# Nano-beam clamping revisited

Ilya Golokolenov  [  Sumit Kumar  [  Baptiste Alperin  [  Bruno Fernandez  [
Andrew Fefferman  [  Eddy Collin  [<EMAIL_ADDRESS>
###### Abstract

Within recent years, the field of nano-mechanics has diversified in a variety
of applications, ranging from quantum information processing to biological
molecules recognition. Among the diversity of devices produced these days, the
simplest (but versatile) element remains the doubly-clamped beam: it can store
very large tensile stresses (producing high resonance frequencies $f_{0}$ and
quality factors $Q$), is interfaceable with electric setups (by means of
conductive layers), and can be produced easily in clean rooms (with scalable
designs including multiplexing). Besides, its mechanical properties are the
simplest to describe. Resonance frequencies and $Q$s are being modeled, with
as specific achievement the ultra-high quality resonances based on “soft
clamping” and “phonon shields”. Here, we demonstrate that the fabrication
undercut of the clamping regions of basic nano-beams produces a “natural soft
clamping”, given for free. We present the analytic theory that enables to fit
experimental data, which can be used for $\\{Q,f_{0}\\}$ design.

###### keywords:

Nano-mechanics (NEMS), mode frequency, mode dissipation, soft clamping

Institut Neel] Univ. Grenoble Alpes, Institut Néel, 25 rue des Martyrs,
Grenoble, 38042, France Institut Neel] Univ. Grenoble Alpes, Institut Néel, 25
rue des Martyrs, Grenoble, 38042, France [RHUL] Royal Holloway University of
London, Egham, Surrey, TW20 0EX, England Institut Neel] Univ. Grenoble Alpes,
Institut Néel, 25 rue des Martyrs, Grenoble, 38042, France Institut Neel]
Univ. Grenoble Alpes, Institut Néel, 25 rue des Martyrs, Grenoble, 38042,
France Institut Neel] Univ. Grenoble Alpes, Institut Néel, 25 rue des Martyrs,
Grenoble, 38042, France Institut Neel] Univ. Grenoble Alpes, Institut Néel, 25
rue des Martyrs, Grenoble, 38042, France NEMS

##  1 Introduction

Doubly-clamped nano-beams are utterly basic, but nonetheless remarkably
versatile. They are routinely used in a broad range of applications, from mass
sensing1 to quantum electronics2. About two decades ago, it had been found
that stoichiometric Silicon-Nitride (Si3N4) thin films grown on Silicon can
store very large tensile stresses, and that subsequently mechanical nano-
structures patterned on this material display very high flexural resonance
frequencies $f_{0}$, and interestingly very high $Q$ factors3. This phenomenon
has been named “dissipation dilution”: it is due to the large stored elastic
energy, as compared to the losses which arise from bending4, 5. Indeed,
bending losses have been found experimentally to be essentially stress-
independent6, 7, 8. The precise amount of stress stored in the structures can
be tuned by stoichiometry, but also by chip-bending4, by the design of the
beam shape9 and by the clamping pillars10.

A precise (yet phenomenological) modeling of the flexure of beams had been
proposed considering ideal clamping11, and then successfully adapted to
membranes12. It is based on a numerical solution of the Euler-Bernoulli
equation which describes the low-frequency dynamics of thin-and-long beams13,
matching the observed $Q$ factor linear increase with beam length $L$, and
decrease with mode number $n$. The key argument is to assume that internal
microscopic friction mechanisms originate from the bending of the material,
whatever they might be. In this sense, the model applies as well to pure
nitride structures11 as to bi-material devices where a metallic layer
dominates the damping14, 12, 15, 8. It also works for both room-temperature11,
12 and low temperature8 experiments. For the former, it has been argued that
losses are dominated by surface effects6; for the latter, a specific mechanism
based on Two-Level-Systems (TLSs) present in the materials is discussed in the
literature16, 17, 18.

Beyond internal damping, the anchoring points appear to play a very important
role in beam dynamics. The vibration of the mechanical modes irradiates
acoustic waves in the supports, which limits the $Q$. This radiation damping
has been modeled in particular for thin supports19, 20, 21, 22, which is
typically the geometry obtained when the fabrication process under-etches the
clamps (so-called undercut). The actual limiting $Q$-value depends on the
precise geometry of the beam (particularly its width $w$ and length $L$), and
of the anchor. Experimentally, for the first flexure $n=1$ of millimeter
length low-stress structures resonating at sub-MHz frequencies, radiation loss
seems irrelevant for widths $w<3~{}\mu$m23; for high-stress devices (tens of
MHz frequencies) with width $w\sim 200~{}$nm, acoustic radiation dominates for
$L<10~{}\mu$m24. This leaves a large playground to experimentalists where
bending is the main source of losses25. On the other hand, when radiation
losses dominate it is possible to suppress them with a clamp structuring that
forbids phonon transport at the specific frequency of the mode $n$ that one
wants to protect: a so-called “phonon shield”24, 26, 27.

But the anchoring does more than enabling irradiation into the bulk: it
defines the precise bending shape at the clamping point. It had been realized
about a decade ago that most of the friction occurs near the beam’s ends,
where the bending is the most dramatic23, 28. As such, beyond creating phonon
gaps in the substrate’s density of states, structuring the clamps has another
(more trivial) effect: it can reduce this bending and mitigate the losses,
which is named “soft clamping”26, 29, 30. In this Letter, we demonstrate
experimentally that a clamp undercut acts essentially as a “soft clamping”
given for free: the quality factor $Q$ of the flexural modes grows with the
beam width $w$, which is the reverse behavior when compared to acoustic
radiation. As well, the clamp modifies substantially the resonance
frequencies31 (we show that it decreases with $w$), which can be used for
design purposes, for instance for nano-beam resonance multiplexing32.

Early beam clamp modeling relied on phenomenological ansatzs23, 28. Instead,
here we present an exact analytic theory which follows the same lines as the
modeling performed on membranes12. It is based on a high-stress Taylor
expansion of the modal parameters (for any mode $n$), introducing as fit
parameters a mass-loaded spring and torque at each of the beam’s ends. We
demonstrate that both the resonance frequency $f_{0}$ and $Q$ can be fit with
a clamp spring coefficient $\alpha_{l,r}$ ($l,r$ for left and right), the
torque being (at lowest order) negligible. $\alpha_{r,l}$ is found to be
proportional to frequency and inversely proportional to width $w$, the
prefactor being a characteristic of the anchor’s shape (and material) solely.
We believe that our results constitute a very useful tool when designing basic
doubly-clamped beams presenting a characteristic fabrication undercut.

##  2 Results

Figure 1:  Left: SEM image of one sample ($400~{}\mu$m long beams), with
device widths $w$ of 50 nm, 100 nm, 200 nm, 500 nm and 1000 nm (from top to
bottom). Bottom-right inset: dark field image of clamping region (suspended
part of length $L_{c}$), and schematic of the layered structure (not to
scale). Top-right: typical phase-resolved linear response line measured for
one of our devices (first flexure $n=1$ of a 100$~{}\mu$m long, 1$~{}\mu$m
large one). Drive current $I_{0}$ of 0.5 nA, $B_{0}$ field 0.2 T. Lines are
Lorentzian fits with normalized heights (see text for details).

###  2.1 Experimental details

Nano-beams of various widths and lengths have been realized from low-stress
Silicon-Nitride. All have a thickness of about 100 nm, and come from the same
wafer. A conductive layer (30 nm Aluminum) has been deposited on top in order
to create electrical contacts (total thickness $e=130~{}$nm). A typical sample
SEM image is shown in Fig. 1 (left), with a zoom-in on one of the clamping
regions (inset, with schematic of the bilayer structure). The design is such
that the beams are all fully suspended within a hollow window; they are
connected to the bulk only through a well-defined over-hanging clamp of length
$L_{c}$ (of order $12~{}\mu$m for all of them). This part has the same
thickness (and same constitution) as the rest of the beam, which guarantees
that all the layer’s stress is transmitted within the structure. The absence
of pedestals holding the beams enables to avoid stress-relaxation effects
occurring when beams are released, and the pedestals bending10. A description
of the fabrication process can be found in Ref.32. Details about sample
characteristics can be found in Supplementary Information.

The measurements are performed using the magnetomotive technique33. A current
$I(t)=I_{0}\cos(\omega t+\varphi)$ is fed into the beam which stands in a
static magnetic field $B_{0}$, thus generating a force $\propto I_{0}B_{0}$ at
frequency $\omega$. Used fields typically range from 0.2 T to 1 T, and care is
taken to take all data in the linear regime, characterizing carefully the
extra damping coming from the electric circuit33 (which needs to be
subtracted). Each sample consists in a set of beams of equal lengths, but with
widths ranging from 50 nm to 1000 nm. For a given sample, all beams can be
connected in series, which enables a straightforward multiplexing (see Fig.
1). Besides, each beam can also be connected independently, in order to
separate its resonances from other devices without ambiguity. As beams move
out-of-plane, a voltage $V(t)$ is induced at their extremities, proportional
to field $B_{0}$ and velocity. Only symmetric modes are detected, the signal
being proportional to the maximal amplitude of motion; anti-symmetric ones
cancel out. Measurements are conducted in cryogenic vacuum at 4.2 K, using a
lock-in amplifier (with in-phase $X$ and quadrature $Y$ components). An
example of resonance peaks obtained via a frequency-sweep is shown in Fig. 1
(right). More details on the setup can be found in Ref32.

###  2.2 Data Analysis and Theory

From the measured peaks one extracts resonance frequency $f_{0}$ (centre
position) and damping $\Delta f$ (full width at half height on $X$ component),
giving us the quality factor $Q=f_{0}/\Delta f$. This is performed on three
different 100$~{}\mu$m long samples, two sets of 300$~{}\mu$m and 400$~{}\mu$m
ones, and one set of 200$~{}\mu$m beams. Only on one set (100$~{}\mu$m long,
1$~{}\mu$m wide) did we measure the mode dependence (with $n=1,3,5$). Only
very few points have been dropped from the statistics, due we believe to
fabrication irreproducibility: the 50 nm beams are not very homogeneous in
width, and even break with thermal cycling. On some beams, defects can be seen
(which look like some sort of filaments, see Fig. 1), which might not be
negligible for the narrowest $w=100~{}$nm devices. Besides, we see a similar
scatter in measured damping (up to a factor 2) as in Ref11. from one sample to
the other. The final error bars in our graphs therefore reflect the
statistical scatter due to the fabrication process. For details, see Ref32.
and Supplementary Information.

Figure 2:  Left, main: dependence of frequency on length $L$ for the first
flexure $n=1$ (all devices having $w=1000~{}$nm). Left, inset: frequency of
100$~{}\mu$m long, $1000~{}$nm wide device normalized to the extrapolated
value $f_{n,0}$ at $w\rightarrow 0$ as a function of mode number $n$. Right:
same normalized frequency for mode $n=1$ of all devices, as a function of
width $w$. The lines correspond to Eqs. (2,34) using a simple ansatz for the
clamp parameter $\alpha$, see text.

The starting point of the modeling is the Euler-Bernoulli equation that
describes the flexure of long-and-thin beams13. As in Ref12. we define as
small parameter:

$a=\sqrt{\frac{E\,I_{z}}{SL^{2}}},$ (1)

with $E$ the Young’s modulus and $S=\sigma\,we$ the tensile force acting on
the beam ($\sigma$ is the in-built axial stress). The second moment of area
$I_{z}=\frac{1}{12}we^{3}$ is a geometrical parameter. The composite nature of
the beam can be accounted for by choosing an effective Young’s modulus $E$
that depends on the $E_{SiN}$ and $E_{Al}$ of the two materials and on their
thickness34 (and taking for $\rho$ the mean density). In practice, this
parameter is close to the $E_{SiN}$ elastic constant, about 200 GPa ($\pm
50~{}\%$) from the literature11, 10. Numerical values for our experiment can
be found in Supplementary Information. The specificity of the approach lies in
the boundary conditions: on each side ($l,r$ for left and right) we impose an
elastic force and an inertial force. They are combined into an effective
spring constant $k_{l,r}=\frac{S}{L}\alpha_{l,r}$. Similarly, a torque (that
combines elastic and inertial contributions) acts also on each end of the
beam, resulting in an effective torsion spring
$\Gamma_{l,r}=SL\,\gamma_{l,r}$. The (dimensionless) constants
$\alpha_{l},\gamma_{l}$ and $\alpha_{r},\gamma_{r}$ fully characterize each of
the $l,r$ clamps, respectively. Ideal clamping is recovered with
$\alpha_{l,r},\gamma_{l,r}\rightarrow\infty$, which leads to the conventional
boundary conditions (no displacement, no bending angle at both ends). The
exact analytical solutions for $f_{0}$ and $Q$ are finally derived to second
order in $a$, and $1/\alpha_{l,r},1/\gamma_{l,r}$ (the explicit damping model
producing $\Delta f$ is discussed thereafter). We fit them on data, and find
out that $\gamma_{l,r}$ can be to first approximation neglected; besides, only
symmetric clamping will be addressed with $\alpha_{l}=\alpha_{r}=\alpha$.
Nonetheless, the full mathematical description is given in Supplementary
Information.

Let us first discuss the resonance frequency parameter $f_{0}$ (given here in
Hz). For any mode $n$, it writes:

$f_{0}(n,a,\alpha)=f_{n,0}\,P_{f}(n,a,\alpha),$ (2)

with $2\pi\times f_{n,0}=\frac{n\pi}{L}\sqrt{\frac{\sigma}{\rho}}$ the usual
$n^{th}$ resonance frequency of a string (of density $\rho$), and $P_{f}$ a
correction function. The latter is found to be:

$\displaystyle P_{f}(n,a,\alpha)$ $\displaystyle=$ $\displaystyle
1-\frac{2}{\alpha}+\frac{4}{\alpha^{2}}$ (3)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\,a\left(2-\frac{8}{\alpha}+\frac{2(12-n^{2}\pi^{2})}{\alpha^{2}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{1}{2}\,a^{2}\left(8+n^{2}\pi^{2}-\frac{2(24+5n^{2}\pi^{2})}{\alpha}+\frac{24(8+n^{2}\pi^{2})}{\alpha^{2}}\right)\\!.$

In Supplementary Information, we compare this result to the exact numerical
Bokaian calculation35 valid for ideal clamping ($\alpha\rightarrow\infty$).
For $a<0.1$ the agreement is very good, but degrades with increasing $n$,
demonstrating that higher orders need to be taken into account in the Taylor
$a$-expansion. Note the difference between Eq. (34) and expressions that can
be found in the literature24, 10.

Experimental data and theory are compared in Fig. 2. On the left panel, we
present the expected $f_{0}$ scalings with $1/L$ (inverse length) and $n$
(mode number; in the normalized plot, $f_{0}(n)/f_{n,0}$ is at first order a
constant). However on the right panel, we demonstrate how the width $w$ of the
beam influences the resonance frequencies: $f_{0}(n=1)/f_{n=1,0}$ decreases
with increasing width, independently of length $L$. This is rather surprising
(since $f_{0}$ is independent of $w$ in the high-stress limit of Euler-
Bernoulli theory), and the effect is less pronounced for higher $n$ (inset
left panel). Inspecting Eq. (34), we see that this behavior can be obtained by
a simple ansatz on the clamp parameter $\alpha$:

$\alpha\propto\frac{n\pi}{w},$ (4)

using the simplest guess. This actually means that the anchor becomes more
stiff at higher frequencies, and for smaller beam widths. Eq. (4) is fit on
data, see lines in Fig. 2, demonstrating very good agreement. The extracted
parameter is characteristic of our clamp geometry, especially its length
$L_{c}$. How the resonance frequency of cantilevers decreases with increasing
$L_{c}$ had been investigated numerically in Ref31. Here, our approach is to
fit this dependence by an effective spring constant $\alpha$; see
Supplementary Information for quantitative parameters, and discussion in the
Conclusion Section.

We now bring our attention to the quality factor $Q$, or equivalently the
damping parameter $\Delta f=f_{0}/Q$. We remind that for our devices,
radiation loss can be safely neglected24. Beyond the nanomechanics literature
which clearly established that friction is directly related to bending11, 12,
we shall discuss internal damping from a materials science perspective. An
ideal solid described in continuum mechanics obeys elasticity theory13: one
introduces strain and stress tensors which are related linearly by two elastic
constants (for isotropic materials), e.g. Young’s modulus $E$ and Poisson’s
ratio $\nu$. When friction mechanisms take place, deviations from this
“Hookean” behavior appear: the solid is called anelastic. This essentially
means that one has to introduce a non-conservative force which acts upon each
elementary volume $\delta\tau$ of the material. The most natural modeling
consists in introducing the rates of change of strain and stress tensors in
the elastic equations, while keeping the hypothesis of linearity36.

The simplest such linear superposition is the Zener model36, and it describes
rather well low-frequency (kHz) elastic properties of conventional materials
(like metals) probed by Dynamic Mechanical Analysis (DMA). More complex models
can be analyzed in order to reproduce the behavior of other materials, or
higher frequencies measurements. Their implication is essentially to generate
a frequency-dependent Young’s modulus $E(f)$ and friction term proportional to
the rate of change of the stress tensor, which can thus be interpreted as an
effective viscosity $\eta(f)$. As for the elastic constants, when introducing
viscosity in a fluid one has to define (in the simplest case) two constants:
$\eta$ and $\zeta$ the second viscosity37. For the sake of simplicity, in our
case we will use the same parametrization as for the elastic properties and we
introduce a damping modulus $E_{p}$ with a damping Poisson’s ratio $\nu_{p}$
(which are both also functions of frequency). As a consequence, when solving
for a harmonic motion using complex forms, linear friction is equivalent to
replacing Young’s modulus by a complex Young’s modulus36 $E\rightarrow
E+iE_{2}$: the linear mechanical response is de-phased from the excitation.
This is a fairly simple writing which is widely used11, 12, but the price to
pay is that all details of the friction model are hidden within the constant
$E_{2}(f)$. Especially, its frequency-dependence (from first principles
$E_{2}$ is function of $f$) is an important ingredient for the understanding
of microscopic processes at stake. The complete mathematical analysis can be
found in Supplementary Information.

Figure 3:  Left, main: dependence of quality factor on length $L$ for the
first flexure $n=1$ (all devices having $w=1000~{}$nm). Left, inset: quality
factor of 100$~{}\mu$m long device as a function of mode number $n$. Right:
$Q$ normalized to the extrapolated value at $w\rightarrow 0$ for mode $n=1$ of
all devices, as a function of width $w$. The lines correspond to Eqs. (5,6,44)
using a simple ansatz for the clamp parameter $\alpha$, see text.

The quality factor $Q$ writes, for any mode $n$:

$Q(n,a,\alpha)=Q_{n,0}\,\frac{P_{f}(n,a,\alpha)^{2}}{P_{Q}(n,a,\alpha)},$ (5)

with $Q_{n,0}=f_{n,0}^{2}/(f_{n,0}\Delta f_{n,0})$ the usual $n^{th}$ mode
quality factor with:

$f_{n,0}\Delta
f_{n,0}=\frac{n^{2}}{4\sqrt{3}}\,\frac{e}{L^{3}}\left(\frac{E_{2}}{\rho\sqrt{E/\sigma}}\right).$
(6)

Eq. (6) has a single material-dependent fit parameter: $E_{2}$ ($\rho$, $E$
and $\sigma$ being known from resonance frequency fits). The $P_{Q}$ function
is obtained as:

$\displaystyle P_{Q}(n,a,\alpha)$ $\displaystyle=$ $\displaystyle
1-\frac{6}{\alpha}+\frac{24-n^{2}\pi^{2}}{\alpha^{2}}$ (7)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\,a\left(6+\frac{1}{2}n^{2}\pi^{2}-\frac{48+6n^{2}\pi^{2}}{\alpha}+\frac{144-8n^{2}\pi^{2}}{\alpha^{2}}\right)\\!.$

For ideal clamping ($\alpha\rightarrow\infty$), it is found experimentally
that $Q$ scales linearly with $L$11, and falls with increasing thickness
(roughly as $1/e$ in Ref23.) and mode number $n$11. This has been fit with
$E_{2}$ essentially frequency-independent11, 12; we reproduce this result in
Supplementary Information by fitting the numerical solutions from Ref11. Note
the difference between Eqs. (5,6,44) and the expressions derived for
membranes12, 24.

Our data and theory are presented in Fig. 3. On the left panel, we show the
conventional dependencies in $L$ and $n$ (the line is calculated with a fit
$E_{2}$ value, see Supplementary Information). On the right panel, we
demonstrate the soft clamping generated by our simple design: $Q$ grows
substantially with increasing width $w$. The dashed upper curve is calculated
for 100$~{}\mu$m length, the lower one for 400$~{}\mu$m and the full line for
200$~{}\mu$m length. The clamp parameter $\alpha$ used here is the same one as
for Fig. 2. The agreement is relatively good, but the remarkable result is
that we predict theoretically the right tendency: the clamp degree of freedom
reduces the bending at the anchor, which increases the $Q$. Introducing a
boundary torsion spring $\gamma_{l}=\gamma_{r}=\gamma$ increases even further
the effect; however, fitted values demonstrate that one requires a higher
order expansion than the order 2 in torque for the quantitative result to be
valid. We therefore preferred to keep this aspect outside of the present
discussion.

##  3 Conclusion

We report on experiments performed on flexural nano-mechanical doubly-clamped
beams, subject to an axial force load. We demonstrate that their suspended
anchoring points act as an “easy soft clamping”: it is responsible for both a
downward resonance frequency shift and an increase in quality factor. We
present an analytic theory that fits the data, based on simple boundary
conditions: forces and torques undergone by the beam’s ends, and due to the
suspended clamp. It turns out that the effective clamp spring constants
$k_{l,r}$ are the dominant ingredient (with $l,r$ standing for left and
right). For symmetric devices with similar clamps, $k_{l,r}\approx k$ is found
to be $k\propto f/w$, with $f$ frequency and $w$ beam width. For the frequency
shift, following Ref31. this can be recast in an effective lengthening
$L\rightarrow L+\Delta L$ with $\Delta L\propto w/f$ at lowest order (and
clamp-parameter dependent). For the quality factor, the fit is reasonably good
but could be improved with a torque spring parameter $\Gamma_{l,r}$. To do so,
the theory would need to be improved; likewise, it could be extended to the
case of beams with no (or very little) axial stress, and even to cantilevers.
The problem addressed here is extremely widespread, and beams with a “natural”
clamp undercut (i.e. due to the fabrication process) are extensively used. For
this reason, we believe that our work is very relevant to nano-mechanical
design, for both defining precisely resonance frequencies (mandatory when
multiplexing is at hand) and quality factors.

##  4 Author Information

Corresponding Author
∗: Eddy Collin<EMAIL_ADDRESS>

Notes
The authors declare no competing financial interest.

We acknowledge the use of the Néel facility Nanofab for the devices
fabrication, and the Néel Cryogenics facility for help with the low
temperature setup. The authors acknowledge support from the ERC CoG grant ULT-
NEMS No. 647917, and ERC StG grant UNIGLASS No. 714692. The research leading
to these results has received funding from the European Union’s Horizon 2020
Research and Innovation Programme, under grant agreement No. 824109, the
European Microkelvin Platform (EMP).

Supplementary Information document: provides details about the samples, and
the complete description of the mathematical modeling. The analytic theory
(derived using Mathematica®) is compared to published theoretical results with
ideal clamping conditions. Numerical finite element simulations (COMSOL®) are
also presented, reproducing qualitatively the data.

##  5 Modal expansion

We start by presenting the theoretical modeling. We consider the case of thin-
and-long beams, i.e. $w\ll L$ and $e\ll L$ ($w$ width, $e$ thickness and $L$
length).

Figure 4: Schematic of (right) clamp with force $F_{r}$ and torque $T_{r}$
that the suspended part (black) exerts on the beam (in blue, see text).

From the well-known Euler-Bernoulli equation (including both bending and
stress terms), we describe the flexure $f(z,t)$:

$EI_{z}\frac{\partial^{4}f(z,t)}{\partial
z^{4}}-S\frac{\partial^{2}f(z,t)}{\partial z^{2}}=-\rho
A\frac{\partial^{2}f(z,t)}{\partial t^{2}}$ (8)

where $A=we$ \- cross-section area, $I_{z}=\frac{1}{12}we^{3}$ \- second
moment of area, $E$ \- material’s Young’s modulus, $S$ \- inside tensile
$(S>0)$ force of the material, $\rho$ \- material density.

We are looking for solution as $f(z,t)=\psi_{n}(z)x_{n}(t)$, where $x_{n}(t)=$
$x_{0}\cos{(\omega_{n}t+\varphi)}$ is the temporal harmonic part (out-of-plane
motion amplitude) and $\psi_{n}(z)$ \- $n$’s mode shape. $\omega_{n}$ is the
mode resonance frequency.

On each side, we introduce a force $F_{l,r}$ and torque $T_{l,r}$, see Fig. 4.
The relaxed boundary conditions with added spring constant $k_{l,r}$ and
inertia $m_{l,r}$ write, for the left $l$ and right $r$ clamps (and similarly
for the torques):

Forces:

$\displaystyle F_{l}=+EI_{z}\frac{\partial^{3}f(z=0,t)}{\partial
z^{3}}-S\frac{\partial f(z=0,t)}{\partial z}$
$\displaystyle=-k_{l}f(z=0,t)+m_{l}\frac{\partial^{2}f(z=0,t)}{\partial
t^{2}}$ (9) $\displaystyle F_{r}=-EI_{z}\frac{\partial^{3}f(z=L,t)}{\partial
z^{3}}+S\frac{\partial f(z=L,t)}{\partial z}$
$\displaystyle=-k_{r}f(z=L,t)+m_{r}\frac{\partial^{2}f(z=L,t)}{\partial
t^{2}}$ (10)

Torques:

$\displaystyle T_{l}=-EI_{z}\frac{\partial^{2}f(z=0,t)}{\partial z^{2}}$
$\displaystyle=-\Gamma_{l}\frac{\partial f(z=0,t)}{\partial
z}+\mathcal{M}_{l}\frac{\partial^{3}f(z=0,t)}{\partial t^{2}\partial z}$ (11)
$\displaystyle T_{r}=+EI_{z}\frac{\partial^{2}f(z=L,t)}{\partial z^{2}}$
$\displaystyle=-\Gamma_{r}\frac{\partial f(z=L,t)}{\partial
z}+\mathcal{M}_{r}\frac{\partial^{3}f(z=L,t)}{\partial t^{2}\partial z}$ (12)

Note signs in forces and torques definition.

The general solution writes as:

$\displaystyle\psi_{n}(z)$
$\displaystyle=C_{n,1}\sin{\left(k_{n+}\frac{z}{L}\right)}+C_{n,2}\cos{\left(k_{n+}\frac{z}{L}\right)}$
$\displaystyle+C_{n,3}\sinh{\left(k_{n-}\frac{z}{L}\right)}+C_{n,4}\cosh{\left(k_{n-}\frac{z}{L}\right)}$
(13)

Considering a shape normalised to $1$: $max[\psi(z)]=1$ at $z=z_{max}$ (we
chose $0<z_{max}\leq L/2$ with no loss of generality).

Now we expand in the high-stress limit, which means having a small parameter
$a=\sqrt{\frac{EI_{Z}}{SL^{2}}}\ll 1$:

$\omega_{n}=\frac{k_{n}(a)}{L}\sqrt{\frac{\sigma}{\rho}}$ (14)

where the stress is defined as $\sigma=S/A$.

$k_{n-}$ and $k_{n+}$ are deduced from $k_{n}(a)$ and they all can be written
as Taylor expansions:

$\displaystyle k_{n}(a)$ $\displaystyle=k_{n}(0)+k_{n}^{\prime}(0)\cdot
a+(1/2)k_{n}^{\prime\prime}(0)\cdot a^{2}+\dots$ (15) $\displaystyle
k_{n+}(a)$ $\displaystyle=k_{n}(0)+k_{n}^{\prime}(0)\cdot
a+(1/2)[k_{n}^{\prime\prime}(0)-k_{n}^{3}(0)]\cdot a^{2}+\dots$ (16)
$\displaystyle k_{n-}(a)$ $\displaystyle=1/a+(1/2)k_{n}^{2}(0)\cdot
a+[k_{n}(0)k_{n}^{\prime}(0)]\cdot a^{2}+\dots$ (17)

Note the $1/a$ in the last Eq. which diverges for small $a$. In the $\cosh$
and $\sinh$ functions, it should be treated with care: we shall neglect
$\exp{\left(-X\right)}$ terms with $X\propto 1/a$ (what we call exponential
approximation), and after take series expansions in $a$ (to lowest order up to
$a^{2}$). Note that the first order terms goes as $a$, while the low-stress
equivalent expansion leads to small term $1/a^{2}=SL^{2}/(EI_{z})$. An
equivalent modeling can be performed in this limit, but is outside of the
scope of the paper.

Solving the problem produces the mode shape with the definition of constants
$C_{n,i}$ and $z_{max}$, with
$k_{n}(0),k_{n}^{\prime}(0),k_{n}^{\prime\prime}(0)$. This is performed using
an extensive Mathematica® code.

Figure 5: Mode shape with ideal clamping in orange, and with
$\alpha_{l}=\alpha_{r}=15.$, $\gamma_{l}=\gamma_{r}=1.$ in blue (see text; in
this case, $z_{max}=L/2$ in both cases). For this graphics, $a=0.05$ and the
$a$ expansion has been pushed to order 4 for the ideal clamping terms.

Regrouping restoring forces and inertial components, we fit with for each side
two fit parameters:

$\displaystyle k_{r,l}$ $\displaystyle\rightarrow$ $\displaystyle
k_{r,l}+m_{r,l}\,\omega^{2}$ (18) $\displaystyle\Gamma_{r,l}$
$\displaystyle\rightarrow$ $\displaystyle\Gamma_{r,l}+{\cal
M}_{r,l}\,\omega^{2}$ (19)

without losses of generality. Introducing dimensionless parameters,
$k_{r,l}=\frac{S}{L}\alpha_{r,l}$ and $\Gamma_{r,l}=(SL)\,\gamma_{r,l}$. The
solutions in the following Sections are given with an expansion at order 2 in
$1/\alpha_{r,l}\ll 1$ and $1/\gamma_{r,l}\ll 1$. As an example, we show in
Fig. 5 the ideal shape with perfect clamping ($\alpha_{l,r}$ and
$\gamma_{l,r}$ being $\infty$), and the one obtained for imperfect clamps (see
caption). The shape is not zero anymore at both ends $z=0,L$, and (more
subtle) the angle $\partial f/\partial z$ is not either.

##  6 Stored energy

Defining energies from forces (and force densities):

$\displaystyle F_{flex}=+EI_{z}\frac{\partial^{3}f}{\partial
z^{3}}\longrightarrow\frac{\partial F}{\partial
z}=+EI_{z}\frac{\partial^{4}f}{\partial
z^{4}}\longrightarrow\mathcal{E}_{flex}$
$\displaystyle=\frac{1}{2}EI_{z}\int_{0}^{L}\left(\frac{\partial^{4}f}{\partial
z^{4}}f\right)dz$ (20) $\displaystyle F_{tens}=-S\frac{\partial f}{\partial
z}\longrightarrow\frac{\partial F}{\partial z}=-S\frac{\partial^{2}f}{\partial
z^{2}}\longrightarrow\mathcal{E}_{tensile}$
$\displaystyle=\frac{1}{2}S\int_{0}^{L}\left(-\frac{\partial^{2}f}{\partial
z^{2}}f\right)dz$ (21) $\displaystyle\rho A\frac{\partial^{2}f}{\partial
t^{2}}\longrightarrow\mathcal{E}_{kin}$ $\displaystyle=\frac{1}{2}\rho
A\int_{0}^{L}\left(\frac{\partial f}{\partial t}\right)^{2}dz$ (22)

Now from integration by parts for flexural energy:

$\displaystyle\left(\frac{\partial^{2}f}{\partial z^{2}}\cdot\frac{\partial
f}{\partial z}\right)^{\prime}$ $\displaystyle=\frac{\partial^{3}f}{\partial
z^{3}}\cdot\frac{\partial f}{\partial z}+\left(\frac{\partial^{2}f}{\partial
z^{2}}\right)^{2}$ (23) $\displaystyle\left(\frac{\partial^{3}f}{\partial
z^{3}}\cdot f\right)^{\prime}$ $\displaystyle=\frac{\partial^{4}f}{\partial
z^{4}}\cdot f+\frac{\partial^{3}f}{\partial z^{3}}\cdot\frac{\partial
f}{\partial z}$ (24)

From which follows:

$\displaystyle\frac{\partial^{4}f}{\partial z^{4}}\cdot
f=\left(\frac{\partial^{3}f}{\partial z^{3}}\cdot
f-\frac{\partial^{2}f}{\partial z^{2}}\cdot\frac{\partial f}{\partial
z}\right)^{\prime}+\left(\frac{\partial^{2}f}{\partial z^{2}}\right)^{2}$ (25)
$\displaystyle\int_{0}^{L}\frac{\partial^{4}f}{\partial z^{4}}\cdot f\
dz=\underbrace{\left[\frac{\partial^{3}f}{\partial z^{3}}\cdot
f-\frac{\partial^{2}f}{\partial z^{2}}\cdot\frac{\partial f}{\partial
z}\right]_{0}^{L}}_{\mathrm{boundary\
term}}+\underbrace{\int_{0}^{L}\left(\frac{\partial^{2}f}{\partial
z^{2}}\right)^{2}\ dz}_{bulk\ term}$ (26)

Same for tensile energy:

$\displaystyle\left(\frac{\partial f}{\partial z}\cdot f\right)^{\prime}$
$\displaystyle=\frac{\partial^{2}f}{\partial z^{2}}\cdot
f+\left(\frac{\partial f}{\partial z}\right)^{2}$ (27)
$\displaystyle\frac{\partial^{2}f}{\partial z^{2}}\cdot f$
$\displaystyle=\left(\frac{\partial f}{\partial z}\cdot
f\right)^{\prime}+\left[-\left(\frac{\partial f}{\partial
z}\right)^{2}\right]$ (28)
$\displaystyle\int_{0}^{L}\left(-\frac{\partial^{2}f}{\partial z^{2}}\cdot
f\right)\ dz$ $\displaystyle=\underbrace{\left[-\frac{\partial f}{\partial
z}\cdot f\right]_{0}^{L}}_{boundary\
term}+\underbrace{\int_{0}^{L}\left(\frac{\partial f}{\partial z}\right)^{2}\
dz}_{bulk\ term}$ (29)

We therefore can define a bulk term and a boundary term for these energies.
The boundary term should be $=0$ for an ideal clamp, i.e. $f=0$ and $\partial
f/\partial z=0$ at $z=0,L$. But here, it is not the case; we define for each
side:

$\displaystyle\mathcal{E}_{flex,bound}=$
$\displaystyle+\frac{EI_{z}}{2}\frac{\partial^{3}f(z=0,t)}{\partial
z^{3}}f(z=0,t)$ (30)
$\displaystyle-\frac{EI_{z}}{2}\frac{\partial^{2}f(z=0,t)}{\partial
z^{2}}\frac{\partial f(z=0,t)}{\partial z}$
$\displaystyle\mathcal{E}_{tens,bound}=$
$\displaystyle-\frac{S}{2}\frac{\partial f(z=0,t)}{\partial z}f(z=0,t)$ (31)

and same for $z=L$, but with reversed signs. One has to pay attention to signs
with the definitions and the integration by parts procedure.

Then we define effective energies for the mode, as a function of the mode
amplitude $x_{n}$:

$\displaystyle\mathcal{E}_{flex}+\mathcal{E}_{tens}$
$\displaystyle=\frac{1}{2}k_{n}x_{n}^{2}(t)$ (32)
$\displaystyle\mathcal{E}_{kin}$
$\displaystyle=\frac{1}{2}m_{n}\dot{x}_{n}^{2}(t)$ (33)

Which include in $\mathcal{E}_{flex}$ and $\mathcal{E}_{tens}$ the bulk and
boundary terms. One should obviously recover $\omega_{n}^{2}=k_{n}/m_{n}$,
where $k_{n}$ is the mode’s spring constant and $m_{n}$ the mode’s effective
mass per definition, computed from the mode shape $\psi_{n}$. This has been
explicitly checked in the Mathematica® code. In the literature, this is
usually not mentioned.

Figure 6: Comparison in the ideal clamp limit of frequency expansion (full
lines) to the exact numerical result by [Bokaian, J. Sound and Vibr. 142, 481
(1990)] (dashed lines). Blue $n=1$, orange $n=2$ and green $n=3$, as a
function of $a$ (see text).

In the first place, the modeling allows us to compute the resonance
frequencies. The full expression at second order in all parameters reads, for
the deviation function $P_{f}$ introduced in the core of the paper:

$\displaystyle P_{f}(n,a,\alpha)$ $\displaystyle=$ $\displaystyle
1-\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)+\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)^{2}$
(34)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\,a\left[2-4\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)+\left(\frac{6-n^{2}\pi^{2}}{\alpha_{l}^{2}}+\frac{6-n^{2}\pi^{2}}{\alpha_{r}^{2}}+\frac{12}{\alpha_{l}\alpha_{r}}\right)\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{1}{2}\,a^{2}\left[8+n^{2}\pi^{2}-(24+5n^{2}\pi^{2})\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.+\left(\frac{48-2n^{2}\pi^{2}}{\alpha_{l}^{2}}+\frac{48-2n^{2}\pi^{2}}{\alpha_{r}^{2}}+\frac{96+28n^{2}\pi^{2}}{\alpha_{l}\alpha_{r}}\right)\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.-2\left(\frac{1}{\gamma_{l}}+\frac{1}{\gamma_{r}}\right)+4\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)\left(\frac{1}{\gamma_{l}}+\frac{1}{\gamma_{r}}\right)\right]$

Note the symmetry $l\leftrightarrow r$ in this expression. A simplified
version of it is in the main paper, Eq. (3). In the case of an ideal clamp
(effective normalised spring constants $\rightarrow\infty$), we compare the
expansion to the exact numerical result from [Bokaian, J. Sound and Vibr. 142,
481 (1990)] in Fig. 6. For the plot, we normalise the frequency to $n\pi$, and
take $1/L\sqrt{\sigma/\rho}=1$. Our computed frequencies match the exact
result for typically $a<0.1$ in the case of ideal clamps, with $n=1$. And it
gets worse for higher modes $n$, which can be compensated for by taking higher
order terms in the $a$-expansion. For the sake of making a good plot, this is
why we used a higher expansion in Fig. 5 (up to fourth order).

##  7 Dissipated energy

We shall use the same approach as the one proposed in [Quirin P.
Unterreithmeier, Thomas Faust, and Jörg P. Kotthaus PRL 105 027205 (2010)].
But we formalise it from the continuum mechanics approach as presented in A.N.
Cleland, Foundations of Nanomechanics, Springer 2003. This implies some minor
modification:

  * •

we make it more generic and exact by starting from stress and strain;
Poisson’s ratios will appear in the expression,

  * •

we start from a proper friction force definition, as in Zener’s model,
generating a friction force proportional to the rate of change of the stress.
This is what is behind the commonly used complex Young’s modulus.

Let us start with the strain 6-component vector for motion of the beam with
mode shape $f(z,t)$ and distortion in the $\vec{x}$ direction (see A.N.
Cleland 2003):

$\varepsilon=(\nu xf^{\prime\prime},\nu
xf^{\prime\prime},-xf^{\prime\prime},0,0,\nu xf^{\prime\prime}).$ (35)

This expression essentially assumes that planes orthogonal to the neutral axis
remain orthogonal to the displaced neutral axis (Euler-Bernoulli
approximation), while it guarantees no lateral stresses (see $\sigma_{el}$
below).

Then the linear response for the stress is:

$\sigma_{el}=(0,0,-Exf^{\prime\prime},0,0,\frac{E\,\nu}{2(1+\nu)}xf^{\prime\prime})$
(36)

Where we have both Young’s $E$ and Poisson’s $\nu$ (elasticity theory). We
assume linearity to apply, such that this stress adds up to the in-built load
$\sigma$ (purely along $\vec{z}$).

The same linear hypothesis can be made for the local friction stress, which at
the macroscopic scale will produce the friction force (prop. to
$\dot{x_{n}}$). But in order to match Zener’s low frequency limit, and
reproduce a standard viscous friction mechanism, this friction component shall
be proportional to the time rate of change of strain $\dot{\varepsilon}$. For
the friction component we thus similarly introduce:

$\displaystyle\sigma_{fr}=E_{p}\cdot\Bigg{(}$
$\displaystyle\frac{\nu-\nu_{p}}{(1+\nu_{p})(1-2\nu_{p})},\frac{\nu-\nu_{p}}{(1+\nu_{p})(1-2\nu_{p})},$
(37)
$\displaystyle-\frac{1-\nu_{p}-2\nu\nu_{p}}{(1+\nu_{p})(1-2\nu_{p})},0,0,\frac{\nu}{2(1+\nu_{p})}\Bigg{)}\cdot
x\dot{f}^{\prime\prime}$

with two parameters describing fricion, $E_{p}$ and $\nu_{p}$ (similar
parametrization of linear response as for stress-strain). Note that the origin
of these terms is outside of the scope of the modeling; but obviously
$E_{p}(\omega)$ and $\nu_{p}(\omega)$ depend on frequency, since they
originate in microscopic mechanisms which should depend on $\omega$ (see
comment below).

The power density lost in friction is then:

$\displaystyle\mathcal{P}=\sigma_{fr}\cdot\dot{\varepsilon}$
$\displaystyle=E_{p}\left(1+\frac{\nu^{2}(5-2\nu_{p})-8\nu\nu_{p}+4\nu_{p}^{2}}{2(1+\nu_{p})(1-2\nu_{p})}\right)x^{2}\left(\dot{f}^{\prime\prime}\right)^{2}$
(38)
$\displaystyle=E_{p}\left(1+\underbrace{\frac{\nu^{2}(5-2\nu_{p})-8\nu\nu_{p}+4\nu_{p}^{2}}{2(1+\nu_{p})(1-2\nu_{p})}}_{\mathrm{small\
parameter\
}o(\nu,\nu_{p})}\right)\omega^{2}x^{2}\left(\frac{\partial^{2}\psi_{n}(z)}{\partial
z^{2}}\right)^{2}x_{n}^{2}(t)$

Now integration over cross-section $x^{2}\rightarrow I_{z}$ and length
$\int_{0}^{L}$:

$\iiint\mathcal{P}\
dxdydz=E_{p}\left[1+o(\nu,\nu_{p})\right]\,\omega^{2}\,I_{z}\underbrace{\int_{0}^{L}\left(\frac{\partial^{2}\psi_{n}}{\partial
z^{2}}\right)^{2}dz}_{\mathrm{same\ integral\ as\ in\ bending\ energy}}\
x_{n}^{2}(t)$ (39)

In terms of mode definition it can be rewritten introducing the friction
force:

$\iiint\mathcal{P}\ dxdydz=\underbrace{\Lambda_{n}\dot{x}_{n}}_{\mathrm{eff.\
friction\ force}}\cdot\dot{x}_{n}=\Lambda_{n}\omega^{2}x^{2}_{n}(t)$ (40)

By definition damping parameter $\Delta\omega_{n}$ is (and
$Q=\omega_{n}/\Delta\omega_{n}$):

$\Delta\omega_{n}=\frac{\Lambda_{n}}{m_{n}}$ (41)

Therefore we can identify:

$\Lambda_{n}=\underbrace{E_{p}\left[1+o(\nu,\nu_{p})\right]}_{\mathrm{fit.\
parameter}}\,I_{z}\int_{0}^{L}\left(\frac{\partial^{2}\psi_{n}}{\partial
z^{2}}\right)^{2}dz$ (42)

There is one formal difference here with the [Quirin P. Unterreithmeier,
Thomas Faust, and Jörg P. Kotthaus PRL 105 027205 (2010)] modeling, which uses
a compex Young’s modulus with imaginary part $E_{2}$: their modeling is
equivalent to the above one with the identity
$E_{p}\left[1+o(\nu,\nu_{p})\right]=E_{2}/\omega$. As a result, with
$\omega\approx\omega_{n}$ for a high-$Q$ resonance we have:

$\omega_{n}\Delta\omega_{n}=\frac{e^{2}E_{2}}{L^{4}\rho}\frac{1}{12}\frac{\int_{0}^{1}\left(\frac{\partial^{2}\psi_{n}[\tilde{z}]}{\partial\tilde{z}^{2}}\right)^{2}d\tilde{z}}{\int_{0}^{1}\left(\psi_{n}[\tilde{z}]\right)^{2}d\tilde{z}}$
(43)

with $\tilde{z}=z/L$. The shape factor defined from $\psi_{n}$ (the ratio of
integrals on the right) tends to $2n^{2}\pi^{2}/a$ at lowest order. This leads
to the result Eq. (6) of the main paper, written in Hz (here, we have all
expressions in Rad/s). From this, one defines the quality factor $Q$, and then
the function $P_{Q}$:

$\displaystyle P_{Q}(n,a,\alpha)$ $\displaystyle=$ $\displaystyle
1-3\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)+\left(\frac{6-\frac{1}{2}n^{2}\pi^{2}}{\alpha_{l}^{2}}+\frac{6-\frac{1}{2}n^{2}\pi^{2}}{\alpha_{r}^{2}}+\frac{12}{\alpha_{l}\alpha_{r}}\right)$
(44)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\,a\left[6+\frac{n^{2}\pi^{2}}{2}-(24+3n^{2}\pi^{2})\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)+\left(\frac{60}{\alpha_{l}^{2}}+\frac{60}{\alpha_{r}^{2}}+\frac{24-8n^{2}\pi^{2}}{\alpha_{l}\alpha_{r}}\right)\right.$
$\displaystyle\left.\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\left(\frac{1}{\gamma_{l}}+\frac{1}{\gamma_{r}}\right)+3\left(\frac{1}{\alpha_{l}}+\frac{1}{\alpha_{r}}\right)\left(\frac{1}{\gamma_{l}}+\frac{1}{\gamma_{r}}\right)\right]$

which represents all deviations from the basic expression (at lowest order in
$a$; note the $l\leftrightarrow r$ symmetry). A simplified version is in the
main paper, Eq. (7). In the limit where the clamp spring constants
$\alpha_{l,r},\gamma_{l,r}$ tend to $\infty$, one recovers the usual
tendencies: a $Q$ factor that grows as $L$, and decreases with mode number
$n$. We demonstrate this by comparing our formulas to the numerical results of
[Quirin P. Unterreithmeier, Thomas Faust, and Jörg P. Kotthaus PRL 105 027205
(2010)]. This is done in Fig. 7, see Caption for the used numerical values (we
take here a $E_{2}$ independent of $\omega$). The agreement is very good; we
believe deviations are due to the finite order of the expansion in $a$.

Further fits of our own data (including the clamp dependence) are discussed in
the next Section.

Figure 7: Numerical calculation of $Q,f_{0}$ from [Quirin P. Unterreithmeier,
Thomas Faust, and Jörg P. Kotthaus PRL 105 027205 (2010)] (open squares, for
different beam lengths $L$) compared to our analytic solution (crosses), in
the case of ideal clamp. Parameters chosen: $\sigma=0.83~{}$GPa,
$E=160~{}$GPa, $\rho=2800~{}$kg/m3, $E_{2}=48~{}$MPa, $e=100~{}$nm,
$w=200~{}$nm.

##  8 Sample Characteristics and clamp fit

The sample fabrication and main characteristics are given in [Golokolenov et
al. JLTP 2022]. The measurements have been performed on two chips, on which
eight sets of beams (2 groups with the 4 different lengths) were present. Not
all of them have been measured (for lack of time), but a good fraction of them
has been characterised. Out of all the data acquired, only few data points
have been excluded, because they seemed to be out-of-statistics (frequency off
by about 20$~{}\%$ max., or $Q$ factor very low and almost independent of
parameters $w,n$). We believe that this comes from the reproducibility of the
fabrication, which remains our limiting parameter.

All data in the core of the paper has been fit with a single set of
parameters, including a single (symmetric) clamp ansatz. We chose:

  * •

$\sigma=0.17~{}$GPa (low-stress sample),

  * •

$\rho=2800~{}$kg$/$m3,

  * •

$E=160~{}$GPa,

  * •

$E_{2}=180~{}$MPa,

that describe the properties of the bilayer structure. Note that our $E_{2}$
is larger than the one of [Quirin P. Unterreithmeier, Thomas Faust, and Jörg
P. Kotthaus PRL 105 027205 (2010)] which had no metal layer. The clamp spring
constant is found to be:

$\alpha_{l}=\alpha_{r}=\alpha\approx 2500.\frac{n\pi}{w},$ (45)

in order to reproduce the properties measured (see discussion in main paper).
We verify $\alpha\gg 1$ in the whole range studied. For the sake of
completeness, we also included a torque spring, following the same ansatz.
However, we realised that in order to influence the fit, this one had to be
particularly small, i.e. out of the validity range of the expansion. We
therefore preferred not to discuss this point further in the paper; because it
would require a more exact expansion to be included. Technically, the graphs
of the paper are obtained with:

$\gamma_{l}=\gamma_{r}=\gamma\approx 300.\frac{n\pi}{w},$ (46)

which has an almost invisible impact on fits (and becomes $\sim 1$ at worst).
This gives us a sort of estimate of the limit within which torques can be
neglected.

Sample name | left clamp ($\mu$m) | right clamp ($\mu$m) | mean ($\mu$m)
---|---|---|---
S2 100$~{}\mu$m assym. | 3.85 | 13.1 | 8.5
S2 100$~{}\mu$m sym. | 7.7 | 11.55 | 9.6
S2 300$~{}\mu$m | 10. | 14. | 12.
S2 400$~{}\mu$m | 5.8 | 13.5 | 9.6
S6 100$~{}\mu$m | 12.5 | 16.7 | 14.6
S6 200$~{}\mu$m | 10.8 | 12.3 | 11.5
S6 300$~{}\mu$m | 13.1 | 16.9 | 15.
S6 400$~{}\mu$m | 11.3 | 16.2 | 13.7
Table 1:  Clamp suspension length $L_{c}$ statistics.

Eqs. (45,46) are characteristics of the clamp. Within our geometry, its
thickness is constant, its composition is the same as for the beam, and we
assume that its width $W$ does not matter either, since $w\ll W$ (at worst
about 20$~{}\mu$m): for the distortion of the anchoring point, the width of
the clamp is essentially infinite. But the length $L_{c}$ of the suspended
part should obviously matter; we give in Tab. 1 the statistics of the clamp’s
suspended region, for all samples. On average, we have $L_{c}\approx
12~{}\mu$m, within about $\pm 50~{}\%$ scatter in the fabrication. We believe
that this scatter could also contribute to the dispersion in the measured
properties.

Figure 8: COMSOL® simulation (first $n=1$ flexure) reproducing the ”easy soft
clamping” effect (see text). Parameters used for simulation given in Figure.

##  9 Comparison to Numerical Simulations

The measured properties can also be qualitatively reproduced by a numerical
simulation, demonstrating the ”easy soft clamping” effect. To demonstrate
this, we perform a simple finite element study on COMSOL®, as shown in Fig. 8.
Parameters used given in the figure, except for width $w$ which is varied.

The calculated resonance frequencies $f_{0}$ and quality factors $Q$ are
presented (in normalised from) in Fig. 9, as a function of $w$. The very same
trend as in the experimental data is visible: the frequency drops with
increasing $w$, while the quality factor grows. It can be fit by a second
order polynomial in $w$, as expected from our analytic theory. The magnitude
of the effect is also correct as compared to experiments, but is not in
perfect quantitative agreement. We believe that this is due to the exact, and
rather precise, choice of parameters that one has to perform to match
simulations on data.

Figure 9: Normalised frequency (left) and quality factor (right) as a function
of beam width. The red point is the $w=0$ extrapolated value, and the dashed
lines simple polynomial (second order) fits (see text).

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