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

44
2
Shockley
Energy (eV)
0
-2
-4
-6
-8
Tamm
-10
FIG. 23: Surface projected bulk band structure for a 12 layer Cu(111) slab as computed from a
plane-wave pseudopotential DFT calculation within the LDA (computed by the authors).
solutions of the Schrödinger equation which would otherwise be imaginary in the bulk
to become real in the semi-infinite crystal. Shockley states appear close to the low
energy (i.e., high binding energy) border of a gap and reveal a free-electron-like behavior
parallel to the surface. A particularly well characterized Shockley state is the one that
occurs on Cu(111). According to ARPES this is a free electron-like state which at
the Γ point of the surface Brillouin zone resides ∼ 0.4 eV below E F [173, 174]. DFT
calculations agree with this finding, as can be seen from the computed band structure
of Cu(111) shown in Fig. 23. The Shockley state can be seen at the Γ point just below E F .
(b) Tamm states: These are characteristic of more tightly bound systems such as the
transition metals in which the valence electrons are d states. Tamm states are split-off
states due to the reduced atomic coordination of the surface and the weaker potential
that arises (which was discussed in section 7.2). They reside also at the low energy
border of gaps and often exhibit negative effective masses. Noble metal surfaces such as
Cu(111) also exhibit Tamm states (in addition to Shockley states) and indeed in Fig. 23
an occupied Tamm state 1.8 eV below E F can be seen at the M point split off from
the top of the Cu d band. Again the computed value of this surface state agrees with
experiment [175, 176].
Finally we mention yet one more type of state that can occur at metal surfaces, these
are so-called image potential states which are localized mostly in the vacuum region of
the metal surface (Fig. 22(d)). Recall from section 7.1 how the image potential rises to
the vacuum level V ∞ as one moves out from the surface into the vacuum (Fig. 19(b)).
This potential can actually support unoccupied bound states, i.e., image potential states.
These states lie above the Fermi energy, in a Rydberg-like series of states converging
towards the vacuum level. Should such image potential states become populated with
electrons these states can have relatively long lifetimes. The reader interested in learning
more about image potential states, in particular, and surface states, in general, should
consult the book by Davison and M. Stȩślicka [161] or some of the excellent reviews on
these topics [164, 177, 178].