Eigenstates, normalisation and state character

The eigensolver returns energies and envelope vectors. This page explains how OghmaNano normalises those envelopes, how it measures the orbital character of each state, and how it keeps track of which subband is which as the in-plane wavevector is scanned — all of which are needed by the optical and transport routines that follow.

1. Normalisation

A multiband envelope has one component per basis function at every node. OghmaNano normalises the total probability, summed over all bands and integrated over the grid, to unity:

\[ \sum_{z}\sum_{b} \big|\psi_b(z)\big|^2\,\Delta z = 1. \]

This physical normalisation makes \(|\psi_b(z)|^2\Delta z\) the probability of finding the carrier in band \(b\) near node \(z\), and it ensures the optical matrix elements and sheet densities have their correct absolute scale. The scalar (parabolic and silicon) solvers use the same rule with a single band component.

2. State character

Because the multiband states mix s-like and p-like orbitals, each eigenstate has a definite character given by the fraction of its probability in each orbital. OghmaNano integrates the squared components,

\[ w_S = \sum_z\big(|\psi_{S\uparrow}|^2+|\psi_{S\downarrow}|^2\big)\Delta z, \quad w_X, w_Y, w_Z \ \text{likewise}, \]

so that \(w_S+w_X+w_Y+w_Z=1\) (plus a nitrogen weight \(w_N\) in the 10-band model). A state with large \(w_S\) is conduction-like (electron); a state dominated by \(w_X, w_Y, w_Z\) is valence-like (hole). The individual \(X, Y, Z\) weights also reveal whether a valence state is heavy-hole-like (mostly \(X, Y\)) or light-hole-like, which controls its optical selection rules.

3. Classifying states by S-character

When the conduction and valence states are solved together in a single spectrum, they must be sorted into electrons and holes. OghmaNano uses the S-character weight \(w_S\) for this: a state is treated as electron-like in proportion to \(w_S\) and hole-like in proportion to \(1-w_S\). This continuous weighting is important near the band edge of narrow-gap materials, where strong conduction–valence mixing makes the electron/hole distinction genuinely fractional. The same weight enters the gain calculation, where it scales the emission and absorption of each transition.

4. Tracking subbands along the dispersion

The in-plane dispersion is built by solving the Hamiltonian at a sequence of \(k_\parallel\) points. The eigensolver returns states in energy order, but energy order can change when subbands cross or anticross, so a naive energy sort would scramble the band labels. OghmaNano instead tracks states by overlap: at each new \(k_\parallel\) it matches each previous eigenvector to the new eigenvector with which it has the largest overlap,

\[ \text{match}(i) = \arg\max_j \Big|\textstyle\sum_z\sum_b \psi^{(k-1)}_{i,b}(z)^\*\,\psi^{(k)}_{j,b}(z)\,\Delta z\Big|^2, \]

so that each subband keeps a consistent identity across the whole dispersion. This continuous labelling is what makes the subband energies \(E_n(k_\parallel)\) smooth functions suitable for the optical and transport integrals.

5. Reading the wavefunctions

The confined envelopes carry direct physical meaning. Their number of nodes gives the subband index; their spatial extent shows how tightly the state is confined and how far it leaks into the barriers; and the electron–hole envelope overlap sets the strength of the optical transition between them. In strained or asymmetric structures the electron and hole envelopes can be pushed to opposite sides of the well, reducing their overlap and the optical efficiency — an effect that the multiband envelopes capture directly and that feeds into the matrix-element calculation.