ELECTRONIC STRUCTURE OF DISORDERED SYSTEMS

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ELECTRONIC STRUCTURE OF DISORDERED SYSTEMS

F. Cyrot-Lackmann

To cite this version:

F. Cyrot-Lackmann. ELECTRONIC STRUCTURE OF DISORDERED SYSTEMS. Journal de

Physique Colloques, 1972, 33 (C3), pp.C3-167-C3-173. �10.1051/jphyscol:1972324�. �jpa-00215058�

JOURNAL DE PHYSIQUE

Colloque C3, suppl&ment au no 5-6, Tome 33, Mai-Juin 1972, page C3-167

ELECTRONIC STRUCTURE OF DISORDERED SYSTEMS

F. CYROT-LACKMANN (*)

Institut Laue-Langevin, Cedex 156, 38-Grenoble-Gare, France

R6sum6. - La densit6 d'Ctats 6lectronique d'un systeme d6sordonnC est Btudite dans l'appro- ximation des liaisons fortes en utilisant un dkveloppement en moments. Nous montrons que les moments de la densite d'Ctats sont directement lies aux inttgrales de recouvrement entre paires d'atomes moyennkes par les fonctions de correlation des atomes dans le systkme. Dans le cas ou les atomes sont situes sur un reseau, les moments se calculent par un dCcompte de chemins sur le reseau. Nous dhcrivons ensuite brikvement certaines des techniques utilisees pour reconstruire la densitt &tats a partir d'un nombre fini de ses moments. Des exemples sont donnCs pour illustrer les diffkrentes possibilitts dtcrites.

Abstract. - The electronic density of states of a disordered system is studied in a tight binding scheme using an expansion in its moments. These moments are directly related to atomic pairs overlap integrals averaged by the correlation functions of the system. When the atoms are located on a lattice, the moments can be computed by a walk counting technique. We describe then briefly some of the methods used to build up some description of the density of states through a knowledge of a finite number of its moments. Some concrete examples are given to illustrate these possibilities.

1. Introduction.

This paper concerns a method recently developed to study the electronic density of states and some related properties of a disordered system in a tight binding scheme. The mean motiva- tion for this stems from the breakdown of the usual approach based on the periodicity of the lattice.

k space, dispersion relation, loose much of their interest in a disordered system as Bloch's theorem no longer holds. Many cases of considerable interest as crystalline lattice with extended defects (stacking faults, surfaces, . . .), amorphous or liquid materials, disordered alloys, ... are relevant from such an approach.

moment's expansion technique. Then we describe the possible methods which can be used to build up the density of states from a knowledge of only a finite number of its moments. Some cases of physical interest are also studied to illustrate these possibilities.

2. Moment's expansion technique.

2 . 1 DEFINI-

TION OF THE MOMENTS. -

Let H be the one electron Hamiltonian of an electron which interacts with N atoms by a set of atomic potentials V(r - Ri) :

The method presented here, is based on a direct

calculation of the moments of the density The states n(E) the system of states. I n a tight binding scheme, their computation is written :

can be done independently of the knowledge of the 1

electronic states and without regard to any crystalline n(E)

Tr 6(E - H) .

N (2)

order. Thus the method is quite general and can be applied to any physical system with any kind of disor-

The moment of order of n(E) is given by : der described in a tight binding scheme. Furthermore,

due to the formal analogy of a tight binding descrip- 1

tion of electron states with the description of phonons ^,up

I E p n ( E ) d E = - T r H P . N (3) and magnons, the moment's expansion method can

be successfully these cases. In fact the The knowledge of all the set of moments of the density moments have been used in the of states determine uniquely n(E), through the charac- case of phonons. Let us also notice that the moment's teristic functionf(x) :

method can be nicely used even for pure crystalline

materials where it is very tedious to calculate the whole

band structure E(k) needed to build up the density of n(E)

_2.n I

^{eixE f ( x ) dx} (4) states n(E).

In a first part, we describe the general features of the where

(*)

On leave from Laboratoire de Physique des Solides, FacultC des Sciences, 91-Orsay.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972324

C3-168 F.

CYROT-LACKMANN 2 . 2 CALCULATION

In a tight

binding scheme, the band is described as a linear combination of a set of atomic orbitals, @(r - Ri) centred on all the atomic sites.

For the sake of clarity, we shall assume that each atom has only one s state, but the calculation can be easily extended to a degenerate band. If Eo is the energy of the s level of the free atom, the atomic orbitals satisfy

[T + V(r - R,)] @(r

R,)

(T

Vi)

=

Eo Gi. ( 5 ) Assuming that atomic orbitals centred on various sites are orthogonal, we can in the tight binding scheme expand the trace in eq. (3) on this set of atomic orbitals

@(r - R,), such that pp is written

... <

I ^H I

> . (6) We neglect all these overlap integrals, but the two centred ones which are of two different types :

-

the off-diagonal elements,

=

B(R) for R

Ri

Rj # 0 . (7) - The diagonal elements,

The first type of overlap integrals, the transfert ones, are related to the width of the band, and the second type of overlap integrals, the crystalline field ones, to its shift.

Using eq. (6), (7) and (8), we can now give the detailed expression used for the computation of the moments of the density of states. Two different cases of disorder have to be considered, one where the atoms lie on lattice sites, and one where the atoms are random- ly distributed in space with some statistics, such as for example in liquids.

a) Lattice case.

When the atoms lie on lattice sites, pp can be computed by a walk counting technique, as eq. (6) writes

pp

1 C HiIj2 Hi,i, ... Hipil . N

The sum is over all chains of p atoms starting at an atom and returning there by a series of p steps. This walk counting technique extends easily to a degene- rate band. The overlap integrals are now matrices

associated to each step between two atoms. One has to evaluate the trace of the product of matrices associated with each closed circuit and sum their contributions.

b) Liquid case. - In liquids or amorphous bodies, the atoms are not located on a lattice, but randomly distributed in space. The probability that the N atoms will be centered at the points R,, R,, ..., RN, defining thus a configuration { R i ) , is given by the density matrix P({ R i 1). The physical properties of the system have to be averaged over all the possible atomic configurations, as they are macroscopic quantities.

Denoting this ensemble average process by bracket, we have for the mean density of states :

Using eq. (6), the moment pp of < n(E) > ^{can be}

written as

The computation of the successive moments needs the knowledge of the successive correlation functions g(P)(R,, . . ., Rp) giving the probability of having a set of p atoms located in R,, R,, ..., R,. If p

N / V is the density of the system, these correlation functions are related to P({ Ri }) through

Taking an s band, and neglecting the a type overlap integrals for a sake of simplicity, the first few centred moments can be written :

x g ( 3 ) ( ~ , , R,, R,) d3 R, d3 R2 d3 R,

Knowing the local range order characterizing the system, the moments can then be computed, either through (13), or numerically through (11) by generat- ing the random system on a computer.

3. Applications.

In Section 11, we showed that it was possible to compute a large number of moments of the density of states of a disordered system, the computation of all the moments being only possible for very simple cases. This section discusses the possible informations obtained on the electronic properties of a system from a partial knowledge of the moments of its density of states. Two different approaches can be given to this problem. The first one concerns exact, but very partial results on the density of states either by the relation of some asymptotic properties of the moments to some peculiar points of the density of states, or by a numerical approach giving some bounds. The second one builds up some analytical models for the density of states and physical properties related, either by some curve fittings methods, or by an approximate summation of the infinite series of the moments.

3.1 EXACT

- a) Characteristic func- tion.

Knowing all the moments of the density of states, eq. (14) gives n(E) through the Fourier trans- form of the characteristic function.

Thus, for example, one can compute exactly all the moments of the density of states of an

band of a perfect simple cubic crystal [I]. The characteristic function f(x) is given by :

where B is the transfer overlap integral and J(2 px) the Bessel function. Figure 1 gives the density of states of a simple cubic crystal using (14). One can also compute exactly all the moments of the density of states of an s band of a simple cubic crystal cut by a (100) plane [I], [2]. The computation is made using eq. (9) and noticing

FIG.

-Density of states of the

band of a simple cubic crystal

6 p)

Exact computation. - - - Edgeworth fit

with

moments.

that there is a one to one correspondence between the paths starting at a point i and returning there after having crossed the surface, and those starting at i and going to its symmetrical point with regard to the first fictitious plane outside the crystal, after having done the same number of stops. One gets for the variations of the characteristic function f (x) due to the cut by the surface

Sf ( x )

5:(2 px) - ~ g ( 2 px) cos2 px . (15) Figure 2 gives the variation of the density of states Sn(E) due to the cut by a (100) plane of a simple cubic crystal using (15).

FIG. 2.

Variation of the density of states of the

band of a simple cubic crystal cut by a

plane

6 p)

Due to some general properties of the Fourier transform, the behaviour of the density of states near its possible singular points is related to the asymptotic behaviour of f(x) [3], [4]. For instance, if f(x) for large x is decreasing faster than any power of x, n(E) has no singular points. If f(x) for large x decreases in oscillating like fAS(x) ei "

where fAs(x) behaves like some power of x, n(E) will have a singularity in E,.

The behaviour of n(E) near the singularity is related to the form offAS(x).

Thus, let us take again as an example the case of a simple cubic crystal, perfect or cut by a (100) plane.

Using the well-known asymptotic expansion of the

Bessel function in eq. (14) and (15), we get easily the

behaviour of the corresponding density of states near

its singular points [I], 141. The density of states of a

perfect simple cubic crystal has four singular points :

the two ends of the spectrum E

1 6 p

2 Em,,

where it behaves like (Em,,

( E OM, and two points

located in E

f E,,,/3 where n(E) has an infinite

discontinuity in its slope. The variation of the density

of states Sn(E) of a simple cubic crystal cut by a (100)

plane has four singular points ; the two ends of the

C3-170 F.

CYROT-LACKMANN spectrum, where n(E) behaves like (Emax

I E 1% and

two points located in E

& Emax/3 where 6n(E) behaves like Log I E Emax/3 1.

This last property is also of some help to study more disordered systems. It relates the shape of the density of states near the band edges to some topological pro- perties of the system through the asymptotic behaviour of the characteristic function, i. e. the behaviour of moments of high order. More precisely, it has been shown [5] that if pp - A(Emax)Pp-s for p

a, then the density of states behaves near its edge like

and vice versa.

Let us again take as an example the case of a simple cubic crystal. The moment q is given by eq. (9), i. e.

by pP Pp, where Pp is the number of closed path with p interatomic jumps. Pp is equal to the number of possible paths of p jumps, i. e. 6P for a simple cubic crystal, multiplied by the probability Wp(0) to return to the origin. For random walks on the lattice, and in the limit of p infinite, the probability Wp(0) can be written for a 3 dimensional lattice, as [6] :

where a is the nearest neighbour distance.

The moment

is then :

The width of the band is 12 P and using relation (16), we get for the behaviour of the density of states near the band edges :

In any other 3 dimensional crystalline structure, Wp(0) would be given by a similar equation to (17) for largep, and the density of states would still behave like (Em,, - E)'I2 near the bottom of the band. For a one dimensional crystalline structure, Wp(0) is propor- tional to p-ll2 for large p, and the density of states behaves like (Em,, - E)-'I2 near the band edge. For a two dimensional structure, Wp(0) is in p - l Ln p and the density of states behaves like Ln(Em, - E).

This reasoning can easily be extended to the case of degenerate bands, replacing the single value /3 by a matrix. The moment pp will be given by the trace of a product of matrices multiplied by the probabi- lity F',(O). W,(O), for large p, is governed only by the topology of the system and behaves for example for a three dimensional lattice always like

Thus the density of states still behaves as (E,,, - E)'I2 at the end of the spectrum. The shape of the density of states near the band edges is therefore directly related to certain topological properties of the system.

Let us now consider the case of amorphous mate- rials, such as amorphous Si and Ge, where recent experiments [7] have shown a sharp optical absorption edge comparable to that found in the corresponding crystals. The results of these experiments suggest that the amorphous state contains a minimum amount of defects, such as dangling bonds, and is close to a perfect or ideal amorphous state. Structural studies are consistent with this picture and suggest a random tetrahedral network model for such solids [8]. Infinite networks can then be constructed such that nearest neighbours are in almost perfect tetrahedral coordi- nation with small distortions from the corresponding crystalline structures and with a highly disordered distribution of further neighbours. In a tight binding description of these amorphous solids we can reason- ably take into account only overlap integrals of the usual sp3 type over pairs of nearest neighbour atoms.

Thus, if the amorphous state is an ideal one, described by a random network model, these overlap integrals will have the same value for any atom of the system.

The previous arguments then apply, and the density of states will behave as (Emax

E ) ' / ~ at the bottom of the band, in agreement with the experimental results on optical absorption.

b) Histogram method.

This method gives the maximal exact information one can get from the knowledge of the first few moments ; it consists of exact bounds of the integrated density of states :

The method, developed in particular by Wall [9]

and Gordon [lo] is described in detail elsewhere [I 11.

Let us just notice here that this method corresponds to a truncation of the infinite continued representation of the Green function G(E) which can be written [lo], as

where the a: can be obtained from the moments. Such

a fraction truncated at any stage if desired, is known

to give a positive definite n(E) if the product

a,,, , ² 0 [12]. More precisely, knowing n moments

of the density of states n(E), the infinite continued

fraction (19) is truncated at the stage cr,. It gives N(E),

the integrated density of states, as an histogram of

n steps with exact bounds for their jumps [12]. Then,

from the knowledge of the histogram of N(E), some

possible curve for n(E) can be obtained by smoothing

N(E) and deriving it. It is also possible to define, in a

way similar to N(E), an histogram for n(E), but mathe-

matics provides no more theorems giving its exact

bounds as for N(E). In fact, the same property is

usually kept on n(E), as shown on several examples [ll].

Figure 3 shows for example the good accuracy of the method applied to the case of a simple cubic crystal with 30 moments. This method can then be quite useful to study the details of some parts of the density of states, such as for example the possible erosion of the impurity level when forming an impurity band in a heavily doped semiconductor, or the extension of the tails of the band in a disordered binary alloy [ll].

FIG.

-Density of states of the

band of a simple cubic crystal obtainedby the histogram method using

moments.

7 C

E max FIG. 4. -A comparison of the exact frequency spectrum for a one dimensional diatomic lattice with a

moment Legendre

polynomial approximation From ref.

3.2 ANALYTICAL

- a) Curve Jitting.

Expe- rimental curves are often, characterized by their first few moments. It is well known that y , gives some information on the width of the curve, the ratio p4/p: on the extension of its wings, the ratio p3/p;12 on its asymmetry, ... Knowing its first n moments, it is then quite reasonable to fit the density of states n(E) by some analytical expression containing n parameters. The approximate function crosses at least n times the exact function [13], and this method can be quite useful when applied to cases where n(E) has some regular shape. Many fittings have been tried. Let us just mention two of them which are commonly used.

A fitting with Legendre polynomials has been extensively used for the study of the phonon spec- trum [4].-It needs a very high accuracy of a large number of moments and the approximate function often oscillates quite in a dramatic manner. Figure 4 shows for example the frequency spectrum of an ordered diatomic one-dimensional lattice calculated with 14 moments, and gives a comparison with the exact spectrum (from ref. [4]).

Another fit uses an Edgeworth series where the trial function is chosen [14] as :

This curve fitting, using an Edgeworth series, is appropriate for describing bell shaped density of states, and physical properties related to it. One can indeed obtain a good estimate of integral properties of the density of states by using for it a curve fitted to its first moments.

This can for example be easily checked on some properties of an s band of a simple cubic crystal, such as its cohesive energy Ec and its surface tension y,, which are respectively given by :

where Eo is the binding energy of the s level and

where n(E) and nt(E) are respectively the density of states of a perfect simple cubic crystal and a semi- infinite one. Figure 5 shows a comparison between the exact result for Ec and one obtained using a density of states fitted to its five first moments. Figure 6 shows a comparison for the surface tension of a simple cubic crystal cut by a (100) plane between the exact result and one obtained with a density of states fitted to the three first moments. The results are in verv good - -

[; Z] ^P(E,

overall agreement, in spite of the simplicity of the n(E)

exp - ₍₂₀₎ model, which can be easily refined by taking into

account a larger number of exact moments.

P(E) being a polynomial of order n - 1 if n moments Let us give some examples of applications of this are taken into account. method which has in particular been used with a fair Figure 1 shows for example the density of states of a success to study a large number of physical properties simple cubic crystal given by an Edgeworth series with of transition metals, related to the formation of their

25 moments. d band, such as cohesive energy, stacking fault energy,

12

C3-172

F. CYROT-LACKMANN

FIG. 5. - Cohesive energy of the

of a simple cubic crystal as a function of the lilling of the band

exact computa- tion. ... approximate computation with

moments using a n

Edgeworth series fit.

FIG. 7. - Theoretical values of the cohesive energy of transition metals using an Edgeworth series curve fitting to five moments.

FIG. 6. - Surface tension of the

band of a simple cubic crystal cut by a (100) plane as a function of the filling of the band

-exact computation. - - - Edgeworth series fit with

moments.

FIG. 8.

Experimental values of the cohesive energy of transition metals. See reference

elastic constants, magnetic properties, surface pro- perties, .. . (see in particular references [2], [13], [16]).

In these cases only a finite number of moments of the density of states can be computed. For example, figure 7 shows theoretical results for the cohesive energy of transition metals obtained with an Edge- worth series curve fitting to five moments 1131. The agreement with the experimental results [15] (Fig. 8) is generally good especially for the 4 d and 5 d periods.

The method is also especially suitable to study proper- ties of the transition metals in the liquid phase as only the three first moments of the density of states can be obtained exactly. Eq. (ll), (12) and (13) indeed show that a computation of the n first moments, p,, ply ...,

p,-, of the density of states would require a knowledge

of the n - 1 first correlation functions. But, at the pre- sent time, we have only very poor informations on the short range order existing in a liquid, as only the pair correlation function can be obtained experimen- tally. The three first moments are thus the only ones which can be computed with a high precision. We have studied for example the heat of fusion L, of the transition metals, which is given by

where the subscript S and L refer respectively to the

solid phase and liquid phase. L, shows the same expe-

C3-173 rimental behaviour as the cohesive energy [15] of

transition metals, but with numerical values much smaller. L, is only a small fraction of the cohesive energy, as the ratio LF/Ec has a value close to 1/20 or 1/30 for nearly all the transition metals. From eq. (21) and (23) this can be easily checked with a simple calculation using for the density of state of transition metals in the solid and liquid phase, curves fitted to the three first moments. Thus, we get

Using the eq. (13) to evaluate p,,, we get for a typical transition metal, a relative variation of the second moment, (p,, - p,,)/pz, of the order of 1/10 [I], leading to a ratio L ~ / E , of the order of 1/20, in agree- ment with the experimental results.

b) Moments and Green's function. - Another way to deal with the description of the density of states knowing exactly only its first few moments is to gene- rate some approximate expressions for the moments of higher order. One then can try to resume these moments to get n(E) through the characteristic func- tion (eq. 4).

It may be of some help to relate these approximate description to results obtained by standard summation techniques. Thus, denoting by G(E) and Go(E) respectively the Green function of the system, and of the isolated atom, one can easily show that

where Eo is the binding energy of the isolated atom.

A useful approach is also to represent the Green function by an infinite continued fraction, such as given by eq. (19). Inserting in this expression approxi- mate moments, leads to an approximate density of

[I] CYROT-LACKMANN (F.), Thkse de Doctorat d'Etat, 1968, Orsay.

[2] CYROT-LACKMANN (F.), J. Physique, 1970, 31, C 1-67.

[3] PERETTI (J.), J. Phys. Chem. Solids, 1960, 12, 216.

[4] MARADUDIN (A. A.), MONTROLL (E. W.) and WEISS (G. H.), Sol. Stat. Physics, 1963, suppl. 3.

[5] DOMB (C.), MARADUDIN (A. A.), MONTROLL (E. W.) and WEISS (G. H.), Phys. Rev., 1959, 115, 18.

[6] CHANDRASEKHAR (S.), Rev. Mod. Phys., 1943, 15, 10.

[7] DONOVAN (T. M.), SPICER (W. E.) and BENNETT (J. M.), Phys. Rev. Letters, 1969, 22, 1058.

[8] POLK (D. E.),

non Cryst. Solids, 1971, 5 , 365.

[9] WALL (H. S.), Analytic Theory of Continued Frac- tions, 1948, D. Van Nostrand, Inc. New York.

[lo] GORDON (R. G.), Adv. in Chemical Physics, 1969, 15, 79 ; and earlier work referred to therein.

[ I l l GASPARD (J. P.) and CYROT-LACKMANN (F.), This Colloquiunl

and to be published.

states, if the necessary positivity on the a'

is verified.

This method has, for example, been used to des- cribe the d band of fcc transition metals with good results [17], with moments of high order approximati- vely generated with small closed chains of nearest neigh- bour atoms. Kirkpatrick has also applied this method in some different way to study the tails of the band of a disordered binary alloy [18], the asymptotic part of the continued function defining G(E) being summed to infinite order through some extrapolation of the a coefficients.

Finally, let us remark that one can also use with great benefit an expansion in function of cumulants.

The density of states is given as a function of the usual Kubo cumulants Kn by [14], [19] :

with

where the cumulants are directly related to the moments.

Contrary to the expansion using the moments charac- teristic function given by eq. (4), one can restrict here the sum to a finite number of cumulants to get some approximate density of states [l]. One can also intro- duce some restricted cumulants associated with some definite class of diagrams used in the standard pertur- bation expansion of the Green's function [20] ; the Kubo cumulants representing in that way the limit which would in principle treat all diagrams. These new types of cumulants seem to be quite useful in practice, either to build up some hierarchy for the moments of high order, or o give some self consistent approach to treat single site effects, pair effects,. . .

[12] SHOHAT (J. A.) and TAMARKIN (J. D.), The Problem of Moments, 1950, Mathematical Surveys 1 (Ameri- can Mathematical Society, Providence, RI).

[13] DUCASTELLE (F.) and CYROT-LACKMANN (F.),

phys.

Chem. Solids, 1971, 32, 285.

[14] KENDALL (M. G.) and STUART (A.), 1963. The Advanc- ed Theory of Statistics, London.

[15] GSCHNEIDER (K. A.), Solidstate Physics, 1964,16,275.

[16] CYROT-LACKMANN (F.) and DUCASTELLE (F.), Phys.

Rev., 1971, B

2406

J. Physique, 1971, 32, C 1-536 ; J. Chem. Solids, 1970,31,1295 ; DUCASTELLE (F.), J. Physique, 1970, 31, 1055.

[17] HAYDOCK (R.) and HEINE (V.), Phys. Rev. Letters, 1971. To be published.

[IS] KIRKPATRICK (S.), J. non Crystalline Solids ; to be published

and this Colloquium.

[19] KUBO (R.), J. Phys. Soc. Japan, 1962, 17, 1100.

[20] GASPARD (J. P.) and CYROT-LACKMANN (F.), to be

published.