2.11. Envelope-function representation
2.11.1. Luttinger-Kohn representation
Hamiltonian operators in nextnano++ are formulated within Luttinger-Kohn representation
\[\begin{aligned}
\LuKoFunction{\alpha}{\WaveK}{\Pos} = \TwoPi{-3/2}\PlaneWave{(\WaveK+\WaveKLattice)}{\Pos} \; \PeriodicFactorSum{\alpha}{\Pos},
\end{aligned}\]
where \(\PeriodicFactorSum{\alpha}{\Pos} \equiv \PeriodicFactor{\alpha}{\WaveKLattice}{\Pos} \equiv \braket{\Pos}{\alpha}\) is a periodic factor at selected wave vector \(\WaveKLattice\), typically a reciprocal lattice vector.
2.11.2. Basis of periodic factors for zincblende
We support outputs in two bases of periodic factors \(\ket{\alpha}\) defined at the \(\Gamma\) point of the first Brillouin zone for 8-band \(\kp\) model, \(\WaveKLattice = \VecZero\).
The main one is the 8-fold basis used by Kane’s formulation of the Hamiltonian without interactions.
\[\begin{aligned}
\BasisKane[8] \equiv \{S\uparrow,\;S\downarrow,\;X\uparrow,\;Y\uparrow,\;Z\uparrow,\;X\downarrow,\;Y\downarrow,\;Z\downarrow\}
\end{aligned}\]
The second one constructed of the basis functions for the \(\Gamma_6\), \(\Gamma_7\), and, \(\Gamma_8\) irreducible representations of the \(T_d\) double group.
\[\begin{aligned}
\BasisTd[8] \equiv \{cb1,\;cb2,\;hh1,\;hh2,\;lh1,\;lh2,\;so1,\;so2\}
\end{aligned}\]
where
\[\begin{split}\begin{aligned}
\ket{cb1} =& \ket{1/2, 1/2}_{\Gamma_6} = \ket{S\uparrow} \\
\ket{cb2} =& \ket{1/2,-1/2}_{\Gamma_6} = \ket{S\downarrow} \\
\ket{hh1} =& \ket{3/2, 3/2}_{\Gamma_8} = \frac{-1}{\sqrt{2}} \left[\ket{X\uparrow}+i\ket{Y\uparrow} \right] \\
\ket{hh2} =& \ket{3/2,-3/2}_{\Gamma_8} = \frac{1}{\sqrt{2}} \left[\ket{X\downarrow}-i\ket{Y\downarrow}\right] \\
\ket{lh1} =& \ket{3/2, 1/2}_{\Gamma_8} = \frac{-1}{\sqrt{6}} \left[\ket{X\downarrow}+i\ket{Y\downarrow}\right]+\sqrt{\frac{2}{3}}\ket{Z\uparrow} \\
\ket{lh2} =& \ket{3/2,-1/2}_{\Gamma_8} = \frac{1}{\sqrt{6}} \left[\ket{X\uparrow}-i\ket{Y\uparrow} \right]+\sqrt{\frac{2}{3}}\ket{Z\downarrow} \\
\ket{so1} =& \ket{1/2, 1/2}_{\Gamma_7} = \frac{1}{\sqrt{3}} \left[\ket{X\downarrow}+i\ket{Y\downarrow}\right]+\sqrt{\frac{1}{3}}\ket{Z\uparrow} \\
\ket{so2} =& \ket{1/2,-1/2}_{\Gamma_7} = \frac{1}{\sqrt{3}} \left[\ket{X\uparrow}-i\ket{Y\uparrow} \right]-\sqrt{\frac{1}{3}}\ket{Z\downarrow}.
\end{aligned}\end{split}\]
The 6-band \(\kp\) models are expressed analogously in the bases
\[\begin{aligned}
\BasisKane[6] \equiv \{X\uparrow,\;Y\uparrow,\;Z\uparrow,\;X\downarrow,\;Y\downarrow,\;Z\downarrow\}
\quad\mathrm{and}\quad
\BasisTd[6] \equiv \{hh1,\;hh2,\;lh1,\;lh2,\;so1,\;so2\}.
\end{aligned}\]
2.11.3. Spinors and envelope components
Any spinor wave function \(\WaveFunction \Arguments{\Pos} \equiv \braket{\Pos}{\WaveFunction}\) can be expressed within the Luttinger-Kohn representation as
\[\begin{aligned}
\WaveFunction\Arguments{\Pos} & = \PlaneWave{\WaveKLattice}{\Pos} \sum_{\alpha \in \Basis} \PeriodicFactorSum{\alpha}{\Pos} \EnvelopeFunction_\alpha\Arguments{\Pos},
\end{aligned}\]
where \(\UnitCell[*]\) is the volume of the reciprocal unit cell (the First Brillouin Zone), \(\WaveFunction_{\alpha}\Arguments{\WaveK}\) is an envelope function factor in the reciprocal space, \(\Basis \in \{\BasisKane[8],\;\BasisTd[8],\;\BasisKane[6],\;\BasisTd[6]\}\) is indexing the considered basis of periodic factors including spin, and
\[\begin{aligned}
\EnvelopeFunction_{\alpha}\Arguments{\Pos} \equiv \int_{\UnitCell[*]} \PlaneWave{\WaveK}{\Pos} \; \WaveFunction_{\alpha}\Arguments{\WaveK} d^3k
\end{aligned}\]
is referred to as an the envelope-function component.
The wave functions \(\ket{\WaveFunction}\) can be conveniently indexed by two quantum numbers, the wave vector in the direction without quantization (e.g., \(\WaveK\) or \(\WaveG\)), and by some ordering number typically related to the order of the energy eigenvalues (e.g., \(\Indexn\) or \(\Indexm\)).
Therefore, the envelope-function representation for \(\ket{\WaveFunction}\) in the coordinate space can be given by
\[\begin{aligned}
\WaveFunction_{\Indexn\WaveK}\Arguments{\Pos} = \PlaneWave{\WaveKLattice}{\Pos} \sum_{\alpha} \EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos} \PeriodicFactorSum{\alpha}{\Pos}.
\end{aligned}\]
In case of simulations in 3D, where there is no band structure, the wave vector \(\WaveK\) is considered zero and sometimes marked as such explicitly, \(\VecZero\).
A set of envelope components \(\{\EnvelopeFunction_{{\alpha\Indexn\WaveK}}| \alpha \in \Basis\}\) provides complete information about the envelope function of the related spinor state \(\ket{\WaveFunction_{\Indexn\WaveK}}\).
These are available in the output files amplitudes_*.* while their square absolute values can be found in probabilities_*.*.
Plane waves can be factored out of the envelope components in the directions where there is no quantization.
Hence the wave functions for simulation in 1D, 2D and 3D are given as
\[\begin{aligned}
\WaveFunction_{\Indexn\WaveK}\Arguments{\Pos} =
\begin{cases}
\TwoPi{-1}\;\;\;
\PlaneWave{\WaveKPar}{\PosPar}\;
\PlaneWave{\WaveKLattice}{\Pos}
\sum_{\alpha}
\EnvelopeFunction_{\alpha\Indexn\WaveKPar}\Arguments{\PosX}\,
\PeriodicFactorSum{\alpha}{\Pos}
&
\mathrm{\quad \mathrm{for\;1D}},
\CaseSpacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\TwoPi{-1/2}
\PlaneWave{\WaveKZ}{\PosZ}\;\;\;\,
\PlaneWave{\WaveKLattice}{\Pos}
\sum_{\alpha}
\EnvelopeFunction_{\alpha\Indexn\WaveKZ}\Arguments{\PosPerp}\,
\PeriodicFactorSum{\alpha}{\Pos}
&
\mathrm{\quad \mathrm{for\;2D}},
\CaseSpacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\qquad\qquad\qquad\;\;\;\,
\PlaneWave{\WaveKLattice}{\Pos}
\sum_{\alpha}
\EnvelopeFunction_{\alpha\Indexn\VecZero}\Arguments{\Pos}\,
\PeriodicFactorSum{\alpha}{\Pos}
&
\mathrm{\quad \mathrm{for\;3D}}.
\end{cases}
\end{aligned}\]
where
\[\begin{aligned}
\WaveK &= \left[\,\WaveKX,\,\WaveKY,\,\WaveKZ\,\right],
\quad
\WaveKPar = \left[\,0,\,\WaveKY,\,\WaveKZ\,\right],
\quad
\WaveKZ = \left[\,0,\,0,\,\WaveKZ\,\right],
\end{aligned}\]
and
\[\begin{aligned}
\Pos &= \left[\,\PosX,\,\PosY,\,\PosZ\,\right],
\quad
\Pos_\perp = \left[\,\PosX,\,\PosY,\,0\,\right],
\quad
\Pos_{||} = \left[\,0,\,\PosY,\,\PosZ\,\right].
\end{aligned}\]
2.11.4. Envelope overlap integrals
Regardless of the dimensionality of the simulation, the envelope functions are normalized such that
\[\begin{aligned}
\sum_{\alpha \in \Basis}
\OverlapIntegralEnv_{\alpha\Indexm\WaveK,\alpha\Indexn\WaveK}
=
\Kronecker{\Indexm\Indexn},
\end{aligned}\]
where \(\OverlapIntegralEnv_{\alpha \Indexm\WaveG,\alpha\Indexn\WaveK}\) are the envelope overlap integrals defined as
\[\begin{aligned}
\OverlapIntegralEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\equiv
\braket{\EnvelopeFunction_{\beta\Indexm\WaveG}}{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
= \;
\begin{cases}
\int_{L}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosX}\,
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosX}\,
\dPosX
&
\mathrm{\quad \mathrm{for\;1D}},
\CaseSpacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iint_{S}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosPerp}\,
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosPerp}\,
\dPosX\,\dPosY
&
\mathrm{\quad \mathrm{for\;2D}},
\CaseSpacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
&
\mathrm{\quad \mathrm{for\;3D}}.
\end{cases}
\end{aligned}\]
Note that the arguments of the envelope-function components \(\PosX\), \(\PosPerp = \left[\,\PosX,\,\PosY,\,0\,\right]\), and \(\Pos = \left[\,\PosX,\,\PosY,\,\PosZ\,\right]\) are spanned over the dimensions where the quantization occurs.
The non-zero elements of the summation in (ref{eq:overlap_integrals_summation}) are describing contribution of corresponding periodic factor to the spinor wave function.
For example, all the overlap integrals \(\OverlapIntegralEnv_{\alpha\Indexn\WaveK,\alpha\Indexn\WaveK}\ne 0\) for the wave function \(\WaveFunction_{\Indexn\WaveK}\).
We refer to them as spinor composition of the wave function, and they are available for outputted all states in the files spinor_composition_*.dat
More related outputs is available in nextnano++.
For selected 1-band and 6-band \(\kp\) models, one can output sums of absolute values of the overlap integrals and square value of such a sum defined as follows
\[\begin{aligned}
|
\sum_{\alpha \in \Basis[6]}
\braket{\EnvelopeFunction_{\beta\Indexm\WaveK}}{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\sum_{\alpha \in \Basis}
\braket{\EnvelopeFunction_{\beta\Indexm\WaveK}}{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2,
\end{aligned}\]
where \(\Basis[6] \in \{\BasisKane[6],\;\BasisTd[6]\}\) is a selected basis of the 6-band model and \(\EnvelopeFunction_{\beta\Indexn\WaveK}\) is the only envelope component available in the 1-band model.
For two different 1-band models, one can output absolute values of the overlap integrals and their square value
\[\begin{aligned}
|
\braket{\EnvelopeFunction_{\beta\Indexm\WaveK}}{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\braket{\EnvelopeFunction_{\beta\Indexm\WaveK}}{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2,
\end{aligned}\]
where \(\EnvelopeFunction_{\beta\Indexm\WaveK}\) and \(\EnvelopeFunction_{\alpha\Indexn\WaveK}\) are the only envelope components obtained in different 1-band models, e.g., for conduction band and for heavy holes.
These are available in the files overlap_integrals_*.*.
Last update: 2025-08-29