Biexciton Hamiltonian Formulation

1. The Biexciton Hamiltonian in the Free-Carrier Basis

The most straightforward way to describe a biexciton is to consider its four constituent particles: two electrons and two holes. Let's denote the electron coordinates as ${\mathbf{r}}_{e1}$ and ${\mathbf{r}}_{e2}$, with effective mass $m_e^{\ast }$, and the hole coordinates as ${\mathbf{r}}_{h1}$ and ${\mathbf{r}}_{h2}$, with effective mass $m_h^{\ast }$.

The total Hamiltonian, $H_{XX}$, is the sum of the kinetic energies of all four particles and the potential energies from all possible pairwise Coulomb interactions. There are $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ such interactions.

The full Hamiltonian can be written as:

$H_{XX}=T+V$

where $T$ is the total kinetic energy operator:
$T=-\frac{{\hslash }^2}{2m_e^{\ast }}{\nabla }_{e1}^2-\frac{{\hslash }^2}{2m_e^{\ast }}{\nabla }_{e2}^2-\frac{{\hslash }^2}{2m_h^{\ast }}{\nabla }_{h1}^2-\frac{{\hslash }^2}{2m_h^{\ast }}{\nabla }_{h2}^2$

And $V$ is the total potential energy, comprising four attractive electron-hole interactions and two repulsive like-charge interactions:

Here, $ϵ$ is the dielectric permittivity of the semiconductor.

For a stable biexciton to form, the ground state energy of this Hamiltonian, $E_{XX}$, must be lower than the energy of two separate, non-interacting excitons, $2E_X$. This energy difference is the biexciton binding energy, $E_B$:

$E_B=2E_X-E_{XX}$

A positive binding energy signifies a stable bound state. This is conceptually analogous to the formation of a diatomic molecule, where the total energy is lower than that of two separate atoms.

The following resource provides a formal presentation of this Hamiltonian, presented in convenient dimensionless units.

This four-body Schrödinger equation, $H_{XX}\Psi =E_{XX}\Psi$, is analytically unsolvable. Its complexity far exceeds the two-body exciton problem. Finding its eigenvalues requires sophisticated numerical techniques, which we will discuss in the next lesson.

2. The Biexciton as a System of Interacting Bosons

The free-carrier formulation is exact but doesn't fully capture the intuitive picture of a biexciton as a bound state of two excitons. Given your background in advanced quantum mechanics and QFT, we can explore a more powerful formalism that works in an exciton basis.

In this view, instead of four fermions, we start with two excitons, which are composite bosons. We use exciton creation operators, $B_i^{†}$, which are themselves composed of electron and hole operators: $B_i^{†}\sim {\sum }_{\mathbf{k},\mathbf{p}}{\varphi }_i(\mathbf{k},\mathbf{p})a_{\mathbf{k}}^{†}{b}_{\mathbf{p}}^{†}$.

A biexciton state is then created by applying two of these operators to the vacuum: $|{\Psi }_{XX}⟩\sim B_i^{†}{B}_j^{†}|0⟩$.

However, excitons are not true bosons. Their constituent fermions obey the Pauli exclusion principle. If two excitons are pushed very close together, their constituent electrons and holes will feel each other's presence. An electron from one exciton cannot occupy the same state as an electron from the other. This leads to an effective repulsion between the excitons at short distances.

This "Pauli repulsion" or "exchange interaction" is a crucial part of the biexciton Hamiltonian in this picture. The formalism handles this through the commutation relations between the exciton operators, which deviate from those of ideal bosons.

To summarize the key idea from the reading: the Hamiltonian for two interacting excitons includes:

1. The energy of two free excitons.
2. Direct Coulomb interaction: The four Coulomb terms between the constituents of the first exciton and the constituents of the second exciton.
3. Exchange interaction (Pauli Scattering): Terms that arise from the antisymmetrization of the total wavefunction with respect to electron exchange and hole exchange. This is a purely quantum mechanical effect with no classical analogue.

3. The Role of Spin and Exchange Symmetry

Just as with trions, the spin configuration of the identical fermions (two electrons, two holes) is fundamentally important. The overall wavefunction must be antisymmetric under the exchange of the two electrons and, separately, under the exchange of the two holes.

Let's consider the spins of the two electrons. They can form a total spin singlet ($S_e=0$, antisymmetric spin wavefunction) or a triplet ($S_e=1$, symmetric spin wavefunction). The same applies to the two holes.

• Spin Singlet ($S_e=0$): Requires a spatially symmetric wavefunction for the electrons to ensure total antisymmetry. This allows both electrons to occupy the same lowest-energy (1s-like) spatial orbital, maximizing their attractive interaction with the holes, but at the cost of higher electron-electron repulsion due to their proximity.
• Spin Triplet ($S_e=1$): Requires a spatially antisymmetric wavefunction. This forces one of the electrons into a higher-energy (e.g., 2p-like) orbital, which has a node at the origin, effectively keeping the electrons apart and reducing their repulsion.

For the biexciton ground state, the energy gained from both electrons occupying the lowest spatial state typically outweighs the cost of increased repulsion. Therefore, the ground state of a biexciton is a spin-singlet for both electrons and holes. This state is often called the "biexciton singlet".

The effective Hamiltonian describing the interaction between the excitons depends on these spin configurations. The paper by Combescot et al. derives the effective Schrödinger equation for different cases, revealing how the exchange interactions contribute.

The key insight from this reading is that the interaction part of the Hamiltonian is not a simple potential. It's a complex operator that includes terms representing direct Coulomb forces and exchange forces. The sign and magnitude of these exchange terms depend explicitly on the spin state (singlet vs. triplet) of the interacting particles, as captured by the (-1)^S factors in the equations. This is a profound consequence of the Pauli exclusion principle in a many-body system.

Conclusion

In this lesson, we have constructed the Hamiltonian for a biexciton, a four-body quantum system. We have seen that this seemingly simple "excitonic molecule" is governed by rich and complex physics.

Key Takeaways:

• In the free-carrier basis, the biexciton Hamiltonian is the sum of the kinetic energies of two electrons and two holes, plus the six pairwise Coulomb potential terms between them.
• A stable biexciton exists if its ground state energy is lower than that of two free excitons, with the difference being the biexciton binding energy.
• In the exciton basis, the system is described as two interacting composite bosons. The Hamiltonian includes direct Coulomb scattering between the excitons and, crucially, exchange interaction (Pauli scattering) terms that reflect the fermionic nature of their constituents.
• The effective Hamiltonian is highly dependent on the spin configuration of the electrons and holes, with the spin-singlet state being the typical ground state.

Preview of the Next Lesson

We have successfully formulated the four-body Hamiltonian, but this is only the first step. The central challenge is to solve the Schrödinger equation to find the biexciton's energy and wavefunction. In the next lesson, "Apply variational methods with appropriate trial wavefunctions to estimate biexciton binding energies," we will dive into one of the most powerful techniques for tackling this problem. We will explore how to construct intelligent guess wavefunctions and apply the variational principle to obtain an upper bound for the ground state energy, allowing us to calculate the binding energy.