Excitons: Binding Energy and Bohr Radius

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to your next lesson on the optical properties of solids.

In our previous lessons, we built a picture of optical absorption where photons create electron-hole pairs, treating them as independent entities. We also explored how causality links the absorptive and dispersive parts of the dielectric function via the Kramers-Kronig relations. Today, we will refine our model by incorporating a crucial piece of physics we've ignored so far: the Coulomb attraction between the newly created electron and hole.

This interaction can lead to the formation of a bound state, a new quasiparticle called an exciton. Our learning outcome for this lesson is to model an exciton as a hydrogenic bound state of an electron-hole pair and calculate its binding energy and Bohr radius. Understanding excitons is fundamental to interpreting the sharp, distinct absorption features that appear just below the band edge in the optical spectra of many semiconductors.

1. The Concept of an Exciton

When a photon with energy $ℏ ω \geq E_{g}$ is absorbed by a semiconductor, it promotes an electron from the valence band to the conduction band, leaving behind a positively charged hole. Instead of moving independently, the negatively charged electron and positively charged hole can form a bound pair via their mutual Coulomb attraction, much like the electron and proton in a hydrogen atom. This electrically neutral electron-hole pair, which can move through the crystal, is the quasiparticle we call an exciton.

There are two primary types of excitons, distinguished by the spatial extent of the electron-hole wavefunction.

Wannier-Mott Excitons: These are found in most common semiconductors (e.g., Si, GaAs, CdS). The electron and hole are separated by many lattice constants, so their interaction is screened by the dielectric response of the crystal lattice. Because they are weakly bound and have a large radius, we can effectively treat them as a "hydrogen atom" embedded in a continuous dielectric medium. This is the type we will focus on today.
Frenkel Excitons: These are found in materials where atoms or molecules are weakly interacting, such as molecular crystals and some insulators (e.g., alkali halides). The electron-hole pair is tightly bound and localized on or near a single atomic or molecular site. Their binding energies are much larger, often on the order of an eV.

To solidify these concepts, please start by reading the introductory section on excitons in the following lecture notes.

Optical Properties of Solids

These notes from R.A. Taylor's course at Oxford provide a concise introduction to excitons, clearly distinguishing between the Wannier-Mott and Frenkel types.

Please read the section titled 'III Excitons' (page 13). Focus on the definitions and characteristics of Wannier-Mott and Frenkel excitons. Pay attention to the typical binding energies and materials where each type is found.

2. The Hydrogenic Model for a Wannier Exciton

Our goal is to model a Wannier exciton. Given your background in quantum mechanics, you'll recognize the setup as a classic two-body central force problem. We have an electron with position $r_{e}$ and effective mass $m_{e}^{*}$ , and a hole with position $r_{h}$ and effective mass $m_{h}^{*}$ . The potential energy of their interaction is the Coulomb potential, but screened by the material's static dielectric constant, $ϵ_{r}$ .

The Hamiltonian for the pair is:

H = - \frac{ℏ ^{2}}{2 m _{e}^{*}} \nabla_{e}^{2} - \frac{ℏ ^{2}}{2 m _{h}^{*}} \nabla_{h}^{2} - \frac{e ^{2}}{4 π ϵ _{0} ϵ _{r} ∣ r _{e} - r _{h} ∣}

Here, $\nabla_{e}^{2}$ and $\nabla_{h}^{2}$ are Laplacians with respect to the electron and hole coordinates, respectively.

Hamiltonian of Hydrogenic Model for Excitons — This image shows the Hamiltonian for a Wannier exciton. The system is modeled as a two-particle electron-hole pair, analogous to a hydrogen atom or positronium, but with the particles' effective masses and a screened Coulomb interaction determined by the material's dielectric constant $\epsilon_r$.

As with the hydrogen atom, we can simplify this by transforming to center-of-mass (CM) and relative coordinates:

CM coordinate: $R = \frac{m _{e}^{*} r _{e} + m _{h}^{*} r _{h}}{m _{e}^{*} + m _{h}^{*}}$
Relative coordinate: $r = r_{e} - r_{h}$

The Hamiltonian separates into two independent parts: $H = H_{C M} + H_{r e l}$ .

Center-of-Mass Motion:
$H_{C M} = - \frac{ℏ ^{2}}{2 M} \nabla_{R}^{2}$
This describes the free motion of the exciton as a whole through the crystal. The total mass is $M = m_{e}^{*} + m_{h}^{*}$ . The eigenstates are plane waves with momentum $ℏ K$ , and the energy is the kinetic energy of the exciton, $E_{k in} = \frac{ℏ ^{2} K ^{2}}{2 M}$ .
Relative Motion:
$H_{r e l} = - \frac{ℏ ^{2}}{2 μ} \nabla_{r}^{2} - \frac{e ^{2}}{4 π ϵ _{0} ϵ _{r} r}$
This describes the internal state of the exciton. The mass here is the reduced mass of the electron-hole pair:
$μ = \frac{m _{e}^{*} m _{h}^{*}}{m _{e}^{*} + m _{h}^{*}}$

This relative Hamiltonian is mathematically identical to that of the hydrogen atom, with two key substitutions:

The electron mass $m_{e}$ is replaced by the reduced mass $μ$ .
The permittivity of free space $ϵ_{0}$ is replaced by the screened permittivity $ϵ_{0} ϵ_{r}$ .

3. Exciton Binding Energy and Bohr Radius

Since the relative Hamiltonian has the same form as the hydrogen atom, we can immediately write down its energy eigenvalues and the characteristic length scale.

Binding Energy

The energy levels of the hydrogen atom are given by $E_{n} = - E_{H} / n^{2}$ , where $E_{H} = \frac{m _{e} e ^{4}}{2 ( 4 π ϵ _{0} ) ^{2} ℏ ^{2}} \approx 13.6$ eV is the Rydberg energy. By making the substitutions $μ \to m_{e}$ and $ϵ_{r} \to 1$ , the energy levels for the relative motion of the exciton are:

E_{r e l, n} = - \frac{μ e ^{4}}{2 ( 4 π ϵ _{0} ϵ _{r} ) ^{2} ℏ ^{2}} \frac{1}{n ^{2}} = - (\frac{μ}{m _{e}} \frac{1}{ϵ _{r}^{2}}) \frac{E _{H}}{n ^{2}} = - \frac{R _{X}}{n ^{2}}

where $n = 1, 2, 3, \dots$ is the principal quantum number.

The quantity $R_{X}$ is the exciton Rydberg, and its value for $n = 1$ is the exciton binding energy, $E_{b} = R_{X}$ . This is the energy required to break the exciton apart into a free electron and a free hole.

Exciton Bohr Radius

Similarly, the Bohr radius of the hydrogen atom is $a_{0} = \frac{4 π ϵ _{0} ℏ ^{2}}{m _{e} e ^{2}} \approx 0.529$ Å. The corresponding characteristic size of the exciton ground state is the exciton Bohr radius:

a_{X} = \frac{4 π ϵ _{0} ϵ _{r} ℏ ^{2}}{μ e ^{2}} = (\frac{m _{e}}{μ} ϵ_{r}) a_{0}

This represents the average separation between the electron and the hole in the exciton's ground state.

The following resource provides a slightly different but complementary derivation of these results.

Lecture note on Physics of Semiconductors (4)

These notes on semiconductor physics explicitly write out the total energy of the exciton state and define the reduced mass, reinforcing the concepts we just discussed.

Read the text associated with equation (3.33) under the heading '3.3.2 Excitons'. This section connects the bandgap energy, the CM kinetic energy, and the hydrogenic binding energy into a single expression for the total exciton energy.

A Concrete Example: Gallium Arsenide (GaAs)

Let's calculate these values for a typical semiconductor, GaAs.

Electron effective mass: $m_{e}^{*} \approx 0.067 m_{e}$
Heavy hole effective mass: $m_{h}^{*} \approx 0.45 m_{e}$
Static dielectric constant: $ϵ_{r} \approx 12.9$

Reduced mass:
$μ = \frac{( 0.067 ) ( 0.45 )}{0.067 + 0.45} m_{e} \approx 0.058 m_{e}$
Binding energy:
$E_{b} = R_{X} = (\frac{0.058 m _{e}}{m _{e}} \frac{1}{12. 9 ^{2}}) \times 13.6 eV \approx 4.7 meV$
Bohr radius:
$a_{X} = (\frac{m _{e}}{0.058 m _{e}} \times 12.9) \times 0.529 A ˚ \approx 118 A ˚ = 11.8 nm$

These results are telling. The binding energy (~5 meV) is small, which is why excitons are typically observed only at low temperatures (since $k_{B} T$ at room temperature is ~25 meV). The Bohr radius (~12 nm) is much larger than the GaAs lattice constant (~0.56 nm), which validates our use of the Wannier model and the continuum dielectric approximation.

4. The Exciton Energy Spectrum and Optical Absorption

The total energy of an exciton with CM momentum $ℏ K$ and principal quantum number $n$ is the sum of the energy needed to create the pair ( $E_{g}$ ), the CM kinetic energy, and the internal binding energy:

E_{e x} (K, n) = E_{g} + \frac{ℏ ^{2} K ^{2}}{2 M} - \frac{R _{X}}{n ^{2}}

What does this imply for optical absorption? Photons carry very little momentum compared to the scale of the Brillouin zone, so optical transitions are "vertical" in an E-k diagram, meaning they create excitons with $K \approx 0$ . This leads to the possibility of absorption at a series of discrete energies below the bandgap:

E_{ab s, n} = E_{g} - \frac{R _{X}}{n ^{2}} for n = 1, 2, 3, \dots

As $n \to \infty$ , the binding energy goes to zero, and the absorption energies converge on $E_{g}$ , the onset of the continuum of free electron-hole pair states. This results in a spectrum featuring a series of sharp "Rydberg" peaks followed by the broad absorption continuum, as shown in the experimental data for GaAs in the resource from R.A. Taylor's notes (Fig. on p.16).

Test your understanding!

Consider a 2D material like a monolayer of MoS₂. The effective masses are $m_{e}^{*} \approx 0.48 m_{e}$ and $m_{h}^{*} \approx 0.54 m_{e}$ . A key difference from a 3D bulk material is that the electric field lines are not fully screened within the material but spread out into the vacuum above and below. This reduces the effective dielectric screening, leading to a much smaller effective $ϵ_{r} \approx 4$ .

How would you expect the exciton binding energy and Bohr radius in monolayer MoS₂ to compare to those in bulk GaAs? Explain your reasoning without performing the full calculation.

Show answer

Binding Energy: The exciton binding energy scales as $E_{b} \propto μ / ϵ_{r}^{2}$ . The reduced mass $μ$ for MoS₂ will be significantly larger than in GaAs because both $m_{e}^{*}$ and $m_{h}^{*}$ are larger. More dramatically, the effective dielectric constant $ϵ_{r}$ is much smaller ( $\approx 4$ vs. $\approx 12.9$ ). Both factors—a larger $μ$ and a much smaller $ϵ_{r}$ —will lead to a significantly larger binding energy in MoS₂ compared to GaAs. (In reality, it's on the order of several hundred meV).
Bohr Radius: The exciton Bohr radius scales as $a_{X} \propto ϵ_{r} / μ$ . The smaller $ϵ_{r}$ and larger $μ$ in MoS₂ will both contribute to making the exciton Bohr radius much smaller than in GaAs. The exciton will be more tightly bound and compact.

Conclusion

In this lesson, we have moved beyond the independent particle picture to account for the Coulomb attraction between electrons and holes.

Key Takeaways:

An exciton is a bound quasiparticle state of an electron and a hole.
Wannier-Mott excitons, common in semiconductors, are weakly bound and have a large spatial extent, allowing them to be modeled as a hydrogen atom in a dielectric medium.
The model predicts a hydrogenic series of energy levels lying just below the semiconductor bandgap $E_{g}$ .
The exciton binding energy ( $E_{b}$ ) and Bohr radius ( $a_{X}$ ) are determined by the material's effective masses (via the reduced mass $μ$ ) and its static dielectric constant ( $ϵ_{r}$ ).
These discrete exciton states give rise to sharp absorption peaks below the continuum band-to-band absorption edge, a key feature in the low-temperature optical spectra of semiconductors.

Preview of the Next Lesson

Now that we have developed a theoretical understanding of the key optical players—free carriers (Drude model), interband transitions, and excitons—we are ready to connect this theory more directly to your experimental work. In the next lesson, we will focus on interpreting photoluminescence (PL) spectra. We will learn how to identify spectral features corresponding to band-to-band recombination, excitonic recombination, and transitions involving defects, providing you with the tools to analyze the kind of data you might encounter in the lab.