142
6 Band Structure
Table 6.1 Representations of the tetraeder group (zincblende structure) in molecular, BSW [462] and Koster [457]
notation and (examples of) corresponding base functions (c.p.: cyclic permutations)
molecular
BSW
Koster
base functions
A 1
1
1
x y z, x 2 + y 2 + z 2
A 2
2
2
x 4 (y 2 − z 2 ) + c.p.
E
12
3
2 z 2 − (x 2 + y 2 ), (x 2 − y 2 )
T 2
15
4
x, y, z, xy, xz, yz
T 1
25
5
z (x 2 − y 2 ) and c.p.
6.2.6 Topological Considerations
Starting with research on the quantum Hall effect and based on previous mathematical theorems, it has
become clear that the band structure of ‘insulators’ has topological properties which in turn lead to a
elegant classification of materials (and many effects/phases) [370, 463, 464]. In this context, the term
‘insulator’ means a material with gap between filled and empty states, i.e. semiconductors are exactly
like this if the temperature is not too high (related to the gap divided by k B ). We recall the discussion
of the diatomic linear chain in Sect. 5.2.3 where the bands turned out to have different topological
properties depending on the ratio of sporing constants.
Topology is a branch of mathematics where objects that are related to each other by a smooth
deformation are classified as the same. For example, a sphere and an ellipsoid are topologically the
same. Also, a doughnut and a cup are the same since they have one hole. A quantity that is independent
of such smooth transformations is termed ‘topological invariant’. Such a number is the genus g of a
surface that counts the number of holes. According to the Gauss-Bonnet theorem, the integral of the
Gaussian curvature K over a closed surface S is given by
S
K d A = 2π (2 − 2 g) .
(6.26)
The Gaussian curvature K = κ 1 κ 2 of a (differentiable) surface in 3D is the product of the principal
curvatures κ 1 and κ 2 (maximum and minimum curvature of the curves from all normal planes that
contain the normal vector intersecting with the surface). For a sphere of radius r , the Gaussian curvature
is 1/r
2 everywhere and the integral in (6.26) is 4 π , making g = 0. For a topologically different example
we look at a torus (all points that have the fixed distance r from a circle of radius R, r < R). It is
parametrized by
r = R
⎛
⎝
cos φ
sin φ
0
⎞
⎠ + r
⎛
⎝
cos φ cos θ
sin φ cos θ
sin θ
⎞
⎠ ,
(6.27)
with both the angles φ and θ running between 0 and 2 π . The principal curvature κ 1 along the θ -
direction is 1/r (for all φ). The other principal curvature κ 2 in azimuthal (φ) direction changes sign
with θ (positive outside, negative inside) and is also independent of φ. Its integral over the outer and
inner part cancel exactly, thus the integral of κ 1 κ 2 over the entire torus is zero and therefore g = 1.
Next, we connect the periodicity of the Brillouin zone in two dimensions with variables on a torus
in 3D as shown in Fig. 6.6 (cmp. Fig. 5.3 for the 1D case). This concept can be generalized for a 3D
band structure and a torus in four dimensions.
If a constant function f = n a b/(2 π) is integrated over the Brillouin zone (X is at ±π/a, Y is at
±π/b), the integral is
6 Band Structure
Table 6.1 Representations of the tetraeder group (zincblende structure) in molecular, BSW [462] and Koster [457]
notation and (examples of) corresponding base functions (c.p.: cyclic permutations)
molecular
BSW
Koster
base functions
A 1
1
1
x y z, x 2 + y 2 + z 2
A 2
2
2
x 4 (y 2 − z 2 ) + c.p.
E
12
3
2 z 2 − (x 2 + y 2 ), (x 2 − y 2 )
T 2
15
4
x, y, z, xy, xz, yz
T 1
25
5
z (x 2 − y 2 ) and c.p.
6.2.6 Topological Considerations
Starting with research on the quantum Hall effect and based on previous mathematical theorems, it has
become clear that the band structure of ‘insulators’ has topological properties which in turn lead to a
elegant classification of materials (and many effects/phases) [370, 463, 464]. In this context, the term
‘insulator’ means a material with gap between filled and empty states, i.e. semiconductors are exactly
like this if the temperature is not too high (related to the gap divided by k B ). We recall the discussion
of the diatomic linear chain in Sect. 5.2.3 where the bands turned out to have different topological
properties depending on the ratio of sporing constants.
Topology is a branch of mathematics where objects that are related to each other by a smooth
deformation are classified as the same. For example, a sphere and an ellipsoid are topologically the
same. Also, a doughnut and a cup are the same since they have one hole. A quantity that is independent
of such smooth transformations is termed ‘topological invariant’. Such a number is the genus g of a
surface that counts the number of holes. According to the Gauss-Bonnet theorem, the integral of the
Gaussian curvature K over a closed surface S is given by
S
K d A = 2π (2 − 2 g) .
(6.26)
The Gaussian curvature K = κ 1 κ 2 of a (differentiable) surface in 3D is the product of the principal
curvatures κ 1 and κ 2 (maximum and minimum curvature of the curves from all normal planes that
contain the normal vector intersecting with the surface). For a sphere of radius r , the Gaussian curvature
is 1/r
2 everywhere and the integral in (6.26) is 4 π , making g = 0. For a topologically different example
we look at a torus (all points that have the fixed distance r from a circle of radius R, r < R). It is
parametrized by
r = R
⎛
⎝
cos φ
sin φ
0
⎞
⎠ + r
⎛
⎝
cos φ cos θ
sin φ cos θ
sin θ
⎞
⎠ ,
(6.27)
with both the angles φ and θ running between 0 and 2 π . The principal curvature κ 1 along the θ -
direction is 1/r (for all φ). The other principal curvature κ 2 in azimuthal (φ) direction changes sign
with θ (positive outside, negative inside) and is also independent of φ. Its integral over the outer and
inner part cancel exactly, thus the integral of κ 1 κ 2 over the entire torus is zero and therefore g = 1.
Next, we connect the periodicity of the Brillouin zone in two dimensions with variables on a torus
in 3D as shown in Fig. 6.6 (cmp. Fig. 5.3 for the 1D case). This concept can be generalized for a 3D
band structure and a torus in four dimensions.
If a constant function f = n a b/(2 π) is integrated over the Brillouin zone (X is at ±π/a, Y is at
±π/b), the integral is