Editing Quantum well (section)

== Physics ==
Quantum wells and quantum well devices are a subfield of [[solid-state physics]] that is still extensively studied and researched today. The theory used to describe such systems uses important results from the fields of [[Quantum mechanics|quantum physics]], [[statistical physics]], and [[Electromagnetism|electrodynamics]].

=== 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.

=== Finite well model ===
The finite well model provides a more realistic model of quantum wells. Here the walls of the well in the heterostructure are modeled using a finite potential <math>V_0</math>, which is the difference in the conduction band energies of the different semiconductors. Since the walls are finite and the electrons can [[Quantum tunneling|tunnel]] into the barrier region. Therefore the allowed wave functions will penetrate the barrier wall.<ref name=":2" />

Consider a finite 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 the carriers experience no potential within the well and potential of <math>V_0</math> in the barrier regions.

The Schrodinger equation for carriers within the well is unchanged compared to the infinite well model, except for the boundary conditions at the walls, which now demand that the wave functions and their slopes are continuous at the boundaries.

Within the barrier region, Schrodinger’s equation for carriers reads:
: <math>-\frac{\hbar^2}{2m_\text{b}^*}\frac{\partial^2\psi(z)}{\partial z^2} + V_0\psi(z)=E\psi(z)</math>
where <math>m^*_\text{b}</math> is the effective mass of the carrier in the barrier region, which will generally differ from its effective mass within the well.<ref name=":0" />

==== Solutions and energy levels ====
[[File:MCM QW FINWELL.jpg|thumb|A sketch of the bound (<math>E < V_0</math>), quantized energy states for carriers in a finite well model. The wave functions are sinusoidal like but decay exponentially in the barrier region. The energies of these states are less than those in the infinite well model.]]
Using the relevant boundary conditions and the condition that the wave function must be continuous at the edge of the well, we get solutions for the wave vector <math>k</math> that satisfy the following [[transcendental equation]]s:
: <math>\tan\left(\frac{k_nd}{2}\right)=\frac{m_\text{w}^*\kappa}{m_\text{b}^*k_n}\quad\text{(even)}</math>
and
: <math>\tan\left(\frac{k_nd}{2}\right)=-\frac{m_\text{b}^*k_n}{m_\text{w}^*\kappa}\quad\text{(odd)},</math>
where <math>\kappa</math> is the exponential decay constant in the barrier region, which is a measure of how fast the wave function decays to zero in the barrier region. The quantized energy eigenstates inside the well, which depend on the wave vector and the quantum number (<math>n</math>) are given by:
: <math>E_n=\frac{\hbar^2 k_n^2}{2m_\text{w}^*}.</math>

The exponential decay constant <math>\kappa</math> is given by:
: <math>\kappa=\frac{\sqrt{2 m_\text{b}^* (V_0 - E_n)}}{\hbar}</math>

It depends on the eigenstate of a bound carrier <math>E_n</math>, the depth of the well <math>V_0</math>, and the effective mass of the carrier within the barrier region, <math>m^*_\text{b}</math>.

The solutions to the transcendental equations above can easily be found using [[Numerical analysis|numerical]] or graphical methods. There are generally only a few solutions. However, there will always be at least one solution, so one [[bound state]] in the well, regardless of how small the potential is. Similar to the infinite well, the wave functions in the well are sinusoidal-like but exponentially decay in the barrier of the well. This has the effect of reducing the bound energy states of the quantum well compared to the infinite well.<ref name=":0" />

=== Superlattices ===
[[File:GaAs-AlAs SL.svg|thumb|A heterostructure made of AlAs and GaAs arranged in a superlattice configuration. In this case, the resulting periodic potential arises due to the difference in band-gaps between materials.]]
A superlattice is a periodic heterostructure made of alternating materials with different band-gaps. The thickness of these periodic layers is generally of the order of a few nanometers. The band structure that results from such a configuration is a periodic series of quantum wells. It is important that these barriers are thin enough such that carriers can tunnel through the barrier regions of the multiple wells.<ref name=":3">Odoh, E. O., & Njapba, A. S. (2015). A review of semiconductor quantum well devices. ''Adv. Phys. Theor. Appl'', ''46'', 26-32.</ref> A defining property of superlattices is that the barriers between wells are thin enough for adjacent wells to couple. Periodic structures made of repeated quantum wells that have barriers that are too thick for adjacent wave functions to couple, are called multiple quantum well (MQW) structures.<ref name=":0" />

Since carriers can tunnel through the barrier regions between the wells, the wave functions of neighboring wells couple together through the thin barrier, therefore, the electronic states in superlattices form delocalized minibands.<ref name=":0" />  Solutions for the allowed energy states in superlattices is similar to that for finite quantum wells with a change in the boundary conditions that arise due to the periodicity of the structures. Since the potential is periodic, the system can be mathematically described in a similar way to a one-dimensional crystal lattice.