REVISTA MEXICANA DE FISICA S53 (7) 74–77 DICIEMBRE 2007

Quantum confined Stark effect inn-type delta-doped quantum wells

A.M. Mitevaa, S.J. Vlaevb,d, V.T. Donchevc, and L.M. Gaggero-Sagerd

aSpace Research Institute, Bulgarian Academy of Sciences,Moskovska Str. 6, 1000 Sofia, Bulgaria.

b Unidad Academica de Fısica, Universidad Autonoma de Zacatecas,Calzada Solidaridad Esquina con Paseo la Bufa, 98060 Zacatecas, Zac., Mexico.c Department of Condensed Matter Physics, Faculty of Physics, Sofia University,

blvd. James Bouchier 5, 1164-Sofia, Bulgaria.dFacultad de Ciencias, Universidad Autonoma del Estado de Morelos,

Av. Universidad 1001, Col. Chamilpa, 62210 Cuernavaca, MOR., Mexico

Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007

Electronic states in Siδ-doped GaAs quantum wells with an electric field applied perpendicular to the layers have been studied. Thomas-Fermi approximation for the self-consistent external potential in the tight binding model and a spin-dependentsp3s∗ basis with nearestneighbours is used. The inhomogeneous Siδ-doped finite region is matched with two semi-infinite homogeneous GaAs barriers with thesurface Green-function matching method. The Stark shifts of the electronic states have been obtained for various Siδ-doped concentrations.The Stark shifts are compared with the available results in the case of rectangular and graded gap quantum wells.

Keywords:Quantum confined Stark effect; Siδ-doped quantum wells; tight-binding method.

Se estudian los estados electronicos en pozos de GaAsδ-dopados con Si en presencia de un campo electrico aplicado a lo largo de ladireccion de crecimiento. Se trabaja en el modelo semi-empırico de enlace fuertesp3s∗ con espin hasta primeros vecinos. El potencial deconfinamiento se encuentra en la aproximacion de Thomas-Fermi y con este potencial considerado como potencial externo se modifican loselementos diagonales del Hamiltoniano. La region inhomogenea del pozo donde se aplica el campo electrico se empalma con las barrerashomogeneas de GaAs usando el metodo SGFM. Los corrimientos de Stark se calculan para diferentes concentraciones del dopaje. Losresultados se comparan con los datos de la literatura para pozos rectangulares y gradados.

Descriptores:Efecto Stark cuantico; pozosδ-dopados con Si; metodo de enlace fuerte.

PACS: 73.21.Fg; 68.65.Fg; 85.35.Be

1. Introduction

Investigation of electric field effects (Stark effects) in theδ-doped semiconductor heterostructures has become possibledue to the abilities of modern crystal growth techniques tocontrol purity and dimensions. Quantum confined Stark ef-fect in quantum wells (QWs) is of great interest due to thefact that most of the devices based on QWs work under ap-plication of an electric field [1,2]. The single and multipleδ-doped structures are experimentally and theoretically in-tensively investigated in order to study their subband spectrawhich is important for the numerous potential applications inoptoelectronic devices. In this work, we present numericalcalculations of electronic energy states in Siδ-doped GaAsQWs at different impurity concentrationsn2D and under ap-plication of an external uniform electric fieldF . The calcu-lations are conducted using the semi-empirical tight binding(TB)method, which is often used to treat field-related effectsin electronic and optical properties of nanostructured materi-als and devices [3-5]. Theoretical works about electric fieldeffect in single and multiple Siδ-doped GaAs systems havebeen done mainly in the effective mass approximation [6-10].To the best of our knowledge, there are no tight binding con-siderations of the Stark effect in theseδ-doped systems. Thedetailed realistic microscopic description based on the TB isnowadays a required step to understand the behavior of mod-

ern semiconductor nanostructures and in such a way to sup-port the project of new promising nanostructures and nanos-tructured devices [11].

2. Model and method

We consider a Siδ-doped GaAs QW with impurity concen-tration n2D from 1×1012 to 10×1012 cm−2 with a step of1×1012 cm−2. The inhomogeneous Siδ-doped finite region(V-shapedδ-doped QW) consists of 250 monolayers (MLs)and is matched with two semi-infinite homogeneous GaAsbarriers. We use thesp3s∗ spin dependent semi-empiricalTB model and the surface Green function matching method,as it is described in [12]. The calculations are performedat the center of the two-dimensional Brillouin zone for theGaAs [100] growth direction. Theδ-potential is treated as anexternal potential in the Thomas-Fermi approximation [13].The external constant electric field is applied between the twosides of the QW in the growth direction. We add it as an ex-ternal potential to all diagonal elements of the Hamiltonianmatrix in each atomic layer:

TBii(n) = TBii(0) + neF, (1)

wheren labels the atomic layer number in the growth direc-tion; e is the electron charge;TBii are the diagonal TB pa-

QUANTUM CONFINED STARK EFFECT INN -TYPE DELTA-DOPED QUANTUM WELLS 75

FIGURE 1. The effective Si δ-potential profile forn2D=5.1012cm−2 at zero electric field and under the applied elec-tric fields F = 5, 10, 15, 20 and 25 kV/cm from top to bottom.

FIGURE 2. The dependence of the electron energy levels (EC0,EC1, EC2, EC3) on the applied electric field strength for a n-typeSi δ-doped GaAs quantum well, as functions of the impurity con-centrationn2D in units1012cm−2. Electron energies: (a) groundenergy levels EC0; and the three excited energy levels (b) - EC1; (c)- EC2; and (d) - EC3. The symbols are:n2D=3 (squares),n2D=5(circles),n2D=8 (open triangles),n2D=10 (solid triangles).

rameters,i is the atomic orbital index;F is the magnitudeof the applied constant electric field. The zero of the exter-nal potential is in the atomic layern = 0. The TB parame-ters used here are determined in Ref. 13.The applied electricfield is directed from the right to the left side of the QW,i.e.it moves downwards the right side of the GaAs conductionband edge. The intensity of the applied positive electric fieldF is zero outside the QW. In the calculations we use electricfields from 0 kV/cm to +25 kV/cm with a 1 kV/cm step. Thezero value of the energy is at the bottom of the GaAs conduc-tion band in the region outside the QW. Figure 1 shows theeffective confining Siδ-potential profile of our QW for donorconcentrationn2D = 5.1012cm−2 and for different appliedelectric fields. For larger concentrations, theδ-potential isdeeper.

FIGURE 3. Stark shifts of the ground (EC0) and three excited(EC1, EC2, EC3) energy levels (respectively (a), (b), (c) and (d))in Si δ-doped GaAs QW for concentrationsn2D(3, 5, 8, and 10).Here: n2D=3 (squares),n2D=5 (circles),n2D=8 (open triangles),n2D=10 (solid triangles).

FIGURE 4. Spatial distributions of the total spectral strengthfor the ground C0 (dotted line) and first excited C1 state (solidline), at given values of applied F in units (kV/cm) and for (a,b)n2D=1.1012cm−2 ,(c,d)n2D=10.1012cm−2.

3. Results and discussion

We have calculated the bound states in theδ-doped QWdescribed above. The number of bound states increaseswith increasing donor concentration. Our calculations givemany more bound states than other calculations based onthe envelope function approximation [6]. For example, forn2D = 10.1012cm−2 andF = 10 kV/cm we found 9 boundstates, while in Ref. 6 only three bound states appear athigh doping. Figure 2 presents the calculated energies ofthe ground (EC0) state and three excited states (EC1, EC2,andEC3) for four different two-dimensional carrier concen-trations(n2D = 3, 5, 8, and10) as a function of the elec-tric field. Here and further, the impurity concentrationn2D

is given in units1012 cm−2. We notice that the electric field

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76 A.M. MITEVA, S.J. VLAEV, V.T. DONCHEV, AND L.M. GAGGERO-SAGER

effects on energy levels are similar for all impurity concentra-tions. Increasing the value of the field decreases the energies.It is seen that at larger donor concentrations the energy lev-els are less sensitive to the electric field influence. Besides,the changes in the energies withn2D are less pronounced forthe ground (Fig. 2a) than for the three excited energy levels(Fig. 2b,c, and 2d). Figure 3 shows the Stark shift of theconsidered energy levels. When increasing the field, all en-ergy shifts increase in absolute value. The Stark shift of theground electron energyEC0 varies linearly with the appliedelectric field and does not depend on the impurity concentra-tionsn2D. The dependence of theEC1 andEC2 Stark shiftson the electric field is almost linear, too, but it slightly in-creases with decreasingn2D. The Stark shift ofEC3 is alsoalmost linear withF and slightly decreases with decreasingn2D, contrary to that ofEC1 andEC2 . We have verified thatthe Stark shift of the considered energy levels has the samemagnitude ifF has an opposite direction. Figure 4 showssome of the total spectral strength spatial distributions for thegroundC0 and first excitedC1 electron energy states, calcu-lated for different electric fields and impurity concentrations

FIGURE 5. Stark shifts (in meV) of the intersubband transitionenergies at differentn2D. Here: (a) - Stark shift of the transitionC0-C1; (b) - the same for C0-C2; (c) - for C0-C3;n2D=3 (squares);n2D=5 (circles);n2D=8 (open triangles);n2D=10 (solid triangles).

indicated on the figure. At zero field, all spatial distribu-tions are symmetrically situated around the center of our QW.At electric fields different from zero, the spatial distributionsare displaced from the center to the right. This is more pro-nounced for states with higher energy. The displacement islarger for lowern2D and largerF , as seen in Fig. 4. Ata givenF value, which depends onn2D, a secondary QWappears on the right side of theδ-doped QW, as also ob-served in [8,10]. With increasingF , the δ-doped QW be-comes narrower and shallower (from the right side), while thesecondary QW gets wider and deeper. As a result, the boundstates spatial distributions increasingly penetrate in the sec-ondary well (Fig. 4a). After a givenF value, the system rep-resents an asymmetric double QW with corresponding boundstates. That value ofF depends onn2D as follows:n2D = 1,F = 8; n2D = 2, F = 14; n2D = 5, F = 25; n2D = 8,F = 40; n2D = 10, F = 60. Further, we have calculated theintersubband transition energies between the statesC0 andCi (i = 1, 2 and 3) for different values of the applied elec-tric field. Figure 5 shows the Stark shifts of the intersubbandtransition energies for four differentn2D values as a func-tion of F . With increasingF , the Stark shifts of these threetransitions increase in absolute value, but in a different waydepending onn2D and the transition. For lowerF , the con-centrationn2D has a weak influence on the curves. For largerF , the increase is faster for lowern2D values in the case of∆E(C0−C1) and∆E(C0−C2), while it is opposite in thecase of∆E(C0− C3).

The above described behavior of the bound states andintersubband transition energies with an increasing elec-tric field is similar to the results reported by other au-thors [6,8,10]. The quantitative comparison of the Stark shiftsshows a little bit lower values compared to the results fromRef. 8 and larger values compared to [10].

In Refs. 4 and 5 we have studied the Stark shift of the in-terband transitions in rectangular and graded gap QWs. Thecomparison with the results presented here concerning the in-tersubband transitions inδ-doped QWs shows that the Starkshift has the same direction and the same order of magnitudefor similar values of the electric field. However, the spatialoverlap between the initial and final state wave functions isbetter preserved in the case ofδ-doped QWs. Therefore,δ-doped QWs are more suitable for device applications at lowfield values. The Stark shift of the intersubband transitionenergies in Siδ-doped GaAs QWs is large enough to be mea-sured experimentally at electric fields above∼ 5 kV/cm. Atlarger fields, the secondary QW dominates the potential pro-file and the initialδ-doped QW gradually vanishes.

4. Conclusions

The first TB calculations of the quantum confined Stark ef-fect in Si δ-doped GaAs QWs is presented. We have stud-ied in detail the Stark shifts of the electronic states and theirspatial distributions, as well as the subband spectra and inter-subband transitions of electrons. The results obtained help to

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QUANTUM CONFINED STARK EFFECT INN -TYPE DELTA-DOPED QUANTUM WELLS 77

better understand the properties ofδ-doped QWs with differ-ent impurity densities subjected to an electric field with dif-ferent magnitudes. Such investigations are very promising inlooking for δ-doped structures that provide good Stark effectcharacteristics for potential device applications, such as FETsand infrared devices, based on the electron intersubband tran-sitions. The results demonstrate that the TB method can beused to investigate the Stark effect in a double asymmetricQW system, which is interesting for coherent intraband radi-ation applications.

Acknowledgements

The support through the project UAZ-2007-35592, as well asfrom the Bulgarian National Fund (contract NoVU-F-203/06)is appreciated. One of the authors (S.J.V.) is especially in-debted to the Universidad Autonoma del Estado de Morelosfor its support.

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