Exact Solutions to Acoustoelectric Interactions in Arbitrary Geometries

Abstract

Acoustoelectric interactions occur when free carriers in a semiconductor interact with the fields of an acoustic wave in a piezoelectric medium. These interactions can amplify acoustic waves, as well as give rise to extremely large phononic nonlinearities and strong non-reciprocal effects. The field of acoustoelectric devices is currently dependent on analytical and perturbative solutions for the two simplest arrangements of piezoelectric-semiconductor materials. While these canonical models have allowed the field to advance substantially, new geometries are arising that do not satisfy assumptions integral to these models. These assumptions include the treatment of the interactions between the acoustic fields and free carriers as weak, the neglect of the tensorial nature of the material properties, the omission of the spatial variations in the phonons' electric field profiles, and the disregard of elastic coupling across material boundaries, among others. We develop, for the first time, a finite element method (FEM) model to solve for acoustoelectric interactions in arbitrary geometries that avoids making the assumptions of the canonical models. We verify the FEM model using results for amplification, dispersion, and non-reciprocity obtained from the canonical models in their regime of validity. We then examine the acoustoelectric effect in two geometries not covered by the canonical models: a thin piezoelectric film placed on a semiconductor substrate and a fully 2D waveguide under a thin semiconductor layer. This work lays the foundation for accurate modeling of arbitrary acoustoelectric geometries such as those currently being developed for all-acoustic radio frequency (RF) signal processing, acoustoelectrically enhanced photonic devices, and quantum acoustoelectric devices.

Figures (7)

• Figure 1: a) Bulk piezoelectric semiconductor where analytical models for the acoustoelectric effect have been derived for plane waves. b) Separated medium heterostructure with an air gap between the piezoelectric medium and the semiconductor designed of Rayleigh wave amplification where a perturbation theory has been developed to model the acoustoelectric effect.
• Figure 2: a) Platform with piezoelectric material on top of a substrate that has been doped to have carriers available for the acoustoelectric effect. This structure is not covered by the canonical models due elastic coupling between the semiconductor and piezoelectric layers. b) Piezoelectric waveguides on a substrate with the semiconductor on top. The canonical models do not take into account rapid transverse horizontal variations in field strength introduced by the acoustic waveguide. c) Optomechanical cavity utilizing the acoustoelectric effect to alter acoustic breathing modes to control phononic properties for enhancement of Brillouin scattering Mack2024. Such an arbitrary geometry is not covered by existing models.
• Figure 3: a) Geometry of the CdS used in the simulation. We find a shear vertical plane wave solved for with a wavelength of $42.5$ microns and a frequency of $42$ MHz. b) Magnitude of the displacement, $|\mathbf{u}|$, for the plane wave. c) $x$-component of the electric field, $E_x$, for the plane wave. d) Gain curves as a function of applied DC electric field with and without trapping. Fits are obtained with the extracted parameters in agreement with the input parameters. e) Absolute error between the simulation and analytical model shown by the discrete points and absolute error between the fit and the analytical model shown by the continuous curves.
• Figure 4: a) Simulation geometry of the separated medium amplifier. b) Magnitude of the displacement, $|\mathbf{u}|$, for the Rayleigh mode. c) $x$-component of the electric field, $E_x$, for the Rayleigh mode. Inset image shows $E_x$ edge on with the surface parallel to $xz$-plane to better illustrate the phase shift in the electric field caused by screening effects. d) Plot of acoustoelectric gain, $\alpha$, as a function of frequency with electron drift velocity fixed at $v_d=3v_a$. Inset shows a portion of the curve corresponding to experimental data from the original work used to verify the perturbation-based model Kino1971. Experimental data points have been estimated utilizing digital extraction tools. e) Plot of $\alpha$ as a function of drift field at $1$ GHz. f) Plot of change in wavenumber, $\Delta\beta$, as a function of drift field at $1$ GHz
• Figure 5: a) Diagram of the simulation geometry for 1 micron thick Al$_{0.6}$Sc$_{0.4}$N grown on 4H SiC. The top layer of SiC with a thickness of $2\lambda$ is doped with a uniform carrier concentration and acts as the semiconductor for acoustoelectric interactions. b) Displacement magnitude, $|\mathbf{u}|$, of the Sezawa mode. c) $x$-component of Electric field, $E_x$, for the same Sezawa mode. d) Filled contour plot showing the gain, $\alpha$, as a function of the carrier concentration, $n_0$, and the applied drift field. e) $\alpha$ as a function of the applied drift field for $n_0=1\times10^{15}$ cm$^{-3}$. An initial estimate and a fit to the perturbation-based model are included. Location on the contour plot is denoted by the dashed line in d. f) Correction ratios, $\eta_K=K^2_{eff}/K^2$ and $\eta_n=n_{eff}/n_0$, depicting the difference between the fit parameters and expected initial estimates. The star indicates the case plotted in e. The dashed lines are the same correction ratios for 2 micron thick film of AlScN.