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

TL;DR

This work introduces a three-dimensional real-space tight-binding model for high-order harmonic generation in confined quantum dots, parameterized from density functional theory via Wannierization to capture finite-size effects while retaining the periodic limit. It develops both a finite-size quantum-dot model and a real-space periodic model, enabling efficient density-matrix propagation on GPUs and reproducing the experimentally observed size dependence of HHG yields and ellipticity-driven suppression in CdSe QDs. The approach is benchmarked against rt-TDDFT and Kubo-based linear response, showing good agreement for linear spectra and qualitative agreement for nonlinear HHG, and it scales to dots up to several nanometers in diameter. By providing ab-initio-derived parameters and GPU-accelerated implementations, the method fills a gap between atomistic/molecular HHG descriptions and bulk solid-state theories, and it can be extended to other nanostructures and driving conditions.

Abstract

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.

Figures (8)

• Figure 1: Examples of considered geometries: a) Optimized $\mathrm{Cd}_{11}\mathrm{Se}_{11}$ cluster (used in rt-TDDFT calculations) b) $\mathrm{Cd}_{198}\mathrm{Se}_{198}$ quantum dot with diameter $2.8\,\mathrm{nm}$ (used in our real-space model) , and c) unit cell of CdSe (wurtzite).
• Figure 2: Comparison of the considered spatial coherences within a) the finite-size model, b) the real-space periodic model, and c) the SBEs along one dimension. The circles illustrate two different sites and their super cell periodic images. A solid (dashed) arrow indicates that the corresponding density matrix element is (not) accounted for during the propagation. The relations between the density matrix elements $\rho_{12}$ and their periodic counterparts are depicted, where $A$ denotes the vector potential and $L$ is the length of the super cell.
• Figure 3: DFT band structures calculated via Quantum Espresso together with their Wannier interpolations for a) CdSe (wurtzite) and b) Si.
• Figure 4: Selected tensor components of the normalized optical conductivity $\sigma(\omega)$ calculated via the real-space periodic model and via Kubo formula for a) Si and CdSe (wurtzite) along the b) a-axis and c) c-axis.
• Figure 5: Emission spectra of our models compared with rt-TDDFT calculations for a) periodic silicon (real-space periodic model) and b) a $\mathrm{Cd}_{11}\mathrm{Se}_{11}$ cluster (finite-size model). The spectra are normalized to their maxima and shifted with respect to each other to improve visibility.