136
6 Band Structure
(a)
(b)
(c)
Fig. 6.1 Zone schemes for a band structure: a extended, b reduced and c repetitive zone scheme
H =
−
2
2m
∇
2
+ U (r)
= E
(6.2)
for an electron. U will be periodic with the lattice, i.e. it will obey (6.1).
Bloch’s theorem says that the eigenstates of a one-particle Hamiltonian as in (6.2) can be written
as the product of plane waves and a lattice-periodic function, i.e.
nk (r) = A exp(i k r) u nk (r) .
(6.3)
The normalization constant A is often omitted. If u nk (r) is normalized, A = 1/
√
V , where V is the
integration volume. The wavefunction is indexed with a quantum number n and the wavevector k. The
key is that the function u nk (r), the so-called Bloch function, is periodic with the lattice, i.e.
u nk (r) = u nk (r + R)
(6.4)
for all vectors R of the direct lattice. The proof is simple in one dimension and more involved in three
dimensions with possibly degenerate wavefunctions, see [451].
If E nk is an energy eigenvalue, then E nk+G is also an eigenvalue for all vectors G of the reciprocal
lattice, i.e.
E n (k) = E n (k + G) .
(6.5)
Thus the energy values are periodic in reciprocal space. The proof is simple, since the wavefunction (for
k + G) exp(i(k + G)r)u n(k+G) (r) is for u n(k+G) (r) = exp(−iGr)u nk (r) obviously an eigenfunction
to k.
A band structure along one k-direction can be displayed in various zone schemes as depicted in
Fig. 6.1. The most frequently used scheme is the reduced zone scheme. In three dimensions, the band
structure is typically shown along particular paths in the Brillouin zone, as depicted, e.g., in Fig. 6.2c.
6.2.2 Free-Electron Dispersion
If the entire wavefunction (from (6.3)) obeys the Schrödinger equation (6.2), the Bloch function u nk
fulfills the equation
6 Band Structure
(a)
(b)
(c)
Fig. 6.1 Zone schemes for a band structure: a extended, b reduced and c repetitive zone scheme
H =
−
2
2m
∇
2
+ U (r)
= E
(6.2)
for an electron. U will be periodic with the lattice, i.e. it will obey (6.1).
Bloch’s theorem says that the eigenstates of a one-particle Hamiltonian as in (6.2) can be written
as the product of plane waves and a lattice-periodic function, i.e.
nk (r) = A exp(i k r) u nk (r) .
(6.3)
The normalization constant A is often omitted. If u nk (r) is normalized, A = 1/
√
V , where V is the
integration volume. The wavefunction is indexed with a quantum number n and the wavevector k. The
key is that the function u nk (r), the so-called Bloch function, is periodic with the lattice, i.e.
u nk (r) = u nk (r + R)
(6.4)
for all vectors R of the direct lattice. The proof is simple in one dimension and more involved in three
dimensions with possibly degenerate wavefunctions, see [451].
If E nk is an energy eigenvalue, then E nk+G is also an eigenvalue for all vectors G of the reciprocal
lattice, i.e.
E n (k) = E n (k + G) .
(6.5)
Thus the energy values are periodic in reciprocal space. The proof is simple, since the wavefunction (for
k + G) exp(i(k + G)r)u n(k+G) (r) is for u n(k+G) (r) = exp(−iGr)u nk (r) obviously an eigenfunction
to k.
A band structure along one k-direction can be displayed in various zone schemes as depicted in
Fig. 6.1. The most frequently used scheme is the reduced zone scheme. In three dimensions, the band
structure is typically shown along particular paths in the Brillouin zone, as depicted, e.g., in Fig. 6.2c.
6.2.2 Free-Electron Dispersion
If the entire wavefunction (from (6.3)) obeys the Schrödinger equation (6.2), the Bloch function u nk
fulfills the equation