Theory

Hamiltonian

The Hamiltonian is derived from the group theory [2]. The k•p Hamiltonian is represented by an 8x8 matrix whose rows and columns are ordered according to the sequence of the basis function set:

(4)

The matrix Hamiltonian with respect to the chosen basis can be written as

(5)

consisting of the intraband matrices H_cc, H_vv and H_ss along the diagonal, that describe conduction band, heavy and light hole valence bands and split-off valence band respectively, interband matrices H_cv, H_cs, H_vs and their Hermitian-conjugative counterparts H_vc, H_sc, H_sv, that describe coupling between different bands. The H_cc matrix is

(6)

where E_g is band gap, m₀ is free electron mass, a_c is deformation potential of the conduction band, ε is a sum of diagonal elements of the strain tensor ε_ij, A' describes contribution to the eight-band Hamiltonian from other band. The H_vv matrix is

(7)

The first and the second terms are also known as Luttinger-Kohn 4x4 Hamiltonian [3] with strain according to the Bir-Pikus description [4]:

(8)

where

(9)

and

(10)

where

(11)

Here a_v, b and d are valence band deformation potentials, γ₁, γ₂ and γ₃ are Luttinger-like parameters and connected to Luttinger parameters γ_i through [5]

(12)

where P is momentum matrix element of the interaction between conduction and valence bands. The Kane P parameter is usually used in form of E_p, where

(13)

The H^vv_kl and H^vv_ek terms describe k-linear and εk-linear coupling between light and heavy hole valence bands:

(14)

(15)

where

(16)

The constant C weights the k-linear terms in the valence bands. The H_ss matrix describing split-off band is

(17)

where Δ is the spin-orbit split-off energy. The H_cv and H_cs matrices are

(18)

where

(19)

and

(20)

Here B is k-square coupling between conduction and valence bands, C₂ is matrix element of simultaneous interaction of conduction and valence bands by the k•p and strain operator. The H_vs matrix consists of four terms:

(21)

The first and the second terms come from Luttinger-Kohn and Bir-Pikus Hamiltonian:

(22)

where

(23)

and

(24)

where

(25)

The H^vs_kl and H^vs_εk terms describe k-linear and εk-linear coupling between valence and split-off bands:

(26)

(27)