The 10-band zincblende / dilute-nitride model

Adding a few percent of nitrogen to GaAs or InGaAs reduces the band gap far more than Vegard-law alloying would predict, and dramatically increases the electron effective mass. This anomaly is not captured by treating nitrogen as an ordinary alloy component. Instead, nitrogen introduces a localised, resonant state above the conduction-band edge that interacts with the extended conduction states through band anticrossing (BAC). The OghmaNano 10-band model builds this physics directly on top of the 8-band zincblende Hamiltonian.

1. Two extra states

The 10-band basis is the eight zincblende states followed by two spin-resolved nitrogen states,

\[ \{\,\underbrace{|S\!\uparrow\rangle,\dots,|Z\!\downarrow\rangle}_{\text{8-band block}},\ |N\!\uparrow\rangle,\ |N\!\downarrow\rangle\,\}. \]

On an \(N\)-node grid the Hamiltonian is therefore a \(10N\) complex Hermitian matrix. Every term of the 8-band model — Kane coupling, valence \(L/M/N\) parameters, spin-orbit coupling, Bir–Pikus strain — is retained unchanged; the model only adds the nitrogen states and their coupling to the conduction states.

2. The band-anticrossing coupling

The localised nitrogen level sits at energy \(E_N\) and couples only to the s-like conduction state. The coupling strength scales with the square root of the nitrogen mole fraction \(x_N\),

\[ V_N = \beta_N\,\sqrt{x_N}, \]

where \(\beta_N\) is a material coupling constant. Physically, \(\sqrt{x_N}\) reflects the coupling of the conduction band to the manifold of randomly placed nitrogen impurities: the matrix element to the symmetric combination grows as the square root of their number density. In the Hamiltonian this appears as

\[ H_{NN} = E_N, \qquad H_{NS} = H_{SN}^\* = V_N, \]

applied identically for each spin. In regions with no nitrogen OghmaNano sets \(V_N = 0\) and moves \(E_N\) far above the spectrum, so the two nitrogen states remain inert and do not perturb the physical bands.

3. Anticrossing eigenvalues

Restricted to one spin, the conduction and nitrogen states form a two-level system,

\[ \begin{pmatrix} E_c & V_N \\ V_N & E_N \end{pmatrix}, \]

whose eigenvalues are

\[ E_\pm = \tfrac{1}{2}\left(E_c + E_N\right) \pm \tfrac{1}{2}\sqrt{(E_c - E_N)^2 + 4 V_N^2}. \]

The lower branch \(E_-\) is pushed down below the original conduction edge \(E_c\), reducing the effective band gap, while the upper branch \(E_+\) appears as a second, nitrogen- derived conduction feature. The repulsion is strongest when \(E_c\) and \(E_N\) are close, so the gap reduction is largest when the resonant level lies just above the band edge. The flattening of \(E_-\) near the anticrossing also increases the electron mass, matching experiment.

4. In the quantum-well solver

In the full quantum-well problem the nitrogen states are added at every node and the \(10N\)-dimensional Hamiltonian is diagonalised with the same shift-invert eigensolver as the 8-band model. When choosing the target energy for the conduction states, OghmaNano applies the two-level anticrossing formula above wherever nitrogen is present, so that the solver aims at the lowered \(E_-\) branch rather than the bare \(E_c\). The state-character analysis reports the nitrogen fraction of each eigenstate, which is largest for the states near \(E_+\) and small but non-zero for the confined \(E_-\) states that carry the laser transition.