Spectroscopy of mechanical dissipation in micro-mechanical membranes
Micro-mechanical membrane oscillators are currently investigated in many optomechanics experiments, where lasers and optical cavities are used for cooling, control, and readout of their mechanical vibrations. Applications lie in the area of precision force sensing and in fundamental experiments on quantum physics at macroscopic scales. The quality factor Q of the mechanical modes of the membranes is a key figure of merit in such experiments. However, the origin of mechanical dissipation limiting the attainable Q is not completely understood and a subject of intense research.
Here we report on an experiment in which we observe a variation of Q by more than two orders of magnitude as a function of the fundamental mode frequency of a SiN membrane. Several distinct resonances in Q are observed that can be explained by coupling to mechanical modes of the membrane frame. The frequency of the membrane modes is tuned reversibly by up to 40% through local heating of the membrane with a laser. This method of frequency tuning has the advantage that the frequency dependence of Q can be studied with a single membrane in situ, resulting in a detailed spectrum of the coupling to the environment of this particular mode. Other methods that compare Q between various structures of different sizes have to rely on the assumption that the environment of these structures is comparable.
Figure 1. Experimental setup. The SiN membrane in a Si frame is glued at one edge to an aluminum holder inside a room-temperature vacuum chamber. The heating laser (red) at 780 nm is power stabilized and focused onto the membrane under an angle. The membrane vibrations are read out with a stabilized Michelson interferometer (blue). The interferometer signal is also used for feedback driving of the membrane with a piezo (PZT).
We investigate "low-stress" SiN membranes that are supported by a Si frame. The frame is glued at one edge to a holder inside a vacuum chamber, see Figure 1. The membranes have frequencies in the 100 kHz to MHz range. To read out the membrane vibrations, a Michelson interferometer is used where one end mirror consists of the membrane. The interferometer is stabilized by the DC to 20 kHz part of the photodiode (PD) signal. The >100 kHz frequency components of the signal are fed into a lock-in amplifier with integrated phase locked loop, which measures the membrane amplitude and drives its motion via a piezo mounted outside of the vacuum chamber. To tune the membrane frequency, a power stabilized 780 nm laser is focused onto the membrane. This laser heats the membrane locally in its center.
Figure 2. Mode spectrum of a membrane (length l=0.5 mm, thickness t=50 nm) as a function of power. At zero power, the lowest 13 modes lie within 2% of the expected frequency. At higher power, anticrossings between higher order modes are visible.
In a first experiment, we demonstrate the tunability of the membrane eigenfrequencies through laser heating. Figure 2 shows the recorded mode spectrum as a function of heating laser power. The spectra are recorded by Fourier transforming the PD signal. One can see a reversible decrease of all mode frequencies with power. The decrease in frequency can be attributed to a thermal expansion of the membrane. A simple model assuming a spatially homogeneous and linear temperature change with power can reproduce the observed dependence within 1 kHz. To model laser absorption in the membrane, we perform a finite element (FEM) simulation of laser heating. We find that a fraction of 1.5x10-3 of the 780 nm laser power is absorbed.
Figure 3. Upper plot: spectrum of membrane dissipation 1/Q of the fundamental membrane mode, showing a variation over two orders of magnitude. Lower plot: vibrations of the frame measured close to the membrane. The resonances in 1/Q coincide with the frame modes.
In a second experiment, we use laser tuning to record a spectrum of the quality factor of the fundamental mode as a function of fundamental mode frequency. We measure the decay time of the membrane amplitude in ring-down measurements after driving it to 0.5 nm. The upper plot in Figure 3 shows the dissipation 1/Q. We observe distinct resonances, changing Q by more than two orders of magnitude. The resonances can be attributed to coupling of the membrane mode to modes of the frame. To prove this, the interferometer is pointed onto the frame next to the membrane and the amplitude response to a driving with the piezo is recorded, as shown in the lower plot in Figure 3. The observed frame modes clearly overlap with the resonances in 1/Q.
Figure 4. Spectrum of membrane dissipation 1/Q for another membrane (l=250 μm, t=50 nm). Away from the frame resonances, the dissipation is independent of frequency.
Besides coupling to frame modes, the frequency dependence of other dissipation mechanisms is of interest. Figure 4 shows the dissipation spectrum of another low-stress membrane. Away from the resonances, we observe a constant baseline, indicating that other dissipation mechanisms are independent of frequency and temperature within our tuning range.
In conclusion, we presented a precise method for laser-tuning of micro-mechanical membrane oscillators and used it for spectroscopy of mechanical dissipation. Resonances in the dissipation were observed and explained as coupling to localized frame modes. Other dissipation mechanisms were found to be independent of membrane frequency and temperature in the measured range.
Reference
[1] A. Jöckel, M. T. Rakher, M. Korppi, S. Camerer, D. Hunger, M. Mader, and P. Treutlein,
Spectroscopy of mechanical dissipation in micro-mechanical membranes,
Appl. Phys. Lett. 99, 143109 (2011), see our list of publications.