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)\[\begin{split}\hat{\mathcal{H}}_{30\mathbf{k}\cdot\mathbf{p}}
=
\begin{bmatrix}
E_{1q} + \hat{\mathcal{H}}_{m_{0}}^{2\times 2} & \hat{\mathcal{H}}_{P_3} & 0 & 0 & 0 & \hat{\mathcal{H}}_{P^{\prime}_1} & & \hat{\mathcal{H}}_{P_2}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& E_{5d} + \hat{\mathcal{H}}_{m_{0}}^{6\times 6} & \hat{\mathcal{H}}_{R_1} & 0 & 0 & \hat{\mathcal{H}}_{Q_1} & \hat{\mathcal{H}}_{P_1} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & E_{3t} + \hat{\mathcal{H}}_{m_{0}}^{4\times 4} & 0 & 0 & 0 & 0 & \hat{\mathcal{H}}_{R_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & E_{1u} + \hat{\mathcal{H}}_{m_{0}}^{2\times 2} & 0 & \hat{\mathcal{H}}_{P_4} & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & E_{1w} + \hat{\mathcal{H}}_{m_{0}}^{2\times 2} & \hat{\mathcal{H}}_{P_5} & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & E_{5c} + \hat{\mathcal{H}}_{m_{0}}^{6\times 6} & \hat{\mathcal{H}}_{P^{\prime}_0} & \hat{\mathcal{H}}_{Q_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & E_{1c} + \hat{\mathcal{H}}_{m_{0}}^{2\times 2} & \hat{\mathcal{H}}_{P_0}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & & E_{5v} + \hat{\mathcal{H}}_{m_{0}}^{6\times 6}
\end{bmatrix}\end{split}\]
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:
\[\begin{split}\hat{\mathcal{H}}_{Q_i} = Q_i\begin{bmatrix}
0 & k_z & k_y & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k_z & 0 & k_x & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k_y & k_x & 0 & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & 0 & k_z & k_y
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & k_z & 0 & k_x
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & k_y & k_x & 0
\end{bmatrix}\end{split}\]
2.15.4. Strain perturbation term
The strain perturbation term can be written as
(2.15.4.1)\[\begin{split}\hat{\mathcal{H}}_{\varepsilon}
=
\begin{bmatrix}
\hat{\mathcal{W}}_{1q}^{2\times 2} & 0 & \hat{\mathcal{W}}_{1q3t}^{2\times 4} & 0 & 0 & \hat{\mathcal{W}}_{1q5c}^{2\times 6} & \hat{\mathcal{W}}_{1q5v}^{2\times 2} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& \hat{\mathcal{W}}_{5d}^{6\times 6} & 0 & \hat{\mathcal{W}}_{5d1u}^{6\times 2} & \hat{\mathcal{W}}_{5d1w}^{6\times 2} & 0 & 0 & \hat{\mathcal{W}}_{5d5v}^{6\times 6}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & \hat{\mathcal{W}}_{3t}^{4\times 4} & 0 & 0 & \hat{\mathcal{W}}_{3t5c}^{4\times 6} & \hat{\mathcal{W}}_{3t1c}^{4\times 2} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & \hat{\mathcal{W}}_{1u}^{2\times 2} & \hat{\mathcal{W}}_{1u1w}^{2\times 2} & 0 & 0 & \hat{\mathcal{W}}_{1u5v}^{2\times 6}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & \hat{\mathcal{W}}_{1w}^{2\times 2} & 0 & 0 & \hat{\mathcal{W}}_{1w5v}^{2\times 6}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & \hat{\mathcal{W}}_{5c}^{6\times 6} & \hat{\mathcal{W}}_{5c1c}^{6\times 2} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & \hat{\mathcal{W}}_{1c}^{2\times 2} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
& & & & & & & \hat{\mathcal{W}}_{5v}^{6\times 6}
\end{bmatrix}\end{split}\]
Note
The wave vector dependent strain terms (linear in \(\mathbf{k}\)) present in [RideauPRB2006] are not included in our implementation.
The \(\hat{\mathcal{W}}_{1i}^{2\times 2}\) terms have the form:
\[\begin{split}\hat{\mathcal{W}}_{1i}^{2\times 2} = a_{1i}\text{Tr}(\hat{\epsilon})\begin{bmatrix}
1 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 1
\end{bmatrix}\end{split}\]
where \(a_{1i}\) is the deformation potential of the corresponding band.
The \(\hat{\mathcal{W}}_{1i1j}^{2\times 2}\) terms have the same form as \(\hat{\mathcal{W}}_{1i}^{2\times 2}\) but with deformation potentials \(a_{1i1j}\).
The \(\hat{\mathcal{W}}_{5i}^{6\times 6}\) is a block diagonal matrix with two identical \(3\times 3\) blocks given by:
\[\begin{split}\hat{\mathcal{W}}_{5i}^{3\times 3} =
\begin{bmatrix}
l_{5i}\epsilon_{xx} + m_{5i}\left(\epsilon_{yy} + \epsilon_{zz}\right) & n_{5i}\epsilon_{xy} & n_{5i}\epsilon_{zx}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n_{5i}\epsilon_{xy} & l_{5i}\epsilon_{yy} + m_{5i}\left(\epsilon_{xx} + \epsilon_{zz}\right) & n_{5i}\epsilon_{yz}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n_{5i}\epsilon_{zx} & n_{5i}\epsilon_{yz} & l_{5i}\epsilon_{zz} + m_{5i}\left(\epsilon_{xx} + \epsilon_{yy}\right)
\end{bmatrix}\end{split}\]
where \(l_{5i}, m_{5i}, n_{5i}\) are the deformation potentials of the corresponding band.
The \(\hat{\mathcal{W}}_{5i5j}^{6\times 6}\) terms have the same form as \(\hat{\mathcal{W}}_{5i}^{6\times 6}\) but with deformation potentials \(l_{5i5j}, m_{5i5j}, n_{5i5j}\).
The term \(\hat{\mathcal{W}}_{3t}^{4\times 4}\) has the block diagonal form with two identical \(2\times 2\) blocks given by:
\[\begin{split}\hat{\mathcal{W}}_{3t}^{2\times 2} =
\begin{bmatrix}
A \varepsilon_{xx} + B (\varepsilon_{yy} + \varepsilon_{zz}) & E (\varepsilon_{yy} - i \varepsilon_{zz})
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E (\varepsilon_{yy} - i \varepsilon_{zz}) & C \varepsilon_{xx} + D (\varepsilon_{yy} + \varepsilon_{zz})
\end{bmatrix}\end{split}\]
where \(A, B, C, D, E\) are the coefficients, that can be derived from the deformation potentials in the following way:
\[\begin{split}\begin{align}
A &= 6(b_{3t} - d_{3t})
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
B &= 3(a_{3t} + b_{3t} - 2c_{3t})
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
C &= 2(a_{3t} - 4c_{3t} + b_{3t} + d_{3t})
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
D &= 5b_{3t} - 2c_{3t} - 4d_{3t} + a_{3t}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E &= \sqrt{3}(3c_{3t} - 2d_{3t} - a_{3t} + b_{3t})
\end{align}\end{split}\]
The \(\hat{\mathcal{W}}_{1i5j}^{2\times 6}\) terms have the form:
\[\begin{split}\hat{\mathcal{W}}_{1i5j}^{2\times 6} = f_{1i5j}\begin{bmatrix}
\varepsilon_{yz} & \varepsilon_{xz} & \varepsilon_{xy} & 0 & 0 & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 & 0 & \varepsilon_{yz} & \varepsilon_{xz} & \varepsilon_{xy}
\end{bmatrix}\end{split}\]
where \(f_{1i5j}\) is the deformation potential of the corresponding bands.
The \(\hat{\mathcal{W}}_{5i1j}^{6\times 2}\) terms are the transposed matrices of \(\hat{\mathcal{W}}_{1j5i}^{2\times 6}\).
The \(\hat{\mathcal{W}}_{3t5c}^{4\times 6}\) is a block diagonal matrix with two identical \(2\times 3\) blocks given by:
\[\begin{split}\hat{\mathcal{W}}_{3t5c}^{2\times 3} = h_{3t5c}\begin{bmatrix}
0 & \sqrt{3} \varepsilon_{xz} & -\sqrt{3}\varepsilon_{xy}
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2\varepsilon_{yz} & -\varepsilon_{xz} & -\varepsilon_{xy}
\end{bmatrix}\end{split}\]
where \(h_{3t5c}\) is the deformation potential of the corresponding bands.
The \(\hat{\mathcal{W}}_{3t1i}^{4\times 2}\) terms has the form:
\[\begin{split}\hat{\mathcal{W}}_{3t1i}^{4\times 2} = g_{3t1i}\begin{bmatrix}
\sqrt{3}(\varepsilon_{yy} - \varepsilon_{zz}) & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2\varepsilon_{xx} - \varepsilon_{yy} - \varepsilon_{zz} & 0
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & \sqrt{3}(\varepsilon_{yy} - \varepsilon_{zz})
\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 2\varepsilon_{xx} - \varepsilon_{yy} - \varepsilon_{zz}
\end{bmatrix}\end{split}\]
where \(g_{3t1i}\) is the deformation potential of the corresponding bands.
The \(\hat{\mathcal{W}}_{1i3t}^{2\times 4}\) terms are the transposed matrices of \(\hat{\mathcal{W}}_{3t1i}^{4\times 2}\).
This page is based on the nextnano GmbH collaboration in the scope of the SiPho-G Project aiming at development of ultrahigh-speed optical components for next-generation photonic integrated circuits, and it is funded by the European Union’s Horizon 2020 research and innovation program under the grant agreement No 101017194.
Last update: 2025-11-04
