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The internal structures of planar polyexcitons, that is, bound states of *n*-body two-dimensional excitons, are theoretically investigated. The diffusion Monte Carlo computations demonstrate that all constituent excitons in the planar polyexcitons are bound by “chemical bonds” of similar strengths.

Various types of few-body bound states are formed in photoexcited semiconductors, such as excitons (electron–hole pairs bound by Coulomb forces) and biexcitons (excitonic molecules). The formation of a bound state of *n*-pairs of excitons was predicted by Wang and Kittel in 1972 [1]. The signature of this complex, called a polyexciton, was found in the photoluminescence spectra of bulk silicon [2] and diamond [3] as a series of emission lines almost equally spaced in energy, although the physical reasons have been elusive. Kaisei Ooe of Osaka University and colleagues revealed the binding properties of planar excitons to a polyexciton [4], thereby explaining earlier observations.

The Pauli blocking effect between two fermions prevents more than two electrons from occupying the same quantum state. Therefore, the binding stability of more than three pairs of excitons in a single bound state is not an evident problem. The formation of polyexcitons requires additional degrees of freedom, such as valley degeneracy. This explains the reason for polyexcitons not being typically observed in direct band gap semiconductors without valley degeneracy.

An exemplary system for planar polyexcitons comprises two sheets of graphene bilayers, each with two-fold valley degeneracies and separated by an interplanar distance *d*. Combined with spin degrees of freedom, the system might allow the formation of trimers and tetramers of excitons, where the electrons reside in one plane and the holes in the other plane. The in-plane repulsive and interplanar attractive Coulomb interactions were included in the calculations. Starting with the wavefunctions optimized by the variational Monte Carlo method, the ground-state energies of the trimers and tetramers of planar excitons were calculated using the diffusion Monte Carlo method that simulates the time-evolving distribution function of assembled classical Brownian walkers.

Ooe and colleagues discovered that the separation energy, *S _{n}*, which is defined as the energy required to remove one exciton from a polyexciton, varies linearly with

*n*for different values of

*d*. This new result, expressed by a relation

*S*=

_{n}*U*(

*n*− 1), predicts equally spaced emission lines in photoluminescence irrespective of the value of

*d*. Coefficient

*U*corresponds to an energy to break one bond inside a polyexciton; this bond energy was observed to be common for all possible exciton pairs and thereby reminiscent of “chemical bonds.” Notably, a narrower electron–electron pair distribution function was obtained for a larger

*n*and smaller

*d*, indicating a shorter bond length (see Fig. 1). This implies that the excitons inside the triexciton and tetraexciton are more closely packed than those in the biexciton. The bond lengths are shorter when the two layers are located closer.

*d* relative to exciton Bohr radius *n* = 2), triexciton (*n* = 3), and tetraexciton (*n* = 4). The inset illustrates configurations of excitons (X’s) inside a biexciton, triexciton, and tetraexciton. The figure is adapted from Fig. 3 in Ref. 4.

A planar tetraexciton cannot take the form of a tetrahedron or square with two longer and four shorter bonds, as we might simply imagine. Instead, the tetraexciton is considered a quantum droplet bound by six equivalent nonrigid bonds, as evidenced by the single-peaked pair distribution function. Although observations in bulk semiconductors with multivalley structures [2,3] already imply the universal properties of the bonds, the present exciting theoretical prediction may ignite experimental searches for polyexcitons in low-dimensional materials.

## References

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**59**, 2899 (1987). 10.1103/PhysRevLett.59.2899 Crossref, Google Scholar - [3] J. Omachi, T. Suzuki, K. Kato, N. Naka, K. Yoshioka, and M. Kuwata-Gonokami, Phys. Rev. Lett.
**111**, 026402 (2013). 10.1103/PhysRevLett.111.026402 Crossref, Google Scholar - [4] K. Ooe, A. Miyamae, and K. Asano, J. Phys. Soc. Jpn.
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## Author Biographies

Nobuko Naka received her Ph.D. in Physics from the University of Tokyo in 2003 and subsequently worked as a Special Postdoctoral Researcher at RIKEN. She was then appointed as a lecturer in the Department of Applied Physics at the University of Tokyo and is currently a professor in the Solid-State Spectroscopy Group in the Department of Physics at Kyoto University. Her research interests include light-matter interactions, carrier transport, and quantum states in photoexcited semiconductors.