ELECTRONIC DENSITY OF STATES FOR LIQUID METALS AND ALLOYS IN THE TIGHT-BINDING APPROXIMATION

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ELECTRONIC DENSITY OF STATES FOR LIQUID METALS AND ALLOYS IN THE TIGHT-BINDING

APPROXIMATION

B. Movaghar, D. Miller, K. Bennemann

To cite this version:

B. Movaghar, D. Miller, K. Bennemann. ELECTRONIC DENSITY OF STATES FOR LIQUID MET-

ALS AND ALLOYS IN THE TIGHT-BINDING APPROXIMATION. Journal de Physique Colloques,

1974, 35 (C4), pp.C4-325-C4-328. �10.1051/jphyscol:1974462�. �jpa-00215653�

JOURNAL

DE PHYSIQUE

Colloque C4, supplbment au no 5, Tome 35, Mai 1974, page C4-325

ELECTRONIC DENSITY OF STATES FOR LIQUID METALS AND ALLOYS IN THE TIGHT-BINDING APPROXIMATION

B. MOVAGHAR, D. E. MILLER and K. H. BENNEMANN Institut fiir Theoretische Physik, Freie Universitat, Berlin

1 Berlin 33, Arnimallee 3, Germany

RBsum6.

-

Nous presentons une methode pour calculer la densite d'btats electroniques des metaux liquides de leurs alliages dans l'approximation des liaisons fortes 2 une orbitale. Nous en dbduisons les expressions pour la densitb d'etats en utilisant l'approximation auto-cohbrente i un site ^(SSSA).

Abstract.

-

A method is presented to calculate the electronic density of states of liquid metals and alloys in the one orbital tight-binding approximation. We derive expressions for the density of states using a self-consistent single-site approximation.

1. Introduction and formulation of the problem.

-

Following Matsubara and Toyozawa 161, Cyrot- Lackmann [3], Roth [7] and other authors, we extend the tight-binding approximation to the electronic structure of liquid metals. The relative simplicity of this model makes it particularly attractive for theoretical investigation of disordered systems [lo].

Matsubara and Toyozawa calculated the electronic density of states N(E) for the special case of a comple- tely random liquid using a diagrammatic analysis.

Cyrot-Lackmann employed a moment technique in the study of the density of states of liquid transition metals. Recently, Roth, using multiple-scattering theory, derived expressions for the electronic density of states corresponding to the quasicrystalline approximation (QCA) of Lax and the self-consistent approximation (SCA) of Schwartz and Ehrenreich.

So far, liquid metal alloys have not been studied using the tight-binding approximation.

In the following, we present a new method for calculating the electronic density of states N(E) within the self-consistent single-site approximation (SSSA) for a liquid metal in the tight-binding approxi- mation. The theory is then extended to liquid binary alloys. We consider a one orbital model and assume that the hopping term does not depend on the type of atom, furthermore, that the atomic distribution functions are independent of constituents.

The problem may be formulated using quasi- orthogonal orbitals (Matsubara and Toyozawa, Cyrot- Lackmann) in which case the Hamiltonian for the electrons may be written

where cia is the energy of an electron with spin a on the

atomic site i ; ci, and ci, are the usual annihilation + and creation operators for electrons in the atomic orbital representation. X i j is given by

and is assumed to be a function of the distance bet- ween the atoms only, Xij = X(Ri

-

Rj). In disordered systems, it would be, in principle, more realistic to allow for non-orthogonality of the atomic orbitals.

Recently, Roth [8] has shown that this may be done by making in (1) the replacement

where E is the energy variable and S i j an overlap integral given by

The electronic density of states for the pure liquid using orthogonal and non-orthogonal orbitals is given respectively by the expressions

and

N ( E ) = - - x I m < 1 S i j S i j > . (5b) nN i,

j

Sij is the one particle Green's function in the orbital representation and < Sij > denotes the liquid average which is defined by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974462

C4-326 B.

MOVAGHAR, D. E. MILLER AND K. H. BENNEMANN

It is also convenient to define some restricted averages an approximation clearly only valid in the high density limit. It follows, therefore, that

< Pij

^{> i}

⁼ [ ^{% P(Ri I R,, ..., RN) JJ d ' ~ , (6b)}

k # i

<

A i > i

= ... ~ ( q ¹

and

j # k # . - - # i E - g o -

< d i

^{> i}

< Gij

^{> i j}

⁼ 5 ^{Sij P(Ri, Rj I R1, ..., RN)}

k + i , j

^{JJ d ' ~ , . x X .}

I k

E - c 0 - ¹ < ^{d i}

^{> i}

^...

x . . . i )

. (13)

i

(6c) Noticing that

P(R1, R,, ..., RN) is the probability of a given confi- < s(R,, RTE) >

⁼

< sii >, ⁼ _- 1

-

<

^{> i}

guration ofions. P(R, ( R,, ..., RN) gives the probability distribution for the remaining (N

-

1) atoms when R,

is held fixed. Similarly, P(R,, Rp 1 R,, ..., R,) is defined. (14)

and writing < A i > i = C(E), we obtain immediately 2. Self-consistent single-site approximation. -Going in the special case of a random liquid,

back to the Hamiltonian given - by - eq. (I), we recall . , that the local Green's function Bit can quite generally be written in the form

(7) with where A i can be expressed, using the renormalized perturbation theory originally due to Watson [ l l ] (see also I?. W. Anderson [I]), as

. ^.

The quantities A:, A:' etc., are given by similar expansions with the restriction that the indices can never be equal to (i, j), (i, j, k), etc.

The essence of the single-site approximation is now to treat the i-th-site exactly while the environment is considered only as an average. Considering, there- fore, the averaged Green's function with the site i held fixed and keeping the exact orbital energy ci we may write

Let us first use this formalism to rederive the SSSA result for the pure liquid, e. g., when ci

= E,

for all i.

In this case, we find from eq. (8)

where n is the concentration and X , is the Fourier transform of X(Ri - Rj).

This is, in fact, the result obtained by Matsubara and Toyozawa [6] and recently by Roth [8] using the self-consistent approximation (SCA) of Schwartz and Ehrenreich [9] in the special case of a random liquid and orthogonal orbitals. The derivation present- ed here is not only simple, but also physically more satisfactory. In fact, one can show that all self- consistent theories obtained so far, including the correlated SCA [8], correspond to special ways of approximating the atomic distribution functions in eq. (13). For example, the treatment given by Ishida and Yonezawa [4] corresponds to approximating the correlations in eq. (13) in the manner illustrated in figure (1). To show that the correlated SCA also

RG.

1.

- C ( E )

or

<

^A*

>*

correspondingto the approximation

Then, extending the single-site argument to every of

Ishida and Yonezawa (1972).

site in the system we have

1

< ~ i > i = ~ ~ .( x i j . . ~

x reduces to a special way of decoupling the correlations

j + k + . . . + i

E - c O - < A:

^{> j}

in eq. (1 3) is more lengthy and will be reported :else-

1 where. We only note here that the SCA involves a

X j k ^"' ~ . . . i )

. (11) rather unsatisfactory approximation of the 3 atom and

E - E ~ - < A?

^{> k i}

higher distribution functions which seriously puts into Self-consistency is achieved by now putting doubt the validity of this scheme. The approximation we propose is illustrated in figure (2). The idea is to keep

> j =

< A? > ,

⁼

⁼ < ^{A ,}

^{> i}

⁽¹²⁾ as close as possible to the Kirkwood superposition

ELECTRONIC DENSITY OF STATES FOR LIQUID METALS AND ALLOYS C4-327 approximation. The self-energy C(E) from figure ( 2 ) is Notice, that Ai only involves paths which never return

given by to the site i in the intermediate steps. Thus, Ai is

d3k independent of the energy of the site i, ci. Comparing (I6) (19) with the multiple-scattering expansion with a

perturbing potential ei at i we find with

S,

⁼

n&,

E -

-

Z(E) + ^{sii = G: +} ci G& G: +

^E:

G: G j G: + ... (20) and thus (19), is equivalent to assuming that

In the solid alloy A, B,-, the CPA corresponds to writing (Brouers et al. [ 2 ] ) .

< ^9.. >.

⁼ I

- - -. . -

-

- g ^{aa a}

E - ci - A[E(E)]

^'

(22) ___----

₊

C: =

1

A[&(E)] is just < Ai

^{> i}

in (13) with all c j replaced

\,

* = ^L by cO(E), and where cO(E) is determined by

E-e,-C(E)

1 -

^X

FIG.

2. - Improved theory for

Z(E)

corresponding to the

E

-

cO(E)

-

A[&(E)] - E

- EA -

A[E(E)] ^-I-

eq. (16) and (17).

1 - x where gk is the Fourier transform of g(r) X(r;, and +

E -

^E~

- A[E(E)] . (23) g(r) is the pair distribution function. From eq. (5b)

we recall that for non-orthogonal orbitals it is also 1, this form we can now generalize the argument to necessary the averaged nOn-lOcal Green's the case of a liquid alloy. We assume that the average function < G i j > respectively <

9ii > i j .

The self- over the liquid distribution and energies of the atoms consistent single-site truncated Gij is easily deduced keeping the site i fixed with energy

E i

can be repre- from the above scheme and corresponds to sented by introducing a medium potential E ( E ) on

1 1 every atom instead of the orkital energies. We, there-

9 . .

=

E

- E~ -

Z(E)

X i j

E

-

c0

-

C(E) + fore, have

where (1 8)

A,[E(E)]

=

C ( E - E(E)) .

Thus, we need only to average over the self-avoiding We have seen how to calculate Z(E) within the SSSA paths leading from i to j. < G i j

^{> i j}

and Z ( E ) must for various approximations of the atomic correlation be evaluated within the same approximation scheme functions. E ( E ) is determined by requiring that the of the atomic distribution functions. This completes average over the orbital energy ci of < Gii > replaces the derivation of the generallized SSSA for the pure

g i

by E(E). Thus, one finds

liquid. We now turn our attention to liquid binary

alloys, and for simplicity, restrict the discussion to 1 -

^X

+

the case when the orbitals are orthogonal. E

-

E ( E )

-

C ( E

-

E(E))

-

E

- EA -

Z ( E

-

E(E)) 3. Liquid binary alloys in the SSSA.

-

For liquid 1 - x

metal alloys, the quantity of physical interest is the

⁺

F - cB

-

Z(E - E(E)) (26) liquid and alloy averaged local Green's function and

< 9,, >. To obtain this quantity within the SSSA,

we first note that within this scheme, it follows from (9) < Sii > ^{= 1}

that the local Green's function with an impurity at E - E(E)

-

Z(E

-

E(E))

^'

(27) the site i and averaged with the site i kept fixed is

given by The average density of states per atom N ( E ) and the

I

local density of states N j ( E ) are given by

C4-328 B. MOVAGHAR, D . E. MILLER AND K. H. BENNEMANN

and the derivation, and can be removed. The problem

1 of clustering is, unfortunately, more difficult to N j ( E ) =

---

Im ( E - E , - Z ( E - E ) ) ^{. (29)} handle. In the first place, one should have to drop

.n the approximation

It is not difficult to see that within this approximation the liquid alloy and solid alloy differ only with respect to the input density of states. This is a result which one intuitively expected.

4. Discussion.

-

In summary, we have introduced a method, with the help of the SSSA for tight-binding liquid metals, which acquires a very simple physical interpretation. The theory was then extended to liquid binary alloys. It would be interesting to extend the theory to several orbitals in order to describe more realistic situations. For example, the d-band in liquid transition metals. For the alloy, the assumption that the atomic distribution functions are independent of constituents is rather restrictive but not essential to

which is hard to do if one insists on self-consistency.

Notice, however, that if the problem of taking into accourt clustering in the pure liquid is solved, it is immediately generalizable to the alloy using the formalism presented above. We hope to apply the theory to the problem of electrical conduction in amorphous and liquid semiconductors and alloys, as well as to calculations of spin susceptibilities in liquid metal alloys.

Ackwnowledgments.

--

The project was supported by D F G under SFB 161.

References

[l] ANDERSON, P. W., Phys. Rev. 109 (1958) 1492-1505. [6] MATSUBARA, T. and TOYOZAWA, Y., Prog. Theor. Phys. 26 [2] BROUERS, F., CYROT, M. and CYROT-LACKMANN, F., Phys. (1961) 739-756.

Rev. B 7 (1973) 4370-4373. [7] ROTH, L., Phys. Rev. Lett. 28 (1972) 1570-1573.

[3] CYROT-LACKMANN, R., Adv. Phys. 16 (1967) 393-400. [8] ROTH, L., Phys. Rev. B 7 (1973) 4321-4337.

[9] SCHWARTZ, L. and EHRENREICH, H., Ann. Phys. 64 (1971) [4] ISHIDA, Q. and YONEZAWA, F., Progr. Theor. Phys. 40 100-148.

(1973) 731-753. [lo] SOVEN, P., Phys. Rev. 178 (1969) 1136-1144.

[5] LAX, M., Phys. Rev. 85 (1952) 621-629. [Ill WATSON, K. M., Phys. Rev. 105 (1957) 1388-1398.