Quantum friction in nanomechanical oscillators at millikelvin temperatures

We report low-temperature measurements of dissipation in megahertz-range, suspended, single-crystal nanomechanical oscillators. At millikelvin temperatures, both dissipation (inverse quality factor) and shift in the resonance frequency display reproducible features, similar to those observed in sound attenuation experiments in disordered glasses and consistent with measurements in larger micromechanical oscillators fabricated from single-crystal silicon. Dissipation in our single-crystal nanomechanical structures is dominated by internal quantum friction due to an estimated number of roughly 50 two-level systems, which represent both dangling bonds on the surface and bulk defects.

Figure 1

(Color online) (a) SEM micrograph of the $7\mu \mathrm{m}$ nanobeam. The diagram shows the measurement circuit. To drive the beam, we pass an alternating current $I\left(\omega \right)$ across the beam of length $L$ in the presence of magnetic field $B$, to set up a force $F_{\mathit{dr}}\left(\omega \right)=I\left(\omega \right)LB$ that is orthogonal to both. The center beam displacement $x$ for the fundamental mode shape in the magnetic field $B$ induces a voltage $V_{\mathit{emf}}\left(\omega \right)=\xi LB{\omega }_{0}x\left(\omega \right)$ in the gold electrode. We detect the response voltage using a RF network analyzer. (b) The fundamental resonance mode of the beam at 14.586 MHz with $Q=33000$. The induced voltage $V_{\mathit{emf}}$ is a Lorentzian peak on top of a white noise background, which is due primarily to the input noise of our preamplifier, and (c) the corresponding phase.

Figure 2

(Color online) (a) The resonance peak, which is Lorentzian in the linear regime, assumes the asymmetric nonlinear shape when the driving power is increased above $-95\mathrm{dBm}$ (trace $D$ on the plot). (b) Linear dependence of the induced power $P_{\mathit{in}}$ on driving power $P_{\mathit{dr}}$ at 4 T. The induced power corresponds to the resonance displacement $x$, shown vs the driving force on a log-log scale. The measured effective spring constant is $k_{\mathit{eff}}=182\mathrm{N}∕\mathrm{m}$. (c) The response of the oscillator increases linearly with magnetic field, as expected when the driving force $F=ILB$ is kept constant at 20 pN. We keep the driving force constant in order to investigate the effect of the magnetic field on energy dissipation. (d) Magnetic field dependence of dissipation. The data set is taken at 60 mK. It exhibits quadratic field dependence for high magnetic fields in both samples. Dissipation saturates in the low field region.

Figure 3

(Color online) (a) Temperature dependence of dissipation for the 12 MHz beam on a log-log scale. The $T^{-2}$ and $\exp (-2e∕k_{B}T)∕k_{B}T$ curves are the predicted acoustic dissipation forms due to TLS in crystals at low and high temperatures, respectively. The best fit to our dissipation data is the $T^{0.36}$ power law. (b) Temperature dependence of the shift in the resonance frequency on a semilogarithmic scale with a guide to the eye (dashed line). According to the acoustic dissipation in crystals, the temperature dependence of the frequency shift at higher temperatures goes as $T^{-1}$.

Figure 4

(Color online) (a) The sharp response of the 12 MHz beam as temperature is increased displays significant frequency shift and dissipation. The shift in the resonance frequency is illustrated by means of a guide to the eye (dashed line), drawn on the raw data. (b) Temperature dependence of the shift in the resonance frequency for both beams on a semilogarithmic scale. The plot for the 14.586 MHz beam has been shifted up for clarity. The data exhibits a peak in the vicinity of 0.7 K in both samples. Below the temperature 0.7 K the dependence is closely logarithmic $\ln T$. However, it is difficult to determine the behavior above 0.7 K due to the sparseness of the data. (c) Below 2 K dissipation shows weak power law dependence $T^{0.36}$. Both samples exhibit saturation of dissipation below 100 mK.