On the Band Structure OP Thin Films and Small Particles

ON THE BAND STRUCTURE OF THIN FILMS AND SMALL PARTICLES L. Valenta Faculty of Mathematics and Physics, Charles University, Prague, Czechoslovakia One of the convincing argument for the existence of the band structure in infinite perfect crystalline media is the following one (see e.g. [1]). The one-electron

Schr~dinger

equation in k-representation is equivalent to

an infinite system of linear homogeneous algebraic equations which goes over in itself if the wave vector is changed (in case of a one-dimensional model of an infinite periodic crystal with the crystal lattice constant a) from

K to k. + 251 j

a.

n

n = 0, : 1, :

Hence.

2

Ej(K)=Ej(K-tz~n

being the band index. It can be shown (as it will be demonstrated in detail elsewhere)

that

this type of reasoning may even be applied to systems with finite dimensions like films and small particles including the presence of impurities and the relaxation of the crystal lattice near the surface. In order to get an insight into the problem, let us sketch our approach on an one dimensional semiinfinite (monoatomic linear chain see the Fig. ).

v (X)

u (X)

~ ~ +

"

=

V(X)

j.

A

X

=

Xo

~.

Let us admit, that the effective potential

vex)

may be treated as con-

sisting of two contributions, namely a strictly periodic part U (X) which is cut off at the surface with

the phase defined by )( 0 (the cutting off being

described by the step function

e (X)

and a potential

V (x)

confined to the

s.urface region of the chain. Hence, corresponding to the situation shown in the Fig.,

V(X): V s(x-A)+8(x-A)U(X-A+X o )

281

Going over from the x-representation to the k-representation, both the Fourier or the Laplace transformations may succesfully be used. Using e.g. the Laplace transformation the representative member of the infinite set of equations takes the form

..,

h~

-2m(K-ik;)dk'2+ co

2: -00

+e- AK

-CD

2$

U e-i,c;:-n(Xo - Al A.. ( I<+ i :;2[ n ) : n

~

~

=£$(1()

Vs(k,,)

Here¢(I<) is the Laplace-transform of the wave functionlf(l() • denotes the Fourier-transform of the surface .,potential VS(x) the coefficients

and Un are

inU(X-A+l(c)=~ Unexp{-~n(x-A+Xo)}' -00

The wave vector k- ~ 1 + i X 2 is complex in general. Our system for A.. "1P '>P 2q! +1 ," '/ 't' (K) goes over in itself if (7\,.2. ~ 171,,2 + am, m - 0, revealing the band structure depending parametrically on ()(1 . A

detailed

discussion and some conclusions are being prepared for pUblication at present. Similar approach may be applied to finite systems.

~

Our conclusions depend sUbstantially on the presence of -c:o in [1J. It means, that conclusions on the existence of some band structure phenomena may be drawn if, at least, a small portion of the medium may be assumed to be periodic, References [1]

Blochincev D.J.: Osnovy kvantovoj mechaniky, GITTL, Moscow 1949

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