Basis states, envelope functions and multiband Hamiltonians

Turning a bulk k·p Hamiltonian into a quantum-well problem requires two ingredients: the envelope-function approximation, which lets a bulk matrix describe a spatially varying structure, and a consistent basis in which to write the matrix. This page assembles both and records the exact basis orderings that OghmaNano uses, so that the model-specific pages can refer back to a single definition.

1. Three kinds of function

It is worth separating three objects that are easily confused. The Bloch function \(\psi_{n\mathbf{k}} = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}\) is the exact crystal eigenstate. The cell-periodic function \(u_{n0}\) at the zone centre is one of the fixed basis functions — s-like for the conduction band, p-like (\(X,Y,Z\)) for the valence band. The envelope function \(F_n(z)\) is a slowly varying amplitude that multiplies each basis function and carries the effect of confinement. In a heterostructure the full state is a sum over basis functions,

\[ \Psi(\mathbf{r}) = e^{i\mathbf{k}_\parallel\cdot\boldsymbol{\rho}} \sum_n F_n(z)\, u_{n0}(\mathbf{r}), \]

and it is the set of envelopes \(\{F_n(z)\}\) that the solver computes.

2. The envelope-function approximation

The envelope-function approximation makes two assumptions: that the same zone-centre basis functions \(u_{n0}\) can be used in every layer, and that the material parameters (gap, masses, Kane and Luttinger parameters) vary with position on a scale much longer than the lattice constant. The bulk k·p matrix then becomes a differential operator acting on the envelopes, with position-dependent coefficients. The essential replacement is that the growth-direction wavevector becomes a differential operator,

\[ k_z \;\longrightarrow\; -i\,\frac{d}{dz}, \]

while the in-plane wavevectors \(k_x, k_y\) remain good quantum numbers because the structure is translationally invariant in the plane. Every \(k_z\) in the bulk Hamiltonian is replaced by this operator, and every \(k_z^2\) by the symmetric second derivative of the previous page. Products such as \(k_x k_z\) become \(k_x(-i\,d/dz)\), symmetrised so that the operator stays Hermitian when the coefficient varies. The result is a matrix differential equation for the envelopes, discretised as described on the finite-difference page.

3. Reading a block Hamiltonian

A multiband Hamiltonian is best read as a small number of blocks before its individual entries are examined. The coarsest division separates conduction-like from valence-like states,

\[ H = \begin{pmatrix} H_{cc} & H_{cv} \\[2pt] H_{vc} & H_{vv} \end{pmatrix}, \]

where \(H_{cc}\) describes the s-like conduction states, \(H_{vv}\) the p-like valence states, and \(H_{cv}=H_{vc}^\dagger\) the Kane momentum coupling between them. The conduction block is essentially a (non-parabolic) effective-mass operator; the valence block contains the heavy-hole, light-hole and split-off physics; and the coupling block is what makes the model "multiband" in the strong sense, tying the conduction non-parabolicity to the valence structure. Strain and spin-orbit terms add to the appropriate blocks.

4. Basis orderings used by OghmaNano

The numerical value of a Hamiltonian entry depends on the order in which the basis states are listed. OghmaNano uses a fixed Cartesian orbital basis for the zincblende and wurtzite models, and the total-angular-momentum basis for Luttinger–Kohn. The orderings are:

• 8-band zincblende and 8-band wurtzite: \(\{\,|S\!\uparrow\rangle,\ |S\!\downarrow\rangle,\ |X\!\uparrow\rangle,\ |Y\!\uparrow\rangle,\ |Z\!\uparrow\rangle,\ |X\!\downarrow\rangle,\ |Y\!\downarrow\rangle,\ |Z\!\downarrow\rangle\,\}\).
• 10-band zincblende: the eight above followed by two localised nitrogen states \(\{\,|N\!\uparrow\rangle,\ |N\!\downarrow\rangle\,\}\).
• 6-band Luttinger–Kohn: \(\{\,\text{HH}\!\uparrow,\ \text{LH}\!\uparrow,\ \text{LH}\!\downarrow,\ \text{HH}\!\downarrow,\ \text{SO}\!\uparrow,\ \text{SO}\!\downarrow\,\}\).
• Silicon Δ-valley: a single (scalar) band per valley, solved once for each of the six valley orientations.

Using the Cartesian \(|X\rangle,|Y\rangle,|Z\rangle\) orbitals rather than the total-angular- momentum \(|J,m_j\rangle\) combinations keeps the momentum and strain couplings real and simple to assemble, at the cost of leaving spin-orbit coupling as an explicit off-diagonal term. The Luttinger–Kohn model instead uses the \(|J,m_j\rangle\) basis, in which spin-orbit coupling is diagonal but the kinetic terms take the classic \(P,Q,R,S\) form. Both choices are physically equivalent; each is convenient for a different model, and both are documented on their respective pages.