6.2 Electrons in a Periodic Potential
141
Fig. 6.5 Theoretical
calculation of the spin
splitting of a the three
lowest conduction bands
(CB1, CB2, and CB3) and
b the top three valence
bands (VB1, VB2, and
VB3) of GaAs. Adapted
from [453]
(a)
(b)
Spin splitting (meV)
120
100
80
60
40
20
0
120
100
80
60
40
20
0
W
K
,
U
X
GaAs
where the arrow refers to the direction of the electron spin. If the crystal is symmetric under inversion,
we have additionally
E n↑ (k) = E n↑ (−k) .
(6.24)
With both time reversal and inversion symmetry the band structure fulfills
E n↑ (k) = E n↓ (k) .
(6.25)
The inversion symmetry is particularly important for the spin-orbit interaction. In the absence of inversion symmetry, e.g. in (non-centrosymmetric) zincblende crystals (Fig. 3.16b) or in heterostructures
(Fig. 12.35b), a spin splitting, e.g. E n↑ (k) = E n↓ (k), is present. It can be thought of as provoked by
an effective magnetic field. Bulk inversion asymmetry (BIA) leads to the Dresselhaus spin splitting
[452, 453] that is shown in Fig.6.5 for GaAs (cmp. Fig. 6.10a). The spin splitting due to structural
inversion asymmetry (SIA) is described by the Bychkov-Rashba Hamiltonian [454, 455]. A review on
these topics can be found in [456].
6.2.5 Symmetry Considerations
In general the symmetry of the lattice is a symmetry of the system’s Hamiltonian and thus transfers into
the electronic (and other) properties of the semiconductor. The means to formulate this mathematically
is group theory and representation theory. At a given reciprocal lattice point, the wave function must
fulfill the given spatial symmetry. Additional symmetry due to spin and spin-orbit interaction enters
via the double-group scheme. This problem has been treated for the 32 point groups (cmp. Table B.2)
in [457] and in [458] particularly for the pc, fcc, bcc and hcp lattices. A detailed treatment for the
zincblende [459] and wurtzite [460] structures have been given. The most popular Hamiltonians are
treated in [461].
The symmetry at particular points in direct or reciprocal space is denoted by the irreducible representations of the symmetry (point) group, e.g. by the i -symbols used in Figs. 6.9, 6.10 or also Fig. 6.44.
As an example, base functions with the symmetry of the irreducible representations of tetraeder group
T d are listed in Table 6.1. With the knowledge of the wave functions at the points of high symmetry, it
is possible to deduce the general nature of the energy bands in the vicinity of such symmetry points.
141
Fig. 6.5 Theoretical
calculation of the spin
splitting of a the three
lowest conduction bands
(CB1, CB2, and CB3) and
b the top three valence
bands (VB1, VB2, and
VB3) of GaAs. Adapted
from [453]
(a)
(b)
Spin splitting (meV)
120
100
80
60
40
20
0
120
100
80
60
40
20
0
W
K
,
U
X
GaAs
where the arrow refers to the direction of the electron spin. If the crystal is symmetric under inversion,
we have additionally
E n↑ (k) = E n↑ (−k) .
(6.24)
With both time reversal and inversion symmetry the band structure fulfills
E n↑ (k) = E n↓ (k) .
(6.25)
The inversion symmetry is particularly important for the spin-orbit interaction. In the absence of inversion symmetry, e.g. in (non-centrosymmetric) zincblende crystals (Fig. 3.16b) or in heterostructures
(Fig. 12.35b), a spin splitting, e.g. E n↑ (k) = E n↓ (k), is present. It can be thought of as provoked by
an effective magnetic field. Bulk inversion asymmetry (BIA) leads to the Dresselhaus spin splitting
[452, 453] that is shown in Fig.6.5 for GaAs (cmp. Fig. 6.10a). The spin splitting due to structural
inversion asymmetry (SIA) is described by the Bychkov-Rashba Hamiltonian [454, 455]. A review on
these topics can be found in [456].
6.2.5 Symmetry Considerations
In general the symmetry of the lattice is a symmetry of the system’s Hamiltonian and thus transfers into
the electronic (and other) properties of the semiconductor. The means to formulate this mathematically
is group theory and representation theory. At a given reciprocal lattice point, the wave function must
fulfill the given spatial symmetry. Additional symmetry due to spin and spin-orbit interaction enters
via the double-group scheme. This problem has been treated for the 32 point groups (cmp. Table B.2)
in [457] and in [458] particularly for the pc, fcc, bcc and hcp lattices. A detailed treatment for the
zincblende [459] and wurtzite [460] structures have been given. The most popular Hamiltonians are
treated in [461].
The symmetry at particular points in direct or reciprocal space is denoted by the irreducible representations of the symmetry (point) group, e.g. by the i -symbols used in Figs. 6.9, 6.10 or also Fig. 6.44.
As an example, base functions with the symmetry of the irreducible representations of tetraeder group
T d are listed in Table 6.1. With the knowledge of the wave functions at the points of high symmetry, it
is possible to deduce the general nature of the energy bands in the vicinity of such symmetry points.