Variational and field-theoretical approach to exciton-exciton interactions and biexcitons in semiconductors

TL;DR

This work addresses the complex problem of exciton–exciton interactions in semiconductors by combining a variational two-exciton framework with a finite-temperature path-integral field theory. The variational approach yields a nonlocal, spin-dependent effective potential $V^{\mathrm{eff}}_{S_c}$ between ground-state excitons, which in the heavy-hole limit reduces to the Heitler–London singlet/triplet potentials and exhibits van der Waals behavior at long range when excited states are included. The field-theoretical formulation recasts excitons as bosonic fields via a polarization field, deriving an effective quartic action $S_{\mathrm{eff}}[X^*,X]$ with a two-body interaction kernel $\mathcal{W}$ that reproduces the variational results on-shell and encodes many-body retardation effects. Together, these results provide a robust, generalizable framework for biexciton spectra and correlated excitonic matter in 2D and 3D semiconductors, with clear pathways to extensions to more complex band structures and multi-exciton sectors.

Abstract

Bound electron-hole pairs in semiconductors known as excitons are the subject of intense research due to their potential for optoelectronic devices and applications, especially in the realm of two-dimensional materials. While the properties of free excitons in these systems are well understood, a general description of their interactions is complicated due to their composite nature, which leads to exchange between the identical fermions of different excitons. In this work we employ a variational approach to study interactions between Wannier excitons and obtain an effective interaction potential between two ground-state excitons in a system of spin-degenerate electrons and holes. This potential is in general nonlocal and depends on the coupled spins of the particles. When particularized to hydrogen-like excitons with a heavy hole, it becomes local and exactly reproduces the Heitler-London result for two interacting hydrogen atoms. Thus, our result can be interpreted as a generalization of the Heitler-London potential to arbitrary masses. Including corrections due to excited states results in a van der Waals potential at large distances, which is expected due to the induced dipole-dipole nature of the interactions. Additionally, we use a path-integral formalism to develop a many-body theory for a gas of excitons, resulting in an excitonic action that formally includes many-body interactions between excitons. While in the field representing the excitons is exactly bosonic, we clarify how the internal exchange processes arise in the field-theoretical treatment, and show that the diagrams corresponding to the interactions between excitons align with our variational calculation when evaluated on shell. Our methods and results lay the groundwork for a generalized theory of exciton-exciton interactions and their application to the study of biexciton spectra and correlated excitonic matter.

Figures (3)

• Figure 1: Schematic illustration of the spin-basis transformation explicitly performed in Appendix \ref{['app: Spin-Basis Transformation']}. The blue and red circles represent the conduction and valence electrons, respectively, and the biexciton eigenproblem simplifies when one considers the coupled electron spins on the one hand and the coupled valence spins on the other.
• Figure 2: Exciton--exciton potential in the heavy hole-limit. The blue curve corresponds to the singlet state of the coupled conduction electrons $(S_{\mathrm{c}} = 0)$ and displays an attractive part with a minimum at around $r = 0.9 a_{X}$. The red curve corresponds to the triplet configuration $(S_{\mathrm{c}} = 1)$ and is repulsive. These functions are exactly the singlet and triplet Heitler--London potentials obtained in the Born--Oppenheimer approximation for the dihydrogen molecule. The radial coordinate and the effective momentum-space potential have been made dimensionless via the mean exciton radius $a_{X}$ and the exciton binding energy $\varepsilon^{\mathrm{b}}_{X}$, respectively, both defined in the main text.
• Figure 3: Convergence of $\mathcal{U}^{\mathrm{c}}_{\mathrm{cc}}(r)$. The top graph shows the convergence as a function of the number of points $N$ on a single segment of the 4D grid at a fixed value of the segment length, here $L = 20$, for different values of $r$. We show the absolute value of the difference between the value obtained for $N < N_{\text{max}}$ and that obtained for $N_{\text{max}}$, where $N_{\text{max}} = 4000$. The bottom plot shows the convergence as a function of the segment length $L$ relative to its value obtained for $L_{\text{max}} = 30$, with $N = N_{L}$ chosen so that the point density stays constant on the segment, here $N_{L} = 20(L - 1)$. Beyond $L = 10$, the difference is smaller than $10^{-12}$ in all cases and stays approximately constant as $L$ becomes larger. We note that for $r = 0.01$, the value of $\mathcal{U}^{\mathrm{c}}_{\mathrm{cc}}$ for $L \ge 8$ becomes the same for all $L$, hence the vertical downward line on the logarithmic plot.