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Tight-binding calculation of orbital relaxations and hopping integral modifications around vacancies and
antisite defects in GaAs
J. van der Rest, P. Pecheur
To cite this version:
J. van der Rest, P. Pecheur. Tight-binding calculation of orbital relaxations and hopping integral modifications around vacancies and antisite defects in GaAs. Journal de Physique, 1983, 44 (11), pp.1297-1305. �10.1051/jphys:0198300440110129700�. �jpa-00209714�
Tight-binding calculation of orbital relaxations and hopping
integral modifications around vacancies and antisite defects in GaAs
J. van der Rest (*) and P. Pecheur (**)
(*) Institut de Physique B5, Université de Liège, 4000 Sart Tilman, Belgium
(**) Laboratoire de Physique du Solide, ENSMIM, Parc de Saurupt, 54042 Nancy-Cedex, France
Résumé. 2014 Nous étudions l’importance des modifications des paramètres de liaisons fortes autour de défauts ponctuels dans les semi-conducteurs sur la structure électronique de ceux-ci et nous appliquons notre méthode
à l’étude des divers états de charge des lacunes et des défauts d’anti-structure dans GaAs. Nous prenons en considé- ration les modifications des intégrales de transfert entre deux atomes premiers voisins d’une lacune, modifications
qui proviennent d’une levée de la condition d’orthogonalité entre les orbitales de l’atome qui a été enlevé et les orbitales de ses voisins. Nous prenons également en considération les modifications des intégrales de transfert entre un défaut substitutionnel et ses premiers voisins, modifications qui proviennent des différences entre les orbitales du défaut substitutionnel et les orbitales de l’atome substitué. Enfin, nous prenons en considération les modifications des niveaux d’ énergie des premiers voisins d’un défaut, modifications qui proviennent du réarran- gement électronique autour du défaut Nous proposons une méthode systématique pour calculer ces modifications
et nous étudions leurs effets sur la structure électronique des divers états de charges des lacunes et des défauts d’anti-structure dans GaAs. Nous comparons nos résultats avec d’autres calculs de liaisons fortes dans lesquelles
ces modifications ont été négligées.
Abstract - We study the influence of changes in the tight-binding parameters around point defects in semiconduc- tors and we apply our method to study the various charge states of the vacancies and anti-site defects in GaAs. We take into account the changes in the hopping integrals between two nearest neighbours of a vacancy which arise from the lifting of the orthogonality condition between the orbitals of the removed atom and the orbitals of the
neighbouring atoms. We take into account the changes in the hopping integrals between a substitutional defect and its nearest neighbours which arise from the differences between the orbitals of the substitutional defect and the orbitals of the substituted atom. Lastly, we take into account the changes in the energy levels of the nearest neigh-
bours of a defect which arise from the electronic rearrangement around the defect We propose a systematic method
to calculate these changes and we investigate their consequences on the electronic structure of the various charge
states of the vacancies and anti-site defects in GaAs. We compare our results with other tight-binding calculations where these changes have not been taken into account
Classification
Physics Abstracts
71.55F
1. Introduction.
The tight-binding approximation is widely used nowadays to study defects in semiconductors. This
approximation leads indeed to numerical programs
relatively simple when compared with a priori methods
like the local density approximation. The tight- binding approximation has nevertheless some intrin- sic difficulties which arise from the fact that it is a
semi-empirical scheme in which the electronic charge
distribution is not self-consistently computed from
the electronic potential. It is therefore impossible to
calculate from first principles the modifications of the tight-binding parameters which are expected to
arise around defects. We propose therefore in this article a systematic method to calculate the most
important changes in the tight-binding Hamiltonian which occur around vacancies and substitutional defects in semiconductors and we discuss their influence on the electronic structure of the defects with application to the GaAs vacancies and anti-site defects.
Around a vacancy, we take into account the modi- fications of the hopping integrals between two nearest
neighbours of the vacancy and the modifications of the energy levels of these nearest neighbours; these
modifications are obviously calculated with respect
to the values in the bulk crystal. When writing down
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198300440110129700
1298
Schrodinger’s determinant for the bulk crystal, we
assume indeed that the orbitals of the neighbouring
atoms are orthogonal to the orbitals of the atom situated on the site where the vacancy will be created.
Upon removal of this atom, this constraint is lifted and the neighbouring orbitals may extend into the vacancy site. The overlaps between these orbitals increase therefore and the hopping integrals between
them increase also. This orbital rearrangement is totaly independent from any atomic rearrangement which may complicate the problem even more. We
study in this paper the orbital rearrangement only
and we assume fixed atomic positions.
We take also into account the changes in the
. energy levels of the first nearest neighbours of the
vacancies. These changes arise because the vacancy states are strongly localized on the orbitals of these
atoms and because these orbitals are themselves modified with respect to their bulk shape as explained
above. This point is treated in the approximation of
local neutrality which allows us to study also the charge states of the vacancies.
Around an anti-site defect, we take into account the modifications of the hopping integrals between
the anti-site atom and its nearest neighbours. These
modifications arise from the differences between the orbitals of the anti-site atom and the orbitals of the substituted atom. We take also into account the
changes in the energy levels of the anti-site atom and of its nearest neighbours which arise from the loca- lization of the defect’s levels not only on the defect
itself but also on its nearest neighbours.
The changes in the hopping integrals are calculated
from atomic orbitals and atomic energy levels via the Wolfsberg-Helmholz formula. This formula is known to give poor results for the bulk hopping integrals but we propose here to use it in an inter-
polation scheme in order to deduce the modified
hopping integrals from the bulk hopping integrals.
The changes in the energy levels of the nearest
neighbours of the defects and, when applicable, of
the anti-site atom are assumed identical for the s
and p orbitals. They are defined from a local neutra-
lity condition which assumes that the total net charge
on the defect and on its nearest neighbours is the
same as the total net charge on the same cluster in
the bulk crystal.
It is obvious that other tight-binding parameters
are modified around point defects; we estimate that
we take into account the modifications which are
the most important for the determination of the defects levels and we do not investigate the changes
in other parameters. Most tight-binding calculation of point defects energy levels neglect indeed these
changes and our aim is to attract the attention on
the need of such changes and to propose a systematic
method to calculate them. Our method is obviously
not based on a priori arguments (how could it be in the tight-binding framework ?) but we explain in
detail what we do, how we do it and what the conse-
quences are.
Let us now summarize this paper. In the next section we compare various sets of tight-binding
parameters for GaAs, paying special attention on
the influence of the value of the parameters on the
position of the A 1 and T2 vacancies levels. We discuss then in the third section the reasons why the hopping integrals between the first neighbours of the vacancy have to be different from the hopping integrals bet-
ween the same atoms in the bulk crystal and we
show how to evaluate their modifications in a simple
way. We apply the method to the GaAs vacancies and we study their various charge states. In the
fourth section we study the antisite defects in GaAs and we show that the hopping integrals between the antisite atom and its first neighbours must differ
from the hopping integrals between two nearest
neighbour atoms in the bulk crystal. We study also
the various charge states of these defects. In the last section we summarize the main conclusions of this work.
2. Tight-binding study of GaAs.
2.1 REVIEW OF PREVIOUS WORK. - The GaAs system has been extensively studied these last years
experimentally as well as theoretically and many sets of tight-binding parameters representing its
bulk electronic structure are therefore available in the literature. We have collected in table I some of the most recent sets together with our own set. These
four sets of parameters give a fairly accurate descrip-
tion of the valence bands and of the bandgap of
GaAs but all of them fail to reproduce even approxi- mately the conduction bands; we consider therefore that there is no argument to prove that one set is
really better than the others.
These sets of parameters have then been used to
study the unrelaxed orbital-quenched vacancies, i.e.
the vacancies obtained by the removal of an atom from the bulk crystal while all the other atoms are
kept at their original positions and all the interactions between these atoms are unaltered. The results are
given in table II. They show a relatively wide uncer- taincy for the position of the levels because they only agree on some very general trends namely that
EGAal is deep in the valence band, that EGa is near the
top of the valence band, that E£%’ is somewhere in the gap and that E.2 is in the upper half of the gap.
These results have been collected in order to
provide enough points of comparison which will
help us to point out more clearly the influence of a
modification of our tight-binding parameters on the electronic levels of the vacancies. We discuss therefore
now how our bulk parameters have been chosen
(§ 2.2) and how these bulk parameters affect the vacancies levels (§ 2.3).
Table I. - Tight-binding parameters for GaAs. The notations are taken from reference 4. N indicates the number
of independent non-zero parameters. All energies are in eV.
Table II. - Comparison between various tight-binding results for the neutral vacancies in GaAs. We give the A,
and T2 levels for both vacancies, the average position of the two levels (E 1 EAt + 3/4 ET2) and the splitting bet-
ween them (AE = ET 2 - E A1). The five first columns correspond to the unrelaxed orbital-quenched vacancies ; the sixth and seventh columns contain our results for the unrelaxed vacancies with rearranged orbitals : in the sixth column the modified hopping integrals are calculated from the perturbation theory (§ 3.1) and in the seventh column they are calculated from the W olfsberg-Helmholz formula (§ 3. 2). The results of ’ the last column have been obtained in the local density approximation. All energies are in eV.
2.2 TIGHT-BINDING PARAMETERS FOR THE PERFECT CRYSTAL. - Our own set of parameters has been obtained as follows. We started from the band struc-
ture calculated by Chelikowsky and Cohen [6] which
is in very good agreement with experimental data [7]
and we turned off the spin-orbit coupling taking the
1300
average value between the two dispersion curves
whenever necessary. We fitted then the energies of
the valence bands and of the first conduction band at the L, A, r, A, X, U and E points. We tried several sets of parameters and we kept finally one with a
limited number of parameters but which gave a
fairly good agreement with the band structure of reference 6 and which gave only a very small charge
transfer (0.1 e) between the Ga and As atoms.
In order to justify this limited number of para- meters, we would like here to comment on the spirit
of the tight-binding approximation. We agree that the tight-binding parameters are disposable constants
as first suggested by Slater and Koster [4] but we
believe nevertheless that these parameters must be choosen in order to optimize the dispersion relations
in the whole Brillouin zone and that great care must be taken therefore when some parameters are deter- mined only from a small number of the fitting points.
Such parameters are then badly determined and one
should check very carefully that their numerical values are physically reasonnable. There is in general
no difficulty with the hopping integrals between
first nearest neighbours : all fittings give similar
values and their orders of magnitude are always
reasonable. This is unfortunately not true for the hopping integrals between more distant neighbours
whose values depend sometimes only on a single fitting point or whose values influence hardly the
overall quality of the fitting. We consider in particular
that the Ga and As atoms are too similar in order to have a Ga-Ga hopping integral differing from the corresponding As-As hopping integral by more than
a factor five; if the fitting procedure yields such a result, we consider that its validity should be checked in the whole Brillouin zone and not only at the
limited number of the usual fitting points. We consider
also that the same care should be taken if the fitting procedure yields large values for hopping integrals
which are identically zero in the two-centre approxi-
mation. In view of this, we decided in particular to
set E°(O11) to zero even if a large value (0.2 eV) could improve the fitting at the L point for the first conduc- tion band; this parameter could indeed not improve
the fit of the first conduction band as a whole. We decided also to take the same values for E,,O.’(-li 2 I -1i)
and for ,. i-l-’i) because they always turned out
to be quite similar and because this simplification
did not alter significantly the description of the band structure. For the same reasons, we assumed also that the Ga-Ga hopping integrals are identical to
the As-As ones and we neglected third nearest neigh-
bour interactions.
With these comments in mind, we found that eleven parameters are enough to reproduce the
valence band dispersion curves along the main symmetry directions (L-F, r-X, X-U, K-E and 1-jT)
and that there is no set of tight-binding parameters
describing adequately the conduction bands. So far
for our comments on the choice of the tight-binding
’
parameters; let us now study the vacancies.
2.3 THE VACANCIES LEVELS. - Like the preceeding
authors we introduce the vacancy by letting the diagonal elements ESS(0 0 0) and E..(O 0 0) go to
’ infinity on the vacancy site; this is equivalent to cutting the bonds between the vacancy and the rest of the crystal [8]. This procedure has been criticized recently [9] but we consider that these arguments are irrelevant. What they really show indeed, is that the limit of an infinitely large potential is not a valid approximation for a vacancy in an arbitrary basis
of wave functions. But we only use here the fact that it is a good approximation for a localized (atomic- like) basis. This point has also been raised recently by Verges [10].
The vacancies levels obtained from our set of parameters are very similar to the average of the other tight-binding results.
Allowing then the orbitals surrounding the vacan-
cies to rearrange themselves, we obtained the results of the sixth and seventh columns of table II : one
notices that the average positions of the levels is
hardly changed but that the spl ting between them is enormously increased. This method and its results will be discussed at length in the next section.
Let us simply conclude this section with a few comments on our numerical precision.B The defect levels are computed via the recursion method of
Haydock et al. [11]; seventeen coefficients of the continued fractions are calculated in a cluster of about 5 000 atoms. This gives a numerical precision
for the position of the levels better than 0.1 eV. We finish our continued fraction with the constant ter-
mination, i.e. we assume that all a’s and all b’s (after
the last calculated coefficients) are constant; this
procedure replaces the gaps by regions of very low densities of states but does not modify significantly
the positions of the bound states. It is possible to improve this procedure and to preserve real gaps [12],
but this would be rather difficult for GaAs because
we have two gaps and because our last calculated coefficients are not yet in the asymptotic regime; it
is therefore nearly impossible, for each defect, to
match the last coefficients of the 18 Green’s functions with their asymptotic form.
3. Orbital rearrangement around a vacancy.
To construct the tight-binding approximation, one
defines a formal set of orthonormal orbitals I t/Ji >
but one works in fact only with the components of the Hamiltonian between any two of these orbitals
Hij = qli H I qlj >. When a vacancy is created in the system, the orbitals of the neighbouring atoms no longer need to be orthogonal to the orbitals of the removed atom; these orbitals are therefore modified with respect to their value in the bulk crystal and the
hopping integrals between them are also modified Our tight-binding formalism does not allow to
compute the potential nor the wave functions and a
direct computation of the modified hopping integrals
is then impossible.
We show therefore in a first subsection 3.1 by a
first order perturbation method that the hopping integrals between two nearest neighbours of a vacancy
are indeed strongly modified by the vacancy and in
a second subsection 3.2 we propose to use the Wolfs-
berg-Helmholz formula as an interpolation scheme
in order to estimate their changes. In a last subsec- tion 3. 3 we study the various charge states of the Ga
and As vacancies.
3.1 FIRST ORDER PERTURBATION METHOD. - In order to calculate the modified hopping integrals
around a vacancy with the perturbation method, let
us call I t/J: > the modified orbitals and Hij = 4,’ I H I t/J; > the hopping integrals between them.
If the two sets of orbitals are not too different, one
can write with
which gives to first order in Sij
where i and j are orbital indices on two first neighbours
of the vacancy and where the summation over k is
over the s and p orbitals of the removed atom. AHij
is therefore a change in second nearest neighbour hopping integrals : two first neighbours of a vacancy
are indeed second nearest neighbours between them.
The modifications to the Slater-Koster E integrals
are given explicitely in appendix A in terms of the
two-centre hopping and overlap integrals between
nearest neighbours. The corresponding numerical
values for the modified E integrals are given in the
fifth column of table I where the components of S have been computed from equation 2, assuming that
the modified wave functions I 4/’> are atomic wave
functions.
These corrections increase noticeably the E inte-
grals around the vacancies which in turn increase
spectacularly the splitting between the vacancies levels (see Table II sixth column).
The importance of these modifications shows
clearly that it is indeed necessary to take into account the changes in the hopping integrals between two
nearest neighbours of a vacancy. But at the same time,
it shows that a first order perturbation method is questionable : the perturbation is indeed too strong
to be handled with a small perturbation method We
consider therefore that it is useless to attempt to improve these results taking into account changes in
the potential and not only changes in the wave func-
tions and/or higher order terms and/or changes in
other hopping integrals. We prefer instead to evaluate directly the modified hopping integrals via an inter- polation scheme which makes use of the Wolfsberg-
Helmholz formula.
3.2 THE WOLFSBERG-HELMHOLZ METHOD. - It is usual in quantum chemistry to evaluate a hopping integral from the overlap integral between the states via the Wolfsberg-Helmholz formula [13] which
writes
where Ei and Ej are the diagonal eigen values of the i and j states and where K is a constant This formula does not provide very good bulk parameters but we
propose here to use it as an interpolation scheme only. Another interpolation scheme has been pro-
posed recently [14] which assumes an exponential decay with distance of the bulk first nearest neighbours hopping integrals in order to provide the hopping integrals between two nearest neighbours of a vacancy.
The basic assumption common to both methods is
to treat the first nearest neighbours of a vacancy as
, remoted first nearest neighbours with the result that the hopping integrals between them are noticeably larger than in the bulk crystal.
Let us now return to expression 4 and discuss of its use. If the overlaps between different orbitals are
taken into account in Schrodinger’s determinant the value of the constant K is often taken of the order of 1.75; on the contrary, if the overlaps are neglected in Schrodinger’s determinant the value of the constant K should be of the order of 0.75 (see appendix B). The precise value of that constant being nevertheless rather difficult to determine, we decided instead to
eliminate it by writing equation 4 twice, once for the
bulk orbitals I t/Jk > and the bulk hopping integral Hkl and once for the modified orbitals I t/J; > and the
modified hopping integral Hi’j which gives
Equation 5 allows then to calculate the modified
hopping integrals Hi’j from the bulk hopping integrals Hkl’ The modified integrals we are interested in are
of course the hopping integrals between two atoms
nearest neighbours of a vacancy i.e. second nearest
neighbours hopping integrals. As discussed in the previous subsection the orbitals of these two atoms are no longer orthogonal to the orbitals of the removed atom and it is therefore reasonable to calculate their
hopping integrals from the bulk hopping integrals
between first neighbours, making use of equation 5
to take into account the change in distance. Secondly,
Ei’ + Ej’
because the
term El k +
Ek + Ela I is found to depend moreupon the type of atoms than upon the type of orbitals
