F. DUJARDIN e t al. : Neutral Bound Excitons in a High Magnetic Field 559

phys. stat. sol. (b) 141, 559 (1987)

Subject classification: 71.35 and 71.55

Centre Lorruin d’0ptique et d ’ ~ E e c t r a ~ i ~ u e des SoEides, Universite‘ cle Metz, Ecole Supkrieure d’Etectriciti TechnopGEe de Metzl) ( a ) aud Laboratoire de Spectroscopie et d’Optique d u Corps Solide, associe‘e au C.N.R.S. nQ 232, Universite‘ Louis Pasteur, Strasbourg2) (b)

Neutral Bound Excitons in a High Magnetic Field

BY P. DUJARDIN (a), B. STBBB (a), and G. MUNSCHY (b)

The influence of a constant magnetic field on the energy of the ground state of an exciton bound t o a neutral shallow impurity is studied in the high field limit. The Schrodinger equation is solved using the adiabatic approximation and considering the Coulomb potential as a perturbation with respect to the fundamental Landau level. The total energy is calculated by means of a combination of four Gaussian type variational wave functions. The results show that the magnetic field enhances the Coulomb interaction and allow to estimate the transition energies.

Nous Btudions l’influence d’un champ magnhtique constant sur 1’6nergie de 1’6tat fondamental d’un exciton li6 A une impuretB peu profonde dans le cas limite de champs magnbtiques intenses. Nous r6solvons 1’6quation de Schrodinger par la mhthode adiatique et estimons le potentiel dinteraction Coulombienne en perturbation par rapport au niveau de Landau fondamental. Nous calculons 1’6nergie totale au moyen d’une combinaison de quatre fonctions d’ondes variationnelles de type Gaussiennes. Nos rksultats montrent que le champ magnetique renforce l’interaction Cou- lombienne e t permettent I’estimation des 6nergies de transition.

1. Introduction

The magneto-optical study of neutral bound excitons has been widely used to charac- terize the impurities in semiconductors [l]. Nevertheless, till now, i t has not been possible to take full advantage of all these experimental results mainly because the lack of theoretical studies of neutral bound excitons in a magnetic field.

In a first approach [2], we have computed the diamagnetic shift of an exciton bound to shallow neutral impurities in the case of relatively low magnetic fields. This ap- proximation breaks down when the ratio of the magnetic and Coulomb energies is of the same order. This restriction can occur for relatively low magnetic fields when the electron or hole effective masses are small. Till now, only the structure of the energy states has been studied in the high field limit [3].

In this paper, we study the influence of an external constant magnetic field on the energy of neutral bound excitons in the case of high effective magnetic fields. We restrict our investigation to shallow impurities.

Without taking into account the Coulomb interaction, the motion of the three mobile particles is free along the direction of the field but is quantized in the plane perpendicular t o i t giving rise to Landau levels. For high enough magnetic fields, the Coulomb interaction is small compared to the magnetic energy. Therefore, we separate the two motions in the sense of the adiabatic approximation [4] and treat the Coulomb potential as a perturbation with respect to the fundamental Landau level. It gives rise to an attractive potential for the z-dependence of the wave function which we

l) 2, rue Edouard Belin, F-57078 Metz C6dex 3, France. 2, 5, rue de l’Universit6, F-67084 Strasbourg CBdex, France.

560 F. DUJARDIN, B. S T ~ B I ~ , and G. MUNSCHY

determine using a four-term Gaussian-type variational wave function. Our results show that the magnetic field enhances, as expected, the Coulomb interaction between the four particles.

2. Hamiltonian

We discuss explicitly the neutral acceptor bound exciton (AO, X), consisting of two holes 1, 2 and an electron e bound to the impurity centre, which is quite analogous to the neutral donor bound exciton (DO, X) by interchanging the electrons and the holes. We assume that the effective-mass approximation is valid and that the constant energy surfaces in the reciprocal space are spherical. The Hamiltonian may be simpli- fied by using the atomic units aA = &h2/mEe2 for length, 2 I EAI = mEe4/e2h2 for energy and the dimensionless effective magnetic field y A = ha1,/(2 lEAl), where m, = = eH/mEc is the effective hole cyclotron frequency related to the constant magnetic field H directed along the z-axis. rnf and rnz are the isotropical electron and hole effec- tive masses. E is an appropriate dielectric constant taking into account the polarisation effects. By using the symmetrical Coulomb gauge Ajr) = + H x r , and neglecting the spins as well as the electron-hole exchange interaction, the Hamiltonian of the system is given by:

;7e = T + V , (2.1)

with 0 = rnf/w& ef = x: + yf and where Li, = --i a/apl, are the projections of the angular momentum operators of the electron and the holes along the direction of the magnetic field. The symmetry properties of the Hamiltonian show that the energies of the two complexes (Ao, X) and (DO, X) are connected by

, (2.4) E(A0, XdU, Ya) - E(W, s)(a-l, Yn) E A ED ~ _ ~ . _ _ _

where the effective fields y,, and y A are assumed to have the same numerical value. For a given magnetic field, we have y D = yA4/02. Because the Hamiltonian is not modified when undergoing an arbitrary rotation @ about the magnetic field direction, the projection of the total angular momentum L, = Ll, + Lpz + L,, along this direc- tion remains constant. Thus the wave function reads

Y = exp ( i M @ ) y ; M = 0, +1, ... (2.5) This function has a definite parity due to the invariance of the Hamiltonian when subjected to the transformations: zi -, - x i , @ -, @ + z. It must also be symmetrical (or antisymmetrical) with respect to the permutation of the two identical particles.

3. Very High Field Limit

For very high effective magnetic fields yA > 1, the Coulomb energies are negligible compared to the magnetic energy tzo~,. I n this limit, the system behaves like three quasi-free particles interacting independently with the field. The energies of t he

Neutral Bound Excitons in a High Magnetic Field 561

carriers become quantized in the plane perpendicular to the magnetic field but remain quasi-continuous in its direction, The Schrodiriger equation of the transverse motion separates into three one-particle equations giving rise to the well-known Landau levels [5, 61. In this case, the projections of the orbital angular momenta Mb (i = 1, 2, e) and of the total orbital angular momentum M = Ml + M2 + Me are constants of motion. They take positive, negative, or zero integer values.

The total transverse wave function reads then M M 11 lu,lu,ul, =

(3.1)

% = c P ~ - P ~ , @ z = ~ 2 - ~ e , @ = y e - (3.2)

~ 2 , ee , @I, @2, @ e ) = - - elsf,@, eiJf.@2 ei.'l@e RyAT1ll, A (el) ~ ~ ~ ~ ~ ' " ( e 2 ) &:(ee)

with

The radial functions R are given by

, lM, l+ l , p Y A li IM?I + FzMz

2 Rg:(et) = exp ( - 2 Q:) 1 ~ 1 (- N , + (3.3)

lFl represents the first kind degenerate hypergeometric function and E~ = 1 (i = 1, 2) or 8% = -1 (i = e). The Landau magnetic quantum numbers N I are positive or zero integers and fulfil the conditions N , 2 -&,Mi. The energy for the transverse motion of the three non-interacting particles is given by

~ N , Z V ~ = ( N + 1) Y A + (Ne + O-'YA (3.4) with N = N , + N2. Each energy eigenvalue corresponds to a large number of de- generate states which differ from one another in the values of the set of quantum numbers MI, M2, and Nl not appearing in the energy expression, 1M, N , and N e keeping fixed values.

4. Adiabatic Approximation

Strictly speaking, the longitudinal and transverse motions cannot be separated when the Coulomb interaction does not vanish. However, for high enough magnetic fields, we can generalize the procedure introduced in the case of hydrogenic problems [7 to 111. In this high field limit, the Coulomb potential (2.3) in the Hamiltonian (2.1) may be considered as a small perturbation and the two motions may be tentatively separated in the sense of the Born and Oppenheimer approximation.

The energies of the transverse motion are given by

(4.1) z , y i v

U M N X e = gXS, f A.&NiV, , where the small corrections &',b7ne are obtained by solving the system of linear equa- tions - - ( F - l i>"S) Xi,Y = 0 . (4.2)

The matrices f' and 3 are defined with respect to the non-perturbated transverse wave functions, symmetrical ( y = 1) or antisymmetrical (Y = -1) with respect to the permutation of the two identical particles,

5G2 F. DUJARDIN, B. STBBE, and G . MUNSCHP

k* denotes the column matrix of the linear coefficients c$$>{f,?’ appearing in the ex- pansion of the wave function

(4.6) c M , A f 2 X ? , I W , M 2 M , v YklY, = c ( ’ Yr;l,,,ye ~lIIM&~l

belonging to the energies U$lNATe (4.1). In the adiabatic approximation, the total wave function reads

(4.7) 1 3 L1’ Zi,j:” 7 v Yai1ds, = M \‘XeYIIlSXe *

The function Z$&v, is either symmetrical (p = 1) or antisymmetrical (p = -1) with respect to the permutation of the two identical particles. Thus, the total wave func- tion (4.7) has a definite symmetry pv. In this approximation we can neglect the deriva- tives of the coefficients cLV,,,,ie with respect to the coordinates z,. The longitudinal wave functions Z$i?,$&e satisfy the equation

A f , LM, If, I’

As a result, the motion of the three particles becomes quantized in the direction of the field due to the Coulomb interaction.

5. Ground State Energy of the Transverse Motion

The ground state transverse energy of the three non-interacting particles (3.5) cor- responds to zero values of the Landau magnetic quantum numbers N,, N,, and N,. Therefore, the normalized non-perturbated transverse wave function (3.1) is given by

pLir,5I2x = XM M o i j 2 IM I l1v21 1 ~ 1 ~ 1 000 ( &f;o)- el I e 2 e e

x exp 1 - (ef + pf + @:) exp i(HlQ1 + MzID2 + MID) (5.1) 4 1

with M , 2 0, M , 2 0, and M e 5 0. We remark that the matrix # (4.5) is diagonal,

In the next step we estimate the Coulomb matrix (4.4) related to the basis functions (4.3) with definite symmetry v,

1 (5.3) (y&f,&f&) - - (p,WlI + v l p w * J l $

000 ) * Oo0 -I/2

The matrix element V T 1 : X ; O = ( y h f : J l ; A l I p p y # Z ’ W )

il!flM20 000

involves six terms. I n particular, we obtain (5.4)

(5.5)

Neutral Bound Excitons in a High Magnetic Field 563

The corresponding matrix is diagonal. We remark that

MnM,O 1 lim (-:) = --, 1 % + M , M , O I Z l l

where represents the gamma function defined by

Solutions of the z-dependence of the wave function are impossible in closed form with (5.5). It is, however, possible [4] to use the approximate expression

The matrix elements ( - l / v2 ) and ( l /ye ) have a similar expression. It is, however, more difficult to compute those relative to the interactions between the mobile particles. In particular, we have estimated the limit of (-l/rle)Mlfif20 when Izl - zeI tends to infinity. The nondiagonal elements are much smaller than the diagonal terms because the presence of the factor [ya(zl - ~ , ) ~ ] - - l ~ : - ~ 1 1 , and can therefore be neglected. On the other hand, i t can be verified that

M;M;O

where

with A, = I MII and A, = lMel and (4n -l)! .

(2% - i)! 24-1' (+ ,2n) = 1 2 2 1 .

The matrix elements may therefore be approximated by

(5.11)

(5.12)

An analogous approximation holds for the matrix elements (-line) and (l/r12). As a result, we obtain the following expression for the adiabatic potential:

(5.13)

(5.14)

564 F. DUJARUIN, B. S T I ~ B ~ , and G. MUNYCHY

I I , '1 t u

-

I I I

Fig. 1. (1) Exact and (2) approximate adia- batic potential for the hole-ionized impurity interaction ( y a = 10)

where PI2 and Dze may be deduced from (5.10) by replacing the indices e and 1 by 2, respectively.

We remark that the coefficients 16, do not depend on the effective mass ratio 0. They have been computed numerically for different values of the effective magnetic field, and the quantum numbers M , and M, (Table 1). Fig. 1 shows that the exact and the approximate adiabatic potentials are very close. They have the same values a t z , = 0 and the same asymptotic forms when z, tends to infinity.

Tab le 1 Values in atomic units of the parameters 0, and BtJ for the adiabatic potential as defined by (5.6) and (5.10) for y A = 1

M% M i B, BU

0 0 0.79788 0.79788 1 0 1.59577 1.06385 1 1 1.59577 1.59577 2 0 2.12769 1.16056 2 1 2.12769 1.82374 2 2 2.12769 2.12769 3 0 2.55323 1.21582 3 1 2.55323 1.96402 3 2 2.55323 2.32112 3 3 2.55323 2.55323 _ _ ____

6. Variational Solution of the Longitudinal Equation

It remains to solve (4.8) for the z-dependence of the wave function (4.7), using the approximate adiabatic potential (5.13). For this purpose, we use the following trial four-term wave function:

P f l k = (1 + pP12) T i j k ;

' p I l k = exp ( - a , z ~ - aizz - bk&, p = + I ,

(6.1) where the parameters ciik, a,, a,, and bl, are determined using the variation method. To get the numerical values of the coefficients cyk for given values of the parameters a,,

Neutral Bound Excitons in a High Magnetic Weld 565

ai, and be, one has to solve the eigenvalue equations

The overlap and kinetic energy matrix elements may be obtained without difficulty

= (vijjty I yijk) = n3i2[(ai + ai,) (a, + aj,) ( b k + bk*) ] - l " , p ; k ' (6.4)

The adiabatic potential matrix elements may all be expressed in terms of the integral co

e-at2 dt = J l+t 9

0

which can be computed by the method of Gauss. Finally, we obtain

The ground state energy is expected to correspond to a symmetrical longitudinal wave function Zp (p = l), and to zero values of the quantum numbers Mi. This assumption has been verified in the special case u = 5, yA = 10.

Fig. 2 reproduces the variations of the ground state total energies versus u for increasing values of the effective magnetic field yA. It appears, as expected, that the magnetic field enhances the binding due to the Coulomb interaction. Our values are in reasonable agreement with those reported in some special cases : the negative hydrogen ion H- [12, 131 corresponding to c = 0 and the hydrogen molecule H, 1141 correspond- ing to infinite values of a. Our results appear to be more adequate for small U-values because our wave function does not take into account the correlations between the mobile particles.

566 F. DUJARDIN e t al. : Neutral Bound Excit.ons in a High Magnetic Field

A -1

I 2', I I 1 1 Pig. 2. Energies of the (A", X) complex vs. the effec-

tive mass ratio B = m$/mz for different values of the \ 2i=l effective magnetic field yA. The zeros of the energies

:;i I \ I

We have restricted the present study to isotropic, spherical, and non-degenerate elec- tron and hole bands. This approximation be- comes questionable for materials withs ( j = = +)-type hole band structure. In this case

d-------, the best results would probably arise from the use of an experimental "mean" hole mass.

Nevertheless, our results are expected to allow the estimation of the transition ener- gy between a neutral acceptor (AO, X),

-6

-il y 0 02 05 I 2 5

hY(i",-\) = hvx t E(.V,X) - E A - Ex (6.8) knowing the fundamental exciton transition energy hv, in a magnetic field. The ac- ceptor and excitori binding energies E , and Ex may be deduced, for instance, from the values reported for the exciton or hydrogenic impurities [15, 161.

References

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(Received February 16, 1987)