Terahertz response of confined electron-hole pair: crossover between strong and weak confinement

We analyze theoretically THz response of an electron-hole pair confined in a semiconductor nanoparticle. We show that the interplay of particle confinement and electron-hole Coulomb interaction leads to significant renormalizations and energy shifts in THz linear conductivity of the nanocrystal. We develop and evaluate models in the strong and the weak confinement regime in order to correctly address the effect of Coulomb interaction. In the weak confinement regime, we find solutions of the problem in a form similar to the Wannier wavefunction whose spatial extent is reduced as a consequence of the confinement. The resulting states are scalable down to the strong confinement regime, enabling a theoretical description of the exciton response for arbitrarily sized nanoparticles.

💡 Research Summary

This paper presents a comprehensive theoretical study of the terahertz (THz) linear response of a single electron‑hole (e‑h) pair confined within a spherical semiconductor nanoparticle (quantum dot). The authors focus on how quantum confinement and the Coulomb interaction between the electron and hole jointly shape the THz conductivity spectrum, an issue that has been largely overlooked in previous works that either treated the carriers independently or used only mean‑field approximations.

The analysis is organized around two limiting confinement regimes, defined by the relationship between the particle radius A and the bulk exciton Bohr radius aX. In the strong confinement regime (SCR), A < aX, the kinetic energy associated with confinement dominates over the Coulomb binding energy. The electron and hole are therefore modeled as independent particles in an infinite spherical well. Their single‑particle eigenstates are expressed in terms of spherical Bessel functions and spherical harmonics, with energies E ∝ k²/(2mj). The two‑particle basis is constructed as direct products of these states, and the full Hamiltonian—including the Coulomb term—is diagonalized within a truncated basis (typically the 1s and three 1p orbitals). This perturbative treatment captures the Coulomb correction to the otherwise particle‑in‑a‑box spectrum. The resulting THz conductivity shows two distinct resonances for very small dots (electron‑ and hole‑specific 1p transitions) that merge into a single resonance as the dot size increases, reflecting the reduced level spacing.

In the weak confinement regime (WCR), A ≫ aX, the e‑h pair can be separated into center‑of‑mass (COM) coordinate R and relative coordinate r. The relative motion follows the familiar Wannier exciton solution (hydrogen‑like wavefunctions with binding energies En ∝ −1/n² and Bohr radii an). However, the finite size of the dot imposes a “hard‑ball” constraint: the exciton cannot approach the boundary closer than a radius ρ(r) = max(μ/me · r, μ/mh · r), where μ is the reduced mass. This leads to an effective dead‑layer of thickness ρ that reduces the available volume for COM motion. The COM kinetic energy becomes χ(r) = ℏ²π²/