COMPLEX ENERGY BAND APPROACH TO THE THEORY OF ELECTRON SURFACE STATES

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COMPLEX ENERGY BAND APPROACH TO THE THEORY OF ELECTRON SURFACE STATES

I. Bartoš

To cite this version:

I. Bartoš. COMPLEX ENERGY BAND APPROACH TO THE THEORY OF ELEC- TRON SURFACE STATES. Journal de Physique Colloques, 1970, 31 (C1), pp.C1-135-C1-137.

�10.1051/jphyscol:1970122�. �jpa-00213753�

JOURNAL DE PHYSIQUE Colloque C 1, supplément au no 4, Tome 31, Avril 1970, page C 1

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COMPLEX ENERGY BAND APPROACH TO THE THEORY OF ELECTRON SUIQFACE STATES

Institute of Solid State Physics, Czech. Acad. Sci., Prague

Résumé. - Les méthodes théoriques utilisées avec succès pour l'étude de la structure de bande dans le volume d'un cristal sont généralisées pour tenir compte des états localisés. La généralisa- tion est basée sur le concept de structure de bande d'énergie complexe (Heine 1963), dans lequel on introduit des vecteurs d'onde complexes. Les solutions propres représentent alors des (( quasi- fonctions de Bloch ⁾⁾ exponentiellement décroissantes. La solution générale en termes de ces fonc- tions à l'intérieur du cristal est raccordée à la solution générale à l'extérieur. Cette méthode peut être utilisée pour des problèmes où l'on rencontre des potentiels périodiques dans une région limi- tée, e. g. états de surface, D. E. L., joints de grain.

Abstract. - Theoretical methods succesfully used for calculation of electron energy bands in the bulk are generalized to include localized states. Generalization is based on the concept of the complex energy band structure (Heine 1963), where in addition to real also complex wave vectors are taken into account. The corresponding eigenvectors represent exponentially damped ⁽⁽ quasi- Bloch functions ^)). In terms of these functions expressed general solution inside thecrystal is matched srnoothly to the general solution outside the crystal. This method may be applied to problems, where limited periodical potential in encountered, e. g. surface states, LEED, twin boundary.

Introduction. - 1 would like to sketch here briefly the basic ideas that enable extending theoretical methods currently used for the calculation of the electron energy bands in the bulk to problems, where the surface has to be taken into account. In this formulation a strong resemblance between the theory of surface states and of the LEED appears.

The crystal structure even near the clean surfaces is not so simple that the effective potential acting on the electron could be assumed strictly periodic up to the surface boundary. Nevertheless there are some reasons why to study model crystals with this kind of potential : The structural changes are limited to very narrow region at the surface and less localized electron there- fore may be only slightly aEected by these changes.

Further for the investigation of real structures it is useful to know what phenomena can be explained in terms of the periodic crystal in order to be able to attribute remaining ones to the deviations from the simple periodic potential.

The matching approach. - Let us study the semi- infinite crystal filling the half-space x > x,. We formulate the problem of electron motion in the following way : for the given energy E we find inside the crystal (for x > x,) such general solution of the Schrodinger equation that remains finite for x ⁴ m.

Then for the same energy we find general solution of the Schrodinger equation in the vacuum (for x < x,) which is finite for x + - m. And finally we try to determine free parameters in both general solutions in such a way that smooth matching of both solutions

is achieved on the boundary plane x = x,. Those energies that admit such smooth matching are the energies of surface states.

As the original translational periodicity is assumed preserved along the surface in Our model, the wave function of the surface state has the Bloch form along the surface only. k, and k, are still good quantum numbers and the matching therefore is to be per- formed a t given k, and k,. These components of the wave vector must be real in order that Y remains finite everywhere.

It is easy to find general solution with given reduced components k,, k, in the vacuum. The main problem is then to determine the general solution inside the crystal

!Ptki.

General formulation of this task is based on the concept of the complex energy band structure.

Complex energy bands.

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Inside the crystal an electron moves in the potential that is periodic in al1 three directions. parti2 solution of the Schrodinger equation can therefore be supposed to have the Bloch form

with u, periodic in al1 three directions. But as now the studied iegion is restricted in the x-direction to x > x,, the x-component of the wave vector k is no more obliged to be real as it is for an infinite crystal.

I t is necessary to generalize the concept of the usual band structure to the complex band structure E(k, ;

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970122

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136 1. BARTOS

k, k,) where k, is allowed to be complex. It can be achieved as follows : Schrodinger equation

H*k ⁼ Ek ^{* k} (2)

can be transformed into the form :

hkp h2 k2 H ( k ) U k = Ek U k H(k) = H

+

^- _m

+

^- _{2 m} (3) E, are eigenvalues of the operator H(k) in (3) and we simply admit complex k, in (3) and look for related eigenvalues.

The general theory of the complex band structure has been presented in 1963 by Heine [l].

At given k,, k, there exists for each energy an infinite number of solutions of the Schrodinger equa- tion

+,

⁽¹⁾ with different k,. Among these solutions there may be a small number of propagating solutions with real k, ; the remaining are exponentially damped solutions with complex k,. 1 would like to stress here that these solutions are not the wavefunctions of the surface state as they do not satisfy conditions of smooth matching a t x = x,.

In terms of these partial solutions, however, is expressed the general solution inside the crystal as a linear combination :

!qkz(r) =

C

^{a(kx) Yk(r)}

.

k,

(4) In the approximate formulation it is sufficient to take only the terms with small imaginary parts of k,.

The figure 1 shows the substantial part of the complex energy bands of Si and Ge in the [100]- direction for several points k,, k,. Full-zone kp- approach has been adopted here.

The matching conditions. - Let us briefly inves- tigate the second half of Our program : matching of general solutions on the boundary x = x,.

Surface states are looked for in the energy region below the vacuum level and therefore only exponen- tially damped solutions are allowed outside the crystal. In the energy gap of the infinite crystal al1 partial solutions have complex k,. Conditions of smooth matching on the surface lead to the homo- geneous system of M linear equations for M unknown coefficients a(k,) from the expansion (4). This system can be non-trivially solved for certain discrete values of energy Es only. When k,, k , is changed then through the corresponding two-dimensional Brillouin zone, Es gets broadened into the band-band of localized or surface states. When investigating the energies above the edge of the allowed band, however, one more partial solution contributing to the general solution inside appears and therefore non-trivial solution of the system exists for al1 such energies. Energy region of allowed bands of the infinite crystal is thus preserved in the semi-infinite crystal, only the wave functions are modified near the surface by the admixture of damped solutions.

FIG. 1. - Cornplex band structure of silicon in the (100)-direction for : a ) k , = O, k z = O ; b) k , = 2 n/a, k z =:O; c) ,k, = a/a, kt = z/a. Dashed lines represent the complex band structure ; the attached numbers denote imaginary part of k , Partial solu- tions of the Schodinger equation at given energy correspond to the point of intersection with the horizontal line at this energy.

COMPLEX ENERGY BAND APPROACH TO THE THEORY C 1 - 137

In the theory of low-energy electron diffraction energy region investigated lies above the vacuum level.

This fact introduces some changes when compared with the theory of surface states. First : it is necessary to respect the energy-dependence of the pseudopo- tential. Second : though now to each plane wave propagating outside the crystal towards the surface corresponds the plane wave propagating in the oppo- site direction, conditions of the LEED admit only one incident plane wave. Therefore in the forbidden energy region - in the energy gap of the infinite crystal - there appears only one additional contri- bution to the general solution outside the crystal as compared with the case of surface states. This ensures the existence of the unique solution for al1 energies.

Coefficients a(k,) determine then the distribution of the incident beam to diffracted beams. In the allowed

band the number of coefficients remains unchanged and we get the same consequences as before. The reason now is that only solutions propagating away from the surface are t o be taken into account.

In both, the theory of surface states and of LEED, the convergence of the method has been investigated. It has been found that just few real lines with smallest imaginary parts of k, are necessary to be taken in order to get the correct results. Moreover it has been shown that the results are not very sensitive t o the choice of the position of the cut x, [3, 41. 1 would like to conclude that this approach seems to be sui- table for extending theoretical methods, successfully used in the bulk, to problems where boundary effects are to be taken into account. Nowadays, the most difficult task consists in the computing of complex energy bands for concrete materials.

References [l] HEINE (V.), Proc. Phys. Soc., 1963, 81, 300.

[2] JONES (R. O.), PYOC. Phys. Soc., 1966,89,443.

[3] BARTOS (I.), Surf. Sci., 1969,15,94.

[4] CAPART (G.), Surf. Sci., 1969,13, 361.