Self-calibrating gas pressure sensor with a 10-decade measurement range

Recent years have seen a rapid reduction in the intrinsic loss of nanomechanical resonators (i.e., chip-scale mechanical oscillators). As a result, these devices become increasingly sensitive to the friction exerted by smallest amounts of gas. Here, we present the pressure-dependency of a nanomechanical trampoline resonator's quality factor $Q$ over ten decades, from $10^{-7}$ to $10^{3}\,\mathrm{mbar}$. We find that the measured behavior is well-described by a model combining analytical and numerical components for molecular and viscous flow, respectively. This model relies exclusively on design and typical material parameters, together with measured values of intrinsic resonance frequency $f_\mathrm{in}$ and quality factor $Q_\mathrm{in}$. Measuring $f_\mathrm{in}$ and $Q_\mathrm{in}$ at a pressure $<\!10^{-7}\,\mathrm{mbar}$ self-calibrates our sensor over its entire measurement range. For a trampoline's fundamental out-of-plane vibrational mode, the resulting deviation between measured and simulated pressure dependencies of the quality factor and resonance frequency is within $15\,\%$ and $4\,\%$, respectively. The resulting error for pressure values inferred from quality factor and frequency measurements is $<10\,\%$, for pressures between $\sim 10^{-6}$ and $\sim 10^{-1}\,\mathrm{mbar}$, and $<25\,\%$ for the complete 10-decade measurement range. Exceptions are two outliers with increased measurement errors, which might be related to the limited accuracy of our commercial pressure gauge. Based on investigations with helium, we demonstrate the potential for extending this sensing capability to other gases, thereby highlighting the practical use of our sensor.