LOCALIZED STATES AND BAND STRUCTURE

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LOCALIZED STATES AND BAND STRUCTURE

F. Bassani

To cite this version:

F. Bassani. LOCALIZED STATES AND BAND STRUCTURE. Journal de Physique Colloques, 1972,

33 (C3), pp.C3-21-C3-25. �10.1051/jphyscol:1972304�. �jpa-00215038�

JOURNAL DE PHYSIQUE

Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-21

LOCALIZED STATES AND BAND STRUCTURE

F. BASSANI

Istituto di Fisica, Universiti di Roma, Italy

Resumb.

On presente la formulation generale du probl6me des Ctats Clectroniques localises sur la base des fonctions de Bloch du cristal parfait.

Des calculs recents sur les Btats d'impuretes dans les semi-conducteurs sont discutCs et les limi- tations dues a l'approximation de la masse effective sont mises en evidence.

On propose pour les excitons une approximation valable lorsque la fonction d'onde est 1ocalisBe sur un petit nombre de cellules elementaires. Cette approximation inclut les effets de la structure de bande non perturbCe et utilise la localisation des fonctions de Wannier. Les separations singulet- triplet et Iongitudinale-transverse sont naturellement incorporees dans l'approximation proposee.

Abstract.

The general formulation of the problem of electronic localized states with point symmetry is presented in the basis of the Bloch functions of the perfect lattice.

The effective mass approximation is discussed and recent calculations are reviewed as well as recent improvements appropriate to impurity and exciton states in semiconductors.

A different approach is proposed for exciton states in insulators, where the localized function extends over a few lattice cells only. The present approach takes into account the unperturbed energy band structure E,(k) and makes use of the localization of the Wannier functions. Singlet- triplet and longitudinal-transverse splittings are naturally incorporated in the present treatment.

I. Introduction.

The purpose of this talk is to describe a number of problems which arise in the computation of impurity states and exciton states in solids and in the interpretation of the optical excitation spectrum. I will not consider those cases which can be treated satisfactorily with the lattice as a perturbation on the free atom or molecule, but rather the more complex situations where the lattice and the impurity must be treated on equal footing.

The problems of impurity states and of exciton states are physically quite different, but their mathe- matical formulation and the computational methods to be used are so similar that one can deal with both problems at the same time, keeping in mind the specific differences. For impurities we consider one- electron states which are IocaIized in space over a number of lattice cells ; for excitons we consider two- particle states (electron and hole) which travel through the lattice as a whole but are also localized over a few lattice cells in their relative position. With the name localized states we indicate both << bound states n, whose wave functions go to zero away from the point defect, and cc resonant states

which behave like scattering states away from the point defect but have a high density amplitude near it. The available expe- rimental results indicate that in general the states are quite different from those or the free impurity atoms or from those of the atoms which compose the crystal, and are essentially determined by the electron band- structure of the crystal [I]. The basic problem in studying the localized states is how to make use of all the information contained in the band structure which is supposed to have been computed. The connection between band structure and impurity or exciton states is the basic point to have in mind and is the problem where modern computational techniques can be used to obtain significant results.

Until now meaningfull results have been obtained only in those cases where a small portion of the band structure and a small section of the Brillouin zone near critical points are sufficient to determine the localized states, i. e. when the effective mass approxi- mation can be applied. Even within the effective mass approximation some problems still remain to be solved, but deep localized states, whose properties are determined by a large fraction of the Brillouin zone and by a large number of energy bands, are an open problem.

I n the following I will describe a general formulation of the impurity and exciton problems. I will point out the conditions for the validity of the effective mass approximation following the procedure of Kohn and Luttinger [2] and will describe some recent improve- ments [3] and some new results. Finally I will indicate a way to overcome the limitations of the effective mass approximation giving a practical procedure for exciton states which can be carried out with modern computa- tional techniques.

11. Formulation of the impurity and of the exciton problem in crystals. - For the impurity case in the one-electron approximation one has to solve the Schrodinger equation :

V, being the crystal potential, and U the impurity potential.

For the exciton states of the lattice one should find the excited states of the Schrodinger equation of the N electron system

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972304

C3-22

F. BASSANI where 2, is the charge of the nuclei located of the points R,.

In the first case the eigenfunctions can be expanded in the complete set of Bloch functions $,(k, r), such that

and we have

In the exciton case the wave functions can be expand- ed in Slater determinants of excited states formed with Bloch functions, where one electron in a valence band v with wave vector k is substituted by an electron in a conduction state c, with wave vector k + ^q,

q being the momentum of the exciton. We can write

+ q) -

Aidk) +

2

Ac,vr(k') [-

6, being 1 or 0 for singlet and triplet states respectively.

The two integral equations (6) and (7) for impurities and excitons are very similar except for the fact that one must consider couple of bands in the latter case.

The sum over the k vectors can be expressed as an integral over the Brillouin zone while the band indices are discrete. The Kernel of the integral equation (7) consists of a Coulomb two-particle matrix element and of an exchange-like term which appears only for singlet states.

To solve the system of integral equations (6) and (7) is a formidable task which has never been attempted for realistic cases. Simplifications have to be made to obtain numerical solutions, but the general approach has always to be kept in mind to establish the connec- tion between band structure and localized states. Some general remarks can be made however from the equations (6) and (7).

a) The exciton and impurity states can be considered as bound states when they are not degenerate with band states, i. e.

They can be considered as resonant and scattering states when the conditions (8) and (9) are not satisfied.

In this case the A(k) of equations (6) and (7) have to be extended into the complex E plane [3]. The eigen-

where the Ket indicates the Slater determinant and the subscript S indicates that a combination of the Slater determinants with total spin 0 or 1 must be considered since the total spin is a good quantum number.

With the above choice of bases the equations (1) and (2) can be transformed into integral equations on the expansion coefficients. For the impurity problem one obtains straight forwardly the integral equation

where the Kernel is the non local matrix element of the impurity potential between Bloch functions. For the exciton problem one obtains after some algebraic manipulations, the equation

values are complex and when the imaginary part is negative and small we have a resonant state, the imaginary part of the complex solution producing a finite but sufficiently long life time.

b) A general way to obtain the solutions of integral equations of type (6) or (7) consists in expressing their kernels as a sum of separable kernels. In this case the expressions in square braket of the above equations become of the general form

and the solutions are given in terms of the zeroes of the corresponding Friedholm determinant [3]. Expan- sion (10) can be obtained in an infinite number of ways, with an infinite sum, but the real problem is to find special expansions for which the Friedholm deter- minant converges rapidly.

c) The convergence properties of secular determi- nants indicate that one can restrict oneself to one band for the impurity case and to two bands for the exciton case if the other bands are well separated in energy from those which produce the dominant contribution.

d ) When A(k) of expansions (4) or (5) are peaked

near extrema or critical points of the band structure,

which is the case for large dielectric constant materials

with weak and slowly varying potentials one can use

the familiar effective mass approximation [2]. This can

be immediately obtained by transforming equation (6)

LOCALIZED STATES AND BAND STRUCTURE C3-23 or (7) into a differential equation in real space on the

Fourier transform

We can expand E(k) and retain only quadratic terms, use plane waves instead of Bloch functions and we must screen the interaction with a dielectric constant

as shown by Kohn [4] and Haken [5]. We obtain for impurity states :

where m: indicates the effective mass tensor referred to its principal axes, and the energy is measured from its value at the critical point of the band considered.

A similar expression results for the exciton states except that in that case one must consider electron and hole coordinates and effective masses. In the case of isotropic effective masses one obtains from equation (7)

where E, is the energy gap between the top of the valence band and the bottom of the conduction band considered. It is immediate to extend equation (13) to the case of anisotropic effective masses and of saddle points, provided one uses an effective mass tensor and allows for negative values of the effective masses.

111. Recent developments within the effective mass approximation.

Though the effective mass approxi- mation was very successful in the fifties in explaining the excited levels of donor impurity states in a number of semiconductors and in explaining the hydrogenic series appearing in the excitation spectrum of some semiconductors [I], [2], its potentialities were not fully exploited until very recently with the availability of large size computers. A few results recently obtained may be mentioned.

i) The effective mass equation (12) for impurity states has been recently solved by Faulkner [6] for the case of anisotropic effective masses in semiconductors like Si and Ge. A variatioanl procedure has been used since the equation is non separable and cannot be integrated. Kohn and Luttinger [2] could perform the calculation only on the ground state, but the agree- ment wlth the experiment could be found on the excited

states because in that case the effective masse theory applies and the Coulomb potential is appropriate.

ii) The exciton effective mass equation for anyso- tropic crystals was similarly solved by Baldereschi and Diaz 171, who obtained good agreement with the excitonic spectrum of GaSe and with the relative intensities of the exciton lines. After reduction of equation (13) to the center of mass, they use an expansion of the exciton envelop function F(r) in spherical harmonics and use variational parameters in the radial part of the function to improve the convergence.

iii) Extensions of equations (12) and (13) have been made to the case of degenerate bands like the valence bands of cubic semiconductors, and reasonable agree- ment with the acceptor levels has been found [8].

iv) Intervally mixing has been computed for equi- valent minima of the Brillouin Zone [9] and for minima of different bands as well [lo]. Altarelli and Iadonisi [lo] have also computed resonant impurity states below secondary minima of the conduction band and their mixing with the states of the lower bands, which is shown to depend on the effective masses and on the separation between secondary minima and the absolute minimum.

v) The lowest energy levels have been computed for excitons in the presence of a constant magnetic field. This requires a modification of equation (13) which amounts to substituting the operator p = - ihV, with p i - - A . The resulting equation for the rela- tive coordinate motion is non separable and has to be solved by sofisticated numerical methods. Some recent results on the lowest eigenvalues [ I l l indicate that the eigenvalues for high values of the magnetic field, obtained by the adiabatic method as a fine struc- ture on the Landau levels, join with those at low values of the magnetic field, obtained with a variational procedure as a mixing of hydrogenic exciton states, according to the non-crossing rule.

Some important problems still remain to be solved even in the effective mass approximation. The devia- tions from coulumb like behavior of the impurity potential and of the electron-hole interaction near the origine for instance has still to be considered in detail in numerical computations, though Phillips has explain- ed the general trend of the chemical shift of the ground state impurity levels [12]. The question of which is the appropriate dielectric function ~ ( q , o) for insulators with a strong frequency dependence in the absorption region, is also a problem which needs to be clarified.

IV. Suggested approach for intermediate binding. - In most experimental situations, particularly in the case of large gap insulators, the effective mass approxi- mation does not apply and one should solve equa- tions (6) and (7) by taking into account the effect of

a large portion of the band structure and not only of

C3 -24

F. BASSANI a small region in k space about the critical point. The problem can be reduced to that of devising a rapidly converging expansion (10) for the Kernel. Altarelli and Bassani 1131 have suggested that this can be accomplished by expressing the matrix elements as sums of integrals on Wannier functions located at the different lattice points.

For the case of impurity states this leads to the results of the method of Slater and Koster [14], who first had the idea to use the localization of Wannier functions for these problems.

For the exciton states, in the optical case where q --- 0, by substituting

into equation (7), and keeping only maximum overlap terms ; we obtain :

with

Jl(R) = 11 dr, dr2 a3r1) a:(r2 - R) x

J2(R) = S dr, dr, a:(r,

R) a:(r2) x

The kernel of the integral equation is now a sum of separable terms. The sum on J1@) can be computed as a sum of dipols ^p,, = e S draz ray, following the method of Heller and Marcus [15] and gives, in the case of cubic lattices,

for transverse excitons, and

for longitudinal excitons, where no is the number of cells per unit volume. With the definition

multiplying equation (15) be eik.RO and suming over k, we obtain the system of liner omogenous equations

where D i refers to the sum of expression ( 1 5 ) which is different for longitudinal and transverse excitons, and is given by (16a) or (16b) in cubic lattices. The condi- tions for non zero solutions give the Friedolm deter- minant equation

and the eigenvalues can be computed provided only a small number of terms are needed to achieve conver- gence.

It is interesting to observe that in equations (18) and (19) both triplet and singlet excitons are included, depending on 6, being 0 or 1, and also the longitudinal- transverse splitting can be obtained depending on the expression used for Di. The convergence should be quite good because A(R) should extend only over a few lattice cells in the case of intermediate binding.

For those exciton states for which A(R) is very extended and the determinant converges too slowly, the effective mass approximation should apply. It can also be observed from (15b) and (15d) that the exchange contribution J2@) goes to zero with increasing R more rapidly than the Coulumb contribution Q(R), and this implies that only for the lowest exciton state which has a small radius there is a singIet-triplet splitting. In general the convergence of the determi- nantal equation (19) gives an estimate of the size of the exciton.

We also wish to remark that the above approach can be easily extended to the case of resonant excitons with the procedure described in reference (3). The case of excitons produced by transitions from core states requires the contribution of more bands since the restriction to a couple of bands in no longer justified.

The main computational problem is to obtain reliable expressions for the parameters (1 5b), which require a good knowledge of Wannier functions. Calculations are in progress on solid Argon where Wannier func- tions have been computed for the conduction band [16].

We plan to extend the method to more complicated

cases and to core excitons.

LOCALIZED STATES AND BAND STRUCTURE

References [I] See for instance :

K o m (W.),

Shallow Impurity States in Si and

Ge >> in Solid State Physics, 5 (Seitz F . and

Turubull D. Eds, Academic Press, New York, 1955).

PHILLIPS (J. C.),

The Fundamental Optical Spectra of Solids >> in Solid State Physics, 18 (Seitz F . and Tuubull D. Eds, Academic Press, New York, 1966).

NIKITINE (S.),

Excitons >> in Optical Properties of Solids, p. 197 (Nudelman S. and Mitra S. S. Eds, Plemun Press, 1969).

KNOX (R. S.),

Theory of Excitons >> in Solid State Physics, Supplement 5 (Academic Press, 1964).

[2] LUTTINGER (J. M.) and KOHN (W.), Phys. Rev., 1955, 97, 869.

KOHN (W.) and LUTTINGER (J. M.), Phys. Rev., 1955, 95, 915.

[3] BASSANI (F.), IADONISI (G.) and PREZIOSI (B.), Phys.

Rev., 1969, 186, 735.

See also :

BASSANI (F.), <<Theory of Impurity States >> in ((Theory of Imperfect Crystalline Solids

p. 265, Interna- tional Atomic Energy Agency, Vienna, 1971 and

PREZIOSI (B.),

Scattering and Resonant States in Crystals >>, ibidem.

[4] KOHN (W.), Phys. Rev., 1957, 105, 509.

[5] HAKEN (H.), ZZ NUOVO Cimento, 1956, 3 , 1230.

[6] FAULKNER (R. A.), Phys. Rev., 1969, 184, 713.

[7] BALDERESCHI (A.) and GORFINKIEL DIAZ (M.), I1 Nuovo Cimento B, 1970, 68, 217.

DEVERIN (J. A.), I1 Nuovo Cimento B, 1969, 63, 1 and related references.

[8] LIPARI (N. 0.) and BALDERESCHI (A.), Phys. Rev.

Lett., 1970, 25, 1660.

See also :

KOHN (W.) and SCHECTER (D.), Phys. Rev., 1955, 99, 1903.

and

MENDELSON (K. S.) and JAMES (H. M.), J. Phys.

Chem. Solids, 1964, 25, 729.

[9] BALDERE~CHI (A.), Phys. Rev. B, 1970, 1, 4673.

KASTNER (T, G.), Phys. Rev. (to be published).

[lo] ALTARELLI (M.) and IADONISI (G.), IZ NUOVO Cimento B, 1971, 5, 21.

ALTARELLI (M.) and IADONISI (G.), I1 NUOVO Cimento B, 1971, 5, 36.

[ l l ] BALDERESCHI (A.) and BASSANI (F.), Proceedings of the X International Conference on Phvsics of Semiconductors, p. 191 (Boston, 1970).

[12] PHILLIPS (J. C.), Phys. Rev., 1970, 1 , 1545.

See also :

HERMANSON (J.) and PHILLIPS (J. C.), Phys. Rev., 1966,150,652.

1131 ALTARELLI (M.) and BASSANI (F.), Journalof Physics C, to appear.

[14] KOSTER (G. F.) and SLATER (J. C.), Phys. Rev., 1954, 96, 439.

KOSTER (G. F.) and SLATER (J. C.), Phys. Rev., 1954,95, 1167.

[15] HELLER (W. R.) and MARCUS (A.), Phys. Rev., 1951, 84, 809.

[16] MIASEK (M.), J. Math. Phys., 1966, 7 , 139.