Damping of Nanomechanical Resonators
We study the transverse oscillatory modes of nanomechanical silicon nitride strings under high tensile stress as a function of geometry and mode index . Reproducing all observed resonance frequencies with classical elastic theory we extract the relevant elastic constants. Based on the oscillatory local strain we successfully predict the observed mode-dependent damping with a single frequency independent fit parameter. Our model clarifies the role of tensile stress on damping and hints at the underlying microscopic mechanisms.
The resonant motion of nanoelectromechanical systems receives a lot of recent attention. Their large frequencies, low damping i.e. high mechanical quality factors, and small masses make them equally important as sensors Jensen2008 (); Lassagne2008 (); LaHaye2009 (); Teufel2009 () and for fundamental studies Aldridge2005 (); LaHaye2009 (); Rocheleau2010 (); Li2008 (); Teufel2009 (); Etaki2008 (); Li2008 (); Eichenfield2009 (). In either case, low damping of the resonant motion is very desirable. Despite significant experimental progress Verbridge2006 (); Huttel2009 (), a satisfactory understanding of the microscopic causes of damping is not yet achieved. Here we present a systematic study of the damping of doubly-clamped resonators fabricated out of prestressed silicon nitride leading to high mechanical quality factors Verbridge2006 (). Reproducing the observed mode frequencies applying continuum mechanics, we are able to quantitatively model their quality factors by assuming that damping is caused by the local strain induced by the resonator’s displacement. Considering various microscopic mechanisms, we conclude that the observed damping is most likely dominated by dissipation via localized defects uniformly distributed throughout the resonator.
We study the oscillatory response of nanomechanical beams fabricated from high stress silicon nitride (SiN). A released doubly-clamped beam of such a material is therefore under high tensile stress, which leads to high mechanical stability Unterreithmeier2009 () and high mechanical quality factors Verbridge2006 (). Such resonators are therefore widely used in recent experiments Eichenfield2009 (); Rocheleau2010 (). Our sample material consists of a silicon substrate covered with 400 nm thick silicon dioxide serving as sacrificial layer and a nm thick SiN device layer. Using standard electron beam lithography and a sequence of reactive ion etch and wet-etch steps, we fabricate a series of resonators having lengths of and a cross section of nm as displayed in Fig. 1a and b. Since the respective resonance frequency is dominated by the large tensile stress Verbridge2006 (); Unterreithmeier2009a (), this configuration leads to resonances of the fundamental modes that are approximately equally spaced in frequency. Suitably biased gold electrodes processed beneath the released SiN strings actuate the resonators via dielectric gradient forces to perform out-of-plane oscillations, as explained in greater detail elsewhere Unterreithmeier2009 (). The length and location of the gold electrodes is properly chosen to be able to also excite several higher order modes of the beams. The experiment is carried out at room temperature in a vacuum below mbar to avoid gas friction.
The displacement is measured using an interferometric setup that records the oscillatory component of the reflected light intensity with a photodetector and network analyser Unterreithmeier2009 (); Azak2007 (). The measured mechanical response around each resonance can be fitted using a Lorentzian lineshape as exemplarily seen in the inset of Fig. 2. The thereby obtained values for the resonance frequency and quality factor for all studied resonators and observed modes are displayed in Fig. 2 (filled circles). In order to reproduce the measured frequency spectrum, we apply standard beam theory (see e.g. Timoshenko ()). Without damping, the differential equation describing the spatial dependence of the displacement for a specific mode of beam at frequency writes (with being the material density MEMShandbook (); , are the (unknown) real Young’s modulus and built in stress, respectively)
Solutions of this equation have to satisfy the boundary conditions of a doubly-clamped beam (displacement and its slope vanish at the supports (, : beam length). These conditions lead to a transcendental equation that is numerically solved to obtain the frequencies of the different modes.
The results are fitted to excellently reproduce the measured frequencies, as seen in Fig. 2 (hollow squares). One thereby obtains as fit parameters the elastic constants of the micro-processed material GPa, MPa, in good agreement with previously published measurements Unterreithmeier2009a ().
For each harmonic, we now are able to calculate the strain distribution within the resonator induced by the displacement and exemplarily shown in Fig. 1c. The measured dissipation is closely connected to this induced strain . As in the case of a Zener model Cleland2003 () we now assume that the displacement-induced strain and the accompanying oscillating stress are not perfectly in phase; this can be expressed by a Young’s modulus having an imaginary part. The relation reads again . During one cycle of oscillation , a small volume of length and cross section thereby dissipates the energy . The total loss is obtained by integrating over the volume of the resonator.
The strain variation and its accompanying energy loss can be separated into contributions arising from overall elongation of the beam and its local bending. It turns out that here the former is negligible, despite the fact that the elastic energy is dominated by the elongation of the string, as discussed below. To very high accuracy we obtain for the dissipated energy . A more rigorous derivation can be found in the Supplementary Information.
The total energy depends on the spatial mode (through , see exemplary Fig. 1c) and therefore strongly differs for the various resonances. To obtain the quality factor, one has to calculate the stored energy e. g. by integrating the kinetic energy . The overall mechanical quality factor is . A more detailed derivation can be found in the Supplementary Information.
Assuming that the unknown value of the imaginary part of the elastic modulus is independent of resonator length and harmonic mode, we are left with one fit parameter to reproduce all measured quality factors and find excellent agreement (Fig. 2, hollow squares). We therefore successfully model the damping of our nanoresonators by postulating a frequency independent mechanism caused by local strain variation. Allowing to depend on frequency, the accordance gets even better, as discussed in detail in the Supplement.
We now discuss the possible implications of our findings, considering at first the cause of the high quality factors in overall pre-stressed resonators and then the compatibility of our model with different microscopic damping mechanisms. In a relaxed beam, the elastic energy is stored in the flexural deformation and becomes for a small test volume . In the framework of a Zener model, as employed here, this result is proportional to the energy loss (see eq. 3) and thus yields a frequency-independent quality factor for the unstressed beam. In accordance with this finding, Ref. Verbridge2006 () reports a much weaker dependence of quality factor on resonance frequency, in strong contrast with the behavior of their stressed beams.
Similar as in the damping model, the total stored elastic energy in a beam can be very accurately separated into a part connected to the bending and a part coming from the overall elongation. The latter is proportional to the pre-stress and vanishes for relaxed beams, refer to the Supplement for details. Assuming a constant , Fig. 3 displays the calculation of the elastic energy and the quality factor for the fundamental mode of our longest () beam as a function of overall built in stress . The total elastic energy is increasingly dominated by the displacement-induced elongation . In contrast the bending energy , which in our model is proportional to the energy loss, is found to increase much slower with . Thus one expects to increase with , a finding already discussed by Schmid and Hierold for micromechanical beams Schmid2008 (). However, their model assumes the simplified mode profile of a stretched string and can not explain the larger quality factors of higher harmonics when compared to a fundamental resonance of same frequency. Including beam stiffness, our model can fully explain the dependence of frequency and damping on length and mode index, as reflected in Fig. 2. It also explains the initially surprising finding Southworth2009 () that amorphous silicon nitride resonators exhibit high quality factors when stretched whereas having Q-factors in the relaxed state that reflect the typical magnitude of internal friction found to be rather universal in glassy materials Pohl2002 (). More generally we conclude that the increase in mechanical quality factors with increasing tensile stress is not bound to any specific material.
Since the resonance frequency is typically easier to access in an experiment, we plot the quality factor vs. corresponding resonance frequency in Fig. 3b; with both numbers being a function of stress. The resulting relation of quality factor on resonance frequency is (except for very low stress) almost linear; experimental results by another group can be seen to agree well with this finding Verbridge2007 (). In addition, we show in the Supplement that although the energy loss per oscillation increases with applied stress, the linewidth of the mechanical resonance decreases.
We will now consider the intrinsic physical mechanisms that could possibly contribute to the observed damping. As explained in greater detail in the Supplement, we can safely neglect clamping losses Hao2003 (); Wilson-Rae2008 (), thermoelastic damping Lifshitz2000 (); Kiselev2008 () and Akhiezer damping Akhieser1939 (); Kiselev2008 () since they all predict damping constants significantly smaller than the ones observed.
Instead, we would like to discuss the influence of localized defect states. Similar to the Akhiezer effect, it is assumed that the energy spectra of defects are modulated by strain Jackle1972 (), which thereby drives the occupancy out of thermal equilibrium. In Ref. Jackle1972 (), this effect is calculated for a broad spectrum of two-level systems. The energy difference of two uncoupled levels as well as their separating tunnel barrier height are assumed to be uniformly distributed, leading to a broad yet not flat distribution of relaxation rates. In the high temperature limit the thus derived energy loss per oscillation becomes frequency-independent as assumed in our model. In addition, published quality factors of relaxed silicon nitride nano resonators Southworth2009 () cooled down to liquid helium temperature display quality factors that are well within the typical range of amorphous bulk materials Pohl2002 (). Moreover, on a different sample chip we measured a set of resonators showing quality factors that are uniformly decreased by a factor of approximately 1.4 compared to the data presented in Fig. 2. Their response can still be quantitatively modeled using now an increased imaginary part of Young’s modulus . We attribute this reduction to a non-optimized RIE-etch step, leading to an increased density of defect states. The corresponding data is presented in the Supplementary Information. These three arguments clearly favor the concept of damping via defect states as the dominant mechanism.
We wish to point out that such a model calculation based on two-level systems cannot be rigourously applied at elevated temperatures, as the concept of two-level systems should be replaced by local excitable systems. However, it seems plausible that such a system still exhibits a broad range of relaxation rates, crucial to explain frequency-independent damping. In contrast, a mechanism with discrete relaxation rates will exhibit damping maxima whenever the oscillation frequency matches the relaxation rate Lifshitz2000 (); Kiselev2008 (); Cleland2003 (). We further notice that in our experiments beams with larger widths exhibit slightly higher quality factors. This indicates an increased defect density near the surface being either intrinsic or caused by the micro-fabrication Yang2002 () (RIE etch). For a fixed cross-section however, the applicability of our model is not affected.
In conclusion, we systematically studied the transverse mode frequencies and quality factors of prestressed SiN nanoscale beams. Implementing continuum theory, we reproduce the measured frequencies varying with beam length and mode index over an order of magnitude. Assuming that damping is caused by local strain variations induced by the oscillation, independent of frequency, enables us to calculate the observed quality factors with a single interaction strength as free parameter. We thus identify the unusually high quality factors of pre-stressed beams as being primarily caused by the increased elastic energy rather than a decrease in damping rate. Several possible damping mechanisms are discussed; because of the observed nearly frequency-independence of the damping parameter , we attribute the mechanism to interaction of the strain with local defects of not yet identified origin. One therefore expects that defect-free resonators exhibit even larger quality factors, as recently demonstrated for ultra-clean carbon nanotubes Huttel2009 ().
Financial support by the Deutsche Forschungsgemeinschaft via project Ko 416/18, the German Excellence Initiative via the Nanosystems Initiative Munich (NIM) and LMUexcellent as well as the Future and Emerging Technologies programme of the European Commission, under the FET-Open project QNEMS (233992) is gratefully acknowledged.
We would like to thank Florian Marquardt and Ignacio Wilson-Rae for stimulating discussions.
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Appendix A Damping Model
In a Zener model, an oscillating strain and its accompanying stress are out-of phase, described by a frequency-dependent, complex elastic modulus . This leads to an energy loss per oscillation in a test volume of cross-section and length .
We now employ this model for our case, namely a pre-stressed, rectangular beam of length , width and height , corresponding here to the x,y,z-direction, respectively. The origin of the coordinate system is centered in the beam. The resonator performs oscillations in the z-direction and, as we consider a continuum elastic model, there will be no dependence on the y-direction. For a beam of high aspect ratio and small oscillation amplitude, the displacement of the -th mode can be approximately written . During oscillation, a small test volume within the beam undergoes oscillating strain .
This strain arises because of the bending of the beam as well as its elongation as it is displaced. The stress caused by the overall elongation is quadratic in displacement, therefore it occurs at twice the oscillating frequency.
Inserting this into eq. 3 and integrating over the cross-section , the accompanying energy losses can be seen to separate into elongation and displacement caused terms.
Integrating over the length yields the total energy loss of a particular mode . In the case that is only weakly frequency-dependent, it turns out that for our geometries the elongation term is several orders of magnitude () smaller than the term arising from the bending. The energy loss therefore may be simplified and writes
Appendix B Elastic Energy of a Pre-Stressed Beam
A volume subject to a longitudinal pre-stress stores the energy when strained; is assumed to be frequency independent in the experimental range (5-100 MHz)
To apply this formula to the case of an oscillating pre-stressed beam, we insert eq. 4 (maximum displacement) and integrate over the cross-section to obtain
Analog to eq. 5 we can omit the first term in the brackets; integrating over the length yields
We can therefore divide the total energy into parts arising from the elongation and the bending of the beam. Depending on the magnitude of the pre-stress, either of the two energies can dominate as seen in Fig. 3a of the main text. We have checked that the kinetic energy ; (: resonance frequency, material density, respectively) equals the total elastic energy, as expected.
Appendix C Frequency-dependent Loss Modulus
There is no obvious reason that the imaginary part of Young’s modulus should be completely frequency-independent. We therefore assume that obeys a (weak) power-law and chose the ansatz:
Fitting our data with the thus extended theory, we achieve a very precise agreement of measured and calculated quality factors, as seen in Fig. S1. The resulting exponent is ; varies therefore by 20% when changes by one order of magnitude.
Appendix D Linewidth of the Mechanical Resonance
The elastic energy of a harmonic oscillator is given by with being effective mass, resonance frequency and displacement, respectively. If we assume the effective mass to be energy-independent, it applies . Recalling the definition of the quality factor , one obtains for the for the Full Width at Half Max (FWHM) of the resonance
As in the main text, the energy depends on the applied overall tensile stress. Figure 6 shows a numerical calculation of the resulting linewidth vs. applied stress; one can see that increase in energy loss per oscillation is dominated by the increase in energy, resulting in a decreased linewidth. The exact effective mass is included in this calculation; as it changes by less than 20%, the above assumption is justified.
Appendix E Microscopic Damping Mechanisms
We start with clamping losses as discussed, e.g., in ref. Hao2003 (); Wilson-Rae2008 (), i. e. the radiation of acoustic waves into the bulk caused by inertial forces exerted by the oscillating beam. With a sound velocity in silicon of km/s, the wavelength of the acoustic waves radiated at a frequency of 10 MHz from the clamps into the bulk will be greater than 500 m, and thus substantially larger than the length of our resonators. Considering each clamping point as a source of an identical wave propagating into the substrate, one would expect that mostly constructive/destructive interference would occur for in-/out-of-phase shear forces exerted by the clamping points, respectively. With clamping losses being important, one would therefore expect that spatially asymmetric modes with no moving center of mass exhibit significant higher quality factors than symmetric ones Ignacio (). Another way to illuminate this difference is that symmetric modes give rise to a net force on the substrate, whereas asymmetric modes yield a torque. Since the measurement (Fig. 2) does not display such an alternating behavior of the quality factors with mode index (best seen for the longest beam), clamping losses are likely to be of minor importance.
The next damping mechanism we consider are phonon-assisted losses within the beam. At elevated temperatures, at least two effects arise, the first being thermoelastic damping: because of the oscillatory bending, the beam is compressed and stretched at opposite sides. Since such volume changes are accompanied by work, the local temperature in the beam will deviate from the mean. For large aspect ratios as in our case, the most prominent gradient is in the z direction. The resulting thermal flow leads to mechanical dissipation. We extend existing model calculations Lifshitz2000 () to include the tensile stress of our beams. Using relevant macroscopic material parameters such as thermal conductivity, expansion coefficient and heat capacity we derive Q-values that are typically three to four orders of magnitudes larger than found in the experiment. Therefore, heat flow can be safely neglected as the dominant damping mechanism. In addition, the calculated thermal relaxation rate corresponds to approximately 2 GHz, so the experiment is in the so-called adiabatic regime. Consequently, one would expect the energy loss to be proportional to the oscillation frequency, in contrast to the assumption of a frequency independent and our experimental findings.
Another local phonon-based damping effect is the Akhiezer-effect Akhieser1939 (); it is a consequence of the fact that phonon frequencies are modulated by strain, parameterized by the Grüneisen tensor. If different phonon modes (characterized by their wave vector and phonon branch) are affected differently, the occupancy of each mode corresponds to a different temperature. This imbalance relaxes towards a local equilibrium temperature, giving rise to mechanical damping. In a model calculation applying this concept to the oscillatory motion of nanobeams Kiselev2008 (), the authors find in the case of large aspect ratios length/height that the thermal heat flow responsible for thermoelastic damping dominates the energy loss by the Akhiezer effect. We thus can safely assume this mechanism to be also negligible in our experiment.
Appendix F Reduced Quality Factor
We fabricated a set of resonators, shown in Fig. S1a, that showed lower quality factors than the ones presented in the main text (Fig. 2); we attribute this reduction to a non-optimized RIE-etch step. As in the main article, it is possible to reproduce the quality factors using a single fit parameter, namely the imaginary part of Young’s modulus . The ratio of the two sets of quality factors is displayed in Fig. S1 b and can be seen to be around 1.4 with no obvious dependence on resonance frequency, mode index or length. A non-optimized etch step causes additional surface roughness and the addition of impurities, thereby increasing the density of defect states. As there is no obvious reason why another damping mechanism should be thereby influenced, we interpret this as another strong indication that the dominant microscopic damping mechanism is caused by localized defect states.
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