Quantum wells and confinement

A quantum well is a thin layer of one semiconductor sandwiched between two layers of a wider-gap semiconductor. When the central layer is thin enough — typically a few nanometres, comparable to the carrier de Broglie wavelength — the component of motion across the layer is quantised into discrete energy levels, while motion in the plane of the layer remains free. The result is a system that is three-dimensional in the material sense but effectively two-dimensional in its electronic dynamics. Quantum wells are the active region of most semiconductor lasers and many photodetectors and modulators, and they are the subject of the OghmaNano quantum-well solvers documented in this part of the manual.

1. How a quantum well is formed

Heterostructures are grown one atomic layer at a time by molecular-beam epitaxy or metal-organic vapour-phase epitaxy. Because the conduction- and valence-band edges of the two materials differ, the spatial profile of the band edges along the growth direction \(z\) has the shape of a well. An electron in the conduction band sees a potential minimum in the low-gap layer bounded by the conduction-band offset \(\Delta E_c\); a hole in the valence band sees a well of depth \(\Delta E_v\). The two offsets together account for the band-gap difference,

\[ \Delta E_c + \Delta E_v = E_{g,\text{barrier}} - E_{g,\text{well}} . \]

How the gap difference divides between the two bands (the band alignment) is a material property and is supplied to OghmaNano through the conduction- and valence-band edges of each layer. The material-parameter page explains the energy references used internally.

2. Why confined states appear

Confinement is a direct consequence of wave mechanics. An electron of effective mass \(m^\*\) moving across a well of width \(L\) behaves like a wave with de Broglie wavelength \(\lambda = h/p\). When \(L\) becomes comparable to \(\lambda\), only standing-wave patterns that fit the well — with the correct decay into the barriers — are allowed. Each allowed pattern is a bound state with a definite energy. For an idealised infinitely deep well the energies follow the familiar particle-in-a-box result,

\[ E_n = \frac{\hbar^2}{2 m^\*}\left(\frac{n\pi}{L}\right)^2, \qquad n = 1,2,3,\dots \]

where \(\hbar\) is the reduced Planck constant. Two features carry over to real wells of finite depth. First, the level spacing grows as the well narrows (\(E_n \propto 1/L^2\)), so thinner wells push their levels further apart and further above the well bottom. Second, lighter carriers (smaller \(m^\*\)) are confined more strongly than heavier ones at the same width. A finite well differs in that it supports only a finite number of bound states and its wavefunctions leak into the barriers as decaying exponentials, as sketched in Figure 1. The next page makes this quantitative through the effective-mass Schrödinger equation.

3. Subbands: quantised across, free in the plane

Confinement quantises only the motion along \(z\). In the plane of the well the carrier is free, so its total energy is the sum of a discrete confinement energy and a continuous in-plane kinetic energy. In the simplest (parabolic) picture,

\[ E_n(k_\parallel) = E_n + \frac{\hbar^2 k_\parallel^2}{2 m^\*}, \qquad k_\parallel = \sqrt{k_x^2 + k_y^2}. \]

Each confined level \(E_n\) therefore becomes a two-dimensional subband: a parabola (more generally a curved sheet) in the in-plane wavevector \(k_\parallel\). The bulk continuum is replaced by a ladder of subbands, each with its own dispersion. This is the single most important structural change caused by confinement and it reshapes the density of states, the optical spectrum and the transport, all of which are treated later in this manual.

4. What OghmaNano computes

Given the layer structure and material parameters, the OghmaNano quantum-well solvers compute the confined states and their in-plane dispersions, and from these the optical and transport properties of the well. The solvers differ in how much band-structure physics they retain:

• A scalar parabolic-band solver for quick, single-band estimates of electron and hole levels.
• Multiband k·p solvers — 8-band and 10-band zincblende, 8-band wurtzite — that capture valence-band mixing, non-parabolicity and conduction–valence coupling.
• A six-valley silicon conduction solver and a six-band Luttinger–Kohn valence solver for Si, Ge and SiGe.
• Shared machinery for strain, the sparse eigensolver, optical matrix elements, gain and scattering.

The model-selection page compares the solvers and recommends which to use for a given material system. All of them start from the same physical idea developed on the next two pages: a confined carrier described by an envelope function obeying a Schrödinger-like equation.