F. DUJARDIN e t al.: Neutral Bound Excitons in a Low Magnetic Field 3 29

phys. stat. sol. (b) 126, 329 (1984)

Subject classification: 13.4 and 13.5.1; 22.2; 22.4

Laboratoire de Physique des Semiconducteurs, Universitd de Metzl) (a ) and Laboratoire de Xpectroscopie et d’0ptique du Corps Xolide, associd au C.N. R.S. no 232, Universitk Louis Pasteur, Strasbourg2) (b )

Neutral Bound Excitons in a Low Magnetic Field BY F. DUJARDIN (a), B. STI~BI~ (a), and G. MUNSCHY (b)

The diamagnetic shift of an exciton bound to a neutral donor or acceptor shallow impurity in direct gap semiconductors is studied using the previously obtained Page and Fraser type wave function. The results are compared with the experimental observations in I n P for the exciton-neutral donor complex and in GaAs and ZnTe for the exciton-neutral acceptor complex. I n all cases qualitative agreement is found while for I n P a very good agreement is obtained if an isotropic experimental effective hole mass is used.

Nous ktudions le dkplacement diamagnktique d‘un exciton li6 ti une impuretk neutre peu profonde de type donneur ou accepteur dans les semiconducteurs ti transitions directes. A cet effet, nous utilisons la fonction d’onde variationelle de Page e t Fraser que nous avons dQtermin6e pr6c6dem- ment. Nous comparons nos rksultats avec les observations exphimentales dans I n P pour le donneur e t dans GaAs e t ZnTe pour I’accepteur. Dans l’ensemble, nous trouvons un accord qualitatif tandis que pour InP, nous obtenons un trAs bon accord ti condition de prendre une valeur moyenne de masse de trou dkduite de l’expkrience.

1. Introduction

Since the pioneering work of Thomas and Hopfield [l], the use of a magnetic field has become a common tool for the identification of excitons bound to impurities. But till now, the lack of accurate enough envelope wave functions has not allowed one to take full advantage of all the information available from the many magneto-optical experiments involving excitons bound to neutral impurities [2]. It is only recently that even in the absence of any external perturbation the stability of these four- particle complexes has been proved by means of strictly variational calculations [3 to 51. Thus only approximate models could be proposed in the past in order to estimate the diamagnetic shift of neutral bound excitons. In particular, when the interaction between an exciton and a neutral acceptor impurity can be represented by an effective short-range potential, the binding energy does not depend on the magnetic field, so that the transition energy shifts of free and bound excitons are equal [6, 71. Experimental observations in CdS [l], GaSb [9], and GaAs [6, 71 seem to confirm this assumption. It has also been suggested that for deep impurities, the transition shift of an exciton bound to a neutral acceptor could be identified with the energy shift of a neutral donor [lo, 111 as it results from more recent experimental data on GaSb [lo] and GaAs [ll]. Finally, other measurements in CdTe [l2] seem to support the fact that the two kinds of neutral bound excitons give rise to a diamagnetic shift similar to that of a free exciton. All these approximations suppose in fact that the magnetic shifts of the electron-hole or the hole-hole correlation energies may be neglected.

l ) Ile du Saulcy, 57045 Metz, France. 2, 5, rue de I’Universitk, 67084 Strasbourg Cedex, France.

330 F. DUJARDIN, B. STEBB, and G. MUNSCHY

In a recent paper [a], we have shown that it is essential to take into account these correlations in order to obtain a stable binding. It seems therefore useful to study the influence of a uniform magnetic field on the ground state of a neutral bound exciton by using our previously obtained [4] fully correlated Page and Fraser type wave function. In this preliminary work, we restrict ourselves to the case of low and inter- mediate fields and to shallow impurities within the effective inass approximation in a simple non-degenerate two-band model. We discuss explicitly the case of the exciton neutral acceptor complex (Ao, X), which is quite analogous to that of the exciton neutral donor complex (DO, X) by interchanging the electrons and the holes.

2. Theory

The effective envelope Hamiltonian may be simplified by using the usual atomic units aA = &h2/mze2 for length, 2 lEAl = mh*e4/~zh2 for energy and the dimensionless effective magnetic field parameter ya = hwc/2 IEAI, where o c = eB/mEc is the effec- tive hole cyclotron frequency related to the magnetic field directed along the z-axis. E is an appropriate dielectric constant taking into account the polarization effects. By using the Lorentz gauge and neglecting the spins as well as the electron-hole exchange interaction [13], the Hamiltonian of the system is given by

(1) H = Ho + H i + H i , where H,, is the zero-field Hamiltonian

Table 1

Effective magnetic fields YA, YD, and yx for acceptors, donors, and excitons correspond- ing to a magnetic flux density B = 10 T in various semiconductors

substance & m%kl Y1 YD YA Yx

AlSb GaP GaAs GaSb InP InAs InSb Ge ZnS ZnSe

12.0 10.75 12.56 15.7 12.4 14.6 17.9 15.36 8.1 9.1

0.18 0.17 0.067 0.045 0.080 0.023 0.014 0.038 0.28 0.14

4.15 4.20 7.65

11.80 6.28

19.67 35.08 13.35 2.54 3.77

0.19 0.17 1.50 5.18 1.02

17.14 69.55 6.95 0.04 0.18

0.11 0.09 0.39 1.46 0.26 3.50

16.77 1.79 0.02 0.05

0.58 0.50 3.43

12.14 2.30

36.15 154.64 15.79 0.10 0.42

The values of E, m,*, and yl = mo/mz, rn: being the isotropic hole mass, are taken from [14 to 161.

Neutral Bound Excitons in a Low Magnetic Field 331

with a = mz/mz, $ = x! + y!, and where Liz are the projections of the angular momentum operators of the three mobile particles. The symmetry properties of the Hamiltonian (1) show that the energies of the two complexes (Ao, X) and (DO, X) are related by

(7) E(An, X)(a, Y A ) - - E ( D o , X ) (a-1* YD)

E A E D

where the effective fields y D and y A are assumed to have the same numerical value. yD is defined in the same manner as yA. I n particular, for a given magnetic field, yn = yA/a2. The linear magnetic term Hi gives rise to the orbital Zeeman splitting, whereas the two terms H i and LIh are in general responsible for the diamagnetic shift. Their influence depends on the value of the effective magnetic field y , which for acceptor impurities may often be one order of magnitude smaller than for donor impurities as results from Table 1.

For small enough values of the effective magnetic field yA, the magnetic terms H i and H i may be computed as perturbations with respect to the zero-field Hamiltonian H,, the eigenvalues of which have been previously determined [4] by using the fol- lowing 35-term variational wave function,

Imnpq) = rpErqer$r,P, exp (-w1 - fir2 - re) , cmnpq, a, @, and k being the variational parameters, Plz the operator of permutation of the indices 1 and 2, and m, n, p , q positive integers with m + n + p + q < 3. Because this function does not depend on the angles, only the quadratic term has to be considered at first order of perturbation, though the two terms H i and H i may give non-zero contributions a t higher orders of perturbation. These latter contri- butions, which need the knowledge of all the discrete and continuous eigenfunctions of the Hamiltonian Ho are expected to be negligible. At the present stage, we restrict ourselves to the first-order contribution of the quadratic term H i which leads to the following total energy shift :

where the function Fper has been determined using the zero-field ground-state wave AEper(YA, 0) = 7; [ E A ~ F p e r ( 0 ) 3 (9)

Fig. 1. Energy shift of the (Ao, X) complex obtained by using the perturbation theory, (1) A E / A E x , (2) AE/(AEx + AEA); and ( 3 ) AB/(AEx + AEA), comparison with the theoretical

l , , , l I I I 1 prediction of Willmann et al. [7] (a = &/rnE) 0.1 05 I 2 510

332 I?. DUJARDIN, B. STBBB, and G . MUNSCRY

1 -"'I1 - .f - Q ! : lL

10

-15

function (8). In the same approximation, the acceptor and exciton diamagnetic shifts are given by AEA = 7; IE,1/2 and A E x = yx IExI/2, with yx = yA(1 + u - ~ ) ~ . Our results reported in Fig. 1 show that in the low-field approximation, the energy shift AE of the neutral bound excitons, may be rather different from that of free excitons and the binding energy shift AW = A E - (AEA + A E x ) is in general non-negligible and depends on the mass ratio, unlike a previous assumption [6, 71. We remark also that the (Ao, X) complex becomes more stable for o-values less than 0.69. Yet, the above conclusions depend on the validity of the perturbation theory. For example the condition AE/\El < 0.1 is realized if y A < 0.1 for u = 0.5 and if y A < 0.4 for f7 = 10.

In order to extend somewhat the validity range of the previous results, we have determined variationally the eigenvalues of the Haniiltonian (1) using the wave func- tion (8) but by varying only the linear parameters cmnpp. The corresponding energy shifts become then

(10) 2 AEVM(yA, 0) = yA lEAl .Fvar(yA, 0) *

In Fig. 2 we have reported the variational as well as the perturbation energies with respect to the first generalized Landau level E L = y A ( 2 + 0-1) lEAl of the complex. In Fig. 3 we compared the (Ao, X) transition energy shift Ahv = AE - AEA to that of a free exciton, Ahvx = AEX and that of the effective mass (Ao, X) pseudo-donor model, Ahv = AEA+ -+ AED - AEA. To determine the energy shifts of the exciton, the donor, and the acceptor impurities, we have used the highly accurate results of Cabib et al. [17] as well as those of Larsen [l8] for that of the positively charged ac- ceptor A+. It appears that, in the effective mass approximation, the exciton and the bound exciton shifts may be equal only in some particular circumstances: i.e. u = 0.5. Moreover, the bound-exciton shifts increase with the effective magnetic field and take

I;.;; I I I ' ' '1

I 2

r 10

-

~

- - - -

- - - - :: -

- - - I I l l 1 1 ) 1 1 1 I

Neutral Bound Excitons in a Low Magnetic Field 333

higher values for the (DO, X) complex than fsr the (A@, X) complex in most of the semiconductors.

3. Comparison with Experiment It may be questionable to compare the results of our present rather crude model with the experimental observations. Nevertheless, in the past, different attempts [6,7,11] have been made to interpret the observed transition shifts in Sn-doped GaAs. The theoretical prediction of Willinann et al. ,[7] is based upon the deuteron model which does not depend on the mass ratio 0 = nz$/m? and is in agreement with the present theory (Fig. 1) only for 0 x 0 . 5 which corresponds roughly to the case of GaAs. From the results shown in Table 2, it can be seen that the agreement between

Table 2 Comparison between theoretical and experimental diamagnetic shifts for the (AO, X) complex in GaAs:Sn (1.507 eV) [SJ and ZnTe:Cu (2.3750 eV) [19 to 221 and the (DO, X) complex in InP [23] for an effective magnetic field of y = 0.1

complex sub- m,*/mo yl m o l r n ~ E B AE, AED Ahvth AhvexP stance (T) (meV) (meV) (meV) (meV)

(AO, X) GaAs 0.0665 7.65

(AO, X) ZnTe 0.16 3.93

(DO, X) InP 0.08 6.28

P I P I

P O I i?01

~141 ~ 4 1 2.02 r231

12.56 2.54 1.04 0.38 1.09 0.39

10.1 14.92 2.30 0.64 2.45 1.61

12.4 0.98 0.114 0.142 0.075

12.1 10.3 0.057 0.078

P I ~ 7 1

c141 c231

c231

[201 P O , 221

theory and experiment is only qualitative, mainly because the effective mass theory is not well adapted to deep impurities, although the shifts observed with different impurities seem to be the same 1171. Indeed, in this approximation, the mean radii of the orbitals are overestimated, so that the theoretical diamagnetic shifts are too large. On the other hand, for deep impurities, the pseudo-donor model seems to be a good approximation, because in this case the transition energy shift Ahv may be identified with the donor energy shift AE,. But this latter shift cannot be identified with the effective mass pseudo-donor transition energy shift because the contribution AEA+ - - AEA x 0.30 meV is not negligible. In the case of ZnTe:Cu [19 to 221 where the Cu-acceptor impurity is less deep, we get a better agreement with the experimental observations [20 to 221. The remaining discrepancy may be due to the fact the valence bands of GaAs and ZnTe are degenerate a t the centre of the Brillouin zone and cannot be accurately described by a simple isotropic hole mass parameter. This can be seen when we compare the results obtained for the (DO, X) complex in I n P (Table 2 ) . By using rnf = l/yl, the results are not very satisfactory but with an experimental

mean” hole mass [23] we find a good agreement between the theoretical and ex- perimental diamagnetic shifts. This fact shows that the degeneracy effects may be very important. On the other hand, it seems necessary to extend the present low-field theory to higher values of the effective magnetic field.

& <

Acknowledgements

We would like to thank Prof. C. Klingshirn, Dr. W. Maier, and Dr. G. Schmieder for giving us information about their results prior to publication.

334 F. DUJARDIN et al. : Neutral Bound Excitons in a Low Magnetic Field

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(Received February 16, 1984)