Coupling Quantum Dots to Elastic Waves in a Phononic Crystal Waveguide

TL;DR

This work addresses the challenge of coupling quantum dot excitons to gigahertz acoustic modes in phononic crystal waveguides. It combines multiband $k\cdot p$-CI modeling of InAs/GaAs and GaAs/AlGaAs QDs with finite-element simulations of snowflake-pattern PnC membranes to quantify deformation-potential and piezoelectric couplings across mode symmetries. The authors demonstrate mode-dependent coupling: diagonal DP yields linear energy shifts under volumetric strain, while shear and piezoelectric interactions produce quadratic and polarizability-driven effects, with energy modulations reaching up to $0.7$ meV for $0.1$ nm displacements and total modulations near $1.5$ meV when all channels are combined. These insights provide design guidelines for QD–phonon hybrids and acousto-optic quantum interfaces, with potential for transduction between microwave and optical regimes and applicability to other phononic geometries.

Abstract

We present a comprehensive study of quantum dot (QD) coupling to various phononic modes in a phononic waveguide, combining multiband kp and configuration-interaction (CI) QD state simulations with finite-element waveguide mode modeling. We consider self-assembled Stranski-Krastanov InGaAs/GaAs as well as local droplet-etched GaAs/AlGaAs structures. Using kp-CI calculations, we quantify the strain and piezoelectric responses of InAs and GaAs QDs. By systematically isolating volumetric/shear deformation-potential and piezoelectric channels, we demonstrate how mode symmetries dictate distinct coupling mechanisms. We identify the dominant coupling channels and characterize their observable signatures in the QD response. We predict strong linear energy shifts under volumetric strain and quadratic behavior under shear strain, especially in GaAs QDs. The piezoelectric effect is dominated by polarizability, which also leads to a quadratic response. The simulations show energy modulations up to 0.7 meV for an acoustic wave with 0.1 nm amplitude. The quadratic response to shear strain and piezoelectric field leads to frequency doubling in the QD response to a mechanical wave and to non-harmonic time traces when linear and quadratic effects contribute to a similar degree. The deep understanding of QD-acoustic couplings opens pathways to the optimal design of QD and waveguide structures, as well as to improved engineering of acousto-optic quantum interfaces.

Figures (17)

• Figure 1: (a) PnC slab with a hexagonal snowflake lattice. (b) Standard W1 waveguide formed by removing a row of holes along the $x$-direction (in $\Gamma-K$ direction in reciprocal space) and shifting the adjacent lattice by $\Delta$. (c) Modified waveguide W1m studied in this work, with two arms of the snowflakes additionally removed. (d) Unit cell of the hexagonal snowflake PnC.
• Figure 2: Supercell of the designed PnC waveguide with boundary conditions used in the simulation. Floquet and periodic boundary conditions are applied along the $x$ and $y$ directions, respectively (green and blue boundaries), while all other surfaces are traction-free (gray surfaces). Symmetry planes are shown at $z=0$ ($\sigma_z$ -- orange plane) and $y=0$ ($\sigma_y$ -- cyan plane).
• Figure 3: The phononic band structure of the snowflake PnC. Band diagrams were calculated for a GaAs structure characterized by parameters ($d$, $r$, $w$, $a$) = ($220$ nm, $0.44a$, $0.19a$, $760$ nm). Under these parameters, a significant phononic band gap (denoted by the gray shaded area) is present for both types of modes. Orange and green curves indicate modes that are symmetric and antisymmetric, respectively, with respect to the $xy$-plane.
• Figure 4: Dispersion curves of guided modes in a W1m waveguide for $\Delta = 0.3a$ (a) and $0.6a$ (b). Point shapes indicate the symmetry of the mode, as defined on the right. The color scale shows the contribution of the respective displacement components, $P_l$, averaged over the volume of a supercell, according to Eq. \ref{['eq:Pl']}.
• Figure 5: Modulation of the excitonic transition for the InAs (left) and GaAs (right) QD as a function of different components of external strain. Dark lines represent fitting functions $f_{ij}^{(\mathrm{InAs})}$ and $f_{ij}^{(\mathrm{GaAs})}$. Pale lines are added as a guide to the eye.