Condensed Matter II
Problem set #7
Spring 2023
1 Donor states in Si
The conduction band of Silicium exhibits six equivalent minima in the direction ∆ in
the first Brillouin zone. The electronic state vectors in such directions are designated by
the following symbols:
• |x⟩ for the minimum in direction [100]
• |y⟩ for the minimum in direction [010]
• |z⟩ for the minimum in direction [001]
• |¯x⟩ for the minimum in direction [¯100]
• |¯y⟩ for the minimum in direction [0¯10]
• |¯z⟩ for the minimum in direction [00¯1]
Figure 1: Regular tetrahedron with vertices Si atoms on the a, b, c, d sites. In the middle
of the tetrahedron is a donor atom.
A donor atom is located at the center of a regular tetrahedron as indicated in Fig 1.
As a reminder, the symmetry of the tetrahedraon is Td, of order 24, with elements:
1
• E (identity)
• 8 rotations C3 about the diagonals of a cube.
• 3 rotations C2 about axes x, y, z.
• 6 improper rotations S4 about axis x, y, z (rotations of angle π/2 followed by a
reflection in a plane perpendicular to the axis of rotation).
• 6 reflections σd in planes containing one edge and the center of the tetrahedron.
1.1 Questions
(i) Apply a symmetry operation of each class of the Td group to the six dimensional
vector








|x⟩
|y⟩
|z⟩
|¯x⟩
|¯y⟩
|¯z⟩








(ii) Use the previous result to establish the character table of the six-dimensional representation
R6 of the group Td.
(iii) Using the previously established (Cf Problem Set #3) character table of the Td
group, decompose R6 into its irreducible components.
(iv) Verify that the following vector states are bases of the corresponding irreducible
representations:
• A1 :
|x⟩ + |y⟩ + |z⟩ + |¯x⟩ + |¯y⟩ + |¯z⟩
√
6
• E :
|x⟩ − |y⟩ + |¯x⟩ − |¯y⟩
2
,
|x⟩ + |y⟩ − 2 |z⟩ + |¯x⟩ + |¯y⟩ − 2 |¯z⟩
√
12
• T2 :
|x⟩ − |¯x⟩
√
2
,
|y⟩ − |¯y⟩
√
2
,
|z⟩ − |¯z⟩
√
2
2