[1]Silvan Schmid

1]Institute of Sensor and Actuator Systems, TU Wien, Gusshausstrasse 27-29,
Vienna, 1040, Austria

# Comparative Analysis of Nanomechanical Resonators: Sensitivity, Response
Time, and Practical Considerations in Photothermal Sensing

Kostas Kanellopulos  Friedrich Ladinig  Stefan Emminger  Paolo Martini  Robert
G. West<EMAIL_ADDRESS>[

###### Abstract

Nanomechanical photothermal sensing has significantly advanced single-
molecule/particle microscopy and spectroscopy, and infrared detection through
the use of nanomechanical resonators that detect shifts in resonant frequency
due to photothermal heating. However, the relationship between resonator
design, photothermal sensitivity, and response time remains unclear. This
paper compares three resonator types — strings, drumheads, and trampolines —
to explore this relationship. Through theoretical modeling, experimental
validation, and finite element method simulations, we find that strings offer
the highest sensitivity (with a noise equivalent power of 280 fW/Hz1/2 for
strings made of silicon nitride), while drumheads exhibit the fastest thermal
response. The study reveals that photothermal sensitivity correlates with the
average temperature rise and not the peak temperature. Finally, the impact of
photothermal back-action is discussed, which can be a major source of
frequency instability. This work clarifies the performance differences and
limits among resonator designs and guides the development of advanced
nanomechanical photothermal sensors, benefiting a wide range of applications.

###### keywords:

nanomechanics, photothermal, resonator, thermal

##  1 Introduction

In nanomechanical photothermal sensing, the resonator detects heat generated
from various processes, including electromagnetic radiation absorption [1, 2,
3, 4, 5, 6, 7] and non-radiative energy transfer from minute samples [8, 9,
10, 11, 12, 13, 14, 15], single molecules [16], single nanoparticles [17, 18,
19, 20, 21, 22, 23], two-dimensional (2D) materials [24, 25], and thin films
[26, 27]. Within this system, the nanomechanical resonator functions as the
sensing element for the detection of energy exchange with the environment via
resonance frequency shifts. As the resonator absorbs heat, its temperature
rises, which decreases the tensile stress, leading to a corresponding
frequency detuning. In other words, the resonator operates as a precise
mechanical thermometer, offering significant advantages in terms of optimal
sensor design and material choice [28]. Notably, an optomechanical
photothermal resonator fundamentally faces limitations from thermomechanical
and photothermal back-action noise, in contrast to the electronic noise that
plagues thermoelectric sensors and detectors [29].

Over the last decade, nanomechanical photothermal sensing has emerged as a
powerful detection approach due to its high temperature sensitivity and
versatility, as testified by the rapidly expanding areas of application, as
outlined in Fig. 1. To date, this technique has shown outstanding performances
in the field of molecular microscopy and spectroscopy, operating in a wide
range of the electromagnetic spectrum, from the visible [17, 16, 20, 21, 22],
near-infrared (IR) [24, 25, 23], to mid-IR [8, 9, 11, 12, 27, 26, 15]. In
addition, the field is making important steps toward bridging the terahertz
(THz) gap with resonant micro- and nanomechanical thermal detectors, offering
a unique approach for room-temperature operation [1, 2, 3, 30, 31, 32, 4, 5,
33, 7]. Light-sound interaction in nanoresonators has been also successfully
employed for enthalpy measurements [34], and detection of near-field heat
radiation transfer [35, 36], as well as phonon heat transfer through vacuum
fluctuations [37].

So far, a variety of mechanical photothermal sensors has been employed for all
this wealth of results, driven by different experimental requisites. For
instance, silicon nitride string resonators have been extensively explored for
photothermal sensing, in particular for microscopy and spectroscopy
applications [38, 17, 8, 18, 10, 9, 11, 13, 14, 20]. Their small cross-
sectional area, together with the mechanical and thermal properties of SiN,
make them extremely sensitive to temperature changes [38]. With their high
surface area, drumheads have been another platform of choice for
nanomechanical photothermal spectroscopy of single nano-absorber [16, 22, 23],
2D materials [24, 25] and thin films [26, 27], as well as for detection of
IR/THz radiation [3, 5, 35, 36, 7]. As it will be shown here, this advantage
comes at the expense of a reduced thermal sensitivity. Recently, trampoline
resonators have also been employed in a variety of works in photothermal
sensing [39, 40, 4, 5, 41], due to their improved thermal insulation compared
to drumheads, leading to competitive sensitivities with respect to the
strings. However, no comprehensive modeling and comparison of different
resonator designs has been performed with respect to their sensitivity,
response time, and practicality.

This study establishes a comprehensive theoretical framework aimed at
assessing the photothermal sensing performance of nanomechanical resonators,
with a focus on noise equivalent power and response time. The analytical
models illuminate, in particular, the interplay between sensor responsivity
and frequency stability. Models for the individual noise components of the
frequency stability are derived, including additive phase noise, temperature
fluctuation noise, and photothermal back-action noise. The models herein are
rigorously validated through comparison with experimental data and finite
element method (FEM) simulations across varied nanomechanical silicon nitride
resonator designs, namely strings, square drumheads, and trampolines, as
schematically depicted in Fig. 2a,e,I.

Figure 1: Nanomechanical photothermal sensing. To date, the photothermal
effect in nanomechanical resonators has been explored in different fields of
application (top row): molecular microscopy (reproduced and cropped from ref.
16–Copyright author(s) 2018, licensed under CC BY-NC-ND 4.0) and spectroscopy
(reproduced and cropped from ref. 25–Copyright 2023 under a CC BY 4.0
license), IR/THz detection (reproduced and cropped from ref. 4–Copyright 2022
under a CC BY 4.0 license), IR-temperature programmed desorption (IR-TPD,
reproduced and cropped from ref. 15–Copyright 2023 under a CC BY 4.0 license),
and radiative heat transfer mechanisms (adapted with permission from ref.
37–Copyright 2019 by Springer Nature), among others. Within this wealth of
studies, different resonator designs have been used (bottom row): strings
(reproduced and cropped from ref. 11–Copyright 2016 under a CC BY-NC-ND 4.0
license), drumheads (adapted with permission from ref. 33–Copyright 2023 by
AIP Publishing), trampolines (reproduced and cropped from ref. 5–Copyright
2023 under a CC BY 4.0 license), phononic crystal (PnC) geometries on
trampolines (reproduced and cropped from ref. 42–Copyright 2020 under a CC BY
4.0 license), and PnC alone (adapted with permission from ref.43–Copyright
2017 by Springer Nature). Depending on the application, addition of a further
layer on top of the sensing area is also possible. Two examples are (central
row): thin-film absorber (reproduced and cropped from ref. 44–Copyright 2020
under a CC BY 4.0 license), and metamaterials (reproduced and rearranged from
ref. 7–Copyright 2024 under a CC BY 4.0 license). Figure 2: FEM validation of
MTF. a. Schematics of a string resonator illuminated by a light source (red)
at the center. At thermal equilibrium, a temperature difference $\Delta T$
from the frame temperature $T_{0}$, will arise upon photothermal heating. b.
String’s thermal conductance $G$ as a function of the point-like heat source
relative position $x/L$, for three different lengths ($0.1$, $1$, and $10\
\mathrm{mm}$). Circles: FEM results of $G$ in the mean temperature framework
(MTF). Solid curve: MTF theoretical calculation (15). c. Comparsion between
FEM results (circles) and model (solid curve) for the relative power
responsivity (2) as a function of the string length. d. Comparison between FEM
results (circles) and model (solid curve) for the thermal time constant (3) as
a function of the string length. Model and FEM used parameters: $\rho=3000\
\mathrm{kg/m^{3}}$, $c_{p}=700\ \mathrm{J/(kg\ K)}$, $\kappa=3\
\mathrm{W/(m\cdot K)}$, $E=250\ \mathrm{GPa}$, $\sigma_{0}=200\ \mathrm{MPa}$,
$\nu=0.23$, $\alpha_{\mathrm{th}}=2.2\ \mathrm{ppm/K}$,
$\epsilon_{\mathrm{rad}}=0.05$, $\alpha=0.5\ \%$, $w=5\ \mathrm{\mu m}$,
$h=50\ \mathrm{nm}$. e. Schematics of a drumhead resonator. f. Circular
drumhead’s MTF thermal conductance for $0.1$, $1$, and $10\ \mathrm{mm}$ side
length. g. Relative power responsivity comparison for drumheads. h. Thermal
time constant comparison for drumheads. Solid curve: theory (3). Blue crosses:
response time of the FEM peak temperature $\Delta T_{FEM}$ for uniform heating
(UH). Black crosses: response time of $\Delta T_{FEM}$ for local heating (LH).
Purple squares: response time of the surface mean temperature $\braket{\Delta
T_{FEM}}$ for LH. Black circles: response time of the resonance frequency
$f_{0_{FEM}}$ for LH. Model and FEM used parameters: $\rho=3000\
\mathrm{kg/m^{3}}$, $c_{p}=700\ \mathrm{J/(kg\ K)}$, $\kappa=2.7\
\mathrm{W/(m\cdot K)}$, $E=250$ GPa, $\sigma_{0}=50\ \mathrm{MPa}$,
$\nu=0.23$, $\alpha_{\mathrm{th}}=1.23\ \mathrm{ppm/K}$,
$\epsilon_{\mathrm{rad}}=0.05$, $\alpha=0.5\ \%$, $h=50\ \mathrm{nm}$. i.
Schematics of a trampoline resonator. j. Trampolines’ MTF thermal conductance
for a frame window side length of $1.1$ mm and five different central pad side
lengths. The model (solid curve) accounts here for a heat source impinging
only in the central pad. k. Relative power responsivity comparison for
trampolines. l. Thermal time constant comparison for trampolines. Model and
FEM used parameters: $\rho=3000\ \mathrm{kg/m^{3}}$, $c_{p}=700\
\mathrm{J/(kg\ K)}$, $\kappa=2.7\ \mathrm{W/(m\ K)}$, $E=250\ \mathrm{GPa}$,
$\sigma_{0}=200\ \mathrm{MPa}$, $\nu=0.23$, $\alpha_{\mathrm{th}}=2.2\
\mathrm{ppm/K}$, $\epsilon_{\mathrm{rad}}=0.05$, $\alpha=0.5\ \%$, $w=5\
\mathrm{\mu m}$, $h=50\ \mathrm{nm}$ [30, 42]. For all the FEM simulations, a
Gaussian beam of waist $w_{0}=1\ \mathrm{\mu m}$ has been used.

##  2 Theory

The figure of merit of a photothermal sensor is the noise-equivalent-power
(NEP) with units [$\mathrm{W/\sqrt{Hz}}$], which for nanomechanical resonators
is defined as [45]

$\text{NEP}=\frac{\sqrt{S_{y}(\omega)}}{\mathcal{R}_{\mathrm{P}}(\omega)},$
(1)

with the one-sided spectral density fractional frequency fluctuations
$S_{y}(\omega)$, with units [$\mathrm{1/Hz}$], and the relative power
responsivity $\mathcal{R}_{\mathrm{P}}(\omega)$, with units [$\mathrm{1/W}$].

The relative responsivity is defined as the fractional detuning of the
resonator eigenfrequency $\omega_{0}$ per absorbed power $P$ and is given by
[45]

$\mathcal{R}_{\mathrm{P}}(\omega)=\frac{\partial\omega_{0}}{\partial\mathrm{P}}\frac{1}{\omega_{0}}\left\lvert
H_{\mathrm{th}}(\omega)\right\rvert=\
\frac{\mathcal{R}_{\mathrm{T}}}{G}\left\lvert
H_{\mathrm{th}}(\omega)\right\rvert,$ (2)

where $\mathcal{R}_{\mathrm{T}}$ denotes the relative temperature responsivity
with units [$\mathrm{1/K}$], $G$ the thermal conductance [$\mathrm{W/K}$], and
$H_{\mathrm{th}}(\omega)=(1+i\omega\tau_{\mathrm{th}})^{-1}$ a low-pass filter
transfer function accounting for the resonators’ thermal response time [46]

$\tau_{\mathrm{th}}=\ \frac{C}{G},$ (3)

with $C$ denoting the resonators’ heat capacity. The absorbed power is the
fraction of irradiated or otherwise introduced power $P_{0}$ that is converted
into heat

$P=\alpha P_{0}.$ (4)

with the absorber- and wavelength-dependent heat conversion factor $\alpha$
($0\leq\alpha\leq 1$).

###  2.1 Temperature responsivity

According to Eq. (2), nanomechanical photothermal sensors are, in essence,
temperature sensors. The temperature responsivity is defined as

$\mathcal{R_{\mathrm{T}}}=\frac{\partial\omega_{0}}{\partial
T}\frac{1}{\omega_{0}}.$ (5)

The eigenfrequency of the resonators considered in this work is a function of
the temperature-dependent tensile stress $\sigma(T)$

$\omega_{0}\propto\sqrt{\sigma(T)},$ (6)

while the effect of bending stiffness is neglected.

####  2.1.1 Strings

In a string resonator with an intrinsic uniaxial tensile stress $\sigma_{0}$
and Young’s modulus $E$, a mean temperature increase $\braket{\Delta T}$
induces a thermal strain along the resonator’s length $L$, resulting in a
stress [45]

$\sigma(T)=\ \sigma_{0}-\ E\ \alpha_{\mathrm{th}}\ \braket{\Delta T},$ (7)

with $\alpha_{\mathrm{th}}$ being the material’s linear coefficient of thermal
expansion. For small temperature changes, the temperature responsivity (5)
together with (6) and (7) approximately is given by

$\mathcal{R}_{\mathrm{T}}=-\frac{\alpha_{\mathrm{th}}}{2}\frac{E}{\sigma_{0}}.$
(8)

The factor $E/\sigma_{0}$ is called the photothermal enhancement factor and is
a unique feature of resonators under tensile stress. For nanomechanical
silicon nitride resonators, the photothermal enhancement factor can reach
values between $10^{2}-10^{8}$.

####  2.1.2 Drumheads

For very thin ($h\ll L$) homogeneous isotropic drumheads, the assumption of
thin shell holds, and the thermal stress is given by [47]

$\sigma(T)=\ \sigma_{0}\ -\ \frac{\alpha_{\mathrm{th}}\ E}{1\ -\
\nu}\braket{\Delta T},$ (9)

with $\nu$ being the resonator’s Poisson’s ratio. Hence, the relative
temperature responsivity is

$\mathcal{R}_{\mathrm{T}}=\ -\ \frac{\alpha_{\mathrm{th}}}{2\ (1\ -\
\nu)}\frac{E}{\sigma_{0}},$ (10)

with the factor ($1-\nu$) accounting for the thermal expansion along the two
in-plane directions (for a detailed discussion about the dependence of
$\mathcal{R}_{T}$ on the heat localization in drumheads, see SI Section S2).

####  2.1.3 Trampolines

The trampolines exhibit a thermal response similar to strings. For a central
pad of area $L^{2}$ and thickness $h$, anchored to the frame via four tethers
of length $L_{\mathrm{t}}$ and rectangular cross-section $w\cdot h$, its
spring constant for the fundamental resonance mode can be expressed as (see SI
Section S1)

$k_{\mathrm{eff}}(T)=\frac{\pi^{2}}{2}\frac{wh}{L_{\mathrm{t}}}(1-\nu)\sigma_{0}\left[1-\frac{\alpha_{\mathrm{th}}E}{\sigma_{0}}\braket{\Delta
T}\right],$ (11)

with the factor $(1-\nu)$ accounting here for the strain release along the
direction perpendicular to the tether length. From the resonance frequency
$\omega_{0}(T)\propto\sqrt{k_{\mathrm{eff}}}$, it is easy to observe that the
temperature responsivity is equal to

$\mathcal{R}_{\mathrm{T}}=-\frac{\alpha_{\mathrm{th}}}{2}\frac{E}{\sigma_{0}},$
(12)

underlining that the thermal expansion at the tethers is the main responsible
for the temperature response (see SI Section S1).

###  2.2 Thermal conductance

Besides the temperature responsivity, the power responsivity ((2)) also
depends on the thermal conductance. The following thermal analysis is carried
out based on the mean temperature framework (MTF) in the steady state that we
introduce here. The model is derived first assuming a point-like heat source.
The case of an evenly spread heat source is discussed in the end of each
subsection.

A resonator of thermal mass $C$ absorbs a power $P$ and dissipates it to the
environment through its conductance $G$, resulting in a mean temperature rise
$\braket{\Delta T}$

$\braket{\Delta T}=\frac{P}{G}.$ (13)

In the MTF, all the resonator thermal properties are defined with respect to
$\braket{\Delta T}$, as this temperature dictates the photothermal response of
a nanomechanical resonator under tensile stress, rather than the local
temperature variations $\Delta T$ (for a detailed discussion, see Fig. 2h and
SI Section S2).

For an isotropic resonator, $C$ is given in the MTF by

$C=\ c_{\mathrm{p}}\ \rho\ V,$ (14)

where $c_{p}$, $\rho$, and $V$ are the specific heat capacity at constant
pressure, mass density, and volume of the resonator, respectively.

As the resonator operates in a vacuum environment, only thermal conduction
$G_{\mathrm{cond}}$ and radiation $G_{\mathrm{rad}}$ contribute to the heat
transfer [48]. In the MTF, the thermal conductance $G$ is given by

$G=G_{\mathrm{rad}}+G_{\mathrm{cond}}=4A_{\mathrm{rad}}\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}+\frac{s_{\mathrm{f}}(\mathbf{r},L,w_{0})}{\beta(\mathbf{r},L,w_{0})}\kappa,$
(15)

where $A_{\mathrm{rad}}$, $\epsilon_{\mathrm{rad}}$, $\kappa$, and
$\sigma_{\mathrm{SB}}$ are the resonator’s radiating surface, its emissivity,
thermal conductivity, and the Stefan-Boltzmann constant, respectively. For the
thermal conduction term $G_{\mathrm{cond}}$, a shape factor $s_{\mathrm{f}}$
is introduced to account for the design geometry via the resonator
characteristic length $L$, the heat source position vector $\mathbf{r}$, and
the heating radius $w_{0}$ [48]. In this way, the dependence of $G$ on the
size of the probing heat source, as well as on its position with respect to
the resonator, e.g. concentric or eccentric to it, are taken into account
within this formalism. The product
$s_{\mathrm{f}}(\mathbf{r},L,w_{0})\cdot\kappa$ is the thermal conduction with
respect to the localized temperature field $\Delta T$. The factor
$\beta(\mathbf{r},L,w_{0})$ denotes the ratio between mean and peak
temperature $\beta=\braket{\Delta T}/\Delta T$, ensuring the correct
description of $G_{\mathrm{cond}}$ in the MTF.

####  2.2.1 Strings

A string (Fig. 2a) of length $L$, width $w$, and thickness $h$ occupies a
volume $V=hwL$, and, assuming $h\ll w$, radiates with an area
$A_{\mathrm{rad}}\approx 2wL$. The factor of 2 accounts for the front and back
surface radiation. This allows the direct evaluation of the thermal
capacitance $C$ and radiative conductance $G_{\mathrm{rad}}$. For
$G_{\mathrm{cond}}$ instead, shape $s_{\mathrm{f}}$ and $\beta$ factors have
to be calculated. In this regard, it has to be noticed that the heating source
can be in any position $x$ along the string length, with the generated heat
flowing along two paths of length $x$ and $L-x$ [28, 46]. For a point-like
heat source ($w_{0}\xrightarrow{}0$), $s_{\mathrm{f}}$ and $\beta$ are given
by

$\displaystyle s_{\mathrm{f}}(x)=\ \frac{4\ w\ h}{L}\ \frac{1}{1\ -\
\left(\frac{2\ x}{L}\right)^{2}},\,\,\text{and}$ (16) $\displaystyle\beta(x)=\
\frac{1}{2}.$ (17)

Eq. (17) is valid as long as the temperature profile is linear, for which
$\braket{\Delta T}=\Delta T/2$ (see SI Section S2).

Fig. 2b shows the overall thermal conductance $G$ (15) as a function of the
relative localized heating position for three different strings. The MTF model
(solid curves) closely aligns with the FEM simulations (circles), where the
conductance has been extract as $G_{\mathrm{FEM}}=P/\braket{\Delta
T_{\mathrm{FEM}}}$. For strings measuring 0.1 mm and 1 mm in length (black and
blue curves, respectively), $G$ strongly depends on the heat source position,
increasing as the latter approaches the frame, due to the enhanced thermal
conduction. This effect is less pronounced for the $10$ mm long string, where
radiative heat transfer dominates. It is worth noting that the $1$ mm long
string shows the best thermal insulation, followed by the $10$ mm long and
$0.1$ mm resonators, consistent with the theoretical and experimentally
determined power responsivity $\mathcal{R}_{\mathrm{P}}$ (see Fig. 2$c\&$3c).

For the case that the heating point source is located in the string center
($x=0$), $G_{\mathrm{cond}}$ can be expressed as

$G_{\mathrm{cond}}=\frac{s_{\mathrm{f}}(x=0)}{\beta(x=0)}\ \kappa=\ \frac{8\
h\ w}{L}\ \kappa.$ (18)

The MTF predicts a factor of 2 higher than what is reported in ref. 45, as
$\braket{\Delta T}$ is considered instead of the peak temperature.

Fig. 2c compares the theoretical power responsivity (2) to the FEM
simulations, showing excellent agreement. For short lengths ($L<1\
\mathrm{mm}$), $\mathcal{R}_{\mathrm{P}}$ increases linearly with $L$, until
it reaches a maximum ($L\approx 1$ mm). In this region, the string is fully
coupled to the thermal bath via thermal conduction ($G=G_{\mathrm{cond}}$). As
the distance between the impinging and anchoring points increases, so does the
power responsivity. For $L>1\ \mathrm{mm}$, the string enters the radiation
limited regime ($G=G_{\mathrm{rad}}$), resulting in a linear reduction of
$\mathcal{R}_{\mathrm{P}}$, due to the increasingly larger emitting surface
area $A_{\mathrm{rad}}\propto L$. This comparison proves the validity of the
responsivity model.

The thermal conductance (15) also depends on the spot size of the probing
laser. FEM simulations for a uniformly heated string shows that thermal
insulation is $1.5\times$ less than under localized heating conditions. Thus,
localizing the heat source at the string center will lead to a $1.5\times$
higher power response (see SI Section S2).

Fig. 2d displays the comparison between FEM (circles) and theoretical response
time (3). Short strings are dominated by conductive heat transfer, with
$\tau_{\mathrm{th}}$ being a linear function of $L$. Conversely, long strings
are dominated by radiative heat transfer and show a time constant independent
of $L$, as both the thermal capacitance $C$ and the conductance
$G=G_{\mathrm{rad}}$ grow linearly with $L$. As can be observed, the model
accurately predicts the string’s time constant. Notably, there is a trade-off
between power responsivity and thermal time constant: for $L\leq 1\
\mathrm{mm}$ higher responsivity corresponds to a slower thermal time
response.

####  2.2.2 Drumheads

A square drumhead resonator of side length $L$ and thickness $h$ (Fig. 2e) has
a volume $V=hL^{2}$ and a radiating surface $A_{\mathrm{rad}}=2L^{2}$. Given
$h\ll L$, no thermal gradient is present along the thickness, and the
eccentric shell model applies for the correct description of
$G_{\mathrm{cond}}$ [48], with the heat being dissipated isotropically to the
frame. For simplicity, the model focuses on a circular drumhead of effective
diameter $D=2L/\sqrt{\pi}$. For this geometry, the shape $s_{\mathrm{f}}$ and
$\beta$ factors are given by (for the derivation, see SI Section S2)

$\displaystyle s_{\mathrm{f}}(\mathbf{r},D,w_{0})=\ \frac{4\ \pi\ h}{2\
cosh^{-1}\left[\frac{D^{2}\ +\ (2\ w_{0})^{2}\ -\ 4\ r^{2}}{2\ D\ (2\
w_{0})}\right]+1},$ (19) $\displaystyle\beta(\mathbf{r},D,w_{0})=\
\frac{1-\frac{1}{2}\left(\frac{2\ w_{0}}{D}\right)^{2}}{1-2\
\mathrm{ln}\left(\frac{2\
w_{0}}{D}\right)}\left[1-\left(\frac{r}{D}\right)^{2}\right].$ (20)

Fig. 2f shows the overall conductance $G$ as a function of a localized heat
source position, for three different circular membranes ($L=0.1$, $1$, and
$10$ mm). The MTF model (solid curves) closely aligns with the FEM simulations
for circular drumheads (circles). The two smaller drumheads ($L<1$ mm, black
and blue curves), primarly coupled to the environment via conduction, exhibit
similar values. Conversely, the larger drumhead in the radiative heat transfer
regime has a constant and worse thermal conductance.

For a focused heat source at the drumhead center ($r=0$, $w_{0}\rightarrow
0$), the conductance becomes

$G_{\mathrm{cond}}=\frac{s_{\mathrm{f}}(r=0)}{\beta(r=0)}\ \kappa=\ 4\ \pi\ h\
\kappa,$ (21)

recovering the same result of ref. 12. Even in the case of a localized heat
source, thermal conduction in drumheads is independent of the side length $L$,
contrary to what happens in strings (18).

Fig. 2g shows the comparison between the theoretical (black solid curve) and
FEM power responsivity (black circle), showing excellent agreement. Small
drumheads ($L<1\ \mathrm{mm}$) shows a responsivity independent of $L$, being
$G=G_{\mathrm{cond}}$ exclusively a function of the material thermal
conductivity $\kappa$ and the resonator’s thickness $h$ (21). Large drumheads
($L>1\ \mathrm{mm}$) enter the radiative regime, and the responsivity drops
down due to the increased surface area. This comparison confirms the validity
of the theoretical responsivity model for drumhead resonators.

Drumheads show a different dependence on heat localization compared to
strings. FEM simulations for a concentric Gaussian beam of varying waist
$w_{0}$ have shown that the power responsivity
$\mathcal{R}_{P}(w_{0}=L/2)\approx\mathcal{R}_{P}(w_{0}\ll 1)/2$, i.e. for a
uniform heating condition (see SI Section S2). As a simple rule here, a point-
like heat source offers a $2\times$ improved photothermal responsivity
compared to uniformly distributed heating.

Fig. 2h compares the theoretical and FEM modeling of the thermal time
constant. The study considers uniform (UH) and local heating (LH) conditions.
The theoretical predictions (black curve) closely align with the scenario of
uniform light illumination (blue crosses), with $\tau_{\mathrm{th}}$ derived
from the temporal evolution of the resonator’s maximum temperature $\Delta T$.
Notably, the thermal equilibrium is reached faster in the case of local
heating (black crosses). For the same scenario, $\tau_{\mathrm{th}}$ has been
additionally estimated through a transient study of the resonance frequency
(black circles), revealing a stronger agreement with the theory. Monitoring
the mean temperature $\braket{\Delta T}$ (dark purple squares) further
supports this result: the two sets of FEM perfectly overlap, indicating that
the resonance frequency is governed by the resultant mean temperature
distribution even in the presence of a local heating source. Opposite to what
has been seen for strings, the most responsive drumheads show the fastest time
response.

####  2.2.3 Trampolines

A trampoline (Fig. 2i) occupies a volume
$V=h\left(L^{2}+4wL_{\mathrm{t}}\right)$ and radiates through its central pad
and tethers with an area
$A_{{\mathrm{rad}}}=2\left(L^{2}+4wL_{\mathrm{t}}\right)$. The 2D heat
conduction problem simplifies here to a 1D scenario, as in strings. Indeed,
heat generated in any position on the central pad conductively dissipates
through the tethers. Since the heat flow is constricted by the tethers, the
resonator can be modeled as a cross-string. According to (16), the resulting
shape and $\beta$ factors are given by

$\displaystyle s_{\mathrm{f}}(x)=\ 2\ \frac{4\ w\ h}{2\
L_{\mathrm{t}}}\frac{1}{1-\left(\frac{x}{L_{\mathrm{t}}}\right)^{2}},\,\,\text{and}$
(22) $\displaystyle\beta(\mathbf{r},L,w_{0})=\ 1.$ (23)

The factor of 2 in Eq. (22) accounts for the two crossing strings, while Eq.
(23) is defined only with respect to the central pad, being the core sensing
area.

Fig. 2j shows the FEM computed values for $G$ for five trampolines of
different central areas $L^{2}$ (circles), together with the MTF predictions
(solid curves). As the heat source moves from the center to the frame along a
tether, the thermal conduction $G_{\mathrm{cond}}$ increases. Moreover, both
$G_{\mathrm{cond}}$ and $G_{\mathrm{rad}}$ rise for increasing area–the former
due to shorter tethers, the latter due to a larger surface. For a tightly
focused beam at the center, the thermal conductance is

$G_{\mathrm{cond}}=4\ \frac{h\ w\ \kappa}{L_{\mathrm{t}}},$ (24)

recovering the results of a cross-string resonator of different tether
lengths.

Fig. 2k displays the theoretical and FEM simulated power responsivity as a
function of the central side length $L$. The model aligns closely with FEM
simulations: resonators with small areas ($L^{2}<100^{2}\ \mathrm{\mu m}^{2}$)
show an almost constant $\mathcal{R}_{\mathrm{P}}$; for larger trampolines
($L^{2}>100^{2}\ \mathrm{\mu m}^{2}$), it decreases linearly as the pad area
grows. The trend is similar to the drumhead case (Fig. 2g). The difference in
orders of magnitude compared to the drumheads relates to the improved thermal
insulation (see Fig. 2f$\&$j). As the window size is kept fixed, the growth of
the central area corresponds to a reduction in tethers’ length. For $L<100\
\mathrm{\mu m}$, long tethers provide high thermal insulation, with
$\mathcal{R}_{\mathrm{P}}$ converging to the cross-string case. As $L^{2}$
approaches $L_{\mathrm{w}}^{2}$, thermal radiation, as well as conduction
increases due to the tethers shortening, with $\mathcal{R}_{\mathrm{P}}$
approaching the drumhead performances. This comparison shows the validity of
the thermal model employed so far.

As only the central pad is here the sensing area, uniform heating would result
in an almost identical mean temperature rise $\braket{\Delta T}$ for this
geometry, leading to no reduction of the power responsivity
$\mathcal{R}_{P}(w_{0}\rightarrow 0)\approx\mathcal{R}_{P}(w_{0}=L/2)$ (see SI
Section S2).

Fig. 2l shows the thermal time constant comparison between the model (solid
curve) and the FEM simulations (circles), showing excellent agreement. For
$L<50\ \mathrm{\mu m}$, the trampoline behaves as a string. For $50\
\mathrm{\mu m}<L<230\ \mathrm{\mu m}$, the resonator thermal capacitance grows
faster than the conductance, increasing the overall response time. For $L>230\
\mathrm{\mu m}$, $\tau_{\mathrm{th}}$ reaches a plateau, to drop down for
increasingly larger central pads. This is explained by the increase in
conduction and radiation: the former, due to the shorter tether length; the
latter, due to a bigger central area $L^{2}$. The interplay between thermal
mass and conductance is the same one observed between the effective mass and
the stiffness for the resonance frequency, as shown in Fig. 6a (see also SI
Section SI).

As for drumheads, the most responsive trampolines exhibit the fastest time
response.

Table 1: Expressions for the relative temperature responsivity $\mathcal{R}_{\mathrm{T}}$ and thermal conductance $G$ for the three designs. The two quantities are used to calculate the relative power responsivity (2), for localized (LH) and uniform (UH) heating. For the drumheads see also SI Section S2. |  $\mathcal{R}_{\mathrm{T}}$ [1/K] | LH: $G$ [W/K] | UH: $G$ [W/K]
---|---|---|---
String | $-\frac{\alpha_{\mathrm{th}}}{2}\frac{E}{\sigma_{0}}$ | $8\frac{wh}{L}\kappa+8wL\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$ | $12\frac{wh}{L}\kappa+8wL\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$
Drumhead | $-\frac{\alpha_{\mathrm{th}}}{2(1-\nu)}\frac{E}{\sigma_{0}}$ | $4\pi h\kappa+4L^{2}\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$ | $8\pi h\kappa+8L^{2}\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$
Trampoline | $-\frac{\alpha_{\mathrm{th}}}{2}\frac{E}{\sigma_{0}}$ | $8\frac{wh}{L_{\mathrm{t}}}\kappa+4(8wL_{\mathrm{t}}+2L^{2})\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$ | $8\frac{wh}{L_{\mathrm{t}}}\kappa+4(8wL_{\mathrm{t}}+2L^{2})\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}$

A summary of the expressions of $\mathcal{R}_{\mathrm{T}}$ and $G$ for the
calculation of the power responsivity (2) is displayed in Table 1, for point-
like source and even illumination.

###  2.3 Frequency stability

High photothermal sensitivity (1) requires also low fractional frequency
noise, as it defines the smallest resonance frequency shift that can be
resolved. In nanomechanical photothermal sensing, the most relevant noise
sources are: i) additive phase noise
$S_{y_{\theta}}(\omega)=S_{y_{\theta_{\mathrm{thm}}}}(\omega)+S_{y_{\theta_{\mathrm{det}}}}(\omega)$,
sum of thermomechanical and detection noise [49]; ii) temperature fluctuation
frequency noise $S_{y_{\mathrm{th}}}(\omega)$ [33]; and iii) photothermal
back-action frequency noise $S_{y_{\mathrm{\delta P}}}(\omega)$

$S_{y}(\omega)=S_{y_{\theta}}(\omega)+S_{y_{\mathrm{th}}}(\omega)+S_{y_{\mathrm{\delta
P}}}(\omega).$ (25)

####  2.3.1 Additive phase noise

Additive phase noise originates from the conversion of thermomechanical
$S_{z_{\mathrm{thm}}}(\omega)$ and detection $S_{z_{\mathrm{det}}}(\omega)$
amplitude noise into phase noise [49]. In the assumption of detection of white
noise, this contribution can be expressed with respect to the thermomechanical
noise peak as [45]

$S_{z_{\mathrm{det}}}(\omega)=\mathcal{K}_{\mathrm{d}}^{2}S_{z_{\mathrm{thm}}}(\omega_{0})=\mathcal{K}_{\mathrm{d}}^{2}\left[\frac{4k_{\mathrm{B}}TQ}{m_{\mathrm{eff}}\omega_{0}^{3}}\right]$
(26)

with $\mathcal{K}_{\mathrm{d}}<1$ for transduction systems able to resolve the
thermomechanical noise. Assuming that the resonator is made to oscillate at an
amplitude $z_{\mathrm{r}}$ by means of a closed-loop frequency tracking
scheme, the resulting fractional frequency noise power spectral density (PSD)
is [45]

$S_{y_{\theta}}(\omega)=\frac{1}{2Q^{2}}\frac{S_{z_{\mathrm{thm}}}}{z^{2}_{\mathrm{r}}}\left[\lvert
H_{\theta_{\mathrm{thm}}}(\mathrm{i}\omega)\rvert^{2}+\mathcal{K}_{\mathrm{d}}^{2}\lvert
H_{\theta_{\mathrm{det}}}(\mathrm{i}\omega)\rvert^{2}\right].$ (27)

$H_{\theta_{\mathrm{thm}}}(\mathrm{i}\omega)$ and
$H_{\theta_{\mathrm{det}}}(\mathrm{i}\omega)$ are the loop-specific transfer
functions for the thermomechanical and detection phase noise. The transfer
functions for an open loop, phase-locked loop, and self-sustaining oscillator
are the same to a good approximation [49]. As an example, for a self-
sustaining oscillator (SSO) scheme, as used in this work, the transfer
functions are [49]

$\displaystyle
H_{\theta_{\mathrm{thm}}}^{\mathrm{SSO}}(\mathrm{i}\omega)=H_{\mathrm{L}}(\mathrm{i}\omega),$
(28) $\displaystyle
H_{\theta_{\mathrm{det}}}^{\mathrm{SSO}}(\mathrm{i}\omega)=\frac{H_{\mathrm{L}}(\mathrm{i}\omega)}{H_{\mathrm{mech}}(\mathrm{i}\omega)}.$

$H_{\mathrm{mech}}(\mathrm{i}\omega)$ and $H_{\mathrm{L}}(\mathrm{i}\omega)$
are the low-pass filter transfer functions of the resonator and system filter,
respectively

$\displaystyle
H_{\mathrm{mech}}(\mathrm{i}\omega)=\frac{1}{1+\mathrm{i}\omega\tau_{\mathrm{mech}}},$
(29) $\displaystyle
H_{\mathrm{L}}(\mathrm{i}\omega)=\frac{1}{1+\mathrm{i}\omega\tau_{\mathrm{L}}}.$

with the resonator time constant $\tau_{\mathrm{mech}}=2Q/\omega_{0}$ and the
filter time constant $\tau_{\mathrm{L}}=1/(2\pi f_{\mathrm{L}})$.

Additive phase noise (27) can be mitigated by actuating the resonator at the
onset of nonlinearity $z_{\mathrm{r_{c}}}$,

$z_{{\mathrm{r_{c}}}}=\ \sqrt{\frac{8}{3\
\sqrt{3}}}\frac{1}{\sqrt{Q}}\sqrt{\frac{m_{{\mathrm{eff}}}\
\omega_{0}^{2}}{\alpha_{{\mathrm{Duff}}}}},$ (30)

with $\alpha_{{\mathrm{Duff}}}$ denoting the effective Duffing term [45]. For
$z_{\mathrm{r}}>z_{\mathrm{r_{c}}}$, additional phase noise of nonlinear
origin could enter the system, worsening the resonator frequency stability at
the integration times of interest in this study [50].

####  2.3.2 Temperature fluctuation frequency noise

Thermal fluctuation fractional frequency noise can also be assumed white. For
a lumped-element model, $S_{y,\mathrm{th}}(\omega)$ is given by [51, 46]

$S_{y_{\mathrm{th}}}(\omega)=\ \frac{4\ k_{B}\
T^{2}}{G_{\mathrm{eff}}}\mathcal{R}_{\mathrm{T}}^{2}\left\lvert\frac{1}{1+i\omega\tau_{\mathrm{th_{eff}}}}\right\rvert^{2}.$
(31)

Here, $G_{\mathrm{eff}}$ and $\tau_{\mathrm{th_{eff}}}$ represent an effective
thermal conductance and time constant, accounting for the fact that
temperature fluctuations occur randomly at any point of the resonator.
$G_{\mathrm{eff}}$ is derived from the integration of the conductance $G$ over
all possible positions of a point-like heat noise source. Since radiation is
heat source position-independent in MTF, only the integration of
$G_{\mathrm{cond}}$ is required. From $G_{\mathrm{eff}}$,
$\tau_{\mathrm{th_{eff}}}=C/G_{\mathrm{eff}}$ can be evaluated.

In a string resonator, thermal noise can enter the system at any point along
its length $L$. Integrating Eq. (16) along $L$ gives the effective conductance

$\displaystyle
G_{\mathrm{eff}}=\left(\frac{1}{\kappa}\frac{1}{L}\int_{0}^{L}{\frac{1}{s_{\mathrm{f}}(x)}}\text{d}x\right)^{-1}+8wL\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}=$
(32) $\displaystyle=\frac{6\kappa
wh}{L}+8wL\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}.$

Eq. (32) results in a higher conductance than (18), as the averaging includes
noise sources closer to the clamping points, where $G_{\mathrm{cond}}$
increases exponentially (see Fig. 2b).

For a circular drumhead, the integration over all the possible noise source
positions gives

$\displaystyle G_{\mathrm{eff}}=\left(\frac{1}{\pi\ (\frac{D}{2})^{2}\
\kappa}\int_{0}^{2\pi}\int_{0}^{(D-d)/2}{\frac{1}{s_{\mathrm{f}}(r,\theta,D,d)}r}\text{d}r\text{d}\theta\right)^{-1}$
(33)
$\displaystyle+8L^{2}\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}=$
$\displaystyle\simeq 4\pi
h\kappa+8L^{2}\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}.$

As with strings, the most important noise contribution is given by the central
region, resulting in $G_{\mathrm{eff}}\simeq G(\mathbf{r}=0)$.

For a trampoline, random energy exchange can occur in both its central pad and
along its four tethers, resulting in

$\displaystyle G_{\mathrm{eff}}=\left(\frac{1}{\kappa}\frac{1}{\sqrt{2}\
L_{\mathrm{w}}}\int_{0}^{\sqrt{2}L_{w}}{\frac{1}{s_{\mathrm{f}}(x)}}\text{d}x\right)^{-1}$
(34)
$\displaystyle+4(8wL_{\mathrm{t}}+2L^{2})\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}=$
$\displaystyle=\frac{6\ \sqrt{2}\ \kappa\ w\
h}{L_{\mathrm{w}}}+4(8wL_{\mathrm{t}}+2L^{2})\epsilon_{\mathrm{rad}}\sigma_{\mathrm{SB}}T_{0}^{3}.$

with $L_{\mathrm{w}}$ denoting the window side length. While trampolines
dissipate $\sqrt{2}\times$ more than strings via conduction, the central pad
will make this geometry extremely sensitive to temperature fluctuations.

####  2.3.3 Photothermal back-action frequency noise

Photothermal back-action frequency noise $S_{y,\mathrm{\delta
P}}(\omega,\lambda)$ originates from the intensity fluctuations of the light
source employed for photothermal sensing, as well as any other light source
used for transduction, such as interferometric lasers. For a continuous wave
(CW) source with an intensity fluctuation PSD $S_{I}(\omega,\lambda)$
[$\mathrm{W^{2}/Hz}$] (also called relative intensity noise, RIN - see Fig.
4c), the resonator fractional frequency fluctuations are given by

$S_{y,\mathrm{\delta P}}(\omega,\lambda)=\ \alpha^{2}(\lambda)\
\mathcal{R}_{\mathrm{P}}^{2}(\omega)\ S_{I}(\omega,\lambda),$ (35)

where $S_{I}(\omega,\lambda)$ typically has the form

$S_{I}(\omega,\lambda)=h_{0}+h_{-1}f^{-1}+h_{-2}f^{-2}$ (36)

for a generic laser source [52]. Here, $h_{0}$ denotes the laser shot-noise
limit $S_{I,\mathrm{sn}}(\lambda)=2hc\braket{P_{0}}/\lambda$, where
$\braket{P_{0}}$ is the average input power; the terms $h_{-1}$ and $h_{-2}$
express the flicker and random walk noise levels, respectively.

Therefore, high optical absorption and responsivity (2) make the resonator
more sensitive to laser intensity noise, highlighting a trade-off between
responsivity and frequency fluctuations. This noise can be mitigated by
selecting materials with low absorption in the targeted spectral range, or by
operating the laser at its shot-noise limit $S_{I,\mathrm{sn}}(\lambda)$.

##  3 Experimental results and discussion

The experimental results focus on low-stress SiN resonators and are compared
with the theoretical model (for details about the measurement procedures, see
Materials and Methods).

###  3.1 Strings

Fig. 3a shows the measured resonance frequency of SiN strings with varying
length. The Q factor of these strings, essential for the theoretical
calculations of the additive phase noise (27), has also been measured (for the
data, see SI Section S3).

Fig. 3b displays the experimental thermal time constant (dark red circles)
compared to theoretical predictions (3), showing excellent agreement.

Figure 3: String design. a Resonance frequency of $56$ nm thick, $5\
\mathrm{\mu m}$ wide string resonators of different lengths. From the
measurements, a stress of 363 MPa is extracted. b Comparison between
theoretical (black solid curve) and measured (dark red circles) thermal time
constant $\tau_{\mathrm{th}}$ for the same set of strings. c Comparison
between theoretical (solid curves) and measured (dark red circles) relative
power responsivity. The error bar indicates the uncertainties in $\kappa$
($2.7-4\ \mathrm{W/(m\ K)}$), $E$ ($200-300\ \mathrm{GPa}$). For these
structures, $\alpha_{\mathrm{th}}=1\ \mathrm{ppm/K}$ has been measured. d
Allan deviation measured for a $2\ \mathrm{mm}$ long string (green solid
curve), driven at the onset of nonlinearity $z_{\mathrm{r_{c}}}$, with low-
pass filter bandwidth $f_{\mathrm{demod}}=2.5\ \mathrm{kHz}$, PLL bandwidth
$f_{\mathrm{pll}}=500\ \mathrm{Hz}$ and optical input power $P_{0}=6\
\mathrm{\mu W}$. The comparison with the theoretical model is also shown
(black solid curve), together with the single contributions (see main text). e
Comparison between the theoretical (black solid curve) and experimentally
extracted (black circles) NEP for strings. The theoretical curve is composed
of two different noise contributions: temperature (red dashed curve) and
thermomechanical (blue dashed curve) fluctuations-induced fractional frequency
noise. For each string’s length, three different resonators were characterized
in terms of NEP. Average and standard mean error for the data points are also
shown for each length.

Fig. 3c compares the theoretical model (2) (black solid lines) with the
measured responsivity (dark red circles). The uncertainty band is defined by
the uncertainty in material parameters $\kappa$ and $E$. All data points fall
within the uncertainty band except for $L=2\ \mathrm{mm}$. This discrepancy
stems from increased radiative losses caused by high probing optical power
($P_{0}=24-40\ \mathrm{\mu W}$). Higher incident powers lead to elevated
temperatures at the string’s center, increasing the radiative heat flux
$\propto(T^{4}-T_{0}^{4})$. This results in a nonlinear reduction of
$\mathcal{R}_{\mathrm{P}}$, as well as a reduction in photothermal response
time $\tau_{\mathrm{th}}$ (see Fig. 3b).

The power responsivity can be enhanced by reducing the resonator’s thickness
$h$ and width $w$. On one hand, thinner strings will improve the thermal
insulation, due to a reduction in cross-sectional area, as well as in
emissivity [53]. On the other hand, narrower strings will reduce the hosting
area for particle and molecule spectroscopy. Hence, the width choice is
critical for photothermal sensing.

Fig. 3d displays the Allan deviation (AD) for a string (green solid curve)
[54]. All the acquired AD have been compared with the theoretical model,
accounting for the transfer functions (27) of the PLL and SSO tracking schemes
[33, 49]. A good match is observed between measurements and theory (black
solid curve) for integration times $\tau<0.1\ \mathrm{s}$, where the main
noise source is additive in-phase (blue solid curve). For $\tau>0.1\
\mathrm{s}$, the data depart from the thermomechanical asymptote, with the
presence of flicker frequency noise for $0.1\ \mathrm{s}<\tau<1\ \mathrm{s}$,
and frequency random walk for $\tau>1\ s$, attributed to photothermal back-
action (see below).

Fig. 3e presents the resulting NEP values, evaluated at
$\tau=\tau_{\mathrm{th}}$. For each length, three different chips have been
analyzed (black circles). The results demonstrate strings’ high photothermal
sensitivity ($0.28-2.5\ \mathrm{pW/\sqrt{Hz}}$). The plot also displays the
theoretical NEP (black solid curve), closely aligning with the measurements.
The sensitivity is mainly limited by thermomechanical noise for almost all the
lengths. The observed deviations must be attributed to the photothermal back-
action.

The positive correlation between noise level measured for long integration
times ($\tau>\tau_{\mathrm{th}}$) and power responsivity is indeed a clear
sign of photothermal backaction (35). To investigate this further, the laser
relative intensity noise $S_{I}(\omega,\lambda)$ has been characterized for
all the optical powers employed in this study and $S_{y_{\mathrm{\delta
P}}}(\omega,\lambda)$ evaluated. The results are displayed in Fig. 3d with the
purple solid curve, showing excellent agreement with the data for $\tau>0.1\
\mathrm{s}$. The observed flicker and random walk frequency noises are due to
the intensity spectral distribution $S_{I}(\omega,\lambda)$, as clearly shown
in Fig.4a, far above the ultimate laser shot noise limit
$S_{I,\mathrm{sn}}(\lambda)$.

Figure 4: Laser intensity fluctuations. a Characterization of the intensity
fluctuations for an average power $\braket{P_{0}}=6\ \mathrm{\mu W}$. The
$f^{-2}$ and $f^{-1}$ noise contributions are shown (dashed red and dashed-
dotted red lines, respectively). b Power spectral density of the
thermomechanical noise for a drumhead resonator for different interferometer
laser powers. c Corresponding measured displacement sensitivity
$\delta_{\mathrm{min}}$, in units [$\mathrm{m/\sqrt{Hz}}$]. It improves
linearly with the laser power, with the effect of simultaneously introducing
higher thermomechanical and laser power instability-induced frequency noise.

Hence, photothermal back-action frequency noise imposes an upper limit on the
probing power used for displacement transduction. On the one hand, high laser
power improves the displacement sensitivity $\delta_{\mathrm{min}}$
[$\mathrm{m/\sqrt{Hz}}$], as shown in Fig. 4b$\&$c [55], reducing the
detection coefficient $\mathcal{K}_{d}$. On the other hand, such a signal
enhancement saturates at higher optical power due to the induced frequency
noise [56], with any low-frequency intensity noise, such as mode hopping [57],
directly impacting the resonator stability [52].

Fig. 4c further shows that the displacement sensitivity is here inversely
proportional to the optical power $P_{0}$, indicating that the laser noise has
a classical (detector and technical noise) and not quantum shot noise origin
[58]. Among the different approaches to mitigate laser classical noise,
feedback intensity noise squashing could offer a simple way to push the laser
to its shot-noise limit [59].

###  3.2 Drumheads

Fig. 5a show the resonance frequency corresponding to the drumheads
characterized experimentally (for the measured Q, see SI Section S3).
Experimental results concerning the thermal time constant are not presented
here, as the photothermal response time of SiN drumheads has been already
experimentally characterized elsewhere [30, 32].

Figure 5: Drumhead design. a Resonance frequency for $50$ nm thick square
drumhead resonator of different side length. From the measurements, a stress
of $150$ MPa is extracted. b Comparison between theoretical (solid curves) and
measured (dark red unfilled dots) relative power responsivity. The error bar
indicates the uncertainties in $\kappa$ ($2.7-4$ W/(m K)), $E$ ($200-300$
GPa), and $\alpha_{\mathrm{th}}(1-2.2\ \mathrm{ppm/K})$. c Allan deviation
measured for a $1\ \mathrm{mm^{2}}$ square membrane (green solid curve),
driven at the onset of nonlinearity $z_{\mathrm{r_{c}}}$, with low-pass filter
bandwidth $f_{\mathrm{demod}}=2.5$ kHz, PLL bandwidth $f_{\mathrm{pll}}=10$ Hz
and optical input power $P_{0}=6\ \mathrm{\mu W}$. The comparison with the
theoretical model is also shown (black solid curve), together with the single
contributions (see main text). d Comparison between the theoretical (black
solid curve) and experimentally extracted (black unfilled dots) NEP for
membranes. Temperature (red dashed curve) and thermomechanical (blue dashed
curve) fluctuations-induced fractional frequency noise are also shown,
together with the single photon noise limited NEP. For each membrane’s length,
three different resonators were characterized in terms of NEP.

Fig. 5b compares the theoretical predictions (2) (solid curves) with the
experimental data (dark red circles) for the relative power responsivity. The
uncertainty band, defined by uncertainties in $\kappa$, $E$, and
$\alpha_{\mathrm{th}}$, encompasses all the experimental points, indicating a
strong agreement between theory and experiments.

Fig. 5c illustrates the AD for a drumhead. In detail, two regimes can be
recognized: i) for different integration times $\tau<0.01\ \mathrm{s}$, the AD
is limited by additive phase noise $S_{y_{\theta}}(\omega)$ (blue solid
curve); ii) $\tau>0.01\ \mathrm{s}$, the noise is dominated by photothermal
backaction $S_{y_{\mathrm{\delta P}}}(\omega)$. Notably, in the absence of
photothermal backaction, temperature fluctuation frequency noise would
dominate. This condition, where a mechanical resonator interacts with the
environment at the single shot noise level, is of significant interest for
micromechanical thermal detectors [46, 29, 5, 7].

Fig. 5d presents the experimental NEP evaluated at $\tau=\tau_{\mathrm{th}}$,
alongside the theoretical sensitivity (black solid curve), closely aligning to
each other. The experimental results of $1-20\ \mathrm{pW/\sqrt{Hz}}$ are one
order of magnitude lower than previously characterized, electrodynamically
transduced drumhead resonators [5], showing the outstanding performances of
pristine SiN structures over integrated nanoelectromechanical systems (NEMS),
where electrodes are an important part of the design [21, 39, 5]. The use of
pure SiN for photothermal sensing applications is enabled by noninvasive
transduction approaches, such as interferometry. In particular, pure optical
transduction offers two key advantages: i) the absence of metal traces
increases the thermal insulation, improving the responsivity (2); ii), the
sensor is not limited by Johnson noise, which usually degrades the frequency
stability (25) of a vast group of NEMS resonators [5]. Conversely, bare SiN
drumheads are mainly affected by temperature fluctuations noise (dark red
dashed curve), as shown for $L>50\ \mathrm{\mu m}$. Moreover, as the resonator
enters the radiation-limited regime, thermal photon shot noise becomes
dominant (dark violet solid curve) [46, 32].

###  3.3 Trampolines

The experimental analysis has been carried out for trampoline resonators with
central pads designed using a Bezier profile [39, 40, 4, 5, 41].

Fig. 6a presents the resonance frequency as a function of the central area
$L^{2}$ (for the Q measurements, see SI Section S3). For small areas
($L^{2}<50^{2}\ \mathrm{\mu m^{2}}$), $\omega_{0}$ can be approximated with
that of a string [45]. In the intermediate range ($50^{2}\ \mathrm{\mu
m}^{2}<L^{2}<500^{2}\ \mathrm{\mu m}^{2}$) the effective mass
$m_{\mathrm{eff}}$ grows faster ($\propto L^{2}$) than the tethers’ effective
stiffness $k_{\mathrm{eff}}$ ($\propto L^{\zeta}$, with $\zeta<2$), leading to
a reduction in resonance frequency $\omega_{0}$. For larger areas ($L^{2}>500\
\mathrm{\mu m}^{2}$) $k_{\mathrm{eff}}$ increases more rapidly than
$m_{\mathrm{eff}}$ ($\zeta>2$), causing $\omega_{0}$ to rise beyond the string
value (see SI Section S1).

Figure 6: Trampoline design. a Resonance frequency for $50$ nm thick
trampoline resonators of different central area side lengths. The window side
length is fixed to $L_{\mathrm{w}}\approx 1$ mm, while the tether width is to
$w=5\mathrm{\mu m}$. From the measurements, a stress of $120\ \mathrm{MPa}$ is
extracted. b Comparison between theoretical (black solid curve) and
experimental thermal time constant $\tau_{th}$. c Comparison between
theoretical (solid curves) and measured (dark red circles) relative power
responsivity. The error bar indicates the uncertainties in $\kappa$ ($2.7-4\
\mathrm{W/(m\ K)}$), $E$ ($200-300\ \mathrm{GPa}$), and $\alpha_{\mathrm{th}}\
(1-2.2\ \mathrm{ppm/K}$). d Allan deviation measured for a $230^{2}\
\mathrm{\mu m}^{2}$ central area trampoline (green solid curve), driven at the
onset of nonlinearities $z_{\mathrm{r_{c}}}$, with low-pass filter bandwidth
$f_{\mathrm{demod}}=2.5\ \mathrm{kHz}$, PLL bandwidth $f_{\mathrm{pll}}=500\
\mathrm{Hz}$ and optical input power $P_{0}=11\ \mathrm{\mu W}$. The
comparison with the theoretical model is also shown (black solid curve),
together with the single contributions (see main text). e Comparison between
the theoretical (black solid curve) and experimentally extracted (dark red
circles) NEP. For each trampoline’s central length, three different resonators
were characterized in terms of NEP.

Fig. 6b compares the theoretical thermal response time (black solid curve)
with the experimental measurements (dark red circles). Discrepancies between
the model and experimental data might be due to variations in specific heat
capacity and mass density from the values used in the model. Nevertheless, a
positive correlation between $\tau_{\mathrm{th}}$ and $L$ is evident. This
slow thermal response significantly impacts the frequency noise in the
experimental setup employed here.

Fig. 6c shows the comparison between the theoretical and measured power
responsivity, exhibiting excellent agreement. As for the other designs, the
shaded band represents uncertainties in $\kappa$, $E$, and
$\alpha_{\mathrm{th}}$.

Fig. 6d shows the AD for a trampoline. Also here, two regimes can be
recognized: an additive phase noise-limited region for integration times
$\tau<0.02\ \mathrm{s}$, and a fully photothermal back-action frequency noise-
dominated region for $\tau>0.02\ \mathrm{s}$. The sum of all the contributions
(black solid curve) matches perfectly the experimental data (green solid). It
is worth noting that $\tau_{\mathrm{th}}$ lies far in the photothermal back-
action dominated region (red dashed vertical line), meaning that, during the
time the resonator takes to reach a new thermal equilibrium, e.g. upon energy
relaxation by a molecule, intensity fluctuations of the probing laser increase
the frequency noise. Conversely, with a shot-noise limited laser, the
temperature fluctuation frequency noise would dominate for
$\tau>\tau_{\mathrm{th}}$.

Fig. 6e displays the experimental sensitivities evaluated at
$\tau=\tau_{\mathrm{th}}$ (dark red circles) compared with the theoretical
calculations (blue and red dashed curves). The plot reveals that the
photothermal back-action (dark violet dashed curve) has degraded the
sensitivity by one order of magnitude compared to the theoretical
expectations. Moreover, this effect is much more pronounced for this design
than for the others. Indeed, the slow thermal response time of trampolines
makes them more sensitive to the laser relative intensity noise (36), where
flicker and random walk noise are present, worsening the corresponding
sensitivity

$NEP_{\mathrm{pba}}=\alpha\sqrt{h_{0}+2\pi\tau_{\mathrm{th}}h_{-1}+(2\pi\tau_{th})^{2}h_{-2}}.$
(37)

However, the data follow the theoretical trend, with the sensitivity worsening
for increasingly larger central areas $L^{2}$. Similar to drumheads,
temperature fluctuations represent the ultimate theoretical limit of the
photothermal sensitivity in the absence of photothermal back-action.

###  3.4 Comparison

To sum up, a theoretical comparison among the three resonator designs of
comparable dimensions is illustrated in the radar chart shown in Fig. 7. The
metrics used for this comparison are the NEP, the thermal time constant
$\tau_{\mathrm{th}}$, and the sensing area $A_{\mathrm{sens}}$, each
normalized to the best-performing value.

The string demonstrates the highest photothermal sensitivity due to its
superior thermal insulation, albeit with the smallest sensing area. It
presents an intermediate thermal response time compared to the other
geometries. The fundamental frequency noise limit for this design is likely
dominated by thermomechanical phase noise. These features make strings an
excellent workhorse for nanomechanical photothermal spectroscopy [28].
Conversely, the drumhead exhibits the lowest sensitivity but offers the
largest sensing area and the fastest thermal response. In particular, the
combination of high speed and optimal sensitivity for this design makes
drumheads ideal for applications requiring quick measurements. Furthermore,
temperature fluctuations are expected to be the ultimate frequency noise
limit.

Drumheads are good candidates for scanning spectromicroscopy, as well as a
promising platform for room-temperature IR/THz detection. In particular, in
the case of single-photon noise-limited detection, the large sensing area
$A_{\mathrm{sens}}$ could enable drumheads to achieve the room-temperature
specific detectivity limit $D^{*}=\sqrt{A_{sens}}/NEP\approx 2\cdot 10^{10}\
\mathrm{cm\sqrt{Hz}}/W$ [29, 32, 5].

Trampolines present a compromise between the highly sensitive strings and the
drumheads with a larger sensing area. As such, trampolines show intermediate
values in terms of power sensitivity and sensing area. Its only drawback is
the slow thermal response, which makes it more susceptible to photothermal
back-action frequency noise than the other designs, as confirmed by
experimental observations. Despite this, its high sensitivity makes this
design a good candidate for photothermal spectroscopy. Moreover, temperature
fluctuations are expected to be the ultimate limiting frequency noise,
therefore making it a promising alternative for IR/THz thermal detection and a
potential competitor for drumheads.

The present study has examined the three main resonator designs exploited so
far in nanomechanical photothermal sensing. Nonetheless, new designs routinely
employed in other fields of nanomechanics, e.g., in optomechanics, could be
explored for photothermal sensing. For instance, phononic crystal (PnC)
engineering could be easily integrated within the sensor, enhancing the
resonator frequency stability, as well as its thermal properties. In
particular, the use of PnC defect modes for sensing applications would boost
the power responsivity due to the increased overlap between the photothermally
induced temperature field and the mechanical mode volume, as already shown
[42].

Figure 7: Performance summary. Radar chart of the nanomechanical photothermal
performances. The chart accounts for the normalized NEP, thermal response time
$\tau_{\mathrm{th}}$, and sensing area $A_{\mathrm{sens}}$. The highest value
for each metric has been used for normalization, with the subscripts referring
to the corresponding design (s, string; d, drumhead; t, trampoline). The
string length, the membrane side length, and the trampoline window side length
are all $1\ \mathrm{mm}$ long. The trampoline has a central area
$L^{2}=230^{2}\ \mathrm{\mu m}^{2}$. All the structures are assumed to be $50\
\mathrm{nm}$ thick.

##  4 Conclusions

In summary, the comparative analysis conducted on three distinct resonator
designs utilized in photothermal sensing—namely strings, drumheads, and
trampolines—has elucidated the relationship between the resonator’s
photothermal sensitivity and its mechanical and thermal properties. Across all
scenarios, the theoretical framework has shown remarkable consistency with
both experimental data and FEM simulations, demonstrating how the resonance
frequency photothermal response is governed by the resultant mean temperature
rise. Overall, strings emerge as the most sensitive design, followed by
trampolines and drumheads. Conversely, drumheads exhibit the fastest thermal
response, followed by strings and trampolines. The analysis has also
highlighted the critical role of photothermal back-action, particularly its
impact on the trampolines’ frequency fluctuations, due to their slowest
thermal response. These findings not only clarify the relative performance of
the resonator designs investigated but also establish a solid groundwork for
the development of next-generation nanomechanical photothermal sensors. This
study represents a significant step forward in the advancement of
nanomechanical sensing technology, offering valuable insights for researchers
seeking to harness the full potential of photothermal sensing in diverse
applications.

##  5 Materials and methods

###  5.1 FEM simulations

FEM analysis (COMSOL Multiphysics, v5.5 and v6.1) has been carried out to
determine the relative power responsivity $\mathcal{R}_{\mathrm{P}}(0)$ and
the thermal time constant $\tau_{\mathrm{th}}$ of the resonators. For
$\mathcal{R}_{\mathrm{P}}(0)$, the Solid Mechanics Physics has been employed
in conjunction with Heat Transfer in Solids. First, a Static Prestress Study
is performed to solve for the resonator stress field, accounting for the
initial prestress $\sigma_{0}$ and the thermal stress components induced by
laser heating. A Gaussian beam with beam waist $w_{0}$ is used as the light
source. The static solutions obtained from this study serve as input
parameters for the Eigenfrequency Study, where the fundamental eigenfrequency
is computed. The procedure is repeated for different input powers $P_{0}$. For
the evaluation of $\tau_{\mathrm{th}}$, a time-dependent study is conducted
with the beam impinging at the resonator center. The temperature of the
resonator is solved for discrete points in time, and the thermal time constant
is evaluated upon fitting with an exponential function.

###  5.2 Experimental setup

The resonators are operated in high-vacuum conditions ($p<10^{-5}\
\mathrm{mbar}$) to minimize gas damping and thermal convection, in a custom-
designed vacuum chamber equipped with a window for optical access to the
chips. The resonators are actuated with a piezoelectric element placed beneath
them. Their out-of-plane displacement is measured with a commercial laser-
Doppler vibrometer (MSA 500, Polytec Gmbh), operated at 633 nm wavelength. The
vibrometer’s signal is sent to a lock-in amplifier (HF2LI, Zurich Instruments)
equipped with a PLL module, or to a frequency counter implemented in a self-
sustained oscillator (PHILL, Invisible-Light Labs GmbH) [49]. The relative
power responsivity $\mathcal{R}_{\mathrm{P}}(0)$ is evaluated upon measurement
of the thermomechanical noise spectrum peak for different input laser powers
[42]. The thermal time constant $\tau_{\mathrm{th}}$ is evaluated with the
90/10 method [5]. For that, the resonator is driven at its resonance frequency
with the PLL or SSO tracking scheme.

###  5.3 Laser intensity fluctuations characterization

The intensity of the probing laser has been recorded for 1 minute with a
silicon photodiode (Thorlabs GmbH S120C, $1\ \mathrm{\mu s}$ response time)
together with a digital power meter console (Thorlabs GmbH PM100D). The
electrical signal is fed to the lock-in amplifier [56]. The recorded intensity
signal is then converted into frequency fluctuations, accounting for the
resonator’s thermal response $H_{\mathrm{th}}(\omega)$, and the corresponding
AD is evaluated (See SI Section S4).

##  6 Acknowledgement

The authors are thankful to Sophia Ewert and Patrick Meyer for the device
fabrication. Furthermore, the authors thank Johannes Hiesberger, Hajrudin
Besic and Niklas Luhmann for useful discussions. This work received funding
from the Defense Advanced Research Projects Agency (DARPA) Optomechanical
Thermal Imaging (OpTIm) Technical Area (TA) 1 Broad Agency Announcement (BAA),
HR001122S0055. This work further received funding from the Novo Nordisk
Foundation under project MASMONADE with project number NNF22OC0077964, and
from the European Innovation Council under the European Union’s Horizon Europe
Transition Open program (Grant agreement: 101058711-NEMILIES).

##  7 Contributions

K.K, and S.S. designed research; K.K., and S.E. performed experiments; K.K.,
and F.L. performed FEM simulations; K.K., F.L., P.M., R.G.W., and S.S.
analyzed the data; K.K., F.L., R.G.W., and S.S. wrote the paper with inputs
from all authors; S.S. supervised the project.

##  8 Conflict of interests

S.S. is co-founder of Invisible-Light Labs GmbH, which provided the PHILL
electronics.

##  9 Supplementary information

The online version contains supplementary materials.

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