2.15.1. The Model
Our implementation of bulk 30-band \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian for zincblende is a simplified version of the model described in [RideauPRB2006] obtained from the one-particle Hamiltonian
\[\begin{align}
\hat{H} &=
\frac{\hat{\mathbf{p}}^2}{2m}
+ V_0\left(\mathbf{r}\right)
+ \frac{\hbar}{4m^2c^2}\left[\hat{\mathbf{\sigma}} \times \nabla V_0\left(\mathbf{r}\right) \right]\circ\hat{\mathbf{p}}
\end{align}\]
Our model is expressed in a basis of 30 Bloch functions at the \(\Gamma\) point of the Brillouin zone.
The Hamiltonian can be represented as:
\[\hat{H} = \hat{H}(\mathbf{k}) + \hat{H}_{\text{SO}} + \hat{H}_{\epsilon}(\hat{\epsilon})\]
where \(\hat{H}(\mathbf{k})\) is the kinetic part of the Hamiltonian, \(\hat{H}_{\text{SO}}\) is the spin-orbit interaction term and \(\hat{H}_{\epsilon}(\hat{\epsilon})\) is the strain perturbation Hamiltonian.
Note
In [RideauPRB2006], the strain Hamiltonian also includes terms linear in \(\mathbf{k}\). These terms are neglected in our implementation.
2.15.2. Hamiltonian without spin-orbit splitting
The Hamiltonian without spin-orbit splitting is a \(30\times30\) matrix, which can be represented in a block form (only the upper triangular is shown, the lower triangular term are complex conjugate of upper triangular terms):
(2.15.2.1)\[\begin{split}\hat{\mathcal{H}}_{30\mathbf{k}\cdot\mathbf{p}}
=
\begin{bmatrix}
E_{1q} + \hat{\mathcal{H}}_{m_{0}} & \hat{\mathcal{H}}_{P_3} & 0 & 0 & 0 & \hat{\mathcal{H}}_{P^{\prime}_1} & & \hat{\mathcal{H}}_{P_2}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& E_{5d} + \hat{\mathcal{H}}_{m_{0}} & \hat{\mathcal{H}}_{R_1} & 0 & 0 & \hat{\mathcal{H}}_{Q_1} & \hat{\mathcal{H}}_{P_1} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & E_{3t} + \hat{\mathcal{H}}_{m_{0}} & 0 & 0 & 0 & 0 & \hat{\mathcal{H}}_{R_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & E_{1u} + \hat{\mathcal{H}}_{m_{0}} & 0 & \hat{\mathcal{H}}_{P_4} & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & E_{1w} + \hat{\mathcal{H}}_{m_{0}} & \hat{\mathcal{H}}_{P_5} & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & E_{5c} + \hat{\mathcal{H}}_{m_{0}} & \hat{\mathcal{H}}_{P^{\prime}_0} & \hat{\mathcal{H}}_{Q_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & E_{1c} + \hat{\mathcal{H}}_{m_{0}} & \hat{\mathcal{H}}_{P_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & & E_{5v} + \hat{\mathcal{H}}_{m_{0}}
\end{bmatrix}\end{split}\]
where \(E_i\) is the energies at \(\Gamma\) point of the corresponding bands, and \(\hat{\mathcal{H}}_{m_{0}}\) is the free electron kinetic energy term:
\[\hat{\mathcal{H}}_{m_{0}} = \frac{\hbar^2}{2m_0}\left(k_x^2 + k_y^2 + k_z^2\right)\]
In contrast with the 8-band model, there are no remote band contributions included i.e. no Luttinger-like parameters of effective mass parameters.
All the off-diagonal terms are linear in \(\mathbf{k}\).
The \(\hat{\mathcal{H}}_{P_i}\) terms have the form:
\[\begin{split}\hat{\mathcal{H}}_{P_i} = P_i\begin{bmatrix}
k_x & k_y & k_z & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & k_x & k_y & k_z
\end{bmatrix}\end{split}\]
where \(P_i\) are the interband momentum matrix elements between the corresponding bands.
The matrix is transposed in upper triangular for the \(\hat{\mathcal{H}}_{P_1}, \hat{\mathcal{H}}_{P^{\prime}_0}\) terms.
The \(\hat{\mathcal{H}}_{R_i}\) terms have the form:
\[\begin{split}\hat{\mathcal{H}}_{R_i} = R_i\begin{bmatrix}
0 & \sqrt{3}k_y & -\sqrt{3}k_z & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2k_x & -k_y & -k_z & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & 0 & \sqrt{3}k_y & -\sqrt{3}k_z
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & 2k_x & -k_y & -k_z
\end{bmatrix}\end{split}\]
where \(R_i\) are the interband momentum matrix elements between the corresponding bands.
The matrix is transposed in upper triangular for the \(\hat{\mathcal{H}}_{R_1}\) term.
The \(\hat{\mathcal{H}}_{Q_i}\) terms have the form:
\[\begin{split}\hat{\mathcal{H}}_{Q_i} = Q_i\begin{bmatrix}
0 & k_z & k_y & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k_z & 0 & k_x & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k_y & k_x & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & 0 & k_z & k_y
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & k_z & 0 & k_x
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & k_y & k_x & 0
\end{bmatrix}\end{split}\]
2.15.3. Spin-orbit splitting term
The spin-orbit splitting part of the Hamiltonian is given below (in the same block form as in equation (2.15.2.1)):
\[\begin{split}\hat{\mathcal{H}}_{SO}
=
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& \hat{\mathcal{H}}^{\Delta}_{d} & 0 & 0 & 0 & 0 & 0 & \hat{\mathcal{H}}^{\Delta}_{vd}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & 0 & 0 & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & 0 & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & \hat{\mathcal{H}}^{\Delta}_{c} & 0 & \hat{\mathcal{H}}^{\Delta}_{vc}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & & \hat{\mathcal{H}}^{\Delta}_{v}
\end{bmatrix}\end{split}\]
where the non-zero blocks are given by:
\[\begin{split}\hat{\mathcal{H}}^{\Delta} = \frac{\Delta}{3}\begin{bmatrix}
-1 & -i & 0 & 0 & 0 & 1
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
i & -1 & 0 & 0 & 0 & -i
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & -1 & -1 & i & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & -1 & -1 & i & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & -i & -i & -1 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1 & i & 0 & 0 & 0 & -1
\end{bmatrix}\end{split}\]
where \(\Delta\) is the spin-orbit splitting energy of the corresponding band.