2.11.2. Envelope dipole-moment matrix elements
The envelope dipole moment matrix elements are vectors defined as
\[\begin{split}\begin{aligned}
\MatDipoleMomentEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\equiv
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \OpDipole \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
= \;
\begin{cases}
\begin{bmatrix}
\int_{L}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosX}\,
\OpDipoleX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosX}\,
\dPosX
\\
0
\\
0
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;1D}},
\CaseSpacingLong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{bmatrix}
\iint_{S}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosPerp}\,
\OpDipoleX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosPerp}\,
\dPosX\,\dPosY
\\
\iint_{S}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosPerp}\,
\OpDipoleY\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosPerp}\,
\dPosX\,\dPosY
\\
0
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;2D}},
\CaseSpacingLong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{bmatrix}
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpDipoleX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\\
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpDipoleY\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\\
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpDipoleZ\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;3D}},
\end{cases}
\end{aligned}\end{split}\]
where \(\OpDipole \equiv \OpDipoleInPositionRepresentation\).
The polarization-dependent envelope dipole moment matrix elements are defined as
\[\begin{aligned}
\polarization\circ\MatDipoleMomentEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\equiv
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \polarization\circ\OpDipole \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
\end{aligned}\]
Similarly as for envelope momentum matrix elements, outputs generated for 8-band \(\kp\) model are the sums
\[\begin{aligned}
|
\sum_{\alpha \in \Basis[8]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\sum_{\alpha \in \Basis[8]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2,
\end{aligned}\]
for 6-band \(\kp\) model
\[\begin{aligned}
|
\sum_{\alpha \in \Basis[6]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\sum_{\alpha \in \Basis[6]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2,
\end{aligned}\]
and for 1-band model
\[\begin{aligned}
|
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \OpDipole
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2.
\end{aligned}\]
with the units \(\mathrm{(e\cdot nm)}\) and \(\mathrm{(e\cdot nm)^2}\), respectively for each model.
2.11.3. Envelope momentum matrix element
The envelope momentum matrix elements are vectors defined as
\[\begin{split}\begin{aligned}
\MatMomentumEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\equiv
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \OpMomentum \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
= \;
\begin{cases}
\begin{bmatrix}
\int_{L}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosX}\,
\OpMomentumX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosX}\,
\dPosX
\\
0
\\
0
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;1D}},
\CaseSpacingLong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{bmatrix}
\iint_{S}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosPerp}\,
\OpMomentumX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosPerp}\,
\dPosX\,\dPosY
\\
\iint_{S}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\PosPerp}\,
\OpMomentumY\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\PosPerp}\,
\dPosX\,\dPosY
\\
0
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;2D}},
\CaseSpacingLong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{bmatrix}
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpMomentumX\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\\
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpMomentumY\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\\
\iiint_{V}
\EnvelopeFunction_{\beta\Indexm\WaveG}^*\Arguments{\Pos}\,
\OpMomentumZ\;
\EnvelopeFunction_{\alpha\Indexn\WaveK}\Arguments{\Pos}\,
\dPosX\,\dPosY\,\dPosZ
\end{bmatrix}
&
\mathrm{\quad \mathrm{for\;3D}},
\end{cases}
\end{aligned}\end{split}\]
where \(\OpMomentum \equiv \OpMomentumInPositionRepresentation\).
The polarization-dependent envelope momentum matrix elements are defined as
\[\begin{aligned}
\polarization\circ\MatMomentumEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\equiv
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \polarization\circ\OpMomentum \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
\end{aligned}\]
where \(\polarization\) is a complex versor describing polarization of related optical filed.
By manipulating with the \(\polarization\) one can extract any linear combination of the components of the envelope momentum matrix elements.
For convenience, nextnano++ calculates these elements without \(-i\hbar\) factor
\[\begin{aligned}
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \polarization\circ\grad \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
=
\frac{i}{\hbar}\cdot
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \polarization \circ (\OpMomentumInPositionRepresentation) \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
=
\frac{i}{\hbar}\cdot
\bra{\EnvelopeFunction_{\beta\Indexm\WaveG}} \polarization\circ\OpMomentum \ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
=
\frac{i}{\hbar}\cdot
\polarization\circ\MatMomentumEnv_{\beta\Indexm\WaveG,\alpha\Indexn\WaveK}
\end{aligned}\]
Related output for 8-band \(\kp\) model contains absolute values and squared absolute values of the sum of such elements with matching wave vectors \(\WaveK\)
\[\begin{aligned}
|
\sum_{\alpha \in \Basis[8]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\sum_{\alpha \in \Basis[8]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2,
\end{aligned}\]
with the units \(\mathrm{\hbar/nm}\) and \(\mathrm{\hbar^2/nm^2}\), respectively, and \(\Basis[8] \in \{\BasisKane[8],\;\BasisTd[8]\}\).
Similarly, the available output for 6-band \(\kp\) model is
\[\begin{aligned}
|
\sum_{\alpha \in \Basis[6]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\sum_{\alpha \in \Basis[6]}
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2.
\end{aligned}\]
As only single envelope components are available for 1-band models, no summation is done in this case.
Here, the output contains
\[\begin{aligned}
|
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|
\quad\mathrm{and}\quad
|
\bra{\EnvelopeFunction_{\alpha \Indexm\WaveK}}
\polarization \circ \grad
\ket{\EnvelopeFunction_{\alpha\Indexn\WaveK}}
|^2
\end{aligned}\]
where both envelopes have the same periodic-factor index \(\alpha\) as they belong to the same 1-band model.