Environmental Engineering Reference
In-Depth Information
In this determinantal equation for the band structure ε j ( k ), the infor-
mation about the structure is separated from that on the potential. The
structure constants
l m ,lm are canonical in the sense that they de-
pend only on the crystal structure and not, for example, on the lattice
constant, as may be seen from the definition (1.3.8), and the potential
function P l ( ε ) is determined entirely by the potential within the atomic
sphere. We shall consider these two terms in turn.
If we include values of l up to 3, i.e. s, p, d ,and f partialwaves,the
structure constants form a square matrix with 16 rows and columns. The
terms with l = l fall into 4 blocks, and these submatrices may be diag-
onalized by a unitary transformation from the lm to an lj representation.
The (2 l +1) diagonal elements
S
S
k
lj of each sub-block are the unhybridized
canonical l bands . The canonical bands for the fcc structure are shown
in Fig. 1.6. If hybridization is neglected, which corresponds to setting
to zero the elements of
k
= l , eqn (1.3.11) takes the simple
S
l m ,lm with l
form
k
P l ( ε )=
S
lj .
(1 . 3 . 12)
Since P l ( ε ) is a monotonically increasing function of energy, as illustrated
in Fig. 1.7, the band energies ε lj ( k ) for the pure l bands are obtained by
a monotonic scaling of the corresponding canonical bands. P l ( ε )does
not, furthermore, depart greatly from a straight line in the energy region
over which a band is formed, so the canonical bands resemble the energy
bands in the solid quite closely, whence the name.
The potential function P l ( ε ) and the logarithmic-derivative function
D l ( ε ) are related to each other through the definition (1.3.10), and this
relationship is shown schematically in Fig. 1.7. It is convenient and
illuminating to parametrize the potential function, when considering the
formation of the energy bands from the canonical bands. The poles of
P l ( ε ), which occur when D l ( ε )= l , divide the energy into regions in
which lie the corresponding atomic energy-levels ε nl . The energies V nl
which separate these regions are defined by
D l ( V nl )= l
(1 . 3 . 13)
and, within a particular region, the energy C nl of the centre of the band
is fixed by the condition that P l ( C nl ) = 0, or
D l ( C nl )=
( l +1) .
(1 . 3 . 14)
The allowed k -values corresponding to this energy are just those for
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