THE BAND STRUCTURE PROBLEM IN TERMS OF SCATTERED WAVES

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THE BAND STRUCTURE PROBLEM IN TERMS OF SCATTERED WAVES

D. Liberman

To cite this version:

D. Liberman. THE BAND STRUCTURE PROBLEM IN TERMS OF SCATTERED WAVES.

Journal de Physique Colloques, 1972, 33 (C3), pp.C3-239-C3-240. �10.1051/jphyscol:1972335�. �jpa-

00215069�

JOURNAL DE PHYSIQUE

Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-239

THE BAND STRUCTURE PROBLEM IN TERMS OF SCATTERED WAVES (*)

D. A. LIBERMAN

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

R6sumB. - La methode de Korringa pour le traitement de problkmes de structure de bandes par l'intermkdiaire des ondes diffuskes par les atomes individuels peut Stre formulee en donnant I'Cqua- tion de ces ondes

oh

est le potentiel associe avec l'atome situe

et

est une onde kmise

ce point du reseau.

n'est pas restreint d'avoir la

muffin-tin

forme, mais obtenir des solutions devient difficile s'il n'a pas cette forme. L'equation ci-dessus peut Stre modifiee facilement pour decrire la structure Clectronique des liquides et des solides desordonnks. Le rksultat est une thkorie approchee Bquivalente

celles proposees par Phariseau et Ziman pour les metaux liquides

par Beeby pour les alliages.

Abstract. - Korringa's method for treating the band structure problem in terms of waves scattered by individual atoms may be formalized by writing an equation for these waves

where

is the potential associated with the atom at

and

is an outgoing wave from that lattice site.

is not restricted to the

muffin-tin

form, but obtaining solutions is difficult unless it is. The equation above is readily modified so as to describe the electronic structure of liquids and disordered solids. The result is an approximate theory which is equivalent to those proposed by Phariseau and Ziman for liquid metals and by Beeby for alloys.

I want to show how a very simple manipulation of Using these two forms for V and $ the Schrodinger the Schrodinger equation for an electron in a crystal equation can be written as follows

lattice can lead to a somewhat different view of the

band theory problem. - - ti2 v2 ei"" cp(r

a ) + z ^{v(r - a ) $(r)}

Suppose our potential function is represented as a

sum -of associated with the individual atoms

V(r)

z ^v(r

^{a )} (1)

is now determined by removing the summation

a

symbols from this equation. Each of the resulting set where the functions

- a ) have a finite range but of equations is equivalent under a translation to each may overlap. Such a representation is of course no other. If

satisfies

limitation a t all on the lattice potential V(r).

Suppose next that the wave function is written in - -

~ " ( r ) + ^{v(r) $(r)}

E q ( r )

2 m (4)

the form familiar in the LCAO approximation,

(2) or, as I prefer to write it,

a

ti2

but let q be a function to be determined so as to give (- ^{2;;; v2 + v(r) - E ) p(r)}

a n exact solution of the Schrodinger equation. This

also can always be done and is no limitation on $.

- v(r) z eik.' q ( r - a )

a # O

(*)

Work performed under the auspices of the

Atomic the Schrodinger equation for $ is satisfied.

Energy Commission. As can be seen the Schrodinger equation has been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972335

C3-240 D. A. LIBERMAN

converted into another equivalent, one which is of

interest for three reasons

1) as a starting point for the development of compu- tational schemes

2) because

admits of interpretation either as an orbital function or a scattered wave

and

3) because we can go from the equation for q, to related approximate ones for disordered systems.

In regard to the first point I only want to mention that for muffin-tin potentials we are led naturally to Korringa's method for solving the Schrodinger equa- tion in a periodic potential [I].

To understand more about the meaning of

consider the equation for it when

is large. Then

is zero and we are left with

If E is negative q, must decay exponentially with the

radius.

is then an atomic-like orbital. If E is positive q, describes a scattered wave. It should be noted that q, does not include the incident wave

that is supplied by the right side of equation (5) which may be regarded as a source term.

Finally suppose we are interested in a disordered system such as a liquid metal. Then we may very easily obtain an approximate theory if we replace the sum over the lattice sites by the obvious integral

where N/V is the number of atoms per unit volume and g(R) is the two particle distribution function. This result is equivalent to the Phariseau-Ziman theory of liquid metals [2]. In a similar way disordered alloys may be treated [3].

References

[I]

KORRINGA

A

PHARISEAU (P.) and ZIMAN

1487.