Mechanical coupling of a suspended structure to its supports is a fundamental energy loss mechanism in micromechanical and nanomechanical resonators1
. Referred to variously as clamping2
or anchor loss3
, this process remains significant even in devices fabricated from high-quality materials operated in vacuum and at cryogenic temperatures, and is in fact unavoidable in any non-levitating system. Although much progress has been made towards the understanding of mechanical dissipation at the microscale and nanoscale2
, obtaining reliable predictions for the fundamental design-limited quality factor, Q
, remains a major challenge while direct experimental tests are scarce5
. At the same time, the implementation of high-quality micromechanical and nanomechanical systems is becoming increasingly important for numerous advanced technological applications in sensing and metrology, with select examples including wireless filters3
, on-chip clocks9
and molecular-scale mass sensing14
, and recently for a new generation of macroscopic quantum experiments that involve mesoscopic mechanical structures16
. Here, we introduce a finite-element-enabled numerical solver for calculating the support-induced losses of a broad range of low-loss mechanical resonators. We demonstrate the efficacy of this approach via comparison with experimental results from microfabricated devices engineered to isolate support-induced losses by allowing for a significant variation in geometry, while keeping other resonator characteristics approximately constant. The efficiency of our solver results from the use of a perturbative scheme that exploits the smallness of the contact area, specifically the recently introduced 'phonon-tunnelling' approach24
. This results in a significant simplification over previous approaches and paves the way for CAD-based predictive design of low-loss mechanical resonators.
The origins of mechanical damping in microscale and nanoscale systems have been the subject of numerous studies during the last decades, and several relevant mechanisms for the decay of acoustic mechanical excitations, that is, phonons, have been investigated2
. These include: (i) fundamental anharmonic effects such as phonon–phonon interactions4
, thermoelastic damping (TED)4
and the Akhiezer effect4
; (ii) viscous or fluidic damping involving interactions with the surrounding atmosphere or the compression of thin fluidic layers29
; (iii) material losses driven by the relaxation of intrinsic or extrinsic defects in the bulk or surface of the resonator32
for which the most commonly studied model is an environment of two-level fluctuators38
and (iv) support-induced losses, that is, the dissipation induced by the unavoidable coupling of the resonator to the substrate3
, which corresponds to the radiation of elastic waves into the supports5
. This last mechanism poses a fundamental limit, as vibrations of the substrate will always be present.
These various dissipation processes add incoherently such that the reciprocals of the corresponding Q-values satisfy 1/Qtot=∑i1/Qi, where i labels the different mechanisms. Thus, in a realistic setting, care must be taken to isolate the contribution under scrutiny. In contrast to all other damping mechanisms (i–iii), which exhibit various dependencies with external physical variables such as pressure and temperature, support-induced dissipation is a temperature- and scale-independent phenomenon with a strong geometric character that is present in any suspended structure. Moreover, its scale independence implies that the same analysis can be applied to both microscale and nanoscale devices. We exploit this geometric character to isolate the support-induced contribution and obtain a direct experimental test of phonon-tunnelling dissipation.
The numerical solver we introduce provides a new technique to efficiently model support-induced losses for a broad class of mechanical structures. Previous approaches have relied on either the direct solution of an elastic wave radiation problem involving the substrate6
or the simulation of a perfectly absorbing artificial boundary5
, with systematic tests as a function of geometry limited to a few specific cases5
. In contrast, our technique represents a substantial simplification in that it reduces the problem to the calculation of a perfectly decoupled resonator mode together with free elastic wave propagation through the substrate in the absence of the suspended structure. A key feature of our method is to combine a standard finite-element method (FEM) calculation of the resonator mode together with the use of an extended contact at the support. This allows us to treat complex geometries, taking proper account of interference effects between the radiated waves.
In summary, we develop and test an efficient method for calculating the clamping loss of high-Q mechanical resonators. Our analysis includes a thorough experimental verification of this theoretical framework by employing resonators that are specifically designed to isolate the clamping-loss contribution to the total dissipation 1/Q. The measured damping in these structures matches the theoretical predictions and demonstrates in a direct manner the strong geometric character of this fundamental dissipation channel.