An Introduction to the Theory of Crystalline Elemental Solids and ...

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7. Surface Electronic Structure
Let’s now consider how the electronic structure at the surface differs from that in the
bulk. We have already seen for semiconductors that the loss of translational symmetry
along the surface normal has important consequences for the electronic structure, notably
through the formation of dangling bonds which, as we have seen, have profound implications
for the surface structures that form. Now we focus on the electronic structures
of metal surfaces and take this as an opportunity to discuss important concepts such
as the surface dipole, the work function, surface core level shifts, and surface states.
First we look at the surfaces of simple metals, where we find that again jellium provides
useful insight, we then move on to the transition metals and tight binding arguments.
7.1 Jellium Surfaces: Electron overspill; Surface Dipole; and Φ
Take the jellium model that we introduced earlier for the infinite crystal and terminate
the positive background (n + ) abruptly along a plane at z = 0, with the positive uniform
background now filling the half-space z ≤ 0 with the form:
n + (z) = n, z ≤ 0
= 0, z > 0 (35)
where n is the mean density of the positive charge in the ionic lattice. For a range of
densities (r s = 2 − 6) Lang and Kohn [148, 149] considered how the density (within the
LDA) would behave at such an interface. The now famous plot displayed in Fig. 19(a)
was obtained, which shows that: (i) the electron density spills into the vacuum; and
(ii) the density within the boundary oscillates in a Friedel manner with an amplitude
that decreases asymptotically with the square of the distance from the surface. The
characteristic wavelength is one half of the Fermi wavelength, k F , where k F = (3π 2 n) 1/3
[150]. The amount of overspill into the vacuum and the amplitude of the Friedel
oscillations depends on r s . The smaller r s is, the larger the overspill. The larger r s is, the
greater the amplitude of the oscillations.
The potentials associated with such density distributions are sketched in Fig. 19(b). In
particular the total effective (Kohn-Sham) potential, V eff , and the electrostatic potential,
V es are plotted. The difference between them is the exchange-correlation contribution to
the total effective potential which one can see comprises the largest part of V eff . This is
generally true for low and intermediate densities. Two aspects of the potentials are worth
commenting upon. First, as a result of the local-density approximation, V eff vanishes
exponentially into the vacuum. This asymptotic behavior is not correct. Instead, since
as an electron moves out of a metal surface its exchange-correlation hole stays behind,
flattening out on the surface, the effective potential should follow an image-like form:
V eff (z) ∼
1
4|z − z image |
. (36)
Here z image is the so-called image plane, which for many purposes is considered the
“effective surface plane” [116, 151]. For typical values of r s the image plane (from LDA
predictions) of clean jellium surfaces resides about 2-3 Å to the vacuum side of the