Chemistry Reference
In-Depth Information
(2)
Empty lattice model
: The wavefunctions for electrons in free space
(where
V
can be chosen to take the form of plane waves, with the
unnormalised wavefunction
(
r
)
≡
0
)
e
i
k
·
r
describing a state with energy
ψ
(
r
)
=
k
2
k
2
E
. If we divide free space into a periodic array of identical
boxes (giving what is referred to as the 'empty lattice'), then we can write
each of the free space wavefunctions as the product of a plane wave times
a constant (and therefore periodic) function:
=
/(
2
m
)
e
i
k
·
r
ψ
k
(
)
=
·
r
1
(3.6)
Hence Bloch's theorem describes wavefunctions which reduce, as one
would hope, to the correct form in the case where the periodic potential
V
(
r
)
→
0.
3.2.2 Electronic band structure
From Bloch's theorem, we can associate a wavevector
k
with each energy
state
E
n
k
of a periodic solid. It is often useful to plot a diagram of the
energies
E
n
k
as a function of the wavevector
k
, which is then referred
to as the
band structure
of the given solid. Figure 3.2(a) shows the band
structure for an electron in free space, which is described by the parabola
E
2
k
2
.
The free electron band structure is modified in several ways in a periodic
solid. In particular, the wavevector
k
associated with a given energy state
is no longer uniquely defined. This can be shown by considering a 1-D
periodic structure, with unit cell of length
L
. We write the wavefunction
for the
n
th state with wavenumber
k
as
=
/(
2
m
)
e
i
kx
u
nk
ψ
(
x
)
=
(
x
)
(3.7)
nk
where e
i
kx
is a plane wave of wavenumber
k
,and
u
nk
(
x
)
is a periodic
function, with
u
nk
. To show that the wavenumber
k
is
not uniquely defined, we can multiply eq. (3.7) by a plane wave with the
periodicity of the lattice, e
i2
π
mx
/
L
, and by its complex conjugate, e
−
i2
π
mx
/
L
,
where
m
is an integer. This gives
(
x
)
=
u
nk
(
x
+
L
)
e
i
kx
e
i2
π
mx
/
L
e
−
i2
π
mx
/
L
u
nk
(
ψ
nk
(
)
=
)
x
x
e
i
(
k
+
2
π
m
/
L
)
x
e
−
i2
π
mx
/
L
u
nk
=
(
x
)
(3.8)
where e
i
(
k
+
2
π
m
/
L
)
x
is a different plane wave to the original choice, and
e
−
i2
π
mx
/
L
u
nk
(
)
=
x
is still a periodic function, with period
L
. We refer to
G
m
2
L
as a
reciprocal lattice vector
, and note that the wavenumber
k
is then
equivalent to the wavenumber
k
π
m
/
+
G
m
in the given 1-D periodic structure.
