Quantum Well Band Structure and the Parabolic-Band Approximation

From Quantum Wells to Electronic Bands

In our introduction to quantum wells, we used Schrödinger's equation to calculate the permitted energies and wavefunctions of an electron confined within a potential well. An example of these solutions is illustrated in ??.

However, this calculation only describes how the electron behaves across the width of the well. A semiconductor quantum well is normally formed from a very thin layer of material extending over a much larger area. If we define the \(z\) direction as being perpendicular to the semiconductor layers, then the electron is confined along \(z\), but remains free to move in the \(x\) and \(y\) directions, parallel to the layers.

In other words, the electron is confined in one dimension but can still move in the other two. This is why quantum wells are commonly described as two-dimensional electronic systems.

To understand the consequences of this, we need to connect the discrete energy levels obtained from Schrödinger's equation to the electronic band structure of the semiconductor.

From Confined Energy Levels to Parabolic Bands

Consider the quantum well shown on the left-hand side of ??. Solving Schrödinger's equation across the well produces a series of confined states, with energies \(E_1\), \(E_2\) and \(E_3\).

These energies describe the electron's confinement in the \(z\) direction. However, the electron can also move parallel to the layers, in the \(x\) and \(y\) directions. As it gains momentum in this plane, its total energy increases.

To describe this motion, we introduce the in-plane wavevector, written as \(k_{\parallel}\). Its magnitude is related to the wavevector components along the two unconfined directions:

\[ k_{\parallel}^{2}=k_x^2+k_y^2. \]

For a simple semiconductor with a parabolic electronic band, the additional energy associated with this in-plane motion is:

\[ E_{\mathrm{parallel}}= \frac{\hbar^2 k_{\parallel}^{2}}{2m^{*}}, \]

where \(m^{*}\) is the electron's effective mass. The effective mass describes how the electron responds to forces within the crystal, accounting for the influence of the semiconductor's electronic structure.

We can now combine the energy obtained from quantum confinement with the additional energy associated with motion parallel to the layers:

\[ \boxed{ E_n(k_{\parallel})= E_n+\frac{\hbar^2 k_{\parallel}^{2}}{2m^{*}} } \]

This equation tells us that each confined energy level becomes the starting point of a parabola, as illustrated on the right-hand side of ??. These parabolas are known as electronic subbands.

At zero in-plane wavevector, the bottom of each parabola corresponds exactly to the energy calculated by solving Schrödinger's equation across the quantum well. As the in-plane wavevector increases, the electron moves up the parabola and gains energy while remaining in the same confined state along \(z\).

We have therefore transformed a set of discrete quantum-well energy levels into a set of continuous, two-dimensional electronic subbands.

Why Do We Use Parabolic Bands?

To understand why semiconductor bands are often represented by parabolas, consider something much more familiar: throwing a cricket ball. The kinetic energy of the ball depends on its mass and velocity:

\[ E_{\mathrm{kinetic}}=\frac{1}{2}mv^2. \]

This tells us that if we throw the ball twice as fast, it has four times as much kinetic energy. Since momentum is given by \(p=mv\), we can also write this relationship as:

\[ E_{\mathrm{kinetic}}=\frac{p^2}{2m}. \]

If we plot the energy of our cricket ball against its momentum, we obtain a parabola. The ball has its lowest kinetic energy when its momentum is zero, and its energy increases quadratically as its momentum increases.

Interestingly, we can use a very similar relationship to describe an electron inside a semiconductor. In quantum mechanics, momentum is related to the wavevector through \(p=\hbar k\). Substituting this into our previous expression gives:

\[ E_{\mathrm{kinetic}}=\frac{\hbar^2 k^2}{2m}. \]

However, an electron inside a semiconductor is not moving through empty space. It interacts with the periodic potential of the crystal, which changes the relationship between its energy and motion. Close to the bottom of many conduction bands, we can account for this by replacing the ordinary electron mass with an effective mass, \(m_e^{*}\).

We also need to include the energy at the bottom of the conduction band, \(E_c\), giving us the parabolic-band approximation:

\[ \boxed{ E(k)=E_c+\frac{\hbar^2 k^2}{2m_e^{*}} } \]

The effective mass determines the curvature of the parabola. A smaller effective mass gives a more strongly curved band, meaning that the electron gains energy more rapidly as its wavevector increases. A larger effective mass produces a flatter band.

The advantage of this approximation is that it allows us to describe an electronic band using just its band-edge energy and effective mass. Once we have calculated the confinement energies using Schrödinger's equation, we can use this relationship to construct the corresponding quantum-well subbands. This calculation can be explored directly in the GaAs/AlGaAs parabolic quantum-well tutorial.

Introducing Electrons and Holes

So far, we have considered only electrons in the conduction band. However, a semiconductor also contains a valence band, in which the absence of an electron can be described as a positively charged particle called a hole.

Close to the conduction-band minimum, the electron energy can be approximated using an upward-facing parabola:

\[ E_c(k)=E_c+\frac{\hbar^2 k^2}{2m_e^{*}}. \]

Similarly, close to a valence-band maximum, we can approximate the valence electron energies using a downward-facing parabola:

\[ E_v(k)=E_v-\frac{\hbar^2 k^2}{2m_h^{*}}. \]

These two parabolas are illustrated in ??. The conduction-band minimum and valence-band maximum are separated by the semiconductor band gap, \(E_g\).

Just as electrons can be confined within a conduction-band quantum well, holes can be confined by a suitable valence-band potential. Solving Schrödinger's equation for both carrier types therefore produces two sets of confined energy levels, from which we can construct the corresponding electron and hole subbands.

From Quantum Wells to Light Emission and Absorption

We have now seen how solving Schrödinger's equation for an electron confined in a quantum well produces a series of energy levels, each of which becomes a parabolic subband when motion parallel to the semiconductor layers is included.

We can perform exactly the same calculation for holes. This is illustrated in ??, where the upper diagrams show the confined electron states in the conduction band, while the lower diagrams show the corresponding confined hole states in the valence band. For clarity, the valence-band well is drawn inverted, with the hole confinement energies measured downwards from the valence-band maximum.

Once we have calculated both sets of states, we can begin to describe the interaction of the quantum well with light. For example, an electron occupying a conduction-band state can recombine with a hole in a valence-band state, releasing energy in the form of a photon. The reverse process is also possible: an incoming photon can be absorbed, exciting an electron from the valence band into the conduction band.

Calculating the Transition Energy

Consider the solid red arrow in ??. This illustrates a transition between the lowest electron and hole subbands at zero in-plane wavevector.

The energy required for this transition is the semiconductor band gap, plus the confinement energies of the electron and hole:

\[ \boxed{ E_{\mathrm{photon}} = E_g+E_{e,n}+E_{h,m} } \]

Here, \(E_g\) is the band gap of the quantum-well material, while \(E_{e,n}\) and \(E_{h,m}\) are the electron and hole confinement energies, measured relative to their respective band edges.

For the example shown in the figure, the band gap is \(1.42\,\mathrm{eV}\), the lowest electron confinement energy is \(39\,\mathrm{meV}\), and the lowest hole confinement energy is \(7\,\mathrm{meV}\). The resulting transition energy is therefore:

\[ E_{\mathrm{photon}} = 1.42+0.039+0.007 = 1.466\,\mathrm{eV}. \]

Notice that this is slightly larger than the band gap of the bulk semiconductor. By confining the electrons and holes within a quantum well, we have increased the energy required for the optical transition. This gives us a way of adjusting the absorption and emission wavelength simply by changing the well width.

What Happens at Higher In-Plane Momentum?

Electrons and holes are not restricted to the bottoms of their respective subbands. They can also move parallel to the semiconductor layers, gaining additional energy as their in-plane wavevector increases.

This is illustrated by the dashed red arrow in the figure. For a direct optical transition, the in-plane crystal wavevector is approximately conserved because the momentum carried by a photon is comparatively small.

In our simple parabolic-band model, the transition energy therefore becomes:

\[ \boxed{ E_{\mathrm{photon}}(k_{\parallel}) = E_g+E_{e,n}+E_{h,m} + \frac{\hbar^2 k_{\parallel}^{2}}{2\mu} } \]

Here, \(\mu\) is the reduced effective mass of the electron-hole pair, defined through:

\[ \frac{1}{\mu} = \frac{1}{m_e^{*}} + \frac{1}{m_h^{*}}. \]

As the in-plane wavevector increases, the transition energy also increases. This means that an optical transition between two subbands is not restricted to a single photon energy: different occupied states along the parabolas can contribute to a wider range of optical transitions.

How Accurate Is the Parabolic Approximation?

The model we have developed is intentionally simple. It assumes that each electronic band can be represented by a parabola with a suitable effective mass. This is often a useful approximation close to a conduction-band minimum, where the relationship between energy and wavevector is approximately quadratic.

However, real semiconductor band structures are more complicated. An example is shown in ??, which illustrates the calculated electronic band structure of GaAs.

Notice that the conduction band is approximately parabolic close to its minimum at the Gamma point, but deviates from this simple shape as we move further away. The valence band is more complicated still, containing heavy-hole, light-hole and split-off bands which can interact with one another.

It is also important to recognise that the horizontal axis of this figure follows a path through reciprocal space, from L through Gamma to X. It is not a plot of the position through a semiconductor device. It describes how the permitted electron energies vary with crystal wavevector.

When band mixing, non-parabolicity, strain or strong confinement become important, we need to go beyond the simple effective-mass approximation. One way to do this is through multiband k·p theory, which allows several electronic bands to be treated together rather than approximating each as an independent parabola.