Quantum Wells and Carrier Confinement
What Is a Quantum Well?
A quantum well is a region of space where the potential energy drops below that of its surroundings, creating an energy barrier that can trap a particle, such as an electron. When the width of the well becomes comparable to the wavelength of the electron, quantum mechanical effects become particularly important.
A typical quantum well is illustrated in ??. In semiconductor devices, such structures can be created by growing a thin layer of one material between two materials with different electronic band edges. A common example is a GaAs quantum well surrounded by AlGaAs barriers.
Describing a Quantum Well with Schrödinger's Equation
When the width of a potential well becomes comparable to the wavelength of an electron, we can no longer adequately describe the electron as a classical particle with a definite position. Instead, we must describe its wave-like behaviour and the probability of finding it at different positions within the well.
To do this, we solve the time-independent Schrödinger equation:
\[ \hat H\psi(x)=E\psi(x). \]
Here, \(\psi(x)\) is the electron's wavefunction, \(E\) is its energy, and \(\hat H\) is the Hamiltonian, which describes the kinetic and potential energy of the electron. The probability density of finding the electron at a particular position is given by \(|\psi(x)|^2\).
For a simple one-dimensional quantum well, Schrödinger's equation can be written as:
\[ \boxed{ -\frac{\hbar^2}{2m^*} \frac{d^2\psi(x)}{dx^2} +V(x)\psi(x) =E\psi(x). } \]
Although this equation may initially look complicated, the individual terms have straightforward physical meanings:
- \(V(x)\) describes the potential-energy profile of the well, including the surrounding barriers.
- \(\psi(x)\) is the wavefunction describing the electron. Its squared magnitude gives the probability density of finding the electron at position \(x\).
- \(E\) is the energy associated with that particular quantum state.
- \(m^*\) is the electron's effective mass, which describes how it responds to forces within the semiconductor. The first term represents its kinetic energy.
By supplying the potential profile \(V(x)\) and solving this equation, we obtain the allowed electron energies and their corresponding wavefunctions. These are precisely the results illustrated in ??: the horizontal energy levels and the coloured wavefunctions associated with each state. In the following section, we will see how these solutions can be calculated for a simple quantum well.
What Does This Equation Mean in Reality?
An intuitive way to understand the solutions to Schrödinger's equation is to consider a violin string fixed at both ends. Such a string can be excited at a discrete set of harmonics. The fundamental harmonic has a single peak, while higher harmonics have additional peaks and nodes. Each harmonic has its own characteristic frequency.
A similar principle applies to an electron confined within a quantum well. The dimensions of the well restrict the electron to a particular set of permitted wavefunctions, each associated with a specific energy. Just as the violin string has a fundamental harmonic and higher harmonics, the quantum well has a lowest-energy state and a series of higher-energy states.
The first three of these states are illustrated in ??. The dashed horizontal lines indicate the permitted energies \(E_1\), \(E_2\) and \(E_3\), while the coloured curves show their corresponding wavefunctions. The lowest-energy state has the simplest shape, with higher-energy states exhibiting progressively more complicated wave patterns.
But what do these wavefunctions actually tell us? Unlike the displacement of a vibrating violin string, an electron's wavefunction has a probabilistic interpretation. The square of its magnitude gives the probability density of finding the electron at a particular position:
\[ \boxed{ P(x)=|\psi(x)|^2 } \]
For example, an electron occupying the lowest-energy state is most likely to be found near the centre of the well. Higher-energy states have more complicated probability distributions, with several peaks separated by nodes where the probability falls to zero. Notice also that the wavefunctions extend slightly into the surrounding barriers, meaning there is a small probability of finding the electron outside the well itself.
Thus, solving Schrödinger's equation tells us two important things: which energies an electron is permitted to occupy, and the probability distribution associated with each of those energies.
A Simple Solution: The Infinite Quantum Well
Let us now try to calculate the wavefunctions of a quantum well. To make the mathematics easier, we will start by imagining that the barriers surrounding the well are infinitely high. This means that the electron cannot exist outside the well, and its wavefunction must fall to zero at both edges.
We will consider a well of width \(L\), extending from \(x=0\) to \(x=L\). Inside the well, we set the potential energy to zero, so \(V(x)=0\). Starting with Schrödinger's equation:
\[ -\frac{\hbar^2}{2m^*} \frac{d^2\psi(x)}{dx^2} +V(x)\psi(x) =E\psi(x), \]
we can remove the potential-energy term, leaving:
\[ -\frac{\hbar^2}{2m^*} \frac{d^2\psi(x)}{dx^2} =E\psi(x). \]
We now need to find a wavefunction that satisfies this equation. A sensible place to start is to guess a sinusoidal solution:
\[ \psi(x)=A\sin(kx). \]
Here, \(A\) determines the amplitude of the wavefunction, while \(k\) describes how rapidly it oscillates with position. A larger value of \(k\) means a shorter wavelength, so more oscillations can fit into a given distance.
We can check whether this is a valid solution by differentiating it twice:
\[ \frac{d^2\psi(x)}{dx^2} =-k^2A\sin(kx) =-k^2\psi(x). \]
Notice that differentiating our guessed function twice simply returns the original function multiplied by a constant. This is precisely the mathematical behaviour required by Schrödinger's equation inside the well.
Substituting our guess into the equation confirms that a sinusoidal wavefunction is a valid solution, provided its associated energy satisfies:
\[ E=\frac{\hbar^2k^2}{2m^*}. \]
We will return to the energy shortly. First, however, there is a more interesting question: which of these sinusoidal solutions can actually exist inside our quantum well?
Which Wavelengths Can Exist Inside the Well?
Although we have found a sinusoidal solution to Schrödinger's equation, not every sinusoid can physically exist inside the well. Remember that our barriers are infinitely high, so the wavefunction must be zero at both edges.
This is very similar to the behaviour of a violin string. As illustrated in ??, a string fixed at both ends can be excited at a discrete set of harmonics. The fundamental harmonic has a single peak, the second has two peaks, the third has three, and so on. Each harmonic satisfies the requirement that the displacement of the string is zero at both ends.
Exactly the same mathematical restriction applies to our infinite quantum well. The permitted wavefunctions must fit between \(x=0\) and \(x=L\), with a value of zero at each boundary.
Our guessed sinusoidal wavefunction already satisfies the first boundary, since \(\sin(0)=0\). To satisfy the second boundary, we require:
\[ \psi(L)=A\sin(kL)=0. \]
A sine function is zero whenever its argument is an integer multiple of \(\pi\). Therefore, the only permitted values are:
\[ kL=n\pi, \qquad n=1,2,3,\ldots \]
Rearranging this gives:
\[ \boxed{ k_n=\frac{n\pi}{L}, \qquad n=1,2,3,\ldots } \]
We have therefore discovered something important: although Schrödinger's equation admits sinusoidal solutions, the boundaries of the well restrict them to a particular set of wavelengths. These are directly analogous to the harmonics of our violin string.
We can write the corresponding wavefunctions as:
\[ \boxed{ \psi_n(x)= \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \qquad 0 \lt x \lt L } \]
The factor in front of the sine function simply normalises the wavefunction, ensuring that the total probability of finding the electron somewhere inside the well is one. Each value of \(n\) gives us a different permitted probability distribution.
Calculating the Energy Levels
Now that we understand which wavefunctions can exist inside the quantum well, we can return to their energies.
There is a useful connection with our violin analogy. The higher harmonics of a violin string have progressively higher frequencies. Similarly, the higher states of a quantum well correspond to progressively higher electron energies.
We can calculate these energies using the relationship obtained earlier when we substituted our sinusoidal solution into Schrödinger's equation:
\[ E=\frac{\hbar^2k^2}{2m^*}. \]
We have already established that only particular values of \(k\) are permitted:
\[ k_n=\frac{n\pi}{L}. \]
Substituting this into the energy expression gives:
\[ E_n= \frac{\hbar^2}{2m^*} \left(\frac{n\pi}{L}\right)^2. \]
Rearranging, we obtain the familiar result for the permitted energies of an infinite quantum well:
\[ \boxed{ E_n=\frac{n^2\pi^2\hbar^2}{2m^*L^2} } \]
This result tells us that the energy increases with the square of the state number \(n\). In other words, the second state has four times the energy of the first, while the third has nine times the energy.
It also reveals something particularly useful for semiconductor engineering: because the energy depends on \(1/L^2\), we can increase the confinement energy simply by making the quantum well narrower. The electron's effective mass also matters, with a smaller effective mass producing higher energy levels for an otherwise identical well.
For example, an infinite well with a width of \(12\,\mathrm{nm}\) and an electron effective mass of \(m^*=0.067m_0\) has a lowest energy of approximately \(39\,\mathrm{meV}\). The first three permitted energies are therefore:
\[ E_1\approx39\,\mathrm{meV}, \qquad E_2=4E_1, \qquad E_3=9E_1. \]
We have now calculated both the permitted wavefunctions and their associated energies. However, our assumption of infinitely high barriers is a mathematical simplification. To describe the more realistic quantum well shown in ??, we next need to consider what happens when the barriers have a finite height.
What Happens When the Barriers Are Not Infinite?
In a real semiconductor quantum well, the surrounding barriers have a finite height. This introduces an important difference: the electron's wavefunction no longer has to fall to zero at the edges of the well. Instead, it can extend a short distance into the surrounding material, even when the electron's energy is below the barrier.
We can describe this using the same Schrödinger equation, but we now need to account for two different regions: the well itself, where \(V(x)=0\), and the surrounding barriers, where \(V(x)=V_0\). For a well centred at \(x=0\), the potential is:
\[ V(x)= \begin{cases} 0, & \lvert x\rvert \lt \dfrac{L}{2}, \\[6pt] V_0, & \lvert x\rvert \geq \dfrac{L}{2}. \end{cases} \]
Let us consider an electron with an energy below the barrier
height, so that \(0
The important difference is that the wavefunction now behaves
differently in the two regions. Inside the well it continues
to oscillate, just as it did in our infinite-well example.
However, inside the barriers it decays exponentially rather
than immediately falling to zero.
This happens because Schrödinger's equation changes when the
potential energy becomes greater than the electron's energy.
The rate at which the wavefunction decays into the barriers
is described by:
\[
\boxed{
\kappa =
\frac{\sqrt{2m^{*}(V_0-E)}}{\hbar}
}
\]
To calculate the permitted energy levels, we must find
solutions for which the oscillating wavefunction inside
the well joins smoothly to the exponentially decaying
wavefunction in the barriers. This is normally done
numerically.
The result is illustrated in
??. Notice how the coloured
wavefunctions extend beyond the edges of the well.
As the square of the wavefunction represents probability
density, this means there is a small probability of
finding the electron within the surrounding barriers.
Because the electron is effectively confined over a slightly
larger region, its energy levels are generally lower than
those predicted by the infinite-well approximation.
Furthermore, a finite well can support only a finite number
of bound states.
The simple infinite-well formula is useful for understanding confinement, but a
realistic device usually needs its actual material layers, band offsets,
effective masses and electric fields included in the calculation. OghmaNano provides
several levels of quantum-well modelling, depending on which effects are important:
🧪 Try this in OghmaNano:
Work through the
GaAs/AlGaAs parabolic quantum-well tutorial
to calculate confined electron and hole states, energy levels and wavefunctions using the ideas developed above.
For more advanced models, see the
k·p band-structure, quantum-well and mobility solver.
The resulting eigenvalues and wavefunctions can then be used in subsequent calculations
of optical transitions, gain and carrier scattering. A practical introduction is provided by the
GaAs/AlGaAs parabolic quantum-well tutorial.
More advanced calculations, including multiband k·p, silicon/SiGe quantum wells,
optical spectra and carrier mobility, are collected in the
k·p band-structure, quantum-well and mobility solver.
One of the most useful properties of a quantum well is that we can
control the energy levels of electrons and holes simply by changing
the width of the well. Making the well narrower generally increases
the separation between the confined energy levels, while making it
wider brings them closer together. This gives us a way of controlling
the electronic and optical properties of a material without necessarily
changing its chemical composition.
Quantum wells are particularly important in semiconductor devices
because they allow us to control where charge carriers are located
and which energy transitions are available to them. Applications include:
The important idea is that a quantum well allows us to design
electronic energy levels through the geometry of a structure.
To understand how these levels arise, we need to solve
Schrödinger's equation for a particle confined within the well.
Calculating Quantum Wells in OghmaNano
Why Are Quantum Wells Interesting?