Optical matrix elements and selection rules

The strength of an optical transition between two states is set by the momentum matrix element between them. In a quantum well this factors into a bulk part — the Kane momentum matrix element — and an envelope part that depends on how well the electron and hole states overlap in space and orbital character. This page derives the matrix element that OghmaNano uses for gain and absorption, and the polarisation selection rules that follow from it.

1. Transitions and the matrix element

The rate of an optical transition between an initial state \(|i\rangle\) and a final state \(|f\rangle\) is proportional to \(|\langle f|\hat{\mathbf{e}}\cdot\mathbf{p}|i\rangle|^2\), where \(\hat{\mathbf{e}}\) is the light polarisation and \(\mathbf{p}\) the momentum operator. For an interband transition the operator connects the conduction and valence Bloch functions through the Kane momentum matrix element \(P\), while the envelopes supply a spatial overlap. The polarisation dependence enters because \(\mathbf{p}\) couples \(|S\rangle\) to different p-orbitals depending on its direction.

2. Parabolic model: envelope overlap

In the scalar parabolic model the electron and hole are single-band envelopes, and the matrix element reduces to their spatial overlap squared,

\[ M^2 \propto \left|\int \psi_e(z)\,\psi_h(z)\,dz\right|^2, \]

with the bulk momentum matrix element factored out separately in the gain prefactor. This captures the essential selection rule that transitions between states of the same parity (for example \(e1\to hh1\)) are strong while opposite-parity transitions are weak or forbidden, but it treats all valence states as identical and so cannot distinguish heavy from light holes.

3. Multiband matrix element

In the multiband k·p models the valence state carries genuine \(X, Y, Z\) orbital character, so the matrix element is built from the overlap of the conduction \(S\) component with the valence \(X, Y, Z\) components (and vice versa). For in-plane (TE) polarisation OghmaNano forms

\[ M_x = P\int\!\big(\psi_{c,S}^\*\psi_{v,X} + \psi_{c,X}^\*\psi_{v,S}\big)\,dz, \qquad M_y \ \text{likewise with } Y, \]

summed over spin, and combines them as

\[ M^2 = \frac{|M_x|^2 + |M_y|^2}{2\,P_{\text{ref}}^2}, \]

normalised by a reference \(P_{\text{ref}}\) taken at the well centre so that the result is a dimensionless overlap-and-character factor. Because \(M_x, M_y\) depend on the valence orbital character, this automatically makes heavy-hole and light-hole transitions differ in strength — physics entirely absent from the parabolic overlap.

4. TE / TM selection rules

The polarisation dependence follows from which p-orbital the momentum operator connects to. Light polarised in the plane (TE) couples \(|S\rangle\) to the in-plane orbitals \(|X\rangle\) and \(|Y\rangle\); light polarised along the growth/c-axis (TM) couples \(|S\rangle\) to \(|Z\rangle\). In the wurtzite model, where the two Kane parameters \(P_1\) (c-axis) and \(P_2\) (basal) differ, OghmaNano computes the two polarisations separately:

\[ M^2_{\text{TE}} = \frac{|M_x|^2+|M_y|^2}{2P_2^2}, \qquad M^2_{\text{TM}} = \frac{|M_z|^2}{P_1^2}, \]

with \(M_z\) built from the \(S\)–\(Z\) overlaps. A valence state with mostly \(X, Y\) character (heavy-hole-like) therefore emits predominantly TE, while a state with strong \(Z\) character emits TM. This is the microscopic origin of the polarisation of quantum-well lasers, and it links directly to the crystal-field and strain ordering of the wurtzite valence bands.