Temperature-dependent refractive index of AlGaAs for quantum-photonic devices near the bandgap

TL;DR

This work addresses the lack of precise cryogenic refractive-index data for the II–IV semiconductor $Al_{x}Ga_{1-x}As$ near the direct bandgap by introducing a cryo-compatible, resonance-based extraction using suspended AlGaAs membranes between gold mirrors. A combination of experimental reflectance spectroscopy and FDTD simulations yields a robust inverse method to determine $n(\lambda,x,T)$ across $x \in [0,0.5]$, $T=4$–295 K, and $\lambda$ up to 1100 nm, with a refined analytical model achieving $R^2 \approx 0.993$ and RMSE ≈ 0.0083. The final model, $n(\lambda,x,T)=n(\lambda,x)+C_0(x)T+D_0(x)T^2$, with explicitly fitted coefficients, enables accurate design of quantum photonic devices and permits inverse parameter estimation to compensate fabrication variations. Demonstrated through applications to hemispherical microlenses and quantum emitters, the method improves predictive accuracy near the band edge and is readily extendable to other semiconductor systems, bridging materials characterization and device engineering in cryogenic photonics.

Abstract

We present an experimental method to determine the refractive index of $Al_{x}Ga_{1-x}As$ (x = 0.0 - 0.5) from 300 K to 4 K across the 500 - 1100 nm wavelength range. The values are extracted from spectroscopically observed microcavity resonances in thin $Al_{x}Ga_{1-x}As$ membranes embedded between fully and partially reflective gold mirrors. Refined Varshni and Paessler models are used to describe temperature-dependent bandgap shifts and material composition. By tracking resonance shifts and benchmarking against finite-difference time-domain simulations, we derive the dispersive optical response with high precision. This yields a quantitatively improved analytical expression for the refractive index of $Al_{x}Ga_{1-x}As$ matching the experimental results with a coefficient of determination as high as $R^2=0.993$, enabling accurate modeling near the band edge at cryogenic temperatures. The method is straightforward and broadly applicable to other semiconductor systems, offering a valuable tool for the design of micro photonic devices such as quantum light sources.

Figures (9)

• Figure 1: Fabrication of planar micro-cavities. a) Membrane on a 100 gold mirror. b) Deposition of semi-reflecting gold squares (350 lateral, 10 thick). For every forth square the top gold layer deposition is omitted, this enables the accurate AFM membrane thickness measurements shown in Tab. \ref{['tab:height']}. c) Photolithographic patterning with 300 resist squares (blue). d) Pattern transfer by reactive ion etching, yielding unstrained, free-standing cavities. e) Optical micrograph of the final structure; white dotted square indicating measurement area.
• Figure 2: White-light reflection spectroscopy at controllable temperature range 295-4. Left: Schematic of the experimental setup, with white-light excitation through optical top-window onto membrane sample consisting of an $\mathrm{Al_{x}Ga_{1-x}As}$ layer enclosed between a fully reflecting (bottom) and a semi-reflecting (top) Au mirror, mounted in a He-flow cryostat with heating element. Live imaging and positioning is performed by a Complementary Metal-Oxide-Semiconductor (CMOS) camera. Reflectance and PL spectra are recorded using a charge couled device (CCD) attached to a 0.5 grating spectrometer. Right: Normalized reflection spectra for $\mathrm{d_{nom}} = \qty{1800}{\nano\metre}$ and $\mathrm{x} = 0.10$ measured from 4-295 in 50 steps, showing resonances and the temperature-dependent band edge. Spectra are vertically shifted for clarity.
• Figure 3: Temperature dependence of the bandgap in $\mathrm{Al_{x}Ga_{1-x}As}$ for nominal aluminium compositions $\mathrm{Al_{nom}} = 0.0\,\text{–}\,0.5$. Experimental data are modeled using the Varshni model (Eq. \ref{['eq:Vurgaftman']}) and the Paessler model (Eq. \ref{['eq:Paessler']}) to extract the band edge and effective aluminium content. Literature reference curves correspond to Eq. \ref{['eq:Vurgaftman']} evaluated at the nominal compositions.
• Figure 4: Refractive index at 295 and 4 extracted from experimental reflectance spectroscopy (dots with error bars). The dashed lines show the modelling according to Adachi et al.adachi1985gaas as given in Eq. (\ref{['eq:Adachi']}), while solid lines represent the refined Model 3 introduced in this work, Eq. (\ref{['eq:Ours']}).
• Figure 5: Temperature-dependent refractive index of $\mathrm{AlGaAs}$ with $\mathrm{Al}_\mathrm{V} = 0.171$, measured from 295 to 4. Experimental data are compared to the Adachi model (black dotted lines), including a band edge shift correction adachi1985gaas. Solid lines show the proposed model across all temperatures, from room temperature (red) to cryogenic temperatures (blue), demonstrating excellent agreement.