Perturbative sensing of nanoscale materials with millimeter-wave photonic crystals

Abstract

We introduce millimeter-wave silicon photonic crystal cavities as a versatile platform for the perturbative sensing of nanoscale materials. This dielectric-based platform is compatible with strong magnetic fields, opening avenues for studying quantum materials in extreme environments where superconducting cavities cannot operate. To establish the platform's performance, we cryogenically characterize a silicon photonic crystal cavity at 4.3 K, achieving a total quality factor exceeding $10^5$ for a 96 GHz mode. As a proof-of-concept for its sensing capabilities, we position a hexagonal boron nitride-multilayer graphene (hBN-MLG) heterostructure at an electric-field antinode of the cavity and measure the perturbative response at room temperature. The heterostructure induces a significant change in the cavity's resonance, from which we extract a total sample conductivity of approximately $5.1\times10^6$~S/m. These results establish silicon photonic crystal cavities as a promising platform for sensitive, on-chip spectroscopy of nanoscale materials at millimeter-wave frequencies.

Figures (7)

• Figure 1: Sensing via cavity perturbation using silicon photonic crystal cavities. (a) Diagram of a sample placed inside a Fabry-Pérot cavity. (b) Diagram of the transmission through the cavity $|\alpha_\mathrm{out}/\alpha_\mathrm{in}|^2$ around one of its modes, measured before ($\omega_\text{c}$) and after the sample has been placed inside the cavity ($\omega_\text{c}'$). The cavity linewidth can also change due to the introduction of the sample $\kappa_\text{i}\to\kappa_\text{i}'$ (blue shading to red shading). (c) Rendering of a millimeter-wave photonic crystal cavity. To maximize the cavity's perturbative response, the sample is introduced at an electric field antinode of the photonic crystal cavity fundamental mode. The solid arrow indicates half of the device's length, 18.74 mm.
• Figure 2: Photonic crystal cavity design and COMSOL simulation results. (a) Geometry of the nominal unit cell, with critical dimensions labeled with parameters and marker styles corresponding to the following subfigure. (b) Distribution of the unit cell parameters along the device, showing the waveguide region at the two ends, the linear taper between the waveguide and the mirror cells, and the cubic interpolation between the mirror cell and the defect cell. (c) Frequency domain simulation of the full device, coupled to WR10 waveguides. We plot the transmission ($S_{21}$) and reflection ($S_{11}$) of the cavity. The fundamental mode is indicated by a blue triangle and the first-order mode is indicated by a yellow diamond. Inset: plot of the fundamental cavity mode, with a 90.958 GHz center frequency and a total linewidth of roughly 450 kHz (dominated by external coupling). (d) The electric field distribution $E_x$ of the fundamental mode (TE0) of the full device.
• Figure 3: Millimeter-wave silicon photonic crystal cavity sensing measurements at room temperature and pressure. (a) Normalized transmission through the photonic crystal cavity with (red) and without (blue) the MLG heterostructure. (b) Close-up of the first harmonic (golden diamond) mode. The black-dash lines represent fits using \ref{['eq:s21']}, fitting $\omega_\text{c}, \kappa$ and $t$. The external coupling is determined from fitting the reflection (\ref{['eq:s11']}). (c) Micrograph showing the flakes after the dry-transfer process. We see multiple flakes which we label A-E, where D and E are hBN and the rest are MLG. The scale bar indicates 100 µ m. (d) Electric-field distribution, $E_x$, of the first-order mode from a frequency domain simulation. The arrow shows where the sample was transferred. (e) An illustration of only conductive flakes, with the electric field distribution exported from (d). (f) The conductivity given by \ref{['eq:cond']} of the samples, using the measured sample volume and positional information. The histogram was generated via sampling the positions of flakes A-C from a normal distribution with a standard deviation of 12.5 μm.
• Figure 4: Cryogenic measurements of the bare mm-wave silicon photonic crystal cavity. (a) Normalized transmission through the setup and photonic crystal cavity at 4.3 K. The shaded region highlights the fundamental mode, where the inset shows the data and the black-dashed line is the corresponding fit. The inferred cavity frequency and linewidth are $\omega_\text{c}\approx 2\pi\cdot 96.261$ GHz and $\kappa \approx 2\pi\cdot 690$ kHz, corresponding to a total quality factor, $Q\approx 1.4\cdot 10^{5}$. (b) Temperature-dependent cavity frequency of the fundamental (blue triangles) and first harmonic (gold diamonds) modes. (c) The total quality factor of the fundamental and first harmonic mode, along with a light-gray line corresponding to the y-axis on the right side indicating chamber pressure in millibar.
• Figure S1: Measurements and modeling of silicon millimeter-wave waveguide losses. (a) Experimental data of 37.487 mm (32.487 mm straight section, 2.5 mm taper sections) tethered silicon waveguide. The black solid line and the black dotted line show the silicon waveguide and a WR10 waveguide of comparable length (25.4 mm), respectively. Normalizing the silicon waveguide to the WR10 waveguide, produces the blue solid line. The green solid line shows a COMSOL frequency domain simulation of the ideal silicon waveguide, corresponding well to the measured data. The black dashed line indicates a fit to the normalized data and has a slope of $-0.02$ dB/GHz and a median value of $-0.67$ dB. (b) Drude model estimates of the complex permittivity, given $\rho_{\text{dc}}= 20$ k-cm. The blue circles indicate $\text{Re}(\tilde{\varepsilon})$ and the red squares indicate $\text{Im}(\tilde{\varepsilon}) \times10^{3}$ to place these quantities on the same scale. (c) Energy confinement factor $\Gamma$, for a silicon waveguide with cross-sectional dimensions $1.1$ mm $\times$ 0.38 mm. The blue circles show the TE0 mode, which is shown in the inset (top left) and the red squares show the TM0 mode (bottom left inset). For the TE0 mode, the colors represent $E_{x}$ and for the TM0 the colors represent $H_{x}$. The black dashed line indicates the TE1 mode, which begins to propagate as the frequency increases (wavelength decreases). (d) The insertion loss of a 37.487 mm silicon waveguide inferred from the imaginary component of the Drude-permittivity of bulk silicon (blue circles) and of a guided TE0 mode, given $\rho_\text{dc} =20$ k-cm.