Quantum well excitons in a high magnetic field

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Quantum well excitons in a high magnetic field B. S. M Physics Department, Marine Technical University, 3 Lotsmanskaya Str., St Petersburg, 190008, Russia

M. J. P, C. A. B, J. L. D Physics Department, University of Nottingham, Nottingham, NG7 2RD, U.K.

An analytical approach to the problem of an exciton with extremely different electron and hole masses in a quantum well (QW) subject to a strong magnetic field is developed. The double adiabatic approximation is used. The dependence of the exciton energy levels upon the magnitude of the magnetic field and the width of the QW is obtained. As the magnetic field increases in magnitude, the exciton binding energy increases. With a narrowing of the QW, the exciton levels shift towards higher energies. However, for a sufficiently strong magnetic field, the size of the QW has little effect of the exciton energy. Our results are shown to be 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 developed for the problem of impurities in semiconductor heterostructures. This subject represents therefore a good starting point. During the last decade, the subject of shallow donor impurities in quasi-twodimensional systems in the presence of an external field has been studied both experimentally and theoretically (see, for example, Shi et al. [1] and in many of the references contained therein). In particular, the theoretical problems of an impurity in a quantum well (QW) subject to a magnetic field has been treated by a number of authors. The majority of papers on the subject consist of numerical calculations which usually rely upon variational or perturbational methods. Greene and Bajaj [2,3] and Greene and Lane [4] used a variational method with a trial function consisting of Gaussian basis sets. In addition to the many variational-type calculations which had been cited in Shi et al. [1], alternative theoretical approaches for the related problem of the shallow donor in single quantum 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 by Dunn and Pearl [5] and extended in Barmby et al. [6–9] for the case of magnetic fields pointing at different angles relative to the QW layers. Another of the authors (BSM) has used analytical methods to 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 impurity in two-dimensional systems is that of a Wannier-Mott exciton. The theoretical problem of the impurity 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 0749–6036/97/020151]13 $25.00/0

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a variational approach. Many authors have used largely numerical methods to solve the problem of excitons; Duggan [16] performed a numerical integration of the Hamiltonian for weak and intermediate magnetic fields whilst Yang and Sham [17] treated the case of a strong field limit. The effect of magnetoexciton 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]. In the latter work, hydrogenic functions were used as bases sets but the degeneracy of the valence band was also included. However, their calculations are strictly valid only for weak and intermediate magnetic fields. An alternative procedure was adopted by Nash et al. [22] who treated the magnetic field as a perturbation valid for fields of up to moderate strength. Hou et al. [23] included some theory with their experimental results for excitons in InGaAs/GaAs strained QWS in which a variational method was employed using hydrogenic wavefunctions as the basis sets for low magnetic fields but with harmonic oscillator wavefunctions for high magnetic fields. Their results were found to remain valid for very high magnetic fields. Independently, Lee et al. [24] devised a modified perturbational method to improve the accuracy of the latter calculations whilst Kavokin et al. [25] used a combination of variational and perturbation techniques to give results valid for weak and moderate strength magnetic fields. Experimental and theoretical work on the exciton problem has expanded a lot within the last few months as evidenced by the work described by Sigrest et al. [26], Xiangdong Zhang et al. [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 WannierMott exciton in a SQW in the presence of a strong magnetic field directed normal to the layers is studied. The most important approximation to be made is that the hole mass is much larger than the electron mass but this approximation is clearly valid for the majority of the III-V group semiconductors. The principle object of this approach is to keep the basic physics of the problem clearly in view throughout the analysis. Numerical calculations are used only in the final stages of the analysis and then only to produce the graphs. The analysis thus remains general and is applicable to many examples until the final computational stage is reached. The approach here follows that of the earlier work of one of us [10] which was devised originally for the impurity electron states in a QW subject to a strong magnetic field. This involves the double adiabatic approximation in order to determine expressions 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 the problem.

2. General theory The z-direction is chosen to lie along the direction of the magnetic field B which is applied perpendicular to the heteroplanes and the SQW is treated as an infinite square well of width d. In previous treatments of the problem of an exciton in a QW with zero magnetic field (for instance Harrison 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 field will mean that the results are less sensitive to the form of the barrier potential than is the case for zero field. Thus the approximation to be used here is qualitatively justified. The centre of the well is taken to be the point z\0. The other parameters relevant to the calculation are the exciton Bohr radius (a ) and the magnetic length (a ). They are defined as usual by a \(4pee ~h2)/(le2) and 0 H 0 0 ~ /eB) where e is the dielectric constant, and l (\[1/m ]1/m ]~1) is the reduced mass of the a \q(h H e h exciton with m the electron effective mass and m the hole effective mass. The strong field limit is e h defined by a @a implying that the motion in the z-direction is slower than that in the heteroplane. H 0

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Furthermore, for many crystals of the type A B and A B , including GaAs, we also have m @m 3 5 2 6 e h implying that the motion of the hole is slower than that of the electron. As a result of these approximations, the double adiabatic approximation can be applied. Excitons induced by the allowed dipole interband optical transitions are considered. With the approximations detailed above, the total exciton momentum K\q^0, where q is the photon momentum. We take the electron and hole bands to be parabolic, non-degenerate and separated by the wide energy gap E . The effective mass Hamiltonian describing the relative motion of the exciton g is given by

A

B

~ h2 d2 ~h2 d2 e2 H (o)[ [ [ ((o,z ,z )\E((o,z ,z ) (2.1) M e h e h 2m dz2 2m dz2 4nee Jo2](z [z )2 e e h h 0 e h where Hr is the Hamiltonian of the exciton in the plane orthogonal to the magnetic field, E is the total exciton energy and W(q,z ,z ) is the exciton wave function. The coordinate q\q [q . e h e h As the QW is approximated by an infinite square well potential, W(q,z ,z ) must vanish at the e h barriers, so that

A

B

d d W q, z \^ , z \^ \0 e 2 h 2

(2.2)

In general, an analytic solution to the Schro¨dinger equation (2.1) cannot be found as the mixing of states caused by the Coulomb potential prevents us from separating the transverse (q) and the longitudinal (z ,z ) states. Also, the boundary condition due to (2.2) is stated in terms of the separate e h coordinates of the hole and electron (z and z , respectively) whereas the Coulomb term in (2.1) h e depends upon the relative coordinate Dz [z D only. e h The first difficulty can be resolved if we concern ourselves only with the case of a high magnetic field as specified above. Under this condition, we can write W(q,z ,z )\Xr (q)W (z ,z ) (2.3) e h N,m N,m e h where N and m are the usual quantum numbers such that (N, DmD\0, 1, 2, . . .) and where Xr (q) N,m is the eigenfunction of the Hamiltonian Hr describing the free motion in a magnetic field in the xy plane and Er is the Landau energy given by N,m ~ ~ heB 1 1 heB (2N]DmD]1)] [ m^(b ^b )B (2.4) E \C ] MN,m e h g 2k 2 m m e h where b are the effective magnetic moments of the carriers. e,h The function U (z ,z ) satisfies the equation N,m e h ~ h2 d2 ~h2 d2 [ [ ]V (z [z ) ' (z ,z )\W ' (z ,z ) (2.5) N,m e h N,m e h N,m N,m e h 2m dz2 2m dz2 e e h h where the longitudinal potential energy V is defined by

A

A

B

B

e2 DX (o)D2 MN,m V (z [z )\[ : do N,m e h 4nee Jo2](z [z )2 0 e h and the energy eigenvalue is given by: W \E[Er N,m N,m Following the assumption made in (2.3), the boundary condition (2.2) becomes:

(2.6)

(2.7)

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A

B

d d U ^ ,^ \0 N,m 2 2

(2.8)

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

A

B

1 o2 X (o)\ exp [ (2.9) M,0,0 4a2 J2na H H Thus we may drop the subscripts from the variables related to the transverse motion and write V for V U for U and W for W . 0,0 0,0 0,0 In order to solve the Schro¨dinger equation (2.5), we assume that the motion of the hole can be separated from the motion of the electron about the hole. Thus we write: U (z ,z )\f (z [z )W (z ) (2.10) j e h j e h j h where f (z [z ) is the wavefunction of the electron about the fixed hole position z , corresponding j e h h to the approximation that the hole mass is infinite (m \O) and k is a label. The boundary conditions h defined by (2.8) become f

j

A

B

d ^ [z \0 2 h

where f (z [z ) is the solution to the equation j e h ~ h2 d2 [ ]V(z [z ) f (z [z )\E (z ) f (z [z ) e h j e h j h j e h 2m dz2 e e In a similar way, the wavefunction for the hole satisfies the differential equation:

A

B

C

[

~ h2

D

d2 ]E (z ) ( (z )\W ( (z ) j h j h j j h 2m dz2 h h

(2.11)

(2.12)

(2.13)

with boundary conditions

A B

d W ^ \0 j 2

(2.14)

3. Calculation In order to solve the Schro¨dinger equation (2.12) for the motion of the electron using the adiabatic approximation subject to the condition a @a , it is convenient to introduce the following H 0 notations. We define 2 2q e2 u\ (z [z ), g\ and E \[ j a k e h a k 8pee a k2 0 0 0 0 so that the equation becomes: d2f (u) 1 j ]kS0D(u2]g2)~1@2D0T f (u)[ f (u)\0 j du2 4 j

(3.1)

(3.2)

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

C A

2 d f [z « j a k h 2 0

BD

\0

(3.3)

Under the condition 2a DuDA HBS0DgD0T a k 0 equation (3.2) transforms to Whittaker’s equation

(3.4)

C D

d2f (u) k 1 j ] [ f (u)\0 du2 u 4 j

(3.5)

The two independent solutions to this equation are the Whittaker functions W and M . The j,1@2 j,1@2 general solution in the region u[0 is given by f (u)\A W (u)]B M (u) (u[0) (3.6) j ` j,1@2 ` j,1@2 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 similar problem of an impurity in a QW subject to a strong magnetic field (Monozon and Zhilich [10]) using the 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 the coefficients is then made between the result of the integration and the standard expansion of the Whittaker functions involved in (3.6) for u@1 (e.g. Gradshtein and Ryzhik [35]). When terms of the same 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 equation for the quantum number k:

A

B S

A

B

W W W W 2 2] 1 ^ Q2(j)]1!2([j) 2[ 1 r(j)]"(j)\[Q(j)]1!([j) 2 4 M M M M 2 2 1 1 where the following definitions have been made

C D

2a2 1 H u(k)\2C[1]w(1[k)] , K(k)\1[1C]1 ln 2 2 a2k2 2k 0

AB C

J2 a C a n 3@2 H]1[ ln Q(j)\ 0 a j a 2 2 0 H with the additional notation

D

where Q(j)\0

(3.7)

(3.8)

(3.9)

2 2 W \W (u ), M \M (u ), u \ (1d[z ), u \ (1d]z ) (3.10) 1,2 j,1@2 1,2 1,2 j,1@2 1,2 1 a k 2 h 2 a k 2 h 0 0 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 field region and this implies u

2a A H@1 1,2 k a 0

(3.11)

The latter condition implies that the theory is only applicable for cases where the hole is separated from the edge of the QW by a distance greater than the magnetic length a . Similarly, the condition H 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 \0), the wavefunctions have a definite h parity. It is clear from (3.10), that for z \0 then u \u , W \W and M \M . The levels of even h 1 2 1 2 1 2 parity correspond to states with the positive sign in front of the radical in (3.7), and the odd parity states correspond to states with the negative sign. However, the classification of the energy levels and states into two groups can be made for the hole in any position in the well (z D0, u Du ); all states with the positive sign in front of the h 1 2 radical in (3.7) are referred to as quasi-even states while those with the negative sign are referred to as quasi-odd states. The quasi-even states have a quantum number k given by k \n]d (n), n g

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

(3.12)

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

(3.13)

while the quasi-odd states have k given by k \n]d (n) n u

The ground level (n\0) is non-degenerate and has k @1. The excited states (n\1, 2, 3, . . .) have a 0 doublet structure consisting of quasi-even and quasi-odd components. It follows from (3.10) and (3.7) that the replacement of z by [z is equivalent to the switch u ½u . Therefore, we see that h h 1 2 k(z )\k([z ) and also, from (3.1), that E(z )\E([z ). This implies that E (z ) is a symmetric funch h h h j h tion of z . In general, there is no exact analytic solution to (3.7) for E (z ); solutions can only be h j h found for the limiting cases of small displacements of the hole from the centre of the well, and for the 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 ground state E (k @1) the inequality 0 0 z h @1 a k 0 0 Furthermore, with the additional assumption that the well is wide, we also have

A B

2 d ^z A1 a k 2 h 0 0

(3.14)

Thus the energy of the ground state is given by 1 E (z )\E (0)] m X2z2 h 0 h 0 2 h where

A

C

d R 1[2exp [ E (0)\[ 0 a k k2 0 0 0

(3.15)

DB

(3.16)

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S

A

B

e2 4R m d e exp [ R\ , )\ . ~ 8nee a hj2 m 2a j 0 h 0 0 0 0 Now k @1 is the smallest root of the equation 0 2a2 1 1 3C H \0 ]w(1[k )] ] ln 0 2k 2 a2k2 2 0 0 0 In the logarithmic approximation, we have and

A B

Dln

A B

2a2 H DA1 a2k2 0 0

(3.17)

(3.18)

(3.19)

so that the solution k to (3.18) is given by 0 2a2 1 H ^[ln (3.20) a2 k 0 0 and the energy E (z ) can be found easily by using (3.15), (3.16) and (3.17). 0 h For the second case of the hole located at the edge of the well, it has been shown (Monozon and Zhilich [10]) that the transcendental equation for k becomes

A B

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

W 0 M 0

(3.21)

where 2d W \W (u ), M \M (u ), and u \ . (3.22) 0 j,1@2 0 0 j,1@2 0 0 a k 0 Again we assume that the well is wide (u [1); the ground state has k \1]d where d@1. The 0 0 expansions for the Whittaker functions for large values of u (Gradshtein and Ryzhik [35]) are used in (3.21) to give the relationship for d namely: 1 1 ]2Q(1)] u2e~u0\0 d d2 0

(3.23)

It follows therefore that

C

D

1 eu 1@ \ 0 J1[8Q(1)u2e~u0[1 0 d 2u2 0 and the expression for E is 0 E

0

A B ^

d R ^[ 2 12

(3.24)

(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(z ); from this result, E (z ) can be calculated using (3.1). Once E (z ) has been found, it is h j h j h possible to solve (2.13) for W (z ), the component of the exciton wavefunction due to the motion of j h the hole, in order to find the total exciton energy W . When substituted into (2.13), E (z ) acts as an j j h effective potential for the hole. The potential energy is expected to consist of a series of wells; the

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Eλ

158

κ=2 κ=1 W1, κ

E1 (zh)

κ=0 κ=4 E0 (zh)

κ=3

κ=2 W0, κ

κ=1

κ=0

zh – d/2

d/2

Fig. 1. A sketch of the effective potential for the hole E (z ) and the resulting energy levels W . j h j,i

bottom of each well is defined by the position of E (z \0) and the top of which (given by j h E (z \^1d)) will coincide approximately with the bottom of the next well (E (z \0)). The specj h 2 j`1 h trum of the total exciton energy will be a sequence of groups of levels W (j\0, 1, 2, . . .). Each of ji the groups is associated with a particular fixed electron state denoted by the quantum number k. A sketch of the expected shape of the effective potential E (z ) and the energy levels W for the ground j h j,i state (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 a and a that satisfy the restriction 0 H Q\0 in (3.9). This condition could only be guaranteed to hold if a /a D0.1 to 0.2. Furthermore, it H 0 is also necessary to choose a value for the well width d; for the above theory to hold it is necessary to have d/a [1. With these parameters fixed within these specified ranges, (3.7) has been solved 0 numerically; the smaller root k \1 was found as a function of the hole position z . As (3.7) is valid 0 h

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Hole at well edge

Curve extrapolated in this region

λ

1

Hole in central region of well

Hole close to well edge

0

zh = 0

zh = d/2 zh

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

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

A B

2z 2z h 0¹ h¹ d d

A B 2z h d

A BA B

a a 0 \1[2 H (4.1) a d 0 0 0 The Whittaker functions in equation (3.6) were calculated using Kummer’s functions M(a,b,z) and U(a,b,z) defined in Abramowitz and Stegun [36] in the form: where

M (z)\e~1@2zz1@2`kM(1]l[k, 1]2l, z) k,k 2

(4.2)

and (4.3) W (z)\e~1@2zz1@2`kU(1]l[k, 1]2l, z) k,k 2 It was found that the first forty terms in each of the expansions of the Kummer’s functions were sufficient to ensure convergence of these functions. As noted above, equation (3.7) is not applicable when the hole is located close to the edge of the well. Instead, we can then use (3.21) which applies when the hole is exactly at the edge of the well; this is solved numerically for the same values of the parameters a /a and d/a to give k(z \1d). Also, if the region where (3.7) is invalid is H 0 0 h 2 sufficiently small, then one expects that it is reasonably accurate to approximate the dependence of k(z ) by a smooth curve connecting the two known regions. A sketch of this is shown in Fig. 2. h Once k(z ) has been calculated numerically, it is easy to apply (3.1) to find the energy of the h electron E (z ) for the fixed hole position z . An illustrative example of the above calculation is shown j h h

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0 aH /a0 = 0.24 d /a0 = 1.6

aH /a0 = 0.24 d /a0 = 1.6

0.8 0.6 0.4

–2

κ=1

–4

κ=0

Eλ

λ

1.0

–1.0

–0.5

0.0

0.5

–6

1.0

–1.0

–0.5

2zh /d

0.0

W0, κ

0.5

1.0

2zh /d

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

1.25

1.25 aH /a0 = 0.1, d /a0 = 2.4 aH /a0 = 0.24, d /a0 = 2.4

1.00

aH /a0 = 0.10, d /a0 = 1.6 aH /a0 = 0.24, d /a0 = 1.6

1.00

λ

0.75

λ

0.75 0.50

0.50

0.25

0.25

0.00

–1.0

–0.5

0.0 2zh /d

Fig. 4.

0.5

1.0

0.00

–1.0

–0.5

0.0

0.5

1.0

2zh /d

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 k (z ), where the dots are the calculated 0 h values and the line representing an extrapolation in the region close to the well edge. The example shown in Fig. 3 has a /a \0.24 and d/a \1.6. It can be seen that the theory is exactly valid over a H 0 0 large part of the QW. The lower graph of Fig. 3 shows a plot of the effective pote ntial E (z ) and j h the resulting energy levels W for the ground state k\0. As can be seen from Fig. 3, the position j,i of the lowest energy level j\0 is insensitive to the accuracy of the extrapolation in the region close to the well edge; in contrast, the higher levels j\1, 2, . . . will be affected by this extrapolation. For this 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(z ) for different h values of the parameters a /a and d/a . The graphs show that the calculation gives results which are H 0 0 in qualitative agreement with expectations; as a /a decreases (corresponding to an increase in the H 0 magnetic field) k also decreases (corresponding to an increase in the magnitude of the energy DED as 0 given in (3.1)); furthermore, a decrease in the well width d results in an increase in k . Closer 0 inspection of Fig. 4 reveals that the variation in k for a change in a /a is less pronounced for a 0 H 0 larger well than for a smaller one; that is, the relative shift in the position of the bottom of the plot of k (z ) with a shift from d/a \0.10 to d/a \0.24 is smaller for a /a \0.10 than for a /a \0.24. 0 h 0 0 H 0 H 0 This behaviour is expected because, for lower values of a /a (a stronger magnetic field) the separH 0 ation of the electron and hole is less, so that the position of the edge of the well has less influence on the electron energy. It is because of this extremely strong magnetic field (a /a ^0.1) that the H 0 electron energy shows practically no dependence upon the displacement of the hole for most of the QW. A comparison of our results with other work would be desirable at this point. As was noted

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–4 –5

W (κ = 0) in units of R

–6 –7 d /a0 = 1.6

–8

d /a0 = 2.0 –9

d /a0 = 2.4

–10 –11 –12

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

aH /a0

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

earlier for the calculation of k (z ), the problem is identical to that of an impurity in a QW. Greene 0 h and Bajaj [2,3], Greene and Lane [4] and Kuhn et al. [37] calculated the energy eigenvalues of a hydrogenic 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 qualitatively confirmed by their numerical calculations. Moreover, it follows from (3.15) that the reciprocal of the derivative of the energy E(z ) with respect to the centre position z is proportional to z~1. This result h h h also agrees with the result of Greene and Bajaj [3, fig. 2]. The energy levels W have also been calculated numerically using the effective potential j,i E (z ). As mentioned earlier, we have restricted our calculations to those for the ground state of the j h exciton (i.e. k\k ). Figure 5 shows the position of this lowest ground state level for different values of 0 a /a and d/a . Figure 5 is seen to have the same qualitative features as expected and the trend H 0 0 observed shows that an increase in the magnetic length (decrease in the magnetic field) corresponds to an increase in the energy. For the smallest values of a (strongest magnetic field), the size of the QW H has a negligible effect on the position of the lowest energy level. It is only for the cases of a /a [0.15 H 0 that one can resolve any difference between the graphs. Narrower wells are seen to have a higher energy than wider wells. Again, the fact that we are using such high magnetic fields makes any direct comparison with earlier work difficult; however, it is worth noting that the flattening of the curves as the 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 considerations of an exciton with extremely different electron and hole masses in the presence of a strong magnetic 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 exciton energy is found to decrease (the ‘binding energy’ increases). Probably, this result has a common feature; it was pointed out by Cen et al. [29] that an increase in the magnetic field leads to an increase in the enhancement of the binding energy of the exciton due to the image charge contribution in the dielectric QW structure. In contrast, with a narrowing of the QW, the exciton levels shift towards higher energies. It has also been shown that, for sufficiently strong magnetic fields (i.e. a /a \0.15 H 0 for the ground state) the size of the QW has little effect on the exciton energy. The results of the double adiabatic approach are in agreement with those obtained earlier by variational-type or perturbation-type calculations. Acknowledgements—One of the authors (BSM) expresses his gratitude to the Royal Society for financial support which has enabled this collaborative programme to proceed.

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