Theoretical model of the density of states of random binary alloys

Solid State Communications, Vol. 85, No. 5, pp. 435-442, 1993. Printed in Great Britain.

0038-1098/93 $6.00 + .00 Pergamon Press Ltd

THEORETICAL MODEL OF THE DENSITY OF STATES OF R A N D O M BINARY ALLOYS N. Zekri,* A. Brezini* and F. Hamdache* International Centre for Theoretical Physics, Trieste, Italy

(Received 4 April 1992; in revisedform 31 July 1992 by E. Molinari) A theoretical formulation of the density of states for random binary alloys is examined based on a mean field treatment. The present model includes both diagonal and off-diagonal disorders and also short-range order. Extensive results are reported for various concentrations and compared to other calculations. 1. INTRODUCTION D U R I N G the last decades improved experimental techniques have followed extensive reports on alkalimetal based random binary alloys [1-7]. Because of their attracting structures considerable attention has been devoted to the study of their electronic states (metal-insulator transition, conductivity . . . . ). Usually, theories involving an effective medium treatment, such Coherent Potential Approximation (CPA) techniques [8-10] which although satisfactory for describing many features of random alloys have a number of shortcomings. Probably, the most significant is its inability to account for local atomic configuration and its inability to describe the localization properties induced by disorder. Therefore, the need to go beyond the single-site CPA has been felt for a long time [9, 10]. Indeed the effects of random clustering are expected to be significant but the incorporation of off-diagonal disorder inherent in any realistic model and shortrange order due to chemical clustering tendencies implies the extension of the single site CPA. In this context, various approaches have been put forward to generalize the CPA in incorporating both off-diagonal disorder and short-range order [11, 12]. Calculations based on these suggestions suffer from main drawbacks: the "correct" formulations are often intractable and simplified approximations lead to unphysical averaged Green function. It is the purpose of this paper via a new formalism which we have recently proposed [13] using the tightbinding approach of the Bethe lattice method to study the electronic density of states (DOS) of

*Present address: Laboratoire de Physique Electronique des Solides, D6partement de Physique, USTO B.P. 1505 Oran E1-M'naouer, Algeria.

substitutional random alloys in the dilute limit, i.e. very low concentrations. We extend this formalism to include effects of local correlations between neighbour atoms by introducing a short-range order (SRO) parameter. Extensive results of DOS are discussed in comparison with other calculations. 2. FORMALISM OF DOS 2.1. Model We consider a random substitutional binary alloy AxBl_x formed of two types of atoms A and B and where x denotes the concentration. The electronic properties may be obtained from Anderson [14] oneelectron one-band tight-binding Hamiltonian: H: ~

e,. Ii)(il + ~ i

vuli)(jl

(1)

j#i

the sites {i} forming a periodic lattice. Here li) describes the atomic orbital on site i and the disorder is usually introduced by assuming that the sites energies {~i} and the hopping matrix elements { Viy} are random variables given by the following joint probability distributions: ?

VAj ) =

,-

V jl-

+ (1 -Pe(',,

Vej) :

V B),

(2)

(IzBjf - v B)

(3)

),5(IVBjf - re,, )

+ (1 - x)

(,i-,a)

We shall be interested in determining the alloy density of states which may be written in terms of the components of the partial densities of states nA(E) and nB(E) as follows:

n(E) = xnA(E) + (1 - x)na(E),

435

(4)

THE DENSITY OF STATES OF R A N D O M BINARY ALLOYS

436

where the quantities nA (E) and ns(E) are given by 71"

= + l l m { ( i a ] ( E q : is - H)-IIiA )},

(5a)

71-

ns(E) = + - l l m {Gss(Eq: is)} 71"

= + l I m {(iB[(Eq: is - H)-llis)}. 7r

(5b)

Here Gii stands for the site diagonal matrix elements of the one-particle Green function in the tightbinding basis. Gii(z) may be expanded in the form

vijaji(z )

(6)

j¢i with

(il z -

g}°i)(z) =

eili)(i[

li) = ( z - e i )

-1

(7) and

Gij(z ) = (i](z - n ) - ' l j ) =g}f)(z)eij+glf)(z) ~ VikGkj(Z ). k¢O

z.~ vL SS--E_eA_SSA

A

Sff-

Z]AV]A E - eA - S] A

s~ - E-

(10)

path in which the lattice site i has been removed and is defined similarly to Si(z). The density of states is related to the average site diagonal Green function of the binary alloy (ll)

with a , ¢ " = [~ - , ~ / ~ - s ~ / . ( z ) ] -~

(12)

denoting the partial Green function associated to either an A atom or B atom occupies the site i.

Sff'

(14)

B

+

zL vL

2

Z,~sVjB

(16)

E - e B - S•'

2

,~ - s ]

2

E - es - S~ '

A

ZBA V~a

S¢ i) standing for the self-energy of states [j ) for a

a . = x a 2 + (1 - x ) a , f

z~B vg. k E-es-

E - eA - S~

kCj ~#j #i

(13)

where Zij is the number o f j atoms surrounding an i atom and the set {s}j)} is defined by

-lVji

× [z - V - s}i/(z)] -~ V~,.

ZAB~

ZA~V)A

SA -- E - eA - S,~ f- E - es -- S~'

aii(Z ) = [Z -- (" i -- Si(z)] -1. (9) Here Si(z) denotes the self-energy associated to the state [i) and its renormalized perturbation series is given by giJ[Z-£i-s)i)(z)]

possible values eA = - w and eB = W if an atom of type A or B respectively occupies the lattice site i. The hopping integrals V/j can take three values V~A, VAB= VBA and VBS depending on the nature of the atom occupying the lattice sites i and j. Because of the computational complexities which occur in studies involving the properties of realistic disorder systems, workers use a Bethe lattice to approximate the underlying structure of the material under consideration. The usual justification is the assertion that since the impurity modes are localized, hence only the local environment is significant. Since the Bethe lattice preserves the constellation of nearest neighbour atoms, it may be used as a description of the structural basis. Random binary alloys [15], and the study of the electronic structure of semiconductors by means of the Bethe lattice has a long tradition and the applications have been shown to be powerful [15, 16]. Taking advantage of the particular topology of the Bethe lattice, namely the absence of closed loops, the self-energy reduces to

(8)

Iterating equation (4) and using equations (5) and (6) enables one to rewrite Gii(z ) in the following way:

Si(Z) : Z

In the present description of random binary alloy

A.~Bt_x the random atomic levels take one of the two

nA(E) = : k l l m {GaA(Eq: is)}

Gii(z) = g}°i)(z) + g}O)(z) ~

Vol. 85, No. 5

2

Z s s VSB + E-

(17)

,. - Sg'

zg. v~.

(18)

where Z~(Bs) describes the average number of B-atoms descendants of an A-atom that has an A(B) parent along the Bethe lattice paths. The quantities Zii and Z~y ) may be substituted by their average value~ i.e

Zij ~ {Zij ) = PijZ,

(19)

z!!J) z!!J)~, = Pij(Z - 1) = Pij K, ,g =-- ,( --,j

(20)

where Z is the coordination number and K the connectivity constant. At this stage we have to introduce approximations

Vol. 85, No. 5

THE DENSITY OF STATES OF R A N D O M BINARY ALLOYS

in order to handle these general equations. Therefore, we consider the situation where S~ = Sff and S~ = Sff. The set of equations (13 and 14) may be treated self-consistently. The probabilistic definitions of the site energies ei and the matrix elements Vij [equations (2) and (3)] implies that the self-energies are also defined by probability density distributions. In the present treatment we consider a self-consistent approximation as usual

z

437

Using the integral definition of delta function: 6(x) =

1 I eitXdt

(25)

--00

and expanding equations (7) and (8) yields

e,(s~) = xZ6(s~ - s~ °)) -'1- 2(1 --

I vii 12

x)xZ-16(SA

--

SA(1)) + . . .

+ xZ(1 - x)Z-16(SA -- S~Z-1)), Z

x H P j [ e j , Vij, Sj]dejdlVijldSj, j#i

+ (1 - x)Z6(sa - S(Z)),

e,(s~) = ( 1 - x)Z~(sB - S(~0)) + Zx(1 - x ) z - 1 6 ( S B - S~ l)) + . . .

where the self-energies at a given site are statistically independent random variables. From the particular topology of the Bethe lattice, Sy is independent of the variables ej and ~j and thus:

PA/B[ej, Vij, Sy] = PA/B[ej, Vij]Ps[Sj]

(26)

(21)

+ Z(1 -- x)xZ-16(SB -- S; Z-t)) + x Z6(se - s~)),

(22)

we get the following expressions of the Fourier transforms #s,(t) and/Ss~(t):

(27)

where 2

(0) _

ZV)A/nn

-AIO - E - eal B - S~IB' /Ss~(t)=

x

dSAPs(SA)exp

-E-eA-SA

(z - 1) V~A/BB 2 s(~'~~ = E -

+oc

,~/~ - S~/o

+

VA2B E--eB/A -- SB/A ,

(28)

+ (l-x) / dSBP/SB) --OO

x exp

i's,(t) = x

(231

it v?,o E - eB - So

dSAP,(SA)exp

where the mean values S,~ and SB are the solutions of the equations (15)-(18). At this stage, we are now in a position to investigate the density of states of a disordered system on a Bethe lattice. The partial densities of states are related to the average diagonal Green's function of the binary alloy by using the self-energy probability distributions given by equations (26) and (28):

Es-e-AA ----SA

+ (1 -- x) ] dSBPs(So) --OO

x exp

itV~n

E - eB - So

ZV]IB AIB = E - eBI~ - SBI,4

s(z)

(24)

nA/B = + i dS,#BPs(SA/t~) Im GA,4/Bn At this stage we have not been able to make any use of these equations and so we are looking for less general solutions. In the following treatment we make use of a mean field approximation, i.e. we consider that the probability distributions P~(Sa) and Ps(S~) are strongly peaked around their mean values (SA) and (SB) to be determined. Such approximation is expected to be valid in the dilute limit, i.e. x - , 0.

(29)

-- fX3

with

GAA/Bn

=

1

E - e~/B - SA/n

(30)

describing the partial Green's function associated to either an A-atom or B-atom. The partial DOS may be

THE DENSITY OF STATES OF R A N D O M BINARY ALLOYS

438

Vol. 85, No. 5

expanded as: l lm hA(E) = ~r

xz ZxZ-l(1 -x) s(Ao ) + E - ea -E - £ A - - SO ) + " "

"4 Z x ( I - x ) z - ' (1 - x) z "[ E - eA -- S(Az-') + E - . . .eA . - S(az) j ' (1 -

n.(e) = ;

x) z

1 Im E - e s

Z x ( 1 - x) z - '

E-es-

Z x Z - Z ( l - x)

(1 - x ) z

(31)

+...

(36)

1 forx>i.'

In the dilute limit, only equation (35) is of interest. The negative values correspond to ordering, a = 0 to complete disorder and positive values to segregation. Under this scheme, the probability distribution for site energies transforms as

P A(£i, Vij) = P AA6((-i -- £A)6( VAj -- VAA )

+ P,~s6(ei- es)b(VAj - VAs),

}

+ E C e--£---S~~-'--)' E - e - s - - S-(Bz) 1' where the quantities S ~ t ( i = O , . . . , Z ) expressions defined by equation (28).

1--X

---
(32)

(37)

PB(£i, Vij) = PBA6(ei -- eA)6(VBj -- VBA )

+ Pss 6(el - e~)6(Vsj - Vss),

are the

(38)

and the equations for self-energies as

2.2. Short-range order parameter The striking difference between amorphous and crystalline materials provides a strong demonstration of the primacy of the short-range order (SRO) in determining the basic nature of their properties. In particular the change with chemical composition in the electronic properties has been attributed to the establishment of a well defined SRO, for instance alkali-metals doped with gold and silver or amorphous transition metal alloys can be checked experimentally by diffraction measurements [2, 3, 6]. From a theoretical point of view, including SRO in disordered alloys remains a challenge [17-21]. The most used theory describing electronic properties of disordered alloys is the Coherent Potential Approximation (CPA) [8, 9, 15]. Despite its many appreciated properties, the CPA approach fails in handling parameters such as SRO. Going beyond these limitations, several theories have been put forward [19-21], at the cost of some computational efforts. We propose here an extension of the previous section including SRO by an analytic method. Towards this end the description of SRO, or equivalently the lack of randomness in the number of like or unlike nearest neighbours around a given atom, may be parametrized by new variables, namely the short-range order parameter a and the probability Pij of a fiatom being next to an/-atom, a and Pij are chosen to be related similarly to Brouers et al. [18], i.e.

PAX = X + (1 -- X)Or;

PBA = X(1 -- a),

(33)

Pss = (1 - x) + xcr;

PAB = (1 -- X)(1 -- a).

(34)

The range of possible values of a is: x l--x--


1 forx
--2'

(35)

ZPAA V2A ZPAB V2B SA -- E - eA - SA + E - es - S s '

(39)

ZPBA V2A ZPBs V~s Ss - E - eA -- S~ + E - eB - S~"

(40)

Using the same mean field treatment as previously yields for the partial density of states

z_eL- '__e..

l Im E - eA PJAS(A °) + E - e A n~(E) = -~ -

"4 l

nB(E)

=

E-e4zPZBIpAASz - t

{

-

PZB ~ } ----s.Z) ,

+ E-

Im .

E - en -

(41)

eA -

PzI~ S ; °) ~ z P Z ; ' PSA

E-,B-

+

S(AI) + . . .

s(z_,) +

E-

,B -

')

- - - - S,7); j , E - en -

(42)

where the set (s (i) , i = O , . . . , Z ) appearing in the denominator has been calculated in taking into account the parameter a via equation (15) (18) and (28). 3. RESULTS AND DISCUSSION In this section, we have reported our numerical results for the density of states (DOS) of Bethe lattice for both diagonal and off-diagonal disorders, parametrized by w = e A - eS and Vii = ~iV~j respectively. Here ~i is equal to ~n or {B depending on the occupation of site i and the constant V is taken as our unit of energy. We have restricted ourselves without loss of generality to Z = 4 in order to compare our data with other theories.

Vol. 85, No. 5

THE D E N S I T Y OF STATES OF R A N D O M B I N A R Y ALLOYS

439

(a) 016 0.14 i

012

i I

010 Z

i

008

I

006

i l

i

004

i

002

i

f

,.-.,

i I

0 -6

i

-4

, 0

-2

I

I

2

4

(b) 016 0.14 0.12 i L

010

Z

1

f~

0 08

I i i

006 i

004

i i

002

i ,

0 -6

-4

0

-2

I

I

2

4

6

Energy

Fig. 1. Density of states for diagonal disorder ( V - - 1) in the case eA = -cB = - 1 . 7 a n d x = 0.01 (full curve), x = 0.1 (broken curve). (a) Present method. (b) CPA method.

3.1. Random systems (~r = O) 3.1.1. Diagonal disorder. In the present situation the eigenstates are bounded within the bands [22, 23] -2v~V

< E < 2x/KV

(43)

- 2 x / K V + w < E _< 2v/-KV + w

(44)

corresponding respectively to the minority and majority bands. In the absence of disorder, the density of states presents a perfect symmetry in respect to the energy E = 0. In the limit of small values of the diagonal disorder parameter w (or equivalently 6 = w/2v/-KV), the density of states appears slightly distorted. As 8 is increased, an impurity resonance is observed and the minority subband becomes apparent. This continuous transition to a split band regime is a typical character of the diagonal disorder. The present model keeps the host and impurity bands split for a critical value 6c ~ 0.85 in close agreement with the CPA results predicting 6c "~ 1.0 [23]. In addition, the sharp spike for the band of states associated to the isolated defects is reproduced quite well. Indeed, the exact value for splitting from the

Saxon-Hunter localization theorem [24] occurs at 6c = 2, i.e. when there are no states from either perfect A- or B-atom crystal at the energy E = 2 v ~ V . The difference is mainly due to the limitation of the present model by means of the mean field treatment similarly to the CPA calculations. Furthermore, as the disorder parameter is increased the majority band is centred around E = w, the sharp spike goes to E - - - 0 and the gap is linear with w. Figure 1 shows the case cA = -~B = 1.7 for the concentrations x = 0.01 and x = 0.1. For the above parameters the bandwidth of the ordered lattice is 4v/3. In particular it is observed that the curves for n(E) exhibit some interesting features as well as the evolution of the structures as a function of the concentration. For comparison the CPA calculations fail in reproducing these structures and the differences are more effective in the regime of low concentration. It is well established that the CPA approach provides a poor description of the minority band although it has some bearing for the majority band [9, 10]. The discrepancy lies in the fact that the CPA is

440

T H E D E N S I T Y OF STATES OF R A N D O M B I N A R Y A L L O Y S 0.25

I

I

I

I

I

0 20,

I

i

Vol. 85, N o . 5

I I

(a)

I

I

I

0.20

i

i,

016

.~

I

i

/ /

0.15

/

012

i

Z

"

Z

0.10

t~

,/ /

~

008

(i

!

II

i !

0.05

0 04

J I I

0 -5

I -I

I

-4

-3

-2

1 0

i I

i 2

t 3

4

0 -4

5

1 I ,

0

-2

Energy 0.2001

o. tso

,

, :ki

o _O°l-

Jiti

Z 0.020

,.

0.015 O.

010

0.005 -60

6

Fig. 3. Density of states for eA = --eS = --2, X = 0.1 and a diagonal disorder V = 1. Random ~r = 0 (full curve) and partially ordered o r = - 0 . 1 (broken curve).

v

li

o.uoo

I

4

Energy v

(b)

I

2

. I ,' ' ' ~ . ,

?

............................. •

.....

-40

:

."~..,~<,:........................

.-t- ' ' ' / l '

I

~;~.;~

""

-20

I 50 I100

l 20

40

f

2.00

60

Energy

Fig. 2. Density of states for non-diagonal disorder ~)~S= ~i V~j, V = 1): ~A = es = 0 and x = 0.1. Total (full curve), partial-A DOS (broken curve) and partial-B DOS (chain curve). (a) ~ = 0 . 8 and ~s = 1.1. (b) ~a = 0.8 and ~s = 4. built on an effective uniform medium for which the entire scattering by all the sites goes to zero implying the tendency to destroy the minority character. In the present model the averaged medium is determined in terms of the probability distributions of the selfenergies Ps(SA) and Ps(Ss) and then translates more precisely the isolated impurities effects at least in the regime of the lower concentrations. Obviously, the alloy density of states calculated within the framework of a Monte-Carlo technique [18] shows a more detailed structure in the minority band. Mainly their self-consistent probability distribution of the self-energy Ps(SA) and P I S s ) are determined from two ensembles of 250 complex selfenergies SA and S a and therefore more accurately than the present approximation.

3.1.2. Off-diagonal disorder. We have considered a binary alloy with a purely off-diagonal disorder. The alloy DOS n(E) for x = 0.1 is shown in Fig. 2 where we have focused our attention to the case eA = es = 0 and ~A = 0.8 and ~s = 1.1 [Fig. 2(a)] and ~s = 4 [Fig. 2(b)]. We also show in Fig. 2 results for the partial components n a ( E ) (broken curve) and no(E) (chain

0.20 Z

~--1...:

I

,

015 f OIO

005 0-6

'~ -4

-2

2

4

Energy

Fig. 4. Partial-A DOS for diagonal disorder (V = l) and for x = 0.1: ~ = 0.2.

curve). The DOS presents a perfect symmetry with respect to the energy E = 0 regardless of the amount of disorder introduced. As expected the DOS on A sites is spread out into the centre of the band. It should be noted that a rather perfect similarity exists with the results o f the r e n o r m a l i z a t i o n - g r o u p approach of d'Alburquerque e Castro [25]. As the off-diagonal disorder parameter measured by the ratio ~S/~A is increased [see Fig. 2(b)] the strength of the maximum of the DOS at E = 0 becomes more effective. It has been argued [26] the appearance of the maximum of the DOS at the band centre is a general feature produced by the offdiagonal disorder. This conclusion describes the signature of delocalization of the band-centre as discussed by Soukoulis et al. [27]. The main difference with the diagonal disorder case is the absence of the splitting and thus no gap is produced by the offdiagonal disorder.

Vol. 85, No. 5

THE DENSITY OF STATES OF RANDOM BINARY ALLOYS

3.2. Short-range order As already mentioned [28], when the system displays some degree of SRO any extensions of the CPA do not always converge and thus becomes inefficient in describing the essential features of the DOS. Therefore, we introduce here the SRO parameter as defined previously to determine the DOS for different values of the band energy parameter and concentration. Figure 3 shows the DOS for x = 0.1, cA - - - e B = - 2 for completely random binary alloys, ~r = 0 (full line) and in partially ordered alloys, i.e. c~= - 0 . 1 (dashed line). The main feature observed is an increase of the gap with increasing ~r and furthermore the structures revealed in the DOS become more pronounced and appear located at the energy E _ -3. Such results are in agreements with conclusions reached from other models calculations [19]. Figure 4 shows partial-A DOS for the case x = 0.1, eA = - e B = - 2 and ~r---0.2, i.e. the value of the SRO parameter corresponding to segregation. Once again the Monte-Carlo simulations [28] based on the tight-binding method for the Bethe cluster model in the limit of large N-atom clusters are more accurate in producing more structure in the DOS due to the limitations of the present mean field treatment.

(b) the extension of the model to higher concentration. (c) the application to realistic semiconducting alloys, where the short-range order plays a significant role, and to liquid alloys.

Acknowledgements I Two of the authors (N.Z. and A.B.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, Italy. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

4. CONCLUSION We have presented a method for studying the electronic structure of random binary alloys within the tight-binding framework developed on a Bethe lattice. The model involves basically a mean field approach treating diagonal and off-diagonal disorder and also short-range order effects. The method is valid only in the limit of low concentration when the mean field approximation becomes accurate. Extensive results have been presented and compared to other theories. Mainly the bands for the Hamiltonian spectrum correspond to the finding of the CPA approach. In particular the present model appears more powerful than the CPA in describing the minority band. However, it appears rather limited against the numerical results obtained by MonteCarlo techniques in the limit of large N-atom clusters. The model can of course be improved as well as applied to specific problems. In particular, we can think of the following lines for the future research: (a) the probability distributions of the self-energies Ps(SA) and Ps(SB) may be improved by including correlations between real and imaginary parts of the self-energies.

441

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

F. Hensel, Adv. Phys. 28, 555 (1970). H. Redslob, G. Steinleitner & W. Freyland, Z. Naturf. a 27, 587 (1982). N. Nicholso, R.W. Schmutzler & F. Hensel, Bet. Bunsenges. Phys. Chem. 82, 621 (1978). H. Hoshino, R.W. Schmutzler & F. Hensel, Phys. Left. ASI, 7 (1975). R.W. Schmutzler, H. Hoshino, R. Fisher, & F. Hensel, Ber. Bunsenges. 80, (1976). W. Martin, W. Freyland, P. Lamparter & S. Steeb, Phys. Chem. Liq. 10, 61 (1980). M. Maret, A. Pasturel, C. Senillou, J.M. Dubois & P. Chieux, J. Phys. 50, 295 (1990). P. Soven Phys. Rev. 156, 809 (1967). D.J. Thouless, Phys. Rep. 13C, 93 (1974). H. Ehrenreich & L.M. Schwartz Solid State Phys. 31, 149 (1976). V. Kumar, A. Mookerjee & V.K. Srivastava, J. Phys. C15, 1939 (1982). V. Kumar & S.K. Joshi, Ind. J. Phys. (commemoration volume 2) 1 (1979). N. Zekri & A. Brezini Solid State Comrnun. 83, 141 (1992). P.W. Anderson, Phys. Rev. 109, 1492 (1958). J.D. Joannopoulos & M.L. Cohen, Solid State Phys. 33, 71 (1976). An informative history of Bethe lattice and calculation techniques is given by M.F. Thorpe (Plenum, New York) (1982) pp. 85-107. J. Franz, F. Brouers & Ch. Holzhey, J. Phys. FIO, 235 (1980). F. Brouers & J. Franz, Phys. Status Solidi (b) 113, 431 (1982). V.V. Garkusha, V.F. Los' & S.P. Repetskii, Theor. Math. Phys. 84, 737 (1990). S . S . Rajput, S.S.A. Razee, R. Prazad & A. Mookerjee, J. Phys. Condens. Matter. 2, 2653 (1990). S. Weinketz, B. Laks & G.G. Cabrera, Phys. Rev. 1343, 6474 (1991). D.J. Thouless, J. Phys. C3, 1559 (1970). R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46, 465 (1974). D.S. Saxon & R.A. Philips, Res. Rep. 4, 31 (1949). J. d'Alburquerque e Castro, J. Phys. C17, 5945 (1984).

442 26. 27.

THE DENSITY OF STATES OF RANDOM BINARY ALLOYS A. Brezini, Phys. Lett. A147, 179 (1990). C.M. Soukoulis, I. Webman, G.S. Grest & E.N. Economou, Phys. Rev. B26, 1838 (1982).

28.

Vol. 85, No. 5

F. Brouers, P. Lederer & M. Heritier, Amorphous Magnetism (Edited by H.O. Hooper), p.151. Plenum Press, New York (1973).