Excitons as Composite Bosons

1. Composite Bosons, or "Cobosons"

An exciton is a prime example of a composite boson. This is a particle that is composed of a definite, even number of fermions. As you have seen in your advanced quantum mechanics and QFT courses, elementary particles are either fermions or bosons, but composite particles can have more complex statistical behavior.

The modern many-body theory of excitons, pioneered by physicists like Monique Combescot, introduces the term "coboson" (a contraction of composite boson) to emphasize this nature. This framework allows for an exact treatment of the fermionic exchanges between excitons.

To understand why this is necessary, let's explore the commutation relations of the operators that create and annihilate excitons. For ideal (or elementary) bosons, their creation ($b^{†}$) and annihilation ($b$) operators obey the canonical commutation relation:
$[b_i,b_j^{†}]={\delta }_{ij}$
Our goal is to derive the actual commutation relation for exciton operators and see how and when it approximates this ideal bosonic form.

2. The Exciton Commutation Relations

Your background in second quantization will be very useful here. We can define an operator that creates an electron-hole pair. The creation operator for an exciton in a specific state $\nu$ (which includes quantum numbers for its internal motion and its center-of-mass motion) is a linear combination of these fundamental pair-creation operators:

$B_{\nu }^{†}={\sum }_{{\mathbf{k}}_{\mathbf{e}},{\mathbf{k}}_{\mathbf{h}}}{\Psi }_{\nu }({\mathbf{k}}_{\mathbf{e}},{\mathbf{k}}_{\mathbf{h}})a_{{\mathbf{k}}_{\mathbf{e}}}^{†}{b}_{{\mathbf{k}}_{\mathbf{h}}}^{†}$

Here, $a_{{\mathbf{k}}_{\mathbf{e}}}^{†}$ creates an electron in the conduction band with wavevector ${\mathbf{k}}_{\mathbf{e}}$, $b_{{\mathbf{k}}_{\mathbf{h}}}^{†}$ creates a hole in the valence band with wavevector ${\mathbf{k}}_{\mathbf{h}}$, and ${\Psi }_{\nu }({\mathbf{k}}_{\mathbf{e}},{\mathbf{k}}_{\mathbf{h}})$ is the exciton wavefunction in momentum space. The corresponding annihilation operator is $B_{\nu }=(B_{\nu }^{†}{)}^{†}$.

The crucial step is to compute the commutator $[B_{\mu },B_{\nu }^{†}]$ using the fermionic anticommutation relations for the electron and hole operators:
$\{a_{\mathbf{k}},a_{{\mathbf{k}}^{\prime }}^{†}\}={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }};\{b_{\mathbf{k}},b_{{\mathbf{k}}^{\prime }}^{†}\}={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}$
and all other anticommutators between different types of operators or two creation/annihilation operators are zero.

The full derivation is algebraically intensive. A detailed, step-by-step derivation is provided in the following resource. We will focus on the process and the final result, which encapsulates the essential physics.

As you've seen in the reading, the calculation leads to a result of the form:

$[B_{\mu },B_{\nu }^{†}]={\delta }_{\mu \nu }-{\Delta }_{\mu \nu }$

The term ${\Delta }_{\mu \nu }$ is an operator that represents the deviation from true bosonic behavior. It is not a simple number but an operator that depends on the number of electrons and holes already present in the system. For a pair operator $b_k=a_{k_e}{\beta }_{k_h}$ (where $\beta$ is the hole annihilation operator), the resource shows this deviation explicitly (Eq. 4.7):
$[b_k,b_{k^{\prime }}^{†}]={\delta }_{k{k}^{\prime }}-a_{k_e^{\prime }}^{†}{a}_{k_e}{\delta }_{k_h{k}_h^{\prime }}-{\beta }_{k_h^{\prime }}^{†}{\beta }_{k_h}{\delta }_{k_e{k}_e^{\prime }}$
The terms like $a^{†}a$ are number operators. This expression reveals that the commutator is modified by the presence of other fermions.

This deviation from bosonic statistics is not just a mathematical curiosity; it has a profound physical meaning.

3. The Physical Origin: Phase-Space Filling

The deviation operator ${\Delta }_{\mu \nu }$ is a manifestation of the Pauli exclusion principle acting on the constituent fermions. It describes a phenomenon known as phase-space filling or Pauli blocking.

Imagine you want to create an exciton in state $\nu$. The operator $B_{\nu }^{†}$ attempts to create an electron in a specific set of momentum states $\{{\mathbf{k}}_{\mathbf{e}}\}$ and a hole in a set of states $\{{\mathbf{k}}_{\mathbf{h}}\}$, weighted by the wavefunction ${\Psi }_{\nu }$. If another exciton is already present, its constituent electron and hole occupy some of these momentum states. Because electrons and holes are fermions, you cannot place a new electron or hole into a state that is already occupied.

Therefore, the presence of existing excitons "fills" the available phase space, making it harder to create a new exciton. The commutation relation reflects this: the act of creating an exciton is affected by the excitons already there, which is fundamentally non-bosonic behavior.

The following resource provides further insight into this concept and quantifies the effect.

For a deeper, more modern perspective, the concept of Pauli blocking is formalized as "Pauli scattering" in the coboson many-body theory. This refers to effective interactions between excitons that arise purely from fermion exchange, even without any Coulomb interaction.

4. The Low-Density Condition

So, under what conditions can we ignore the deviation term ${\Delta }_{\mu \nu }$ and treat excitons as ideal bosons?

The answer is when the density of excitons is sufficiently low. The magnitude of the deviation term ${\Delta }_{\mu \nu }$ is proportional to the density of excitons. If excitons are, on average, very far apart from each other, the probability that the electron of one exciton occupies a state needed by another is very small.

We can quantify this. The "volume" occupied by a single Wannier-Mott exciton is of the order of $a_B^3$, where $a_B$ is the exciton Bohr radius. Let $n_{ex}$ be the volume density of excitons. The quantity $n_{ex}{a}_B^3$ represents the fraction of the total crystal volume that is "filled" by excitons. The bosonic approximation is valid when this fraction is much less than 1.

The condition to treat excitons as bosons is the low-density limit:
$n_{ex}{a}_B^3\ll 1$
This is equivalent to saying that the mean inter-exciton separation, $d_{ex}\approx n_{ex}^{-1/3}$, is much larger than the exciton radius $a_B$.
$d_{ex}\gg a_B$
When this condition holds, the excitons are dilute and rarely interact or overlap, so their internal fermionic structure becomes irrelevant, and they behave as point-like elementary bosons.

Conclusion

Today, we have rigorously established the statistical nature of excitons and the conditions required to treat them as bosons, a crucial step for exploring their many-body physics.

Key Takeaways:

• Excitons are composite bosons (or cobosons), not elementary bosons. Their statistics are only approximately bosonic.
• The deviation from ideal bosonic behavior arises from the Pauli exclusion principle acting on their constituent fermions (electrons and holes). This effect is known as phase-space filling or Pauli blocking.
• The exciton commutation relation is $[B_{\mu },B_{\nu }^{†}]={\delta }_{\mu \nu }-{\Delta }_{\mu \nu }$, where ${\Delta }_{\mu \nu }$ is an operator term that depends on the density of existing excitons.
• Excitons can be treated as ideal bosons only in the low-density limit, where the average inter-exciton distance is much larger than the exciton Bohr radius ($n_{ex}{a}_B^3\ll 1$).

Preview of the Next Lesson

Now that we have established the conditions for treating an exciton gas as an ideal boson gas, we can apply the tools of statistical mechanics. In our next lesson, we will do just that, moving on to the exciting topic of exciton condensation. We will address the learning outcome: Derive the critical temperature for Bose-Einstein condensation (BEC) of a 3D ideal exciton gas.