Environmental Engineering Reference
In-Depth Information
Fig. 1.3.
The hcp and dhcp crystal structures. In the latter, the
B and C sites have hexagonal symmetry, while the A sites have local
cubic symmetry, for an ideal
c/a
ratio.
To determine the eigenstates for the conduction electron gas, we
adopt the same procedure as that outlined for atoms in the previous
section. The external potential
v
ext
(
r
) in (1.2.2) is now the Coulomb
attraction of the nuclei situated on the crystal lattice, shielded by the
electrons of the ionic core, which areusuallytakentohavethesame
charge distribution as in the atoms. The potential consequently has the
translational symmetry of the periodic lattice, and so therefore does the
effective potential
v
eff
(
r
), which arises when we make the single-particle
approximation (1.2.5) and the local-density approximation (1.2.9). In
the atom, the eigenfunctions are determined by the boundary condition
that their amplitude must vanish for large values of
r
and, when (1.2.12)
is integrated numerically, they are automatically continuous and differ-
entiable. The translational symmetry of the solid is expressed in
Bloch's
theorem
:
ψ
(
r
)=
e
i
k
·
R
ψ
(
r
−
R
)
,
(1
.
3
.
1)
and this boundary condition gives rise to eigenfunctions
ψ
j
(
k
,ε,
r
)and
eigenvalues
ε
j
(
k
) which are functions of the wave-vector
k
in reciprocal
space. All the electron states may be characterized by values of
k
lying
within the
Brillouin zone
, illustrated for the hexagonal and fcc structures
in Fig. 1.4, and by the band index
j
defined such that
ε
j
(
k
)
ε
j
+1
(
k
).
The determination of the eigenstates of the Schrodinger equation,
subject to the Bloch condition (1.3.1) is the central problem of energy-
band theory. It may be solved in a variety of ways, but by far the most
≤
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