Parabolic bands and the effective-mass approximation

The simplest useful description of a quantum well treats each band as an isolated parabola characterised by a single effective mass. This parabolic-band or effective-mass approximation reduces the many-electron crystal problem to a one-particle Schrödinger equation for a slowly varying envelope function. It is the basis of the OghmaNano parabolic solver and the conceptual starting point for the multiband k·p models that follow.

1. The effective-mass idea

Near a band extremum the energy of a bulk carrier varies quadratically with wavevector,

\[ E(\mathbf{k}) = E_0 + \frac{\hbar^2 k^2}{2 m^\*}, \]

where \(E_0\) is the band edge and \(m^\*\) the effective mass, defined through the band curvature \(1/m^\* = \hbar^{-2}\,\partial^2 E/\partial k^2\). The effective mass packages the influence of the crystal potential into a single number: a carrier in the crystal responds to external forces as though it were a free particle of mass \(m^\*\). The curvature of the band therefore sets the mass — light bands are strongly curved, heavy bands nearly flat.

2. The envelope-function Schrödinger equation

In a heterostructure the band edge \(E_c(z)\) and the effective mass \(m^\*(z)\) both vary with position along the growth direction. The full wavefunction is written as a rapidly oscillating Bloch function modulated by a slowly varying envelope \(\psi(z)\). The envelope obeys a one-dimensional Schrödinger equation in which the band edge acts as the potential. Writing the in-plane motion as a plane wave \(e^{i\mathbf{k}_\parallel\cdot \boldsymbol{\rho}}\), the envelope satisfies

\[ \left[ -\frac{\hbar^2}{2}\,\frac{d}{dz}\!\left(\frac{1}{m^\*(z)}\frac{d}{dz}\right) + \frac{\hbar^2 k_\parallel^2}{2 m^\*(z)} + V(z) \right] \psi(z) = E\,\psi(z). \]

Here \(V(z)\) is the band-edge profile — \(E_c(z)\) for electrons, or the reflected valence profile for holes — and the first term is the across-well kinetic energy. Two points deserve emphasis. First, the kinetic operator is written with the mass inside the derivatives, as \(\tfrac{d}{dz}\tfrac{1}{m^\*}\tfrac{d}{dz}\), rather than \(\tfrac{1}{m^\*}\tfrac{d^2}{dz^2}\). This symmetric (BenDaniel–Duke) form keeps the operator Hermitian when the mass jumps at an interface and conserves probability current across it. Second, the in-plane term simply adds \(\hbar^2 k_\parallel^2/2m^\*\) to every level, reproducing the subband parabolas of the previous page.

3. The parabolic approximation and its limits

The parabolic model assumes the effective mass is a constant of each material, independent of energy and of the state considered. This is accurate close to a band edge but degrades as carriers are pushed to higher energy, for three reasons. Real conduction bands are non-parabolic: the mass grows with energy because of coupling to the valence band. The valence band is not a single parabola at all but several strongly coupled bands (heavy hole, light hole, split-off) whose masses depend on direction and mix at finite \(k_\parallel\). And strain and confinement can reorder and warp the valence bands beyond any single-mass description. When these effects matter — which is almost always the case for the valence band and for narrow-gap or highly confined systems — a multiband k·p model is required.

4. The OghmaNano parabolic solver

OghmaNano builds two independent scalar problems, one for electrons and one for holes. It places the valence-band maximum of the structure at the energy zero and measures both band edges from that common reference, so that electron and hole levels are reported on the same scale. The electron problem uses the conduction-band profile \(E_c(z)\) as its potential and the electron effective mass \(m_e(z)\); the hole problem uses the reflected valence profile so that confinement corresponds again to a potential minimum, together with the hole mass \(m_h(z)\). Each problem is discretised on a uniform grid and solved with the shared block shift-invert eigensolver. The in-plane dispersion is then generated by scanning \(k_\parallel\) and adding the parabolic in-plane term, and the electron–hole envelope overlap is evaluated for the gain calculation.