2.12. Matrix elements

2.12.1. Calculated within quantum{} group

These matrix elements can be calculated using keywords dipole_moment_matrix_elements{ } and momentum_matrix_elements{ }.

Attention

These matrix elements are not used in calculation of optical spectra in optics{ quantum_spectra{ } }.

2.12.2. 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.

2.12.3. 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.

2.12.4. Kinematic-momentum matrix elements

These matrix elements are calculated using Hellman-Feynman theorem as described in [Eissfeller2008]. They are always outputted for \(\WaveK=\VecZero\) to bias/OpticsQuantum/[QUANTUM_REGION_NAME]/kp8_kp8/transitions_*.dat. They can be calculated for all integrated wave vectors \(\WaveK\) and output by setting output_transitions to yes and stored in bias/OpticsQuantum/[QUANTUM_REGION_NAME]/kp8_kp8/transition_dispersion_*.fld.

Attention

These matrix elements are used in calculation of optical spectra in optics{ quantum_spectra{ } }.


Last update: 2025-08-29