Wannier-Mott vs. Frenkel Excitons: A Criterion for Distinction

Introduction

In our last lesson, we explored the significant impact of dielectric screening on the properties of Wannier-Mott excitons. We established that these excitons are well-described as hydrogen-like quasiparticles, but their binding energy and spatial extent are heavily modified by the material's dielectric constant and the effective masses of the charge carriers. Their defining feature is that the electron and hole are weakly bound and delocalized over many lattice sites.

Today, we will complete our foundational picture by introducing the other major class of excitons: the tightly-bound Frenkel exciton. Our goal is to derive and apply the criterion that distinguishes Wannier-Mott excitons from Frenkel excitons. By the end of this lesson, you will be able to identify which model is appropriate for a given material and understand the physical justification behind this distinction.

This lesson bridges our phenomenological understanding of excitons and prepares us for the more fundamental many-body treatment we will tackle in the next module.

Two Limiting Cases: A Tale of Two Excitons

Excitons exist on a spectrum, but it's incredibly useful to understand the two idealized limits.

As the diagram and our previous lessons have shown, the Wannier-Mott exciton is a creature of semiconductors, where moderate to high dielectric constants screen the Coulomb interaction effectively. The Frenkel exciton is its counterpart in materials with poor screening.

The key takeaway is that the fundamental distinction lies in the spatial extent of the exciton relative to the crystal's lattice constant, $a$.

• Wannier-Mott Exciton: The electron-hole separation, given by the exciton Bohr radius $a_B^{\ast }$, is much larger than the lattice constant ($a_B^{\ast }\gg a$).
• Frenkel Exciton: The electron and hole are confined to the same or adjacent atoms/molecules. Here, the concept of a Bohr radius is less meaningful; the exciton's size is comparable to the lattice constant ($a_B^{\ast }\lesssim a$).

The Frenkel Exciton: A Tight-Binding Approach

To understand the Frenkel exciton more deeply, we must move away from the effective mass approximation (which assumes slowly varying wavefunctions over many unit cells) and adopt a picture of localized molecular excitations. The appropriate theoretical tool here is the tight-binding model, which you've likely encountered in your condensed matter courses for describing electron bands in solids. Here, we apply it to the transport of a neutral excitation.

The starting point is a crystal composed of $N$ weakly interacting molecules. The excitation is not seen as a separate electron and hole, but as an excited state of a single molecule. Due to intermolecular interactions, this localized excitation can "hop" from one molecule to another, becoming delocalized throughout the crystal.

Let's walk through the derivation of the Frenkel Hamiltonian, which formalizes this picture. The following resource provides a rigorous derivation based on the Heitler-London approximation.

Let's summarize the key steps and the resulting Hamiltonian.

1. Starting Point: The full electronic Hamiltonian for an aggregate of $N$ molecules, which includes all intra- and inter-molecular interactions.
2. Approximation 1 (Hartree Product): We assume the molecules are far enough apart that their electronic wavefunctions do not overlap. The total wavefunction is simply a product of the individual molecular wavefunctions. This is the cornerstone of the tight-binding picture.
3. Approximation 2 (Singly Excited Subspace): We restrict our basis to states where the crystal as a whole contains only one excitation (i.e., only one molecule is in its excited state at any time). This is valid for low-intensity optical excitation.
4. Approximation 3 (Neglecting certain interactions): Interactions involving nuclei between different molecules are often neglected or treated as part of a surrounding environment. The dominant intermolecular term is the electron-electron Coulomb interaction.

After these approximations, and subtracting the ground state energy, we arrive at the effective Frenkel-Heitler-London Hamiltonian:
${\hat{H}}_{\text{eff}}={\sum }_{m=1}^N{\epsilon }_m{\sigma }_m^{†}{\sigma }_m+{\sum }_{m\ne n}^{N,N}{J}_{mn}{\sigma }_m^{†}{\sigma }_n$
Let's break this down:

• ${\sigma }_m^{†}$ and ${\sigma }_m$ are operators that create and annihilate an excitation on molecule $m$, respectively. They act on the two-level system of each molecule (ground or excited). Given your familiarity with quantum mechanics, you can think of them as Pauli raising and lowering operators acting on a pseudo-spin system.
• ${\epsilon }_m$ is the on-site energy: the energy required to excite an isolated molecule $m$.
• $J_{mn}$ is the transfer integral or hopping term. It represents the coupling that allows an excitation to move from molecule $m$ to molecule $n$. This coupling arises from the Coulomb interaction between the transition dipole moments of the two molecules.

The eigenstates of this Hamiltonian are not localized on a single molecule but are coherent superpositions of the excitation being on any molecule in the crystal—a Frenkel exciton band.

Applying the Criterion: When Does Each Model Apply?

We now have two distinct models. The criterion for choosing between them is a comparison of length scales. Let's make this concrete.

The Criterion: Calculate the Wannier-Mott Bohr radius $a_B^{\ast }=a_0\left(\frac{{ϵ}_r}{\mu /m_e}\right)$ and compare it to the material's lattice constant $a$.

• If $a_B^{\ast }\gg a$: The electron-hole pair is delocalized over many unit cells. The crystal appears as a continuous dielectric medium. The Wannier-Mott model is appropriate. This is characteristic of typical semiconductors (Si, GaAs, etc.).
• If $a_B^{\ast }\lesssim a$: The calculated Bohr radius is unphysically small, suggesting the electron and hole are confined within a single unit cell. The continuum approximation breaks down. The Frenkel model is the correct starting point. This is true for molecular crystals (e.g., anthracene) and many large-gap ionic insulators (e.g., NaCl).

Beyond the Ideal Limits: Model Breakdown

Nature is rarely so clean-cut. The Frenkel and Wannier-Mott models represent idealized limits, and understanding when they fail is as important as knowing when they apply.

The same resource we used to derive the Frenkel Hamiltonian also provides an excellent computational analysis of its limitations. The authors use Time-Dependent Density Functional Theory (TDDFT) to test the validity of the Frenkel model's approximations for a dimer of coumarin molecules.

The key findings, which you can see in the 'Conclusions' (Section 3.4) of resource LINK, are:

• For intermolecular distances greater than ~8 Å, the tight-binding description works reasonably well. The excitation can be described as hopping between distinct molecules.
• For distances smaller than this, the model breaks down. The assumption of non-overlapping molecular wavefunctions fails. The TDDFT calculations show that the character of the excited state changes, and it can acquire charge-transfer character, where the electron hops to the neighboring molecule while the hole remains. This is a step towards a Wannier-Mott picture, where the electron and hole are spatially separated.
• Simple approximations for the coupling term $J_{mn}$, like the ideal dipole approximation (IDA), fail at short distances. More accurate methods that account for the full shape of the transition charge densities are needed.

This reinforces our main theme: the physics of excitons is governed by the interplay between localization and delocalization, determined by screening and intermolecular distance.

A related concept is the Mott transition, where at high exciton densities, the excitons get so crowded that their wavefunctions overlap significantly. The screening from neighboring excitons breaks them apart, and the system transitions from an insulating gas of excitons to a conductive electron-hole plasma. This occurs when the average distance between excitons becomes comparable to the exciton Bohr radius, another example of a length-scale comparison dictating the system's fundamental nature. We will explore this in detail in a future lesson.

Conclusion

This lesson has established the two fundamental archetypes of excitons and the criterion to distinguish them.

Key Takeaways:

• Excitons fall on a spectrum between two limits: the delocalized, weakly-bound Wannier-Mott exciton and the localized, tightly-bound Frenkel exciton.
• The Wannier-Mott model uses the effective mass approximation, treating the crystal as a dielectric continuum. It is suitable for most semiconductors.
• The Frenkel model uses a tight-binding (Heitler-London) approach, describing the coherent hopping of a local molecular excitation. It is suitable for molecular crystals and many insulators.
• The primary criterion for distinguishing them is the ratio of the exciton's spatial extent (Bohr radius, $a_B^{\ast }$) to the lattice constant ($a$). If $a_B^{\ast }\gg a$, the Wannier-Mott model applies; if $a_B^{\ast }\lesssim a$, the Frenkel model is more appropriate.

Preview of the Next Lesson

We have now built a solid phenomenological foundation by examining the Wannier-Mott and Frenkel models. However, these are distinct pictures. In the next module, we will begin our journey into the many-body formulation of excitons. We will learn to describe excitons using the language of second quantization, which provides a more powerful and unified framework. Our first step will be to formulate the electron-hole pair Hamiltonian using creation and annihilation operators. This will eventually lead us to the Bethe-Salpeter equation, the cornerstone of modern exciton theory.