Editing Quantum well (section)

=== Infinite well model ===
The simplest model of a quantum well system is the infinite well model. The walls/barriers of the potential well are assumed to be infinite in this model. In reality, the quantum wells are generally of the order of a few hundred milli[[electronvolt]]s. However, as a first approximation, the infinite well model serves as a simple and useful model that provides some insight into the physics behind quantum wells.<ref name=":0" />

Consider an infinite quantum well oriented in the ''z''-direction, such that carriers in the well are confined in the ''z''-direction but free to move in the ''x''–''y'' plane. we choose the quantum well to run from <math>z = 0</math> to <math>z = d</math>. We assume that carriers experience no potential within the well and that the potential in the barrier region is infinitely high.

The [[Schrödinger equation]] for carriers in the infinite well model is:
: <math>-\frac{\hbar^2}{2m_\text{w}^*}\frac{\partial^2\psi(z)}{\partial z^2} = E\psi(z)</math>
where <math>\hbar</math> is the [[reduced Planck constant]] and <math>m^*_\text{w}</math> is the [[Effective mass (solid-state physics)|effective mass]] of the carriers within the well region. The effective mass of a carrier is the mass that the electron "feels" in its quantum environment and generally differs between different semiconductors as the value of effective mass depends heavily on the curvature of the band. Note that <math>m^*_\text{w}</math>  can be the effective mass of electrons in a well in the conduction band or for holes in a well in the valence band.

==== Solutions and energy levels ====
[[File: MCM QW INFWELL.jpg|thumb|The first two energy states in an infinite well quantum well model. The walls in this model are assumed to be infinitely high. The solution wave functions are sinusoidal and go to zero at the  boundary of the well.]]
The solution [[wave function]]s cannot exist in the barrier region of the well, due to the infinitely high potential. Therefore, by imposing the following boundary conditions, the allowed wave functions are obtained,
: <math>\psi(0) = \psi(d) = 0.</math>

The solution wave functions take the following form:
: <math>\psi_n(z) = \sqrt{\frac{2}{d}} \sin(k_nz) \qquad k_n=\frac{n\pi}{d}.</math>

The subscript <math>n</math>, (<math>n > 0</math>) denotes the integer [[quantum number]] and <math>k_n</math> is the [[wave vector]] associated with each state, given above. The associated discrete energies are given by:
: <math>E_n=\frac{\hbar^2k_n^2}{2m_\text{w}^*} = \frac{\hbar^2}{2m_\text{w}^*} \left(\frac{n\pi}{d}\right)^2.</math>

The simple infinite well model provides a good starting point for analyzing the physics of quantum well systems and the effects of quantum confinement. The model correctly predicts that the energies in the well are inversely proportional to the square of the length of the well. This means that precise control over the width of the semiconductor layers, i.e. the length of the well, will allow for precise control of the energy levels allowed for carriers in the wells. This is an incredibly useful property for [[band-gap engineering]]. Furthermore, the model shows that the energy levels are proportional to the inverse of the effective mass. Consequently, heavy holes and light holes will have different energy states when trapped in the well. Heavy and light holes arise when the maxima of valence bands with different curvature coincide; resulting in two different effective masses.<ref name=":0" />

A drawback of the infinite well model is that it predicts many more energy states than exist, as the walls of real quantum wells, are finite. The model also neglects the fact that in reality, the wave functions do not go to zero at the boundary of the well but 'bleed' into the wall (due to quantum tunneling) and decay exponentially to zero. This property allows for the design and production of superlattices and other novel quantum well devices and is described better by the finite well model.