Surface States and Leed
The following remarks, which introduce the session on localized states in solids, concern the simplest kind of localized state and its close relationship to the scattering states used in low-energy electron diffraction (LEED). These simple localized states are the so-called intrinsic surface states, which can exist on the surface of an ideal crystal and are localized in the vicinity of that surface; they are described by a wave function which has the usual Bloch periodicity parallel to the surface (with real wave numbers kx and ky), but attenuates exponentially both going into the crystal and into the vacuum outside the crystal, hence these states are localized in one direction - the z direction. The idea of such states goes back a long time, to Tamm1 (1932), and has been discussed by many workers since.2 In fact such states appeared in the calculations of George Watkins, on the energy level of a finite cluster, described at this Conference. In Watkins’ calculations they were a nuisance, since he was trying to simulate the behavior of a bulk crystal, and he tried to get rid of them, whereas now we are specifically interested in these states and concerned with getting an accurate description for the case of the semi-infinite crystal.
KeywordsPlane Wave Surface State Incident Plane Wave Bloch Wave Bloch Function
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References and Footnotes
- 2.For recent reviews see, for example, S. G. Davison and J. D. Levine “Surface States” in Solid State Physics, Vol. 25, p. 2. (H. Ehrenreich, F. Seitz, D. Turnbull, eds.) Academic Press, 1970, Peter Mark, W. R. Bottoms “Electronic States of Free Surfaces” in Progress in Solid State Chemistry, Vol. 6, p. 17 (H. Reiss, J. O. McCaldin, eds. ), Pergamon Press 1972.Google Scholar
- 3.For convenience, all plane waves and Bloch waves are normalized to unit flux at the surface plane.Google Scholar
- 4.The discrete set of diffracted plane waves or beams has k and k values differing from the incident plane wave by reciprocal lattice vectors of the surface net.Google Scholar
- 6.An equation equivalent to (1) is the matrix equation (10) of reference 5, on introducing the general amplitudes of αi. and γj. and bringing the T’s and R’s to the same side.Google Scholar
- 7.There is no difficulty in generalizing the model and Eq. (1) to the case of any finite number of layers differing from bulk between vacuum and bulk, including the transition layer as one leaves vacuum. Essentially a product of different transfer matrices is used to carry the wave function from vacuum to the beginning of bulk crystal, where the matching to a superposition of Bloch waves is made.Google Scholar
- 9.For LEED spectra calculations, we actually use an artificial boundary condition instead of exact matching of solutions at a surface plane. This boundary condition, called the no- reflection boundary condition, suppresses all reflections at the surface plane, hence is more realistic than matching wave functions at a discontinuity of potential since the true, more gradual, change of potential between vacuum and crystal will give small reflection. We find that the no-reflection boundary condition makes little difference in LEED spectra, but it could have a drastic effect on surface states.Google Scholar
- 10.The calculations for Figs. 2 and 3 required N ⩾ 21 beams to achieve an accuracy of better than 0.01 eV, hence the representation required 18 or more attenuating waves to describe the region of the d bands accurately; at N=17, splittings of order 0.06 eV appeared in the energies of states that should have been degenerate at the center of the Brillouin zone, indicating the accuracy of the representation at that N (note that the calculation of the energy bands is not variational).Google Scholar
- 12.The determinant of M becomes very large (~10150) when N=21 and many attenuating waves are included in the representation, which inclusion, as footnote 10 points out, is needed for adequate accuracy. This large magnitude complicates the determination of the zeros.Google Scholar