0.001 eV depending

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Figure 1: Total density of states for bulk Al(x)Ga(1 x)As, GaAs and AlAs in the conduction band.

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1362 PIERS Proceedings, Beijing, China, March 2327, 2009

on the special points number. To determine the confinement conditions we calculated the total(integrated) density of states for the homogeneous bulk materials GaAs and AlGaAs. A quantumwell for all electron K-vectors appears with a barrier height of 150 meV. (See Figs. 1(a), 1(b)).

We can see in the Fig. 1 that the AlGaAs conduction band starts at energy 150 meV higher thanthe GaAs conduction band. This means that the electrons will be confined for all energies in theenergy interval between the band edges and for all K-vectors, if a finite GaAs slab is sandwichedbetween AlGaAs barriers.

3. RESULTS AND DISCUSSION

In Fig. 2(a) the total density of states in the energy interval of the confinement is presented for aquantum well of 15 monolayers.

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Figure 2: Integrated density of states and spatial distributions in the conduction band of rectangularAl(x)Ga(1 x)As/GaAs, x = 0.35 quantum well of 15 monolayers. (a) Integrated (total) density of states.(b) Integrated spectral strengths of the first sub-band. (c) Integrated spectral strengths of the second sub-

band. The zero of the energy is fixed at the top of the AlAs valence band in Gamma point.

The integration is conducted summing in 2080 special points of the Brillouin zone. There aretwo sub-bands of the same height with step-like behavior. It indicates that for all K-points thereare no more than two bound states. The spatial distributions of the first and second sub-band areshown in Figs. 2(b), (c). The spectral strengths are localized in the well region. The features inFigs. 2(a), (b), (c) correspond to the basic properties of the rectangular quantum wells of infinitebarriers [2]. There are contributions from the bulk density of states and the curves in Fig. 2 arenot completely as in the ideal case. Fig. 3 presents similar results for the density of states in wellsof 11 and 21 monolayers when only one bound states exists for all K-points.

The zero of the energy is fixed at the top of the AlAs valence band in Gamma point.

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2.20 2.22 2.24 2.26 2.28 2.300.00

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Figure 3: Total density of states for a single rectangular Al(x)Ga(1 x)As/GaAs, x = 0.35 quantum wellintegrating in 2080 special points over the 2DBZ. The well width is (a) 11 and (b) 21 monolayers and theprojection of the density of states is done in the middle of the well.

4. CONCLUSIONS

We have studied the problem of the electron confinement in rectangular quantum wells formedfrom the INTEGRATED density of states of the bulk materials GaAs and AlGaAs. We have foundenergetic intervals where the density of states has two-dimensional properties. The integratedspectral strengths for these energies are localized in the well region.

ACKNOWLEDGMENT

We deeply oblige the support of the Autonomous University of Zacatecas and PROMEP throughthe projects UAZ-2007-35592 and UAZ-2007-35523.

REFERENCES

1. Singh, J., Physics of Semiconductors and Their Heterostructures, McGraw-Hill, Inc., NewYork, 1993.

2. Harrison, P., Quantum Wells, Wires and Dots, John Wiley & Sons Ltd., 2000.3. Harrison, W. A., Elementary Electronic Structure, World Scientific, Singapore, 1999.4. Hughes, H. P. and H. I. Starnberg, Electron Spectroscopies Applied to Low-dimensional Mate-

rials, Kluwer Academic Publishers, 2000.5. Cunningham, S. L., Special points in the two-dimensional Brillouin zone, Phys. Rev. B,

Vol. 10, No. 12, 49884994, 1974.6. Vlaev, S., V. R. Velasco, and F. Garca-Moliner, Electronic states in graded-composition

heterostructures, Phys. Rev. B, Vol. 49, No. 16, 1122211229, 1994.