Summary of Q Measurements, Comparison to Clamping Models

8 6 4 2 0 Qua lit y fa ct or (x 10

6

)

10 8

6 4

2 0

frequency (MHz)

d_m = 30 nm d_m = 50 nm d_m = 100 nm d_chip

d_chip

mm mm

= 0.5

= 0.2

d wm

dchip wchip

m

Figure 4.9: Compilation of Q measurements for different membrane and chip thicknesses. Here we have fixed the square chip and membrane dimensions at w_m = 500µm and w_chip = 5 mm, respectively. The thickness of the film is varied between d_m = 30 nm, 50 nm, and 100 nm. The thickness of the chip is varied betweend_chip= 200 µm and 500µm. For a given membrane mode, the quality factor scales roughly inversely with membrane thickness. Marginal improvement is also seen for thicker chips. In all cases, the membrane is mounted by resting on a planar surface under its own weight.

products. For free-standing chips, most of our study as been limited towm= 0.25 mm and 0.5 mm membrane windows. We have found that at similar mechanical frequencies, Qm for the modes of thewm= 0.25 mm membrane are roughly half that of the wm = 0.5 mm membrane. A subset of results ford_m= 50 nm membranes in which the chip dimensions has been fixed atd_chip= 0.2 mm thick andw_chip= 5 mm wide is shown in Figure 4.10.

4.8 Summary of Q Measurements, Comparison to Clamping

8 6 4 2 0 Qua lit y fa ct or (x 10

6

)

10 8

6 4

2 0

frequency (MHz)

w _m = 0.25 mm w _m = 0.50 mm

d w_m

d_chip w_chip

m

Figure 4.10: Compilation ofQmeasurements for different membrane sizes. Here we have membrane thickness at at dm = 50 nm and the chip dimensions at{dchip, wchip} = {0.2 mm,5 mm}. The widthwmof the square membrane is varied between 0.25 mm, 0.5 mm, and 1 mm. As an important caveat, for 1 mm membranes we have only made measurements for rigidly mounted chips— we believe this explains the discrepancy from the trend inferred from the 250 µm and 500µm trials.

For these two cases the membrane chip was allowed to rest on a planar surface under its own weight.

vs. “free-standing” under its own weight. Note that the best results are obtained for the thinnest (30 nm) membranes deposited on the thickest (500µm) chips. The scatter in measuredQ_mvalues extends two orders of magnitude and is predominantly associated with lower order modes of rigidly mounted membrane chips. This scatter is diminished for higher order membrane modes, which we may interpret as having weaker coupling to the substrate. At high frequencies, these quality factors appear to asymptotically approach aQ×f product of several 10¹³Hz. Higher frequencies were not measured for lack of ability to drive them with the piezo.

It is interesting to compare these qualitative trends to loss mechanisms often quoted in NEMs and MEMs literature. A tremendous amount of work has been dedicated to this subject, so we confine ourselves to the loss mechanisms that have been suggested for transverse vibrations of structures based on thin films. Surprisingly little is understood physically about the role of tensile stress in these types of resonators [41]. For doubly clamped nano-beam type geometries at room temperature, for example, it has been suggested that primary loss mechanisms include viscous air damping, thermoelastic loss, and external (clamping) loss due to surface layers and acoustic coupling to the support [39, 59]. We touch on these subjects below.

10

¹⁰

10

¹¹

10

¹²

10

¹³

10

¹⁴

Q*f (Hz )

12 10

8 6

4 2

0

green:dm= 30 nm red:dm= 50 nm blue:dm= 100 nm circle: w _m square: w

triangle: w _m = 1.00 mm solid:free-standing chip open:clamped chip

frequency (MHz)

m

m= 0.25 mm

= 0.50 mm

d wm

dchip wchip

Figure 4.11: Summary of Q measurements. Here we highlight the results for differing membrane thicknessdm, membrane sizewm, and mounting technique (“clamped” vs. free-standing). Different chip size and thicknesses are lumped together, though we note that moderately improved quality factors are recorded for thicker chips. The highest Q×f products were obtained for dm =30-nm- thick,wm= 500-µm-wide membranes on adchip = 500-µm-thick, wchip = 5-mm-wide chip resting on three corners under its own weight. Membranes with different dimensions appear to exhibitQ×f with scaling roughlywm/dm.

4.8.1 Clamping Mechanisms

Air damping for a thin beam of thicknesstobeys a relation [40]

Q_air =ρtΩ_m 4

rπ 2

rRT M

1

P ≈6×10⁸× t

50 nm

Ω_m/2π 1 MHz

10⁻⁶ mbar P

r T

298 K (4.21) in the molecular flow regime, characterized by a mean free path ` between gas particles (with diameter d0) that is greater than the largest dimension of the oscillator. For an ideal gas, ` = k_BT_room/√

2πd²₀P >1 mm impliesP <6×10⁻² mbar for this condition to hold for 50-nm-thick films at room temperature withd₀≈0.4 (typical for gas particles in air). For our operating pressure of P ∼ 10⁻⁶ mbar neither the absolute value nor the thickness dependence associated with air damping seems to match the measuredQ_mdata.

A second candidate is thermoelastic loss, which arises when elastic strain produced by vibration induces local temperature gradients. Equilibration of these gradients draws energy from the vibra- tion. This form of loss can dominate in room temperature NEMS/MEMs systems, where the small dimension of the oscillator can lead to thermal relaxation rates as fast or faster than the vibrational

frequency. A formula which is often applied to resonators under tensile stress but which is derived for purely bending-type resonators in is [60]

QT ED= cv

Eα²T 6

ξ² − 6 ξ³

sinhξ+ sinξ coshξ+ cosξ

−1

; ξ=d

rΩmρcm

2κ_c (4.22)

where dis the thickness of the structure. c_v is the specific heat per unit volume, E is the Young’s modulus, αis the linear thermal expansion coefficient, c_m is the specific heat per unit mass, κ_c is the thermal conductivity, andT is the ambient temperature. For parameters{d, ρ, E, κ_c, α, c_m, c_v}

= {50 nm,2.7g/cm³,210 GPa,30 W/m/K,2.3×10⁻⁶ m/K,710.6 J/kg/K}, one predicts Q×f ≈ 6×10¹³Hz (the second term in (4.22) is negligible in our case, sinceξ1). The main uncertainty in this number comes from the value ofκc, which varies by as much as a factor of ten in the literature (see [39] and references in Table 4.1). This result suggests that thermoelastic losses may explain the highestQ×f products we’ve achieved. However, we have several reasons to believe that this is not the case. First of all,ξ1 suggests QT ED∝d⁻²_m; if this were the case and 30 nm membranes were thought to be limited by thermoelastic losses, then results for our 100-nm-thick membranes would exceed the predicted thermoelastic limit. More importantly, our resonators are decidedly far from

“bending”-type resonators. The derivation ofQ_{T ED} in [60] assumes a relationship between internal strain and mechanical frequency which significantly overestimates the curvature of the transverse vibrations associated with a given mechanical frequency (it is this curvature which leads to volume change and thermoelastic heating for a thick 1D or 2D string). A more rigorous calculation which takes into account tension has been carried out by our colleague Darrick Chang. He predicts that the thermoelastic QT ED×f limit for our square drum resonators can be as high as ∼ 10¹⁹ Hz.

This prediction is in qualitative agreement with the results of a calculation we’ve been told has been independently carried out by the Yale group [61].

We have yet to identify a smoking gun from among the various clamping loss mechanisms nor- mally cited for NEMs structures. Phenomenologically, what we observe is a quality factor that scales roughly inversely with membrane thicknessd_m, roughly linearly with membrane square dimensions wm, roughly linearly with chip thickness dchip, and roughly independent of frequency for higher harmonics. By contrast, models for surface loss [59] in planar resonators predict that thicker films should exhibit higherQm, since fractionally less of the vibrational energy is shared with the lossy surface layer. Models for acoustic coupling of doubly clamped beams and cantilevers to their support substrates generally predict that Qshould scale as a power law greater than or equal to linear in the length of the resonator, depending on the geometry of the substrate [39]. These models assume, however, that T = 0, that the beam width is much smaller than its length (in our case the two are equal), and that the wavelength of an acoustic phonon propagating in the substrate material at the vibrational frequency of the resonator is much different than relevant dimensions of either

the resonator or the substrate. In our case, a 1–10 MHz frequency phonon propagating in silicon with elastic wave speed ≈ 5000 m/s has a wavelength of 0.5 - 5 mm, which is of the same order as the square dimensions of the membrane (wm = 0.25–1 mm), the thickness of the chip (dchip = 0.2–0.5 mm), and the square dimensions of the chip (wchip = 5 mm–10 mm)! Recent work by the Cornell group with Ignacio Wilson-Rae [58] has taken some of these considerations into account.

They predict that for square and circular membrane geometries similar to ours, the quality factor limit due to acoustic coupling to the support should indeed scale as wm/dm. They compare their prediction to measurements made on dm= 112.5 nm, wm = 250-µm-square membranes developed in-house with T = 1200 MPa and using a somewhat different processing method than Norcada.

Their Q prediction is a factor 10–100 higher than measured. Accounting for this discrepancy by adding an ad-hoc internal loss term, they are able to predict to within factor of∼2 the differences in the quality factors of different higher harmonics for the same square membrane. It is interesting to note that the quality factors they observe agree to within a factor of∼2 with the value of Q≈10⁶ that our results would phenomenologically predict for a membrane of their dimensions.