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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 Hcc, Hvv and Hss along the diagonal, that describe conduction band, heavy and light hole valence bands and split-off valence band respectively, interband matrices Hcv, Hcs, Hvs and their Hermitian-conjugative counterparts Hvc, Hsc, Hsv, that describe coupling between different bands. The Hcc matrix is
(6)
where Eg is band gap, m0 is free electron mass, ac is deformation potential of the conduction band, ε is a sum of diagonal elements of the strain tensor εij, A' describes contribution to the eight-band Hamiltonian from other band. The Hvv 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 av, b and d are valence band deformation potentials, γ1, γ2 and γ3 are Luttinger-like parameters and connected to Luttinger parameters γi 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 Ep, where
(13)
The Hvvkl and Hvvek 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 Hss matrix describing split-off band is
(17)
where Δ is the spin-orbit split-off energy. The Hcv and Hcs 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 Hvs 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 Hvskl and Hvsεk terms describe k-linear and εk-linear coupling between valence and split-off bands:
(26)
(27)

 

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