Binding energy of hydrogenic impurities in a quantum well under the tilted magnetic field

Binding energy of hydrogenic impurities in a quantum well

under the tilted magnetic field

E. Kasapoglua,*, H. Sarıa, I. Sokmenb

aDepartment of Physics, Cumhuriyet University, 58140 Sivas, TurkeybDepartment of Physics, Dokuz Eylul University, Izmir, Turkey

Received 31 October 2002; received in revised form 8 November 2002; accepted 10 November 2002 by K.-A. Chao

Abstract

This paper treats theoretically the angle dependence of the ground state binding energy of a shallow donor impurity in

semiconductor quantum-well systems on the tilted magnetic field. By making an appropriate coordinate transform we have

calculated the ground state binding energy of a shallow donor impurity at the center of GaAs/Ga12xAlxAs quantum well in the

effective-mass approximations and variationally. We show that the binding energy depends strongly not only on quantum

confinement, but also on the direction of the magnetic field. For example; for L0 ¼ 100 A, the change of the binding energy

between u ¼ 15 and 458 approximately is 2:5Ry (,13 meV). We expect that this change will be useful in designing the

quantum-well structure in which the impurity effects play important role.

q 2003 Elsevier Science Ltd. All rights reserved.

PACS: 71.55.Eq; 71.55. 2 i

Keywords: A. Quantum wells; C. Impurities in semiconductors

1. Introduction

With the development of several experimental tech-

niques, such as molecular beam epitaxy and metal organic

chemical–vapor deposition, there has been a lot of work

devoted to the understanding of hydrogenic impurity states

in low-dimensional semiconductor heterostructures such as

quantum wells [1–6], quantum-well wires [7–13], and

quantum dots [14–16]. Studies of semiconductor multilayer

quasi-two dimensional system as well as single quantum

wells of a GaAs/GaAlAs crystal type shows that the carriers

caught by impurity centers effect essentially on the

electronic properties of such system. Magnetic or electric

fields are effective tools for studying these properties. A

number of papers are devoted to the theoretical studies of

the impurity states in the quantum wells when the external

fields are applied. The use of the tilted magnetic fields is of

interest theoretically as it illustrates confinement effects. If

the magnetic field is tilted in respect to the interface, the

variables in Schrodinger equation cannot be seperated and

variational [17,18] or perturbation [19,20] methods have

been used. So far only the eigenenergies of two-dimensional

electrons subjected to a tilted magnetic field have been

solved analytically using a parabolic potential well [21]. In

our previous studies [22,23], however, we have completely

solved the Schrodinger equation using a square well

potential as confining potential and obtained analytical

solutions without making any approximations for two-

dimensional semiconductor heterostructures under the tilted

magnetic field.

In this study, we report a calculation, with the use of a

variational approximation, of the ground state binding

energy of a hydrogenic donor impurity at the center of a

GaAs quantum well in the presence of a magnetic field

applied tilted to the growth direction. To solve the

Schrodinger equation we apply an orthogonal transform-

ation, and then we use a tricky substitution into the potential,

that makes the Hamiltonian seperable in terms of the new

coordinates. The general solution smoothly goes to the

0038-1098/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

PII: S0 03 8 -1 09 8 (0 2) 00 7 74 -3

Solid State Communications 125 (2003) 429–434

www.elsevier.com/locate/ssc

* Corresponding author. Tel.: þ90-346-21910101937; fax: þ90-

346-219 11 86.

E-mail addresses: [email protected] (H. Sari),

[email protected] (E. Kasapoglu).

http://www.elsevier.com/locate/ssc

results of two limits where the magnetic field is either

parallel or perpendicular to the layers. Our results are given

as a function of the well width, magnetic field strength and

tilt angle. An interaction of carrier with the impurity centre

is considered to be one of the Coulomb potential type.

2. Theory

We define the z-axis to be along the growth axis, and take

the magnetic field to be applied in the x–z plane at angle-u

to the x-axis. We choose a gauge for the magnetic field in

which the vector potential A is written form A ¼ ð0; xB �

sin u2 zB cos u; 0Þ using the 7·A ¼ 0 gauge, where B ¼

ðB cos u; 0;B sin uÞ and u is the angle between the direction

of the magnetic field and x-axis.

Within the framework of an effective-mass approxi-

mation, the Hamiltonian of a hydrogenic donor in a GaAs

quantum well, in the presence of an applied magnetic field,

can be written as

H ¼1

2me

~p þe

c~A

� �2

2e2

10l~re 2 ~rilþ VðzeÞ; ð1Þ

where me is the effective mass, e is the elementary charge,

~p is the momentum, 10 is the dielectric constant, and VðzeÞ

is the confinement potential profile for the electron in the

z-direction. The functional form of the confinement

potential is given as

VðzeÞ ¼ V0½SðzL 2 zeÞ þ Sðze 2 zRÞ�; ð2Þ

where S is the step function, and the left and right

boundaries of the well are located at z ¼ zL ¼ 2L0=2

and z ¼ zR ¼ L0=2; respectively. By using the following

transformation,

z0

x0

!¼

cos u 2 sin u

sin u cos u

!z

x

!ð3Þ

the Hamiltonian can be written as below

H ¼1

2me

ðp2x0 þ p2

z0 Þ þ1

2me

p2y þ

e2B2

2mecz02 þ Vðx0e; z

0eÞ

2e2

10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx0e 2 x0iÞ

2 þ ðy 2 yiÞ2 þ ðz0e 2 z0iÞ

2q ; ð4Þ

where Eq. (4) does not contain the term ðeB=mecÞz0py

because, the expectation value of this term is identically

zero for the chosen trial wave function in Eq. (11).

After the coordinate transformation, the left and right

boundaries of wells on the x0 and z0 axes are

x0L;R ¼ zL;R sin uþ x cos u;

z0L;R ¼ zL;R cos u2 x sin u;

ð5Þ

respectively. The solution of the corresponding Schrodinger

equation is not straightforward since, after the coordinate

transformation the potential energy of the electron in the

well Vðx0e; z0eÞ couples x0 and z0 variables. In order to

decompose the potential energy of the electron, we rewrite

the step functions in Eq. (2) as follows:

SðzL 2 zeÞ ¼ cos2 uSðz0L 2 z0eÞ þ sin2 uSðx0L 2 x0eÞ;

Sðze 2 zRÞ ¼ cos2 uSðz0e 2 z0RÞ þ sin2 uSðx0e 2 x0RÞ:

ð6Þ

By considering the above equations, we can separate the

potential as (see Appendix A)

Vðx0e; z0eÞ ¼ Vðx0eÞ þ Vðz0eÞ ð7Þ

where

Vðx0eÞ ¼ V0 sin2 u½Sðx0L 2 x0eÞ þ Sðx0e 2 x0RÞ�;

Vðz0eÞ ¼ V0 cos2 u½Sðz0L 2 z0eÞ þ Sðz0e 2 z0RÞ�:

ð8Þ

Notice that applied magnetic field is parallel to the

growth direction at the u ¼ 908 and the electron becomes

free in the z0 direction and the eigenvalues do not depend on

z0, and that applied magnetic field is perpendicular to the

growth direction at u ¼ 08 and the electron becomes free in

the x0 direction and the eigenvalues do not depend on x0. As

known, for these values of u, the Schrodinger equation can

be solve exactly and we not need such a transformation to

solve the problem.

By scaling all lengths in effective Bohr radius ðaB ¼

10"2=mee2Þ; and energies in effective Rydberg ðRy ¼

mee4=2120"

2Þ; and considering above results, we can rewrite

the dimensionless Hamiltonian of the system as,

~H ¼ 2d2

d~x0 2þ ~Vð~x0eÞ2

d2

d~z02þ ~Vð~z0eÞ þ

e2B2"2

4m2ec2R2

y

~z02

2d2

d~y22

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~x0e

2 þ ~y2 þ ~z0e2

p ; ð9Þ

where ~x0i; ~y0i and ~z0i is equal the zero since, donor impurity is

located on center of the well.

We propose the following variational trial wave function

for the electron bound to impurity

c ¼ cð~x0Þcð~z0Þwðy;aÞ ð10Þ

where the wave function in the y-direction wðy;aÞ is chosen

to be Gaussian-type orbital function:

wðy;aÞ ¼1ffiffia

p2

p

� 1=4

e2y2 =a2

; ð11Þ

in which a is a variational parameter, cðx0Þ is the wave

function of the electron in the x0 direction which is exactly

obtained from the Schrodinger equation in the x0 direction.

cðz0Þ; the wave function of the electron in the z0 direction. To

solve the Schrodinger equation in the z0 direction, we choose

as base the eigenfunction of the infinite potential well with

the Lb width. We have also used this technique in our

previous studies [22,23]. In calculating the wave functions

cðz0Þ; we have ensured that the eigenvalues are independent

E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434430

of the choice infinite potential well width Lb and that the

wave functions are localized in the well region.

The total energy of the system is evaluated by

minimizing the expectation value of the Hamiltonian in

Eq. (9) with respect to a:

mina

kcl ~Hlcl ¼ ~E: ð12Þ

The binding energy of the donor impurity ground state is

given by

~EB ¼ ~E0 2 ~E; ð13Þ

where ~E0 is the lowest electron total subband energy in the x0

and z0 directions, respectively. Substituting the expectation

value of the Hamiltonian into the Eq.(12), we get the ground

state binding energy of the donor impurity.

3. Results and discussions

The values of the physical parameters used in our

calculations are me ¼ 0:0665m0 (m0 is the free electron

mass), 10 ¼ 12:58 (static dielectric constant is assumed to

be same GaAs and GaAlAs), V0 ¼ 228 meV. These

parameters are suitable in GaAs/Ga12xAlxAs heterostruc-

tures with an Al concentration of x ø 0:3: Without losing

generality, and for simplicity in numerical calculations, we

have chosen the boundaries of the well at x0LðRÞ ¼ ^L0=2 �

sin u and z0LðRÞ ¼ ^L0=2 cos u; and after the coordinate

transformation which satisfies the following equation:

z0R 2 z0L ¼ L0 cos u; x0R 2 x0L ¼ L0 sin u; ð14Þ

derived from Eq. (5).

In Fig. 1, we display the variation of the binding energy

of the ground state for a donor at the center of a GaAs

quantum well as a function of the well width for different

magnetic field values and u ¼ 158: As seen in this figure

impurity binding energy increases as the well size increases

as the independent of all magnetic field values, since the

geometric confinement predominates at small L0 values

(100 # L0 # 200 A). For B ¼ 1T ; the binding energy

increases as L0 increases and reaches a maximum value.

Where the binding energy is maximum the system has

quasi-two-dimensional character. After the certain L0 value

(L0 ø 250 A), impurity binding energy decreases as L0

increases, since the confinement of the electron in the z0

direction decreases i.e. the influence of the Coulomb field of

the impurity center on the electron weakens. This behaviour

reproduces several results previously reported [3,4,11]. For

L0 . 200 A, at large magnetic field values, magnetic

confinement becomes stronger and the impurity binding

energy increases as the magnetic field increases. In this

limit, the extension of the wave function in the plane which

is perpendicular to the magnetic field is determined

primarily by the magnetic field and the barrier potential is

a small perturbation on the magnetic term. Also from Fig. 1

it can be observed that, for strong magnetic fields the

binding energy reaches a constant value for large well

width. The results in the large L0 limit and for magnetic

fields in the experimental range are compared with the

hydrogenic atom limit [24]. The comparison shows that in

the range studied the present calculation is quite accurate.

In Fig. 2, we show the variation of the impurity binding

Fig. 1. The variation of the binding energy of the ground state for a

donor at the center of a GaAs quantum well as a function of the well

width for u ¼ 158 and three different magnetic field values.




E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434 431

energy versus the well width for u ¼ 308 and different

magnetic field values. Impurity binding energy decreases for

all magnetic field values as L0 increases. When we compare

the results are obtained in this case with that of u ¼ 158, we

see that the binding energy of impurity at the center of the

well increases with increasing tilt angle, since the

localisation of the electron increases in both x0 and z0

direction, and the electronic probability density around the

impurity is higher than in the previous case.

In Fig. 3, we present the variation of the impurity binding

energy versus the well width for u ¼ 458 and different

magnetic field values. For u ¼ 458; binding energy becomes

maximum, since the effective well widths and potential

heights of electron in both x0 and z0 directions are equal,

electron is under the effect of the same geometric

confinement in both directions. If we compare the results

of the binding energy for u ¼ 458 with that of u ¼ 158; we

see that the binding energy changes from , 1:5Ry to , 4Ry

for L0 ¼ 100 A and all magnetic field values. So, tilt angle is

a good tunable parameter providing a change on the

impurity binding energy for especially small L0 values.

In Figs. 4 and 5, we present the variation of the impurity

binding energy versus the well width for u ¼ 608 and u ¼

758; respectively. For these tilt angles, the well width in

which the binding energy begins to be sensitive to the

magnetic field becomes smaller than the previous tilt angles.

This case is evident in Fig. 5. In these tilt angles values, the

well width in which the electron is confined in the z0

direction became so small that the electron is delocalised in

all L0 values and an interaction of the electron with the

impurity centre is completely provided with the magnetic

confinement.

The variation of the impurity binding energy versus the

tilt angle for different magnetic field values and L0 ¼ 200 A

is given In Fig. 6. As seen in this figure, in the range of

0 # u # 458 impurity binding energy increases as tilt angle

increases and reaches a maximum value at u ¼ 458; and then

in the range of 45 # u # 908 the binding energy decreases

up to u ¼ 758 and for further tilt angle value it converges to

a constant value. The observations are that; in the range











0 # u # 458, as the localization of the donor electron in the

z0 direction and Coulombic interaction increases, the

binding energy increases, and in the range of

45 # u # 908, the well width becomes smaller in the z0

direction and interaction decreases between donor electron

and ion as delocalisation for donor electron begins, and thus

the binding energy also decreases. It is important to observe

that very similar magnetic field dependence has the ground

state exciton binding energy in quantum wells [22]. As a

consequence of this analysis it can be seen that the direction

of the magnetic field plays an essential role in the

determination of the binding energy.

In conclusion, this paper presents the solution of the

Schrodinger equation of a square well potential problem

under the influence of an externally applied tilted magnetic

field by making the Hamiltonian separable via a coordinate

transformation. In this study, we calculated by using a

variational approximation the ground state impurity binding

energy in the single square quantum well under the

externally applied tilted magnetic field as a function of the

tilt angle, magnetic field and the well width. It is seen that

the direction of the magnetic field causes important changes

in the binding energy. For example; for L0 ¼ 100 A, the

change of the binding energy between u ¼ 15 and 458

approximately is 2:5Ry (,13 meV). So, we can say that the

quantum well structure is reduced to the quantum wire from

the results are obtained at u ¼ 458, since the system is under

the effect of the same geometric confinement in both x0 and

z0 directions. To the best of our knowledge, so far studies

about the shallow donor impurity binding energy under the

tilted magnetic field have not been reported. We expect that

these results will be of importance in the understanding of

experimental studies related with donor impurities in GaAs

quantum wells under the external tilted magnetic field.

Appendix A

In order to decompose the potential energy Vðx0e; z0eÞ

which is defined in Eq. (4), let us introduce a function

f ðx0; z0Þ ¼ cos2 uSðz0L 2 z0Þ þ sin2 uSðx0L 2 x0Þ ðA1Þ

By using the variables x0, and z0 in terms of x and z we can

write the step function Sðz0L 2 z0Þ as follows:

Sðz0L 2 z0Þ ¼ SðzL cos u2 x sin u2 z cos uþ x sin uÞ

¼ SððzL 2 zÞcos uÞ ðA2Þ

and since cos u . 0; one can rewrite the last term in Eq.

(A2) as SððzL 2 zÞcos uÞ ¼ SððzL 2 zÞÞ: Similarly, by con-

sidering the above results we can write Sðx0L 2 x0Þ ¼

SððzL 2 zÞsin uÞ ¼ SððzL 2 zÞÞ: Thus f ðx0; z0Þ takes the fol-

lowing form:

f ðx0; z0Þ ¼ ðcos2 uþ sin2 uÞ SðzL 2 zÞ ¼ SðzL 2 zÞ ðA3Þ

By using the same procedure for Sðz 2 zRÞ we can write

the step function as in Eq. (6), and by considering the above

equations we can decompose the potential Vðx0e; z0eÞ as Eq.

(7).

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Documents

Binding energy of hydrogenic impurities in a quantum well under the tilted magnetic field