Modeling high-order harmonic generation in quantum dots using a real-space tight-binding approach

Recently, the size-dependence of high-order harmonic generation (HHG) in quantum dots has been investigated experimentally. In particular, for longer driving wavelengths and QDs smaller than 3,nm, HHG was strongly suppressed, however, there is no computational model capable of describing the strong-field response of such systems. In this work, we introduce a computationally efficient three-dimensional real-space tight-binding model specifically designed for the simulation of HHG in confined systems. The model parameters are meticulously derived from density functional theory (DFT) calculations for the semiconductor bulk, followed by a process of Wannierization. Our findings demonstrate that the proposed model accurately captures the observed dependency of the HHG yield on the quantum dot size. Additionally, we simulate the HHG yield for elliptically polarized pulses for different QD-sizes and driving wavelengths up to $5,μ{\mathrm{m}}$. The herein proposed model fills the theoretical void in simulating HHG within medium-sized nanostructures, which cannot be described by methods applied for periodic solids or small molecules or atoms.

💡 Research Summary

In this paper the authors address a pressing gap in the theoretical description of high‑order harmonic generation (HHG) from semiconductor quantum dots (QDs). Recent experiments on CdSe nanocrystals have shown that when the dot diameter falls below roughly 3 nm, the HHG yield drops dramatically, especially for driving wavelengths in the mid‑infrared (2–5 µm). Existing computational tools—real‑time time‑dependent density‑functional theory (rt‑TDDFT) for molecules and atoms, and semiconductor Bloch equations (SBEs) or bulk tight‑binding models for periodic solids—cannot simultaneously capture the finite size, the discrete electronic structure, and the strong‑field dynamics of medium‑sized QDs containing hundreds of atoms.

To overcome these limitations the authors develop a three‑dimensional real‑space tight‑binding (TB) model built on maximally‑localized Wannier functions (MLWFs) derived from density‑functional theory calculations of bulk CdSe. By constructing separate Wannier manifolds for the valence and conduction bands, they obtain compact Hamiltonian matrices H c, H v and dipole matrices D cv, D cc, D vv that are strongly localized in real space. The QD geometry is then generated simply by cutting a spherical region out of the bulk lattice; all Wannier sites whose unit‑cell centers lie inside the sphere are retained, while those outside are discarded. This “real‑space Wannier truncation” preserves the correct bulk limit and automatically yields a finite‑size Hamiltonian without the need for additional surface passivation terms (although the authors note that surface relaxation could be added via Harrison scaling if desired).

The many‑body Hamiltonian is expressed in terms of electron (e†) and hole (h†) creation operators acting on the Wannier states. The field‑matter interaction is introduced in the length gauge as E(t)·r, with the position operator r expanded in the Wannier basis using the pre‑computed dipole matrices. Coulomb interactions are omitted on the basis that, for the dot sizes considered, confinement energies and field‑induced Stark shifts (hundreds of meV) dominate over exciton binding energies, and experimental HHG spectra show no clear excitonic signatures.

Time propagation is performed on the single‑particle density matrix ρ, which is vectorized and evolved according to the von‑Neumann equation i ∂t ρ =