New theoretical method to study densities of states of tetrahedrally coordinated solids

Solid State Communications,Vol. 15, pp. 6 17—620, 1974.

Pergamon Press.

Printed in Great Britain

NEW THEORETICAL METHOD TO STUDY DENSITIES OF STATES OF TETRAHEDRALLY COORDINATED SOLIDS F. Yndurain*t, J.D. Joannopoulos~,Marvin L Cohen~and LM. Falicovt Department of Physics, University of California, Berkeley, California 94720, U.S.A. (Received 24 April 1974 by A.A. Maradudin)

A new simple method is proposed to calculate local densities of states of arbitrary tetrahedrally coordinated solids. It involves the selection of a finite cluster of atoms connected to an infinite Bethe lattice of coordination four. The method is accurate, is easily handled numerically, and converges fast. Low-order approximations yield sufficient information which is susceptible to consistent physical interpretation. This has been made for the diamond, BC—8 and ST—12 structuresin terms of the ring topology around a given atom. Comparison with exact calculations is very good and the ring interpretation is physical and conceptually appealing.

THE STUDY of covalent semiconductors, especially Ge and Si, in their various structural forms, both crystalline polymorphs and amorphous solids, is presently the subject of intensive efforts. These have been prompted by the experimental determination of electronic densities of states by means of ultraviolet and X-ray photoemission spectroscopy.1 .2 The observed densities of states present features which can qualitively be assigned as due to either long-range effects (bandwidths, van-Hove singularities) dr short-range characteristics. The influence of short-range order in tetrahedrally coordinated solids can be seen to come from two distinctly different causes: (i) bond-angle and bond-length distortions and (ii) the local topology of the atomic configuration. We are primarily interested in this latter aspect.

considerably more difficult when the periodicity of the system is lost or when the primitive unit cell of a periodic system contains very many atoms.3 Various approaches have been suggested to overcome-such difficulty: (1) Expansion of the density of states using moments4 these are directly related to the counting of closed paths which connect the atoms in the structure; (2) Calculation of the discrete spectrum of a finite cluster of atoms5’6 (3) Study of exactly soluble mathematical models7 (Bethe lattice, Husumi cacti) which have very simple topological properties. —

None of these methods is very satisfactory: method (1) is not very useful due to poor convergence of the series; (2) requires very large clusters to yield results independent of the boundary conditions and (3) produces relatively featureless densities of states which provide little insight into the problem.

Theoretical calculations of densities of states, even with a simple model Hamiltonian, become * Fulbright fellow under the auspices of the “Program of Cultural Cooperation between the U.S.A. and Spain”.

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We propose here a new method for calculating densities of states. It is based on a reasonable model, which is exactly soluble, avoids the difficulties of the other methods, is very easy to handle numerically and can in principle, by successive approximations,be made to converge to the exact result. We believe in addition that the low-order approximations yield accurately and rather effortlessly, most of the important information

the National Science

~ Work supported in part by the National Science Foundation through Grant GH 35688. 617

618

DENSITIES OF STATES OF TETRAHEDRALLY COORDINATED SOLIDS

related to the problem.

and

The method is for calculating the local density of states (LDOS) of an atom in terms of the local environment of that atom. The atom is assumed to be in an infinite connected network with tetrahedral coordination throughtout. The main feature of the model consists of replacing the real network by a suitable finite cluster of atoms; this cluster is connected out side to a Bethe lattice of coordination number four, It is now possible to calculate exactly the mattrix element of the local Green function in which the central atom is the reference point. The Bethe lattice guarantees a reasonable boundary condition (infinite system) and provides a smooth featureless background. Any feature obtained in the calculations can thus be attributed to the short-range (local) configuration around the pertinent atom.

a

=

~ I~’~ili>
(1)

where i> represents a local s-like orbital centered a~ atom i and. l’~,= V for nearest neighbors, and J’~,= otherwise. This Hamiltonian has eigenvalues {e,. } which are related by an analytic transformati n to the eigen~ values of a more realistic four-state sp3 Hamiltonian which is itself a good approximation to the “s-like” states in the valence band of the group IV elements.8 In the diamond structure we choose a cluster of (1 + 28) atoms which contains 12 six-fold rings passing through the central atom; each of the 28 non-central atoms is in at least one of the 12 rings. There are 5 inequivalent classes of atoms: the central atom, 4 nearest-neighbors, 12 second-nearest-neighbors, 12 third-nearest-neighbors and an infinite number outside the cluster in the Bethe lattice. The diagonal element (01gb) of the Greens function in this model is given by (01gb)

=

{e ~

where E

=

1

—

4V2(e

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—

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—

3V2 (

Vci)’ (2)

}_1

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—

—

Va)1,

(3)

—l2V2)~2]/6V.

(4)

These equations are obtained by solving a set of (4 X 4) linear equations in the unknown (I jg 10); a given by (4) is the contribution from the Bethe lattice.8

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For the purpose of calculation we have chosen a one-state tight-binding Hamiltonian H

[~_(~2

=

Vol. 15, No. 3

-2

01

I 0

1:11101 2 —10

_____

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ENERGY

FIG. 1. Density-of-state calculations for the diamond and one-class-ring structures. (a) One-orbital Hamiltonian in the diamond structure: Bethe lattice (dashed line),exact calculation reference 7 (light fullsp3 line) and our results (heavy from full line). (b) Four-orbital Hamiltonian in the diamond structure: Bethe lattice (dashed line), exact calculation from reference 8 (light full line) and our results (heavy full line). (c) Structure with 6 n-fold rings around the central atom in the oneorbital Hamiltonian: 1 (full line~n = 6’ 2 (dashed line) n = 5; 3 (dotted line) n = 7; 4 (broken’line) n = 8. (d) Structure with 6 n-fold rings in the four-orbital sp3 Hamiltonian. Notation as in (c). (e) The orbital energies for isolated six-fold rings (full lines), five-fold rings (dashed seven-fold fold ringslines), (broken lines). rings (dotted lines) and eightThe LDOS obtained from (2) is given in Fig. 1(a). Superimposed we show the densities of states of the Bethe lattice and the diamond structure. In Fig. 1(b) we show similar results obtained by using the fourstate Hamiltonian. It is easily seen from these figures that the local environment of the atoms gives the main contribution to the density of states. In particular the association of structure in the LDOS with the ring statistics of the cluster is shown in Figs. 1(c)—(e). Here we have constructed 5 clusters which are made

Vol. 15, No.3

DENSITIES OF STATES OF TETRAHEDRALLY COORDINATED SOLIDS

up of 6 rings of only one type (i.e., five fold, six-fold, seven-fold and eight-fold rings respectively). Each pair of bonds of the central atom is part of a ring. In Fig. 1(c) we have plotted the LDOS for these ringcluster—Bethe systems corresponding to the Hamiltonian (1). In Fig. 1(d) we show the equivalent calculations for the four-state sp3 Hamiltonian. The structure in these densities of states can be easily identified with the eigenvalues of isolated rings as shown in Fig. 1(e). The agreement is excellent and indicates that the ring-like nature of the local environment is paramount in determining type of structureoffound in the density of states. Athe close examination 1(c) shows that the strength of peaks is larger the Fig. smaller

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the ring. This indicates the importance of the small rings in a cluster.

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We have also applied our method to examine the total density of states BC—8important and ST—12 3 which have(TDOS) provenof tothe be very structures in the study of the structural aspects of amorphous Ge and Si. In the BC—8 structure we only have one type of atom and consequently, as in diamond, LDOS and TDOS are equivalent. We have chosen a cluster in the same manner as described previously: it contains 26 atoms. The results for the BC—8—Bethe system are shown in Figs. 2(a) and (b) along with the crystalline BC—8 spectrum. ~ Again, the agreement between the BC—8 and BC—8—Bethe spectra is good, indicating the importance of a local configuration. A comparison of Figs. 2(a) and (b) with Figs. 1(c) and (d) reveals the six-fold ring character of the BC—8 structure, which is caused by the 9 six-fold rings passing through each atom. The ST—12 structure is quite interesting since it provides us with a system which exhibits five, six, seven and eight-fold ring character. There are two types of atoms in the primitive cell with 4 atoms of type 1 and 8 atoms of type 2. Consequently, we have two types of clusters and the LDOS for these ST—l2—Bethe systems are shown in Figs. 2(c) and (d). In the cluster with a type I central atom we have 4 five-fold, 2 sixfold, 4 seven-fold, and 3 eight-fold rings of bonds with a total of 27 atoms. Comparing this spectrum with that of Figs. 1(c) and (d) we find that the peak at 1.5 in Fig. 2(c) is mainly caused by a six-fold ring peak and the overlap of a five and seven-fold ring peak. The shoulder around 0.3 seems tobe caused principally by an overlap of a five— and eight-fold ring peak.

a

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__________________

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2

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—10

ENERGY

—8

—6

—4

FIG. 2. Density-of-state calculations for the BC—8 [(a) and (b)]calculation and ST—l2for[(c) through (a) Our BC—8 with(I)] the structures. one-orbital Hamiltonian. (b) Our calculation for BC—8 with four-orbital sp3 Hamiltonian (full line) and the exact calculation from reference 3 (dashed line). (c) The LDOS for the two different atoms in the one-orbital Hamiltonian for ST—l2. (d) The LDOS forTDOS ST—12 the four-orbital sp3 Hamiltonian. (e) The forand ST—l2 in our calculation for the one-orbital Hamiltonian. (f) The TDOS (ST—l2 structure, four-orbital sp3 Hamiltonian) according to our calculation (full line) and to the exact results of reference 3 (dashed line). Similarly in the cluster with a type 2 central atom, we have 31 atoms with 3 five-fold, 2 six-fold, 5 seven-fold and 8 eight-fold rings of bonds. Again a comparison of this spectrum with Figs. 1(c) and (d) reveals that the peak at 2.1 is mainly caused by a seven-fold ring peak and the overlap of a six- and eight-fold peak. The hump near 0 is due primarily to an eight-fold ring peak and the overlap of a five- and seven-fold ring peak. Finally, the little bump at (—1.4) is caused mostly by a six-fold ring peak and the peak at (—2.7) is mostly due to a five-fold ring peak and the overlap of a seven- and eightfold ring peak. In Figs. 2(e) and (f) we show the TDOS of the ST—12—Bethe system as obtained from a weighted average over the LDOS spectra. We also show the

620

DENSITIES OF STATES OF TETRAHEDRALLY COORDINATED SOLIDS

crystalline ST—12 spectrum3 which is considerably more complicated than the BC—8 and diamond spectra. Nevertheless we are still able to say that the TDOS is mainly due to the local configuration. We have found that for all these structures the local densities of states are nearly independent of how the rings are arranged in the clusters and depend mainly on the number and type of rings.

Vol. IS, No. 3

In conclusion we believe that our method provides a powerful way of studying the TDOS of an infinite system of atoms in terms of the local environment of each atom. In particular, this enables us to calculate the TDOS of any system given the percentage of atoms with the same ring statistics. Conversely, given a TDOS it may be possible to distinguish between possible ring statistics. This method is presently being extended to dt..i,with problems of impurities and surface effects.

REFERENCES 1.

DONOVAN T.M. and SPICER W.E., Phys. Rev. Lett. 21, 1572 (1968).

2.

LEY L., KOWLACZYK S., POLLAK R. and SHIRLEY D.A., Phys. Rev. Lett. 29, 1088 (1972).

3. 4.

JOANNOPOULOS J.D. and COHEN M.L., Phys. Rev. B7, 2644 (1973). GASPARD J.P. and CYROT—LACKMANN F., J Phys. C: Solid State Ph vs. 6, 3077 (1973).

5.

SLATER J.C. and JOHNSON K.H., Phys. Rev. B5, 844 (1972); JOHNSON K.H. and SMITH F.C., Phys. Rev. B5, 831 (1972).

6.

McGILL T.C. and KLIMA J.,Phys. Rev. B5, 1517 (1972).

7.

THORPE M.F., WEAIRE D. and ALBEN R., Phys. Rev. B7, 3777 (1973).

8.

THORPE M.F. and WEAIRE D.,Phys. Rev. B4, 3518 (1971).

Wir schiagen eine neue einfache Methode vor zur Berechung lokaler Zustandsdichten in beiebigen Festkorpern mit der Koordinationszahl 4 der Atome. Hierbei wird em endliches gluster von Atomen gewahit und an em ünendliches Bethe—Gitter mit Koordinationszahl 4 angeschlossen. Die Methode ist sehr präzis, kann numerisch leicht durchgefuihrt werden und konvergiert rasch. Einfache Näherungen ergeben bereits genugend Information zu einer physikalischen Interpretation des Systems. Testergebnisse werden präsentiert für Diamand, BC—8 und ST—12—Gitter. Sie werden diskutiert unter besonderer Berucksichtigung der Ring—struktur un die jeweiligen Atome. Der Vergleich-mit exakten Rechnungen ist sehr gut; die Interpretation der Ergebnisse niittels Ring—Formationen ist physikalisch seh.r zufriedenstellend.