Symmetry and Symmetry-Breaking in Semiconductors by Bernd Hönerlage & Ivan Pelant

Author:Bernd Hönerlage & Ivan Pelant
Language: eng
Format: epub
ISBN: 9783319942353
Publisher: Springer International Publishing

(5.1.1)

Considering the total-angular momentum and its projection component onto the z-axis , the functions Eq. (5.1.1) are also eigenfunctions of the total-angular momentum operator of the valence-band states with = 1/2, respectively. By analogy to the conduction band discussed above, because of its twofold degeneracy, the total angular-momentum operator is now used for the pseudo-spin description of the valence-band subspace. Using Eq. (4.​1.​3) or Eq. (4.​1.​4) we can then introduce hole states and establish their correspondence to the valence-band states of the split-off band using Eq. (4.​1.​5).

As discussed in Chap. 4, excitons are formed in the direct product space of electron and hole states, i.e. we build the Kronecker product . The exciton ground state is thus only fourfold degenerate. We now construct in this subspace an effective exciton Hamiltonian, which has the same symmetry properties as the full Hamilton operator, i.e. which remains invariant under all symmetry operations of the point group of the crystal. It transforms as a scalar (which has symmetry) and is an even function under time reversal () . Since the conduction band is only twofold degenerate, we choose, as discussed in Chap. 2, the Pauli-spin matrices, which are the matrices , , and together with the unit matrix as a basis to span the matrix describing the interacting conduction-band states. These matrices are given in Eqs. (2.​2.​3) and (2.​2.​5) by

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