Quantum well excitons in a high magnetic field

Quantum well excitons in a high magnetic field

B. S. MPhysics Department, Marine Technical University, 3 Lotsmanskaya Str., St Petersburg, 190008, Russia

M. J. P, C. A. B, J. L. DPhysics Department, University of Nottingham, Nottingham, NG7 2RD, U.K.

An analytical approach to the problem of an exciton with extremely different electron andhole masses in a quantum well (QW) subject to a strong magnetic field is developed. Thedouble adiabatic approximation is used. The dependence of the exciton energy levels uponthe magnitude of the magnetic field and the width of the QW is obtained. As the magneticfield increases in magnitude, the exciton binding energy increases. With a narrowing of theQW, the exciton levels shift towards higher energies. However, for a sufficiently strong mag-netic field, the size of the QW has little effect of the exciton energy. Our results are shown tobe in agreement with previous numerical work on the problem.

( 1997 Academic Press Limited

1. Introduction

The approach to the exciton problem described in this paper closely resembles that developedfor the problem of impurities in semiconductor heterostructures. This subject represents therefore agood starting point. During the last decade, the subject of shallow donor impurities in quasi-two-dimensional systems in the presence of an external field has been studied both experimentally andtheoretically (see, for example, Shi et al. [1] and in many of the references contained therein). Inparticular, the theoretical problems of an impurity in a quantum well (QW) subject to a magneticfield has been treated by a number of authors. The majority of papers on the subject consist ofnumerical calculations which usually rely upon variational or perturbational methods. Greene andBajaj [2,3] and Greene and Lane [4] used a variational method with a trial function consisting ofGaussian basis sets. In addition to the many variational-type calculations which had been cited inShi et al. [1], alternative theoretical approaches for the related problem of the shallow donor in singlequantum well (SQW) and multi-quantum well (MQW) systems have been given. Two of the authors(JLD and CAB) have been involved in a matrix diagonalization procedure developed originally byDunn and Pearl [5] and extended in Barmby et al. [6–9] for the case of magnetic fields pointing atdifferent angles relative to the QW layers. Another of the authors (BSM) has used analytical methodsto study shallow donor impurity states and energy levels in cases of large magnetic fields [10,11].

From a mathematical point of view, a very similar problem to that of a shallow donor im-purity in two-dimensional systems is that of a Wannier-Mott exciton. The theoretical problem of theimpurity states in zero magnetic field was considered originally by Bastard [12] and Bastard et al.[13]. Exciton states for a QW in zero magnetic field were also considered by the latter authors [14]whilst the effect of applying an electric field was included in the work of Brum and Bastard [15] using

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a variational approach. Many authors have used largely numerical methods to solve the problem ofexcitons; Duggan [16] performed a numerical integration of the Hamiltonian for weak and intermedi-ate magnetic fields whilst Yang and Sham [17] treated the case of a strong field limit. The effect ofmagnetoexciton mixing in a QW in semiconductors was studied experimentally by Iimura et al. [18]and Potemski et al. [19] and theoretically using numerical techniques by Bauer and Ando [20,21]. Inthe latter work, hydrogenic functions were used as bases sets but the degeneracy of the valence bandwas also included. However, their calculations are strictly valid only for weak and intermediatemagnetic fields. An alternative procedure was adopted by Nash et al. [22] who treated the magneticfield as a perturbation valid for fields of up to moderate strength. Hou et al. [23] included some theorywith their experimental results for excitons in InGaAs/GaAs strained QWS in which a variationalmethod was employed using hydrogenic wavefunctions as the basis sets for low magnetic fields butwith harmonic oscillator wavefunctions for high magnetic fields. Their results were found to remainvalid for very high magnetic fields. Independently, Lee et al. [24] devised a modified perturbationalmethod to improve the accuracy of the latter calculations whilst Kavokin et al. [25] used a combina-tion of variational and perturbation techniques to give results valid for weak and moderate strengthmagnetic fields. Experimental and theoretical work on the exciton problem has expanded a lot withinthe last few months as evidenced by the work described by Sigrest et al. [26], Xiangdong Zhang etal. [27], Oelgart et al. [28], Cen et al. [29], Reynolds et al. [30], Harris et al. [31] and Bar-Ad et al.[32], for example.

In this paper, an alternative but entirely analytical approach to the problem of a Wannier-Mott exciton in a SQW in the presence of a strong magnetic field directed normal to the layers isstudied. The most important approximation to be made is that the hole mass is much larger than theelectron mass but this approximation is clearly valid for the majority of the III-V group semiconduc-tors. The principle object of this approach is to keep the basic physics of the problem clearly in viewthroughout the analysis. Numerical calculations are used only in the final stages of the analysis andthen only to produce the graphs. The analysis thus remains general and is applicable to manyexamples until the final computational stage is reached. The approach here follows that of the earlierwork of one of us [10] which was devised originally for the impurity electron states in a QW subjectto a strong magnetic field. This involves the double adiabatic approximation in order to determineexpressions analytically for the exciton energy levels. Despite the large differences in the calculations,the results obtained will be shown to be in good agreement with previous numerical work on theproblem.

2. General theory

The z-direction is chosen to lie along the direction of the magnetic field B which is appliedperpendicular to the heteroplanes and the SQW is treated as an infinite square well of width d. Inprevious treatments of the problem of an exciton in a QW with zero magnetic field (for instanceHarrison et al. [33]) it has been found to be more accurate to take a well with finite barriers. However,for the case of a very high magnetic field, it is expected that the extra confinement caused by the fieldwill mean that the results are less sensitive to the form of the barrier potential than is the case forzero field. Thus the approximation to be used here is qualitatively justified. The centre of the well istaken to be the point z\0. The other parameters relevant to the calculation are the exciton Bohrradius (a

0) and the magnetic length (a

H). They are defined as usual by a

0\(4pee

0h~2)/(le2) and

aH\q(h~/eB) where e is the dielectric constant, and l (\[1/m

e]1/m

h]~1) is the reduced mass of the

exciton with me

the electron effective mass and mh

the hole effective mass. The strong field limit isdefined by a

H@a

0implying that the motion in the z-direction is slower than that in the heteroplane.

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Furthermore, for many crystals of the type A3B5

and A2B6, including GaAs, we also have m

e@m

himplying that the motion of the hole is slower than that of the electron. As a result of these approxi-mations, the double adiabatic approximation can be applied.

Excitons induced by the allowed dipole interband optical transitions are considered. With theapproximations detailed above, the total exciton momentum K\q^0, where q is the photonmomentum. We take the electron and hole bands to be parabolic, non-degenerate and separated bythe wide energy gap E

g. The effective mass Hamiltonian describing the relative motion of the exciton

is given by

AHM

(o)[h~2

2me

d2

dz2e

[h~2

2mh

d2

dz2h

[e2

4nee0Jo2](z

e[z

h)2 B((o,z

e,z

h)\E((o,z

e,z

h) (2.1)

where Hr is the Hamiltonian of the exciton in the plane orthogonal to the magnetic field, E is thetotal exciton energy and W(q,z

e,z

h) is the exciton wave function. The coordinate q\q

e[q

h.

As the QW is approximated by an infinite square well potential, W(q,ze,z

h) must vanish at the

barriers, so that

W Aq, ze\^

d2, z

h\^

d2B\0 (2.2)

In general, an analytic solution to the Schrodinger equation (2.1) cannot be found as the mixing ofstates caused by the Coulomb potential prevents us from separating the transverse (q) and thelongitudinal (z

e,z

h) states. Also, the boundary condition due to (2.2) is stated in terms of the separate

coordinates of the hole and electron (zh

and ze, respectively) whereas the Coulomb term in (2.1)

depends upon the relative coordinate Dze[z

hD only.

The first difficulty can be resolved if we concern ourselves only with the case of a high mag-netic field as specified above. Under this condition, we can write

W(q,ze,z

h)\XrN,m

(q)WN,m

(ze,z

h) (2.3)

where N and m are the usual quantum numbers such that (N, DmD\0, 1, 2, . . .) and where XrN,m(q)

is the eigenfunction of the Hamiltonian Hr describing the free motion in a magnetic field in the xyplane and ErN,m

is the Landau energy given by

EMN,m

\Cg]

h~eB2k

(2N]Dm D]1)]h~eB

2 A1

me

[1

mhBm^(b

e^b

h)B (2.4)

where be,h

are the effective magnetic moments of the carriers.The function U

N,m(z

e,z

h) satisfies the equation

A[h~2

2me

d2

dz2e

[h~2

2mh

d2

dz2h

]VN,m

(ze[z

h)B'

N,m(z

e,z

h)\W

N,m'

N,m(z

e,z

h) (2.5)

where the longitudinal potential energy V is defined by

VN,m

(ze[z

h)\[

e24nee

0

:DX

MN,m(o) D2

Jo2](ze[z

h)2

do (2.6)

and the energy eigenvalue is given by:

WN,m

\E[ErN,m(2.7)

Following the assumption made in (2.3), the boundary condition (2.2) becomes:

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Superlattices and Microstructures, Vol. 21, No. 2, 1997 153

UN,mA^

d2

,^d2 B\0 (2.8)

To simplify the calculation, we consider only the ground transverse state for which N\m\0. It hasthe form:

XM,0,0

(o)\1

J2naH

expA[o2

4a2HB (2.9)

Thus we may drop the subscripts from the variables related to the transverse motion and write V forV

0,0U for U

0,0and W for W

0,0.

In order to solve the Schrodinger equation (2.5), we assume that the motion of the hole canbe separated from the motion of the electron about the hole. Thus we write:

Uj(ze,zh)\fj(ze[zh)Wj(zh) (2.10)

where fj(ze[zh) is the wavefunction of the electron about the fixed hole position z

h, corresponding

to the approximation that the hole mass is infinite (mh\O) and k is a label. The boundary conditions

defined by (2.8) become

fjA^d2[z

hB\0 (2.11)

where fj(ze[zh) is the solution to the equation

A[h~2

2me

d2

dz2e

]V(ze[z

h)B fj(z

e[z

h)\Ej(z

h) fj(z

e[z

h) (2.12)

In a similar way, the wavefunction for the hole satisfies the differential equation:

C[h~2

2mh

d2

dz2h

]Ej(zh)D(j(z

h)\Wj(j(z

h) (2.13)

with boundary conditions

WjA^d2 B\0 (2.14)

3. Calculation

In order to solve the Schrodinger equation (2.12) for the motion of the electron using theadiabatic approximation subject to the condition a

H@a

0, it is convenient to introduce the following

notations. We define

u\2

a0k

(ze[z

h), g\

2qa0k

and Ej\[e2

8pee0a0k2

(3.1)

so that the equation becomes:

d2fj(u)du2

]kS0D(u2]g2)~1@2D0T fj(u)[14

fj(u)\0 (3.2)

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where S0D . . . D0T is an average with respect to the function Xr0,0(q). The label k introduced above

is equivalent to a quantum number which determines the states of the motion along the z-axis.The transformation of coordinates affects the boundary condition which thus becomes

fjC2

a0kA[z

h«

d2B D\0 (3.3)

Under the condition

DuDA2a

Ha0kBS0DgD0T (3.4)

equation (3.2) transforms to Whittaker’s equation

d2fj(u)du2

]Cku[

14D fj(u)\0 (3.5)

The two independent solutions to this equation are the Whittaker functions Wj,1@2 and Mj,1@2. Thegeneral solution in the region u[0 is given by

fj(u)\A`

Wj,1@2(u)]B`

Mj,1@2(u) (u[0) (3.6)

where A`

and B`

are constants. For the region u\0, the general solution can be found directly from(3.6) by making the substitutions A

`]A

~, B

`]B

~and u]s\[u.

Equations like (3.2) have been studied in detail previously for the mathematically similarproblem of an impurity in a QW subject to a strong magnetic field (Monozon and Zhilich [10]) usingthe Hasegawa–Howard [34] method. This method may be summarized as follows. In the region u@1,a double integration method is performed on (3.2) using a trial function. A comparison of thecoefficients is then made between the result of the integration and the standard expansion of theWhittaker functions involved in (3.6) for u@1 (e.g. Gradshtein and Ryzhik [35]). When terms of thesame order are equated, a set of linear equations are found. The boundary conditions defined by (3.3)are then introduced and the system of equations that result are solved by determinantal methods.This method has been used to find solutions to (3.2) to give the following transcendental equationfor the quantum number k:

r(j)]"(j)\[Q(j)]12!([j) A

W2

M2

]W

1M

1B^SQ2(j)]1

4!2([j) A

W2

M2

[W

1M

1B2

(3.7)

where the following definitions have been made

u(k)\2C[1]w(1[k)]1

2k, K(k)\1[1

2C]1

2lnC

2a2H

a20k2D (3.8)

Q(j)\a0

aHA

n2 B

3@2

C lnJ2 a

Ha0j

]1[C2 D where Q(j)\0 (3.9)

with the additional notation

W1,2

\Wj,1@2(u1,2

), M1,2

\Mj,1@2(u1,2

), u1\

2a0k

(12d[z

h), u

2\

2a0k

(12d]z

h) (3.10)

C is the Euler constant (\0.577), C(x) is the gamma function, and w(x) is the psi function (the

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logarithmic derivative of the gamma function). We note that (3.7) is only valid in the strong fieldregion and this implies

u1,2

A2k

aH

a0

@1 (3.11)

The latter condition implies that the theory is only applicable for cases where the hole is separatedfrom the edge of the QW by a distance greater than the magnetic length a

H. Similarly, the condition

Q(k)\0 (see equation 3.9) is also valid only for a sufficiently strong magnetic field.When the hole is situated at the centre of the well (z

h\0), the wavefunctions have a definite

parity. It is clear from (3.10), that for zh\0 then u

1\u

2, W

1\W

2and M

1\M

2. The levels of even

parity correspond to states with the positive sign in front of the radical in (3.7), and the odd paritystates correspond to states with the negative sign.

However, the classification of the energy levels and states into two groups can be made forthe hole in any position in the well (z

hD0, u

1Du

2); all states with the positive sign in front of the

radical in (3.7) are referred to as quasi-even states while those with the negative sign are referred toas quasi-odd states. The quasi-even states have a quantum number k given by

kn\n]d

g(n), (n\0, 1, 2, . . .) (3.12)

while the quasi-odd states have k given by

kn\n]d

u(n) (n\1, 2, 3, . . .) (3.13)

The ground level (n\0) is non-degenerate and has k0@1. The excited states (n\1, 2, 3, . . .) have a

doublet structure consisting of quasi-even and quasi-odd components. It follows from (3.10) and (3.7)that the replacement of z

hby [z

his equivalent to the switch u

1½u

2. Therefore, we see that

k(zh)\k([z

h) and also, from (3.1), that E(z

h)\E([z

h). This implies that Ej(zh) is a symmetric func-

tion of zh. In general, there is no exact analytic solution to (3.7) for Ej(zh); solutions can only be

found for the limiting cases of small displacements of the hole from the centre of the well, and forthe hole situated close to the edge of the QW.

For the first case of the hole located near to the centre of the well we have for the groundstate E

0(k

0@1) the inequality

zh

a0k0

@1

Furthermore, with the additional assumption that the well is wide, we also have

2a0k0A

d2^z

hBA1 (3.14)

Thus the energy of the ground state is given by

E0(z

h)\E

0(0)]

12

mhX2z2

h(3.15)

whereE

0(0)\[

Rk20A1[2expC[

da0k0D B (3.16)

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R\e2

8nee0

a0

, )\4Rh~j2

0S

me

mh

expA[d

2a0j0B .and (3.17)

Now k0@1 is the smallest root of the equation

3C2

]w(1[k0)]

12k

0

]12

lnA2a2

Ha20k20B\0 (3.18)

In the logarithmic approximation, we have

DlnA2a2

Ha20k20B DA1 (3.19)

so that the solution k0

to (3.18) is given by

1k0

^[lnA2a2

Ha20B (3.20)

and the energy E0(z

h) can be found easily by using (3.15), (3.16) and (3.17).

For the second case of the hole located at the edge of the well, it has been shown (Monozonand Zhilich [10]) that the transcendental equation for k becomes

u(k)\[2Q(k)]C([k)W

0M

0

(3.21)

where

W0\Wj,1@2(u

0), M

0\Mj,1@2(u

0), and u

0\

2da0k. (3.22)

Again we assume that the well is wide (u0[1); the ground state has k

0\1]d where d@1. The

expansions for the Whittaker functions for large values of u (Gradshtein and Ryzhik [35]) are usedin (3.21) to give the relationship for d namely:

1d]2Q(1)]

1d2

u20e~u0\0 (3.23)

It follows therefore that

1@1d\

eu02u2

0CJ1[8Q(1)u2

0e~u0[1D (3.24)

and the expression for E0

is

E0A^

d2B^[

R12

(3.25)

which is in qualitative agreement with the results of Greene and Bajaj [3].For hole positions other than the above limiting cases, (3.7) must be solved numerically to

find k(zh); from this result, Ej(zh) can be calculated using (3.1). Once Ej(zh) has been found, it is

possible to solve (2.13) for Wj(zh), the component of the exciton wavefunction due to the motion ofthe hole, in order to find the total exciton energy Wj. When substituted into (2.13), Ej(zh) acts as aneffective potential for the hole. The potential energy is expected to consist of a series of wells; the

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κ = 2

W0,κ

– d/2

κ = 1

κ = 0

κ = 4

κ = 3

κ = 2

κ = 1

κ = 0

E1(zh)

E0(zh)

d/2

zh

Eλ

W1,κ

Fig. 1. A sketch of the effective potential for the hole Ej(z

h) and the resulting energy levels W

j,i.

bottom of each well is defined by the position of Ej(zh\0) and the top of which (given byEj(zh\^1

2d)) will coincide approximately with the bottom of the next well (Ej`1

(zh\0)). The spec-

trum of the total exciton energy will be a sequence of groups of levels Wji(j\0, 1, 2, . . .). Each ofthe groups is associated with a particular fixed electron state denoted by the quantum number k. Asketch of the expected shape of the effective potential Ej(zh) and the energy levels Wj,i for the groundstate (k^0) and the first excited state (k^1) is shown in Fig. 1.

4. Numerical results and discussion

The first stage of the calculation was to choose values for a0

and aH

that satisfy the restrictionQ\0 in (3.9). This condition could only be guaranteed to hold if a

H/a

0D0.1 to 0.2. Furthermore, it

is also necessary to choose a value for the well width d; for the above theory to hold it is necessaryto have d/a

0[1. With these parameters fixed within these specified ranges, (3.7) has been solved

numerically; the smaller root k0\1 was found as a function of the hole position z

h. As (3.7) is valid

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0

zh

λ

1

Hole at well edge

zh = 0 zh = d/2

Curve extrapolated in this region

Hole in central region of well

Hole close to well edge

Fig. 2. A sketch of k(zh) for different regions of the hole position.

only when the hole is separated from the edge of the well by a distance greater than the magneticlength, solutions were sought subject to the condition

0¹2z

hd

¹A2z

hd B

0

where A2z

hd B

0

\1[2AaH

a0B A

a0

d B (4.1)

The Whittaker functions in equation (3.6) were calculated using Kummer’s functions M(a,b,z) andU(a,b,z) defined in Abramowitz and Stegun [36] in the form:

Mk,k(z)\e~1@2zz1@2`kM(1

2]l[k, 1]2l, z) (4.2)

and

Wk,k(z)\e~1@2zz1@2`kU(1

2]l[k, 1]2l, z) (4.3)

It was found that the first forty terms in each of the expansions of the Kummer’s functionswere sufficient to ensure convergence of these functions. As noted above, equation (3.7) is not appli-cable when the hole is located close to the edge of the well. Instead, we can then use (3.21) whichapplies when the hole is exactly at the edge of the well; this is solved numerically for the same valuesof the parameters a

H/a

0and d/a

0to give k(z

h\1

2d). Also, if the region where (3.7) is invalid is

sufficiently small, then one expects that it is reasonably accurate to approximate the dependence ofk(z

h) by a smooth curve connecting the two known regions. A sketch of this is shown in Fig. 2.

Once k(zh) has been calculated numerically, it is easy to apply (3.1) to find the energy of the

electron Ej(zh) for the fixed hole position zh. An illustrative example of the above calculation is shown

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1.0

1.2

0.4–1.0

1.0

0.8

0.6

–0.5 0.0 0.5

2zh/d

λ

aH /a0 = 0.24 d /a0 = 1.6

1.0

0

–6–1.0

–2

–4

–0.5 0.0 0.5

2zh/d

Eλ

aH /a0 = 0.24 d /a0 = 1.6

W0, κ

κ = 1

κ = 0

Fig. 3. An example of two stages of the calculation of the exciton energy levels.

1.0

1.25

0.00–1.0

1.00

0.50

0.25

–0.5 0.0 0.5

2zh/d

λ

aH /a0 = 0.1, d /a0 = 2.4

0.75

aH /a0 = 0.24, d /a0 = 2.4

1.0

1.25

0.00–1.0

1.00

0.50

0.25

–0.5 0.0 0.5

2zh/d

λaH /a0 = 0.10, d /a0 = 1.6

0.75

aH /a0 = 0.24, d /a0 = 1.6

Fig. 4. The electronquantum number of k plotted as a function of the hole position.

in Fig. 3. The upper graph of Fig. 3 shows the dependence of k0(z

h), where the dots are the calculated

values and the line representing an extrapolation in the region close to the well edge. The exampleshown in Fig. 3 has a

H/a

0\0.24 and d/a

0\1.6. It can be seen that the theory is exactly valid over a

large part of the QW. The lower graph of Fig. 3 shows a plot of the effective pote ntial Ej(zh) andthe resulting energy levels Wj,i for the ground state k\0. As can be seen from Fig. 3, the positionof the lowest energy level j\0 is insensitive to the accuracy of the extrapolation in the region closeto the well edge; in contrast, the higher levels j\1, 2, . . . will be affected by this extrapolation. Forthis reason, we will concentrate henceforth on the lowest energy level j\0.

Figure 4 shows the results of a calculation of the electron quantum number k(zh) for different

values of the parameters aH

/a0

and d/a0. The graphs show that the calculation gives results which are

in qualitative agreement with expectations; as aH

/a0

decreases (corresponding to an increase in themagnetic field) k

0also decreases (corresponding to an increase in the magnitude of the energy DE D as

given in (3.1)); furthermore, a decrease in the well width d results in an increase in k0. Closer

inspection of Fig. 4 reveals that the variation in k0

for a change in aH

/a0

is less pronounced for alarger well than for a smaller one; that is, the relative shift in the position of the bottom of the plotof k

0(z

h) with a shift from d/a

0\0.10 to d/a

0\0.24 is smaller for a

H/a

0\0.10 than for a

H/a

0\0.24.

This behaviour is expected because, for lower values of aH

/a0

(a stronger magnetic field) the separ-ation of the electron and hole is less, so that the position of the edge of the well has less influenceon the electron energy. It is because of this extremely strong magnetic field (a

H/a

0^0.1) that the

electron energy shows practically no dependence upon the displacement of the hole for most of theQW.

A comparison of our results with other work would be desirable at this point. As was noted

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0.24

–4

–120.10

W (

κ =

0) in

un

its

of R

0.14

–9

–5

–6

–7

–8

–10

–11

0.12 0.16 0.18 0.20 0.22

aH /a0

d /a0 = 1.6

d /a0 = 2.0

d /a0 = 2.4

Fig. 5. The energy of the exciton ground state plotted as a function of scaled magnetic field for different wellwidths.

earlier for the calculation of k0(z

h), the problem is identical to that of an impurity in a QW. Greene

and Bajaj [2,3], Greene and Lane [4] and Kuhn et al. [37] calculated the energy eigenvalues of ahydrogenic impurity in a QW as a function of the position of the impurity for different (moderate)magnetic fields. All our conclusions which are valid for the strong magnetic field case are qualitativelyconfirmed by their numerical calculations. Moreover, it follows from (3.15) that the reciprocal of thederivative of the energy E(z

h) with respect to the centre position z

his proportional to z~1

h. This result

also agrees with the result of Greene and Bajaj [3, fig. 2].The energy levels Wj,i have also been calculated numerically using the effective potential

Ej(zh). As mentioned earlier, we have restricted our calculations to those for the ground state of theexciton (i.e. k\k

0). Figure 5 shows the position of this lowest ground state level for different values of

aH

/a0

and d/a0. Figure 5 is seen to have the same qualitative features as expected and the trend

observed shows that an increase in the magnetic length (decrease in the magnetic field) corresponds toan increase in the energy. For the smallest values of a

H(strongest magnetic field), the size of the QW

has a negligible effect on the position of the lowest energy level. It is only for the cases of aH

/a0[0.15

that one can resolve any difference between the graphs. Narrower wells are seen to have a higherenergy than wider wells. Again, the fact that we are using such high magnetic fields makes any directcomparison with earlier work difficult; however, it is worth noting that the flattening of the curves asthe field tends towards moderate strength was also observed in Maan et al. [38] for fields up to 23T.

5. Conclusion

It has been shown that the double adiabatic approximation can be used for analytical con-siderations of an exciton with extremely different electron and hole masses in the presence of a strongmagnetic field. The dependence of the exciton energy levels upon the magnitude of the magnetic field

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and width of the QW has been obtained. As the magnetic field increases in magnitude, the excitonenergy is found to decrease (the ‘binding energy’ increases). Probably, this result has a commonfeature; it was pointed out by Cen et al. [29] that an increase in the magnetic field leads to an increasein the enhancement of the binding energy of the exciton due to the image charge contribution in thedielectric QW structure. In contrast, with a narrowing of the QW, the exciton levels shift towardshigher energies. It has also been shown that, for sufficiently strong magnetic fields (i.e. a

H/a

0\0.15

for the ground state) the size of the QW has little effect on the exciton energy. The results of thedouble adiabatic approach are in agreement with those obtained earlier by variational-type or per-turbation-type calculations.

Acknowledgements—One of the authors (BSM) expresses his gratitude to the Royal Society for finan-cial support which has enabled this collaborative programme to proceed.

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Quantum well excitons in a high magnetic field