Editing Quantum well (section)

=== 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" />