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### Submitted on 1 Jan 1972

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**APPLICATION OF THE MOMENT’S METHOD TO** **THE CALCULATION OF THE DENSITY OF STATES**

**OF THE IMPURITY BAND**

### J. Gaspard, F. Cyrot-Lackmann

**To cite this version:**

### J. Gaspard, F. Cyrot-Lackmann. APPLICATION OF THE MOMENT’S METHOD TO THE CAL-

### CULATION OF THE DENSITY OF STATES OF THE IMPURITY BAND. Journal de Physique

### Colloques, 1972, 33 (C3), pp.C3-175-C3-178. �10.1051/jphyscol:1972325�. �jpa-00215059�

**JOURNAL **DE **PHYSIQUE ** **Colloque C3, supple'ment ****au ****no 5-6, Tome 33, Mai-Juin 1972, page (23-175 **

**APPLICATION OF THE MOMENT'S METHOD TO THE CALCULATION ** **OF THE DENSITY ** **OF STATES OF THE IMPURITY BAND **

J. P. GASPARD (*) and

**I?. **

CYROT-LACKMANN (**)
Institut Laue-Langevin, Cedex 156, 38-Grenoble-Gare, France
Rksumk. - Le formalisme des moments et la methode de I'histogramme sont utilisks pour d6crire la densite d'ktats 6lectroniques d'une bande d'impuretes dans Ie domaine mktallique. Par simulation du dksordre sur ordinateur, on etudie la possibilite d'une Qosion de la densite d'ktats au voisinage du niveau d'impuret6s, comme l'a prkvu Lifshitz.

Abstract.

### -

The formalism of the moments and the histogram method are used to describe the electronic density of states of an impurity band for concentrations in the metallic range. By simu- lating the randomness on a computer, we study the possibility of an erosion >> of the density of states near the impurity level, as foreseen by Lifshitz### .

**1. Introduction. **

### -

The aim of this paper is to describe a method to build up the density of states*moments and to apply it to the study of impurity band.*

**n(E) from the knowledge of a finite number of its**In the case of imperfect or amorphous materials, because of the lack of periodicity, it is quite impossible to compute n(E) directly by means of the dispersion relations. For these materials, in the tight binding approximation, one way to deal is to compute the moments of the density of states. In practice one can compute only a finite number of moments, so it is important to be able to build up an approximate function for n(E) from its few first moments. All the curve fitting methods are rather imprecise and often give rise to spurious effects. The histogram method is the only way to provide exact bounds on the integrated . . -

density of states.

In a first part we introduce the formalism of the moments ; then we review some empirical methods, while in the third part we describe the histogram method. The fourth part is devoted to the study of the center of the electronic density of states of an impurity band.

**2. Formalism of the moments. **

### -

Let*H be the*hamiltonian of an electron interacting with a set of N atoms located on the sites

**Ri**

**H **

= **H**

**T**

## +- 2

^{V(r }^{- }

^{R ~ ) }^{= }

^{T }^{i- }

### C **qg; **

(I)
**i ****i **

(*) And Universit6 de Likge B-4000 Litge (Belgique).

(**) And Laboratoire de Physique des Solides, Facult6 des Sciences, 91, Orsay (France).

The centred moments of the density of states are defined by

Usually the density of states is normed to unity, thus
* ,uO *= 1. In a tight binding scheme, F. Cyrot-Lack-
mann [I] has shown that the moments can be written
as

where

**fl **

are the usual two centres overlap integrals.
**fl**

The N x N matrix /3 pIays the same part as the adjacency matrix in the graph theory [2].

**3. Curve fitting method. **- In all fitting methods,
we assume * n(E) to be approximated by an analytical *
expression containing

*; the*

**n parameters***moments*

**n first**

^{p0, }### .. .,

^{- }

### ,,

are computed as functions of these parameters. These latter can be determined by solving a system of n equations with n unknowns. It has been demonstrated [3] that the approximate function crosses at least n times the exact function. This result is the only and rather poor information we can get by using this method, if we have a*no other infor- mation on the shape of the function. Furthermore, in most cases, the approximate function oscillates in a dramatic manner and is imprecise in the vicinity of an mportant discontinuity. Let us say*

**priori****a few words on**the usual methods and let us apply them to the square function case.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972325

**APPLICATION OF THE MOMENT'S METHOD TO THE CALCULATION ** **C3-177 **

the exact value of n(E) crosses all discontinuities of n,(E). Figure 3 shows the good accuracy of the method applied to the square function with 10 moments.

Figure 4, illustrates the example of the simple cubic crystal with 30 moments.

the second one will be developed in a forth coming paper.

For building up our model, we generate at random
the positions of the atoms in agreement with the
fixed concentrations ; then we compute the **j? **matrix by
formula (11) and its successive powers. The model
is made of 50 atoms, and the 12 first moments are
calculated and averaged over 6 randomly generated
configurations. The histogram method seems the most
efficient method to construct n(E) and to test whether
or not the <(erosion ^{>) } announced by Lifshitz [lo]

appears. We present now the first results obtained ; the concentration varies from 0.3 x loi6 to 1018 impurities

~ m - ~ . At low concentration, ** ^{c }**= 0.3

### x

1016 i ~ m - ~ , the impurity band is narrow and shows an asymmetry due to the fact that the odd moments are negative, with a tail in the negative energies range and an abrupt drop in the positive energies range, as*shown in figure 5. When the concentration increases,*c = 1.0 x lo1' i ~ r n - ~ no <<erosion

^{)) }appears in the centre of the band, as it can be seen in figure 6. In order to appreciate exactly the possibility of this

^{(( }ero- sion

**B,**it is necessary to average over a large number

**5. ****Impurity band. **

### -

We apply here the histogram method to the study of the electronic density of states of an impurity band in a semiconductor in the tight*impurities, located in a volume*

**binding approximation. We consider N identical****V, **

with concentration
c **V,**

^{= }

*N/V such as to be in a metallic range just above*the Mott transition, for example for phosphorus impurities in germanium

*c*= [loi6, 101*] i ~ m - ~ . The impurities are distributed randomly but it is possible to give a minimum separation. The overlap integrals are given in a hydrogen-like model by

^{o }*where Vo is twice the ionization energy of the funda- *
mental level and

**R, **

is the Bohr radius (65-70 **R,**

### A

for Ge).It is possible to solve the problem in two different
ways : the first one would be to generate, by Monte
Carlo calculations, a random system of impurities,
and to calculate its properties ; the second way would
be to make analytic calculations in order to get the
moments of the density of states, averaged on all *^{o }*
possible configurations of the system. Preliminary
results of the first method are presented here, while

**C3-178 ** **J. P. GASPARD AND ** **F. CYROT-LACKMANN **

of configurations as we have done. For the highest **Conclusion. **

### -

The possibility to calculate the concentration*=*

**c****lo1',**the band is broadened and still moments of the density of states of disordered systems

no **<c **erosion), appears, as shown in figure 7 ; this is of primary interest and nearly the only way to

is due to the large importance of quadruplets, calculate * n(E) *and integral quantities involving

**n(E).**quintuplets

### ...

contributions. Indeed, as demonstrated We have shown that the only method to study the in [lo], the^{(( }erosion >> is mainly due to doublets and local properties of

**n(E)**### -

in our case the centre of the triplets contributions. impurity band### -

is the histogram method.**References **

[I] CYROT-LACKMANN * (F.), Adv. in Phys., 1967, 16, 393. * [6] WALL (H. S.), Analytic Theory of ContinuedFractions

*@. van Nostrand, Inc. New York, 1948).*

**J.****Physique,****1970,31, C 1-67.**[2] KASTELEYN (P. W.), Graph Theory and Theoretical

Physics, Ed. by F. Harary (Acad. Press, London

### 17]

DELTOUR**(J'),**

*19687 397 413'*

**Physica9**1967). [8] GORDON * (R. G.), *J.

*1968, 9, 655.*

**Math. Phys.,**## r3]

DucAsTELLE**(F.) and**CyROT-LAcKMANN (F.),

**J .***[9] SHOHAT (J. A.) and TAMARKIN (J. D.), The Problem of*

**Phys.**

**Chem.****Solids, 1971, 32, 285.**[4] MARADUDIN **(A.), MONTROLL **(E.) and WEISS (G.), Moments, Mathematical Surveys 1 (American

* Sol. Stat. Physics, *suppl. 3. Mathematical Society, Providence, RI, 1950)

**151 **KENDALL (M. G.) and STUART **(S.), **The Advanced 2nd edition.

Theory of Statistics Vol. 1 (C. Griffin, London,

### [lo]

LIFSHITZ (I.*1964, 13, 483.*

**M.), Adv. in****Phys.,**1969), 3rd edition.