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Physica E 27 (2005) 198–203

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Shallow donor impurity binding energy in the V-shapedquantum well under the crossed electric and magnetic fields

E. Kasapoglua,�, I. Sokmenb

aPhysics Department, Cumhuriyet University, 58140 Sivas, TurkeybPhysics Department, Dokuz Eylul University, Izmir, Turkey

Received 1 December 2003; accepted 9 November 2004

Available online 8 December 2004

Abstract

We have calculated variationally the ground state binding energy of a hydrogenic donor impurity in V-shaped

quantum well (VQW) or full-graded GaAs=Ga1�xAlxAs quantum wells in the presence of crossed electric and magnetic

fields. These homogeneous crossed fields are such that the magnetic field is parallel to the heterostructure layers and the

electric field is applied perpendicular to the magnetic field. The dependence of the donor impurity binding energy on the

well width and the strength of the electric and magnetic fields are discussed. We hope that obtained results will provide

important improvements in device applications, especially for narrow well widths and for a suitable choice of both

fields.

r 2004 Elsevier B.V. All rights reserved.

PACS: 71.55.Eq; 71.55.�i; 73.20.Dx

Keywords: Crossed electric and magnetic field; Impurity binding energy

1. Introduction

In recent years many theoretical and experi-mental investigations have been performed on theissue of the hydrogenic binding of an electron to adonor impurity which is confined within low-dimensional heterostructures [1–14]. The under-

e front matter r 2004 Elsevier B.V. All rights reserve

yse.2004.11.002

ng author. Tel.: +903462191010;

1186.

ss: ekasap@cumhuriyet.edu.tr (E. Kasapoglu).

standing of the electronic and optical properties ofimpurities in such systems is important because theoptical and transport properties of devices madefrom these materials are strongly affected by thepresence of shallow impurities. And also, magneticand electric fields are effective tools for studyingthe properties of impurities in heterostructures. So,the use of the combined effects of an electric andmagnetic field on impurity states is also of interest,but little has been studied on this subject. Cen andBajaj [4,15,16] have used a variational method to

d.

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E. Kasapoglu, I. Sokmen / Physica E 27 (2005) 198–203 199

consider the impurity states in symmetric andasymmetric quantum wells subjected to parallelelectric and magnetic fields directed perpendicularto the heteroplanes. Wang and Chuu [17,18] havecalculated free electron energy levels in single anddouble asymmetric quantum wells in the presenceof a crossed electric field perpendicular to theheteroplanes and a magnetic field in the plane.Monozon et al. [19] have shown that the analyticalapproach can be used for the consideration of animpurity electron (hole) in an infinite quantumwell in the presence of crossed electric andmagnetic fields. Among the various systems undercurrent investigation, the quantum wells (QWs)have attained considerable theoretical and experi-mental attention. In order to get sufficient tuningrange, various QW structures such as stepped QW,graded-gap, two-step and coupled asymmetric QW[20–22,23, and references therein] have beeninvestigated in an effort to enhance the electric-field-induced changes. In these structures, theStark shifts, the changes in oscillator strengthsand absorption coefficients were predicted theore-tically and confirmed experimentally to be largerthan changes that occur in conventional squarepotential QWs [23]. V-shaped quantum wells(VQWs) have some unique properties, whichcannot be obtained with conventional QWsincluding the sensitivity of the intersubbandabsorption for the normal incident lights [24].Furthermore, VQWs have potential applicationsin novel semiconductor diode laser [25] as well asin infrared optoelectronics based on intersubbandQW transitions [26].

In this study, we report a calculation, withthe use of a variational approximation, of theground state binding energy of a hydrogenicdonor impurity in VQW or full-gradedGaAs=Ga1�xAlxAs quantum wells in the presenceof crossed electric and magnetic fields. Electricfield is parallel—applied to the growth direction(z-direction), magnetic field is perpendicular—applied to the growth direction or electric field.In our calculations, we obtained the explicitdependencies of the impurity binding energy onthe well width, the strengths of the electric andmagnetic fields. The V-shaped or full-gradedpotential profile is obtained by changing linearly

from 0 to 0.3 the aluminium concentration—x inthe Ga1�xAlxAs layer. As known, one importantaspect of electronic band structure engineering isthe realization of graded heterostructures, in whichthe composition is varied continuously in space.Electronic and optoelectronic devices, which ex-ploit these effects include, to date: graded-baseheterostructure bipolar transistors, which promotethe egress of carriers through the base; gradedseparate confinement heterostructure laser activeregions, which not only confine light to thequantum wells, but may also promote transportwithin the active region and increase devicebandwidth [26,27].

2. Theory

We define the z-axis to be along the growth axis,and take the electric field to be applied to thegrowth direction, and the magnetic field to beapplied to the x-axis, i.e. ~B ¼ ðB; 0; 0Þ:We choose avector potential A in written form ~A ¼ ð0;�Bz; 0Þto describe the applied magnetic field.Within the framework of an effective—mass

approximation, the Hamiltonian of a hydrogenicdonor impurity in VQW or full-gradedGaAs=Ga1�xAlxAs quantum well in the presenceof crossed electric and magnetic fields, can bewritten as

H ¼1

2me

~pe þe

c~Að~reÞ

h i2þ V ðzeÞ þ eFze

�e2

�oj~re �~rij; ð1Þ

where me is the effective mass, e is the elementarycharge, ~pe is the momentum, �o is the dielectricconstant, ~r ð¼~re �~riÞ is the distance between thecarrier and the donor impurity site, F is the electricfield strength and V(ze) is the confinementpotential profile for the electron in the z-direction.VQW potential profile and amplitude of nor-

malized subband wave function of electron—jcð~zÞj2 versus the normalized position—~z ¼ z=L

are given in Figs. 1A and B for F ¼ 0; B ¼ 0 andFa0; Ba0; respectively (vertical lines indicate theedges of barriers). The functional form of the

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Fig. 1. VQW potential profile and amplitude of normalized

subband wave function of electron—jcð~zÞj2 versus the normal-

ized position—~z ¼ z=L (vertical line indicates the edges of

barriers): (A) for case F ¼ 0 and B ¼ 0; and (B) for case Fa0

and Ba0:

E. Kasapoglu, I. Sokmen / Physica E 27 (2005) 198–203200

confinement potential is given by

V ðzeÞ ¼

Vo; zeo� L=2;

2Vo

Ljzej; jzejoL=2;

Vo; ze4L=2:

8>><>>: (2)

By using cylindrical coordinates x ¼ r cos f; y ¼

r sin f; we obtained the Hamiltonian as follows:

H ¼ �_2

2me

q2

qr2þ

1

rqqr

þ1

r2q2

qF2

� �

�_2

2me

q2

qz2eþ

e2B2

2mec2z2e þ V ðzeÞ þ jejFze

�e2

�o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðze � ziÞ

2q ; ð3Þ

where the term r ð¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pÞ is the distance

between the electron and impurity in the (x-y)

plane. Eq. (3) does not contain the termððeB=mecÞzepyÞ because the expectation value ofthis term is identically zero for the chosen trialwave function in Eq. (4).We propose the following variational trial wave

function for the electron bound to impurity:

CðrÞ ¼ cðzÞjðr; lÞ; (4)

where cðzÞ is the eigenfunction of the Hamiltonianin Eq. (3) without the electron–donor interactionterm and the one-particle envelope function cðzÞdescribes the motion of the electron in the z-direction with respect to the quantum well. Tosolve the Schrodinger equation in the z-direction,we take as base the eigenfunctions of the infinitepotential well with the Lb width. Lb is well widthof the infinite well at the far end of VQW with L

width ðLb4LÞ and its value is determined accord-ing to the convergence of the energy eigenvalues.These bases are formed as

cnðzÞ ¼

ffiffiffiffiffiffi2

Lb

scos

npLb

z � dn

�; (5)

where

dn ¼

0 if n is odd;p2

if n is even;

8<:

and so, the wave function in the z-direction isexpanded in a set of basis function as follows:

cðzÞ ¼X1n¼1

cncnðzÞ: (6)

In calculating the wave function cðzÞ; weensured that the eigenvalues are independent ofthe chosen infinite potential well width Lb and thatthe wave functions are localized in the well region.This method, which gives accuracies greater than0.001meV, is well controlled, gives the QWeigenfunctions, and is easily applied to situationsof varying potential and effective mass. Thistechnique allows one to follow the developmentof the QW eigenstate outside the well and todetermine the validity of quasi-bound stateapproximation. As with many numerical ap-proaches, the algorithm can only be applied overa finite region of space, which means that we must

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0 50 100 150 200 250 30010

11

12

13

14

- - - -

F = 0

- - - B = 0

F= 75 kV/cm

L (Å)E

( m

eV )

b

B = 1 TB = 15 T

Fig. 2. The variation of the ground state binding energy of a

hydrogenic donor impurity located in the center of VQW under

the crossed electric and magnetic field as a function of the well

width for different electric and magnetic field values.

E. Kasapoglu, I. Sokmen / Physica E 27 (2005) 198–203 201

place arbitrary boundaries and boundary condi-tions on the solution. For states that are suffi-ciently quasi-bound, then if the boundary issufficiently far away from the quantum well, itshould have no effect on the eigenstates. In fact,this can be used as reasonable criteria for having awell-defined quasi-bound state. For detailed in-formation please check Ref. [28]. The relativemotion between the electron and donor is con-tained in j ðr; lÞ: Because of the Coulombic natureof the electron–donor interaction, the relative partof the wave function in the (x-y) plane—jðr; lÞcan be chosen to be of the form of the hydrogen-like atomic wave function in two dimension. Wehave also used these functions (Eqs. 4 and 7) in ourprevious studies, respectively [29,30 (A) andreferences therein, 12,13]:

jðr; lÞ ¼1

l2

p

� �1=2

e�r=l (7)

in which l is a variational parameter.The ground state impurity binding energy Eb as

a result of the electron–donor Coulomb interac-tion is obtained as follows:

Eb ¼ Ez �minl

hCjHjCi; (8)

where Ez are the ground-state energies of electronsobtained from the Schrodinger equation in the z-direction.

3. Results and discussions

The values of the physical parameters used inour calculations are me ¼ 0:0665mo (mo is the freeelectron mass), �o ¼ 12:58 (static dielectric con-stant is assumed to be same GaAs and GaAlAs),Vo ¼ 228meV: These parameters are suitable inGaAs=Ga1�xAlxAs heterostructures with an Alconcentration of x ffi 0:3:

The binding energies of a hydrogenic donorimpurity located in the center of V-shaped QWunder the crossed electric and magnetic fields as afunction of the well width for different electric andmagnetic field values are given in Fig. 2. As seen inthis figure, when the dimension of QW increases,the impurity binding energy increases until it

reaches a maximum value, and then decreases.This behavior is related to the change of theelectron confinement in QW. When the dimensionof the well decreases, the confinement of electronsis strengthened, and therefore the impurity bindingenergy increases. When the dimension of QW isreduced to a small limited value, most of theelectronic wave function begins to leak out of thewell and therefore the impurity binding energydecreases. As the electric field increases, for caseFa0; the Coulombic interaction between theelectron and a donor impurity located in thecenter of the well decreases since the electron shiftsto the left side of the well and so, for F ¼

75 kV=cm; the impurity binding energy becomesweaker than that at F ¼ 0 (see Fig. 2). Further-more, the magnetic field effect or magneticconfinement is more evident especially for bothFa0 and large well widths, since the geometricconfinement in the large well widths becomes weakand magnetic confinement becomes predominant.It is expected that the enhancement of the bindingenergy in the presence of a magnetic field shouldbe large. However, the effect of the magnetic field

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Fig. 3. The variation of the VQW potential profile and

amplitude of normalized subband wave function of electron—

jcð~zÞj2 versus the normalized position—~z ¼ z=L for L ¼ 40 (A;F ¼ 150kV=cm; for (A) B ¼ 0; (B) B ¼ 15T; (C) B ¼ 20T and

(D) B ¼ 25T:

E. Kasapoglu, I. Sokmen / Physica E 27 (2005) 198–203202

also depends on the geometric shape of thestructure. As seen in Figs. 1 and 3, the magneticfield does not have a significant effect on the VQWstructure. Impurity binding energy in the VQW ismore sensitive to the geometric confinement andelectric field than the magnetic field. Resultsobtained in this paper for B ¼ 0 are the same asresults of VQW in Ref. [29].

To see the combined effects of the electric andmagnetic fields in the narrow wells, the variationof the VQW potential profile and amplitude ofnormalized subband wave function of electron—jcð~zÞj2 versus the normalized position— ~z ¼ z=L;for L ¼ 40 (A; F ¼ 150 kV=cm and different mag-netic field values, are given in Figs. 3(A)–(D). Theeffects of the crossed electric and magnetic fields inthe narrow wells lead to the interesting results: (A)for B ¼ 0; electron is in the delocalization regimedue to the well width and electric field value.Electron localizes in the triangle well not in theVQW and, thus, the Coulombic interactionbetween the electron and donor impurity locatedin the center of the VQW decreases; (B) for B ¼

15T; the probability finding the electron in theVQW increases with magnetic field. In this case,impurity binding energy becomes larger than thatof case (A); (C) for B ¼ 20T; the new shape of thepotential profile becomes very interesting with thecombined effects of the electric and magnetic fieldsin the narrow wells. As seen in this figure, electronpenetrates to the left side of the VQW by the effectof the electric field and localizes in the barrier,which forms into a parabolic well by the effect ofthe magnetic field. In this case, the behavior ofelectron is very similar to Landau-like states; (D)for B ¼ 25T; by increasing the magnetic fieldvalue, we can increase the probability of findingthe electron in the VQW again. Thus, one particlethat delocalizes in the narrow wells, can belocalized again by increasing the magnetic fieldvalue. In the case of quantum well under thecrossed electric and magnetic fields, there aresignificant modifications of the electrical states ofthe quantum well, due to the competing effects ofthe confining potential and the potential resultingfrom the applied magnetic and electric fields andalso the optical properties that have been changedsignificantly. This indicates potential applications

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E. Kasapoglu, I. Sokmen / Physica E 27 (2005) 198–203 203

in infrared detectors and modulators. So, we hopethat these results will provide important improve-ments in device applications, especially for narrowwell widths and for a suitable choice of both fields.

As a result, we have studied theoretically theimpurity binding energy in the VQW under thecrossed electric and magnetic fields with the use ofa variational approximation. Electric field isparallel—applied to the growth direction (z-direc-tion), magnetic field is perpendicular—applied tothe growth direction or electric field. To the best ofour knowledge, this is the first study for hydro-genic donor impurities in VQWs under the crossedelectric and magnetic fields. We expect that thispaper will be of great help for theoretical studies ofthe physical properties of VQWs.

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