2.2.1. Hamiltonian: 8-band model for zincblende

The Model

Hint

This model can be triggered for any point of the simulation using classical{ bulk_dispersion{KP8{}}}. See the Bulk Electronic Band Structure section for reference on syntax.

Our implementation of the 8-band \(\mathbf{k} \cdot \mathbf{p}\) model for bulk crystals is a simplified version of the matrix Hamiltonian described in a PhD thesis [AndlauerPhD2009] obtained from the one-particle Hamiltonian

\[\begin{align} \hat{H} &= \frac{\hat{\mathbf{p}}^2}{2m} + V_0\left(\mathbf{r}\right) + \frac{\hbar}{4m^2c^2}\left[\hat{\mathbf{\sigma}} \times \nabla V_0\left(\mathbf{r}\right) \right]\circ\hat{\mathbf{p}} \end{align}\]

The description below contains also definitions and relations that can be found in [BirnerPhD2011] and [BahderPRB1990].

Warning

The Hamiltonian below does not contain terms related to the presence of the magnetic field. Therefore, proper operator ordering is neglected to keep formulas as simple as possible. Also, parameters \(N^+\), \(N^-\), \(\kappa\), and \(g\) are not included here. Comprehensive documentation will be published elsewhere.

Our model is expressed in a basis of class \(\mathcal{A}\) functions:

\[\{\ket{s\uparrow},\;\ket{s\downarrow},\; \ket{x_1\uparrow},\;\ket{x_2\uparrow},\;\ket{x_3\uparrow},\; \ket{x_1\downarrow},\;\ket{x_2\downarrow},\;\ket{x_3\downarrow}\;\}\]

The Hamiltonian can be concisely written in a block form as follows.

\[\begin{split}\hat{\mathcal{H}}_{\mathbf{k}\cdot\mathbf{p}} = \begin{bmatrix} \hat{\mathcal{H}}_{\mathrm{cc}}\left(\mathbf{k},\,\hat{\epsilon}\right) & 0 & \hat{\mathcal{H}}_{\mathrm{cv}}\left(\mathbf{k}\right) & 0 \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0 & \hat{\mathcal{H}}_{\mathrm{cc}}\left(\mathbf{k},\,\hat{\epsilon}\right) & 0 & \hat{\mathcal{H}}_{\mathrm{cv}}\left(\mathbf{k}\right) \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hat{\mathcal{H}}_{\mathrm{vc}}\left(\mathbf{k}\right) & 0 & \hat{\mathcal{H}}_{\mathrm{vv}}\left(\mathbf{k}\right) + \hat{\mathcal{H}}_{\mathrm{vv}}\left(\hat{\epsilon}\right) + \hat{\mathcal{H}}_{\mathrm{so}\uparrow\uparrow} & \hat{\mathcal{H}}_{\mathrm{so}\uparrow\downarrow} \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0 & \hat{\mathcal{H}}_{\mathrm{vc}}\left(\mathbf{k}\right) & \hat{\mathcal{H}}_{\mathrm{so}\downarrow\uparrow} & \hat{\mathcal{H}}_{\mathrm{vv}}\left(\mathbf{k}\right) + \hat{\mathcal{H}}_{\mathrm{vv}}\left(\hat{\epsilon}\right) + \hat{\mathcal{H}}_{\mathrm{so}\downarrow\downarrow} \\ \end{bmatrix}\end{split}\]

where \(\mathbf{k}\) is a wave vector and \(\hat{\epsilon}\) is a strain tensor.

Diagonal elements for the conduction band are defined as

\[\hat{\mathcal{H}}_{\mathrm{cc}}\left(\mathbf{k},\,\hat{\epsilon}\right) = E_{\mathrm{c}} + A_c k^2 + a_{\mathrm{c}} \mathrm{Tr}\{\hat{\epsilon}\},\]

where \(k\) is length of the wave vector, \(E_\mathrm{c}\) is conduction-band edge, \(a_c\) is absolute hydrostatic deformation potential for the conduction band, \(\mathrm{Tr}\{\hat{\epsilon}\}\) is trace of the strain tensor, \(A_\mathrm{c}\) is defined as

\[A_\mathrm{c} = A' + \frac{\hbar^2}{2m_0}.\]

\(A'\) is one of Kane parameters. It contains interactions between the conduction band and the remote bands \(\mathcal{B}\) with \(\Gamma_5\) symmetry

\[A' = \frac{\hbar^2}{m_0^2} \sum_{nj}^{\mathcal{B} } \frac{\left|\bra{s} \hat{p}_1 \ket{n \Gamma_5 j}\right|^2}{E_\mathrm{c}-E_{n,\Gamma_5}}.\]

Blocks introducing interaction between conduction and valence bands are given by

\[\hat{\mathcal{H}}_{\mathrm{cv}}\left(\mathbf{k}\right) = \begin{bmatrix} \imath P_0 k_1 + B k_2k_3 & \imath P_0 k_2 + B k_1k_3 & \imath P_0 k_3 + B k_1k_2 \end{bmatrix}\]

and

\[\begin{split}\hat{\mathcal{H}}_{\mathrm{vc}}\left(\mathbf{k}\right) = \begin{bmatrix} -\imath P_0 k_1 + B k_2k_3 \\ -\imath P_0 k_2 + B k_1k_3 \\ -\imath P_0 k_3 + B k_1k_2 \end{bmatrix},\end{split}\]

where \(k_1,\,k_2,\,k_3\) are three components of the wave vector of interest, \(P_0\) is a Kane parameter describing interactions between conduction band and valence bands within the \(\mathcal{A}\) basis

\[P_0 = - \imath \frac{\hbar}{m_0} \bra{s} \hat{p}_1 \ket{x_1},\]

and \(B\) is a Kane parameter including interaction between the all the bands in class \(\mathcal{A}\) and remote bands \(\mathcal{B}\) of \(\Gamma_5\) symmetry

\[B = 2 \frac{\hbar^2}{m_0^2} \sum_{nj}^{\mathcal{B} } \frac{ \bra{s} \hat{p}_1 \ket{n \Gamma_5 j} \bra{n \Gamma_5 j} \hat{p}_1 \ket{x_3}} {\left[E_\mathrm{c} +E_\mathrm{v}\right]/2-E_{n,\Gamma_5}}.\]

with top valence band energy \(E_\mathrm{v} = E_\mathrm{v,av} + \Delta_0\).

Blocks for the valence bands without the strain included are defined as

\[\begin{split}\begin{aligned} \hat{\mathcal{H}}_{\mathrm{vv}}\left(\mathbf{k}\right) =& \begin{bmatrix} E_\mathrm{v,av} + \frac{\hbar^2}{2 m_0} k^2 & 0 & 0 \\ 0 & E_\mathrm{v,av} + \frac{\hbar^2}{2 m_0} k^2 & 0 \\ 0 & 0 & E_\mathrm{v,av} + \frac{\hbar^2}{2 m_0} k^2 \\ \end{bmatrix}\\ +& \begin{bmatrix} L' k_{1}^2 + Mk_{2}^2 + Mk_{3}^2 & N' k_1 k_2 & N' k_1 k_3 \\ N' k_1 k_2 & M k_{1}^2 + L'k_{2}^2 + Mk_{3}^2 & N' k_2 k_3 \\ N' k_1 k_3 & N' k_2 k_3 & M k_{1}^2 + Mk_{2}^2 + L'k_{3}^2 \\ \end{bmatrix}, \end{aligned}\end{split}\]

where \(E_\mathrm{v,av}\) is average energy of valence bands at \(\Gamma\) point, \(M\), \(N'\), and \(L'\) are Kane parameters introducing interactions between the valence bands in \(\mathcal{A}\) and remote bands \(\mathcal{B}\) of \(\Gamma_1,\,\Gamma_3,\,\Gamma_4,\,\Gamma_5\) symmetries

\[\begin{split}\begin{aligned} M & = H_1 + H_2\\ N' & = F' - G + H_1 - H_2\\ L' & = F' + 2G\\ \end{aligned}\end{split}\]

where

\[\begin{split}\begin{aligned} G & = \frac{\hbar^2}{2 m_0^2} \sum_{nj}^{\mathcal{B} } \frac{\left|\bra{x_1} \hat{p}_1 \ket{n \Gamma_3 j}\right|^2}{E_\mathrm{v}-E_{n,\Gamma_3}}\\ F' & = \frac{\hbar^2}{2 m_0^2} \sum_{nj}^{\mathcal{B} } \frac{\left|\bra{x_1} \hat{p}_1 \ket{n \Gamma_1 j}\right|^2}{E_\mathrm{v}-E_{n,\Gamma_1}}\\ H_1 & = \frac{\hbar^2}{2 m_0^2} \sum_{nj}^{\mathcal{B} } \frac{\left|\bra{x_1} \hat{p}_1 \ket{n \Gamma_5 j}\right|^2}{E_\mathrm{v}-E_{n,\Gamma_5}}\\ H_2 & = \frac{\hbar^2}{2 m_0^2} \sum_{nj}^{\mathcal{B} } \frac{\left|\bra{x_1} \hat{p}_1 \ket{n \Gamma_4 j}\right|^2}{E_\mathrm{v}-E_{n,\Gamma_4}}\\ \end{aligned}\end{split}\]

Spin-orbit interaction within the valence bands is introduced by

\[\begin{split}\hat{\mathcal{H}}_{\mathrm{so}\uparrow\uparrow} = \frac{\Delta_0}{3} \begin{bmatrix} 0 & -\imath & 0 \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \imath & 0 & 0 \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0 & 0 & 0 \\ \end{bmatrix} = \left[\hat{\mathcal{H}}_{\mathrm{so}\downarrow\downarrow}\right]^\dagger \quad &\mathrm{and} \quad \hat{\mathcal{H}}_{\mathrm{so}\uparrow\downarrow} = \frac{\Delta_0}{3} \begin{bmatrix} 0 & 0 & 1 \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0 & 0 & -\imath \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -1 & \imath & 0 \\ \end{bmatrix} = \left[\hat{\mathcal{H}}_{\mathrm{so}\downarrow\uparrow}\right]^\dagger,\end{split}\]

where spin-orbit interaction energy \(\Delta_0\) is defined by

\[\begin{aligned} \frac{\Delta_0}{3} & = - \imath \frac{\hbar}{4m_0^2c^2} \bra{x_1} \left[\nabla V_0\left(\mathbf{r}\right) \times \hat{\mathbf{p}}\right]_2 \ket{x_3}. \end{aligned}\]

The strain is introduced to the valence bands by

\[\begin{split}\hat{\mathcal{H}}_{\mathrm{vv}}\left(\hat{\epsilon}\right) = \begin{bmatrix} l \epsilon_{11} + m \epsilon_{22} + m \epsilon_{33} & n \epsilon_{21} & n \epsilon_{31} \\ n \epsilon_{21} & m \epsilon_{11} + l \epsilon_{22} + m \epsilon_{33} & n \epsilon_{32} \\ n \epsilon_{31} & n \epsilon_{32} & m \epsilon_{11} + m \epsilon_{22} + l \epsilon_{33} \\ \end{bmatrix}\end{split}\]

where \(\epsilon_{ij}\) are elements of the strain tensor \(\hat{\epsilon}\) and \(m,\,n,\,l\) are matrix elements of a strain-dependent interaction operator, further defining deformation potentails for the valence bands.

Note

All sections below may be moved elswhere in near future

Offsets

\[E_\mathrm{c} = E_{\mathrm{g}}^{\mathrm{(db)}} + E_\mathrm{v,av}^{\mathrm{(db)}} + \frac{1}{3}\Delta_0^{\mathrm{(db)}} \quad,\quad\ E_\mathrm{v,av} = E_\mathrm{v,av}^{\mathrm{(db)}} \quad,\quad\ \Delta_0 = \Delta_0^{\mathrm{(db)}}\]

Where the following mapping to our database is applied.

Table 2.2.1.1 Mapping of offsets to the database

parameter

value in the database

\(E_{\mathrm{g}}^{\mathrm{(db)}}\)

database{ ..._zb{ conduction_bands{ Gamma{ bandgap } } } }

\(E_\mathrm{v,av}^{\mathrm{(db)}}\)

database{ ..._zb{ valence_bands{ bandoffset } } }

\(\Delta_0^{\mathrm{(db)}}\)

database{ ..._zb{ valence_bands{ delta_SO } } }

Attention

If temperature dependence is triggered then the Varshni formula is applied to the energy gap such that \(E_\mathrm{c}\rightarrow E_\mathrm{c}(T)\) and \(E_\mathrm{v,av}\rightarrow E_\mathrm{v,av}(T)\).

Deformation potentials

\[\begin{split}\begin{aligned} a_{\mathrm{c}} & = a_{\mathrm{c}}^{(\mathrm{db})}, \\ m & = a_{\mathrm{v}}^{(\mathrm{db})} - b^{(\mathrm{db})}, \\ n & = \sqrt{3} \,d^{(\mathrm{db})}, \\ l & = a_{\mathrm{v}}^{(\mathrm{db})} + 2 b^{(\mathrm{db})}, \\ \end{aligned}\end{split}\]

Where the following mapping to our database is applied.

Table 2.2.1.2 Mapping of deformation potentials to the database

parameter

value in the database

\(a_{\mathrm{c}}^{(\mathrm{db})}\)

database{ ..._zb{ Gamma{ defpot_absolute } } }

\(a_{\mathrm{v}}^{(\mathrm{db})}\)

database{ ..._zb{ valence_bands{ defpot_absolute } }

\(b^{(\mathrm{db})}\)

database{ ..._zb{ valence_bands{ defpot_uniaxial_b } }

\(d^{(\mathrm{db})}\)

database{ ..._zb{ valence_bands{ defpot_uniaxial_d } }

k.p parameters

Attention

In this section we assume that rescale_S_to is not defined in the input file at all, like in the examples below. The topic of rescaling S parameter and it’s influence on the Hamiltonian will be discussed elsewhere.

As the \(\mathbf{k} \cdot \mathbf{p}\) models have been derived in the literature on numerous ways, there are couple of parameterisation standards available of which preference is not clear. Also, depending on the method applied to obtaining parameters some of them are easier accessible that the others. Therefore, depending on the source and the material of interest different schemes of parametrisation may be preffered by the user. For this purpose multiple possibilities of connecting our database to this model are available.

Default settings

The default settings are equivalent to setting all the attrubutes use_Luttinger_parameters, from_6band_parameters, approximate_kappa, evaluate_S to no.

Examples
  1. Controlling parameters of the Hamiltonian for computation of electronic energy dispersion for a bulk crystal

classical{
    bulk_dispersion{
        KP8{
            from_6band_parameters = no
            use_Luttinger_parameters = no
            approximate_kappa = no
            evaluate_S = no
        }
    }
}
  1. Controlling parameters of the Hamiltonian for which h Schrödinger equation is solved

quantum {
    region{
        kp_8band{
            kp_parameters{
                from_6band_parameters = no
                use_Luttinger_parameters = no
                approximate_kappa = no
                evaluate_S = no
            }
        }
    }
}

Then the Kane parameters are defined by

\[M = \frac{\hbar^2}{2 m_0} M^{\mathrm{(db)}} \quad,\quad\ N' = \frac{\hbar^2}{2 m_0} N'^{\mathrm{(db)}} \quad,\quad\ L' = \frac{\hbar^2}{2 m_0} L'^{\mathrm{(db)}}\]
\[A_\mathrm{c} = \frac{\hbar^2}{2 m_0} S^{\mathrm{(db)}} \quad,\quad\ B = \frac{\hbar^2}{2 m_0} B^{\mathrm{(db)}} \quad,\quad\ P = \sqrt{\frac{\hbar^2}{2 m_0} E_{\mathrm{p}}^{\mathrm{(db)}}}\]

where the following mapping to our database is applied.

Table 2.2.1.3 Mapping of Kane parameters to the database

parameter

value in the database

\(M^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ M } } }

\(L'^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ L } } }

\(N'^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ N } } }

\(S^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ S } } }

\(B^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ B } } }

\(E_{\mathrm{p}}^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ E_P } } }

Luttinger parameters and electron effective mass

One needs to set all three parameters from_6band_parameters, use_Luttinger_parameters, evaluate_S to yes to use the Luttinger parameters (as defined for 6-band \(\mathbf{k} \cdot \mathbf{p}\) model) and the effective mass of electrons.

Examples
  1. Controlling parameters of the Hamiltonian for computation of electronic energy dispersion for a bulk crystal

classical{
    bulk_dispersion{
        KP8{
            from_6band_parameters = yes
            use_Luttinger_parameters = yes
            approximate_kappa = no
            evaluate_S = yes
        }
    }
}
  1. Controlling parameters of the Hamiltonian for which h Schrödinger equation is solved

quantum {
    region{
        kp_8band{
            kp_parameters{
                from_6band_parameters = yes
                use_Luttinger_parameters = yes
                approximate_kappa = no
                evaluate_S = yes
            }
        }
    }
}

Then the Kane parameters are defined by

\[\begin{split}\begin{aligned} M & = \frac{\hbar^2}{2 m_0}\left[-\gamma_1^{\mathrm{(db)}}+2\gamma_2^{\mathrm{(db)}}-1\right]\\ N' & =\frac{\hbar^2}{2 m_0}\left[-6\gamma_3^{\mathrm{(db)}}\right] + \frac{E_{\mathrm{p}}^{\mathrm{(db)}}}{E_{\mathrm{g}}}\\ L' & = \frac{\hbar^2}{2 m_0}\left[-\gamma_1^{\mathrm{(db)}}-4\gamma_2^{\mathrm{(db)}}-1\right] + \frac{E_{\mathrm{p}}^{\mathrm{(db)}}}{E_{\mathrm{g}}}\\ A_c & = \frac{\hbar^2}{2 m_0} \left[\frac{1}{m_\mathrm{e}^{\mathrm{(db)}}} - \frac{2 E_\mathrm{p}^{\mathrm{(db)}}}{3 E_{\mathrm{g}}} - \frac{E_\mathrm{p}^{\mathrm{(db)}}}{3\left[E_{\mathrm{g}}+\Delta_0^{\mathrm{(db)}}\right]}\right]\\ B & = \frac{\hbar^2}{2 m_0} B^{\mathrm{(db)}}\\ P & = \sqrt{\frac{\hbar^2}{2 m_0} E_{\mathrm{p}}^{\mathrm{(db)}}},\\ \end{aligned}\end{split}\]

where the following mapping to our database is applied.

Table 2.2.1.4 Mapping to the database

parameter

value in the database

\(\gamma_1^{\mathrm{(db)}}\)

database{ ..._zb{ kp_6_bands{ gamma_1 } } }

\(\gamma_2^{\mathrm{(db)}}\)

database{ ..._zb{ kp_6_bands{ gamma_2 } } }

\(\gamma_3^{\mathrm{(db)}}\)

database{ ..._zb{ kp_6_bands{ gamma_3 } } }

\(m_e^{\mathrm{(db)}}\)

database{ ..._zb{ conduction_bands{ Gamma{ mass } } } }

\(\Delta_0^{\mathrm{(db)}}\)

database{ ..._zb{ valence_bands{ delta_SO } } }

\(E_{\mathrm{p}}^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ E_P } } }

\(B^{\mathrm{(db)}}\)

database{ ..._zb{ kp_8_bands{ B } } }

Rescaling S

One of ways to get rid of spurious solutions in quantum structures is to rescale \(S\) parameter to 0 or 1. The \(S\) defines \(A_c\) as

\[A_c = \frac{\hbar^2}{2 m_0} S\]
Examples
  1. Rescaling \(S\) in the Hamiltonian for computation of electronic energy dispersion for a bulk crystal

classical{
    bulk_dispersion{
        KP8{
            rescale_S_to = 1
        }
    }
}
  1. Rescaling \(S\) in the Hamiltonian for which the Schrödinger equation is solved

quantum {
    region{
        kp_8band{
            kp_parameters{
                rescale_S_to = 1
            }
        }
    }
}

The initial value of \(S\) is determined according to choices described before. If one chose evaluate_S = no then

\[\begin{aligned} S = S^{(db)}, \end{aligned}\]

otherwise, if one chose evaluate_S = yes then

\[\begin{aligned} S = \frac{1}{m_\mathrm{e}^{\mathrm{(db)}}} - \frac{2 E_\mathrm{p}^{\mathrm{(db)}}}{3 E_{\mathrm{g}}} - \frac{E_\mathrm{p}^{\mathrm{(db)}}}{3\left[E_{\mathrm{g}}+\Delta_0^{\mathrm{(db)}}\right]}. \end{aligned}\]

In the input file, one can request consistent rescaling the model such that \(S \rightarrow S^{\mathrm{(new)}}\) resulting in

\[A_c = \frac{\hbar^2}{2 m_0} S^{\mathrm{(new)}}.\]

A rescaled Kane energy \(E_\mathrm{p}^{\mathrm{(new)}}\) is evaluated to ensure that the model gives the same electronic band structure (ideally) as before the rescaling, but without spurious solutions. It is done directly from the assumption the \(S = S^{\mathrm{(new)}}\).

\[\begin{split}\begin{aligned} E_\mathrm{p}^{\mathrm{(new)}} & = E_\mathrm{p}^{\mathrm{(db)}} + \left[S-S^{\mathrm{(new)}}\right] \frac {E_{\mathrm{g}}\left[E_{\mathrm{g}}+\Delta_0^{\mathrm{(db)}}\right]} {E_{\mathrm{g}}+\frac{2}{3}\Delta_0^{\mathrm{(db)}}}\\ \end{aligned}\end{split}\]

After the rescaled Kane energy is evaluated, it is used to update or redefine other relevant Kane parameters entering the model.

\[\begin{split}L' & \rightarrow L' + \frac{E_\mathrm{p}^{\mathrm{(new)}}-E_\mathrm{p}^{\mathrm{(db)}}}{E_{\mathrm{g}}},\\ N' & \rightarrow N' + \frac{E_\mathrm{p}^{\mathrm{(new)}}-E_\mathrm{p}^{\mathrm{(db)}}}{E_{\mathrm{g}}},\\ P & = \sqrt{\frac{\hbar^2}{2 m_0} E_{\mathrm{p}}^{\mathrm{(new)}}}.\end{split}\]

Where \(L'\) and \(N'\) are initially evaluated according to the choices in the kp_parameters{} group as described in previous sections.