# Temperature mediated back-action in micro- and nanomechanical resonators

L. Bellon  0000-0002-2499-8106<EMAIL_ADDRESS>CNRS, ENS de Lyon,
Laboratoire de Physique, F-69342 Lyon, France  P. Verlot  0000-0002-5105-3319
Université Paris-Saclay, CNRS, ENS Paris-Saclay, CentraleSupélec, LuMIn,
91405, Orsay, France  Institut Universitaire de France, 1 rue Descartes, 75231
Paris, France

###### Abstract

We theoretically investigate the thermally induced back-action effects in
absorption-sensitive micro- and nanomechanical resonators. We propose a
unified approach, enabling to simultaneously address both the effective
dynamics and non-equilibrium phononic state, depending on the position of a
confined sensing (and heating) probe at the surface of the mechanical device.
We present an analytical solution for a unidimensional resonator whose
thermomechanical deformation profile generally follows that of the mechanical
losses. In particular, we find that both the dynamics and the mechanical
fluctuations strongly depend on the loss distribution. The effect of the
quantum fluctuations of the heat source is also discussed. Our approach
provides a thorough, general platform for analyzing thermal back-action
effects and their consequences, which may be of significance for future
development in ultrasensitive nanomechanical research.

##  I Introduction

Micro- and nanomechanical resonators have recently surged as a prominent class
of ultrasensitive probes, both in Science and Technology [1, 2, 3]: Owing to
the unique combination of a ultra-low mass and high mechanical Q-factor, these
devices have the ability to convert very small external driving phenomena into
large displacements, which are to be further readout with a high precision,
using a localized high energy probe [4].

An essential task in mechanical sensing is to accurately determine both the
mechanical response and mechanical noise, the former bridging the measured
motion to the driving phenomenon of interest, while the latter setting the
sensitivity of the measurement [5]. Due to their reduced size and high aspect
ratio, micro- and nanomechanical transducers generally display higher thermal
resistances, resulting in much enhanced responses to thermal variations [6].
In this context, the impact of absorption-induced thermal gradients has long
been known through the mechanism of dynamical back-action [7]. The principle
is for thermal gradients to create a strain-dependent stress inside the
resonator, which feeds back to the deformation, resulting in both a modified
effective mechanical response and phonon occupancy [8, 9]. While providing a
successful, qualitative description of the dynamics associated with thermal
gradients, this approach often relies on phenomenological descriptions of the
thermal-to-motion conversion mechanisms [10, 11, 12]. More accurate models
have been introduced [13, 14], but are based on numerical simulation
describing specific resonators, and have not been extended to generic cases.
Most importantly, the existing approaches leave aside a fundamental aspect
related to the presence of thermal gradient, that is the out-of-equilibrium
nature of mechanical states subjected to thermal gradients [15, 16, 17].

In this paper, we propose a unified theoretical approach that describes both
the dynamical response and the fluctuation distribution of a nanomechanical
system crossed by a large, measurement-induced thermal gradient. We
specifically present an analytical solution of a unidimensional resonator,
whose thermomechanical deformation profile is generally assumed to follow its
thermoelastic losses. The resulting thermal back-action force is derived and
computed as a function of the position of the probe along the resonator, and
the associated fluctuations are analyzed, both in the thermal and quantum
regimes. Our work aims at representing a general, thermal back-action analysis
platform for ongoing and future ultrasensitive nanomechanical research.

##  II General hypothesis

We consider a one-dimensional mechanical resonator, such as a cantilever, beam
or rod, operated in vacuum. Typical examples include micro-cantilevers [18]
(such as those used for atomic force microscopy [19]) and micro- and nanowires
[20, 21, 11] (NW). We will use the latter denomination in this article without
loss of generality. The NW can oscillate in the transverse direction $x$.
$\xi(y)$ depicts its small amplitude deflection as a function of its
longitudinal coordinate $y$. The computation is performed in the clamped-free
geometry, but extension to double-clamped NW is straightforward. A spatially
confined probe beam is used to measure the deflection in $y=y_{p}$, e.g. a
laser [13, 20, 21, 11] or an electron beam [22, 9, 23, 24]. This probe is
focused in $(x_{p},y_{p})$, in a fashion such that the transverse vibrations
produce a modification of the beam-NW interaction that can be detected by a
suitable sensor (e.g. split photodetector [25], interferometric sensor [26],
secondary electron detector [23], on-chip electrodes [27], etc.). As a side
effect of the detection scheme, a fraction $a$ of the incoming probe beam
power $P$ is absorbed, resulting in a temperature rise of the NW. We are
interested here in computing the back-action forces associated with this
process and how it affects the resonance [8, 10, 28, 13, 29].

Figure 1: Methodology: we consider a 1D mechanical oscillator of length $L$
along the $y$ axis, vibrating along the transverse $x$ axis at its resonance
angular frequency $\omega$, with an amplitude $\Xi_{n}\phi_{n}(y)$
proportional to the eigenmode shape $\phi_{n}$. This motion is probed with a
light or electron beam focused close to its edge, sketched here with a
gaussian shaped red spot centered in $(x_{p},y_{p})$. A fraction $a$ of the
beam energy is absorbed by the resonator, leading to a dynamic temperature
field $T(y)$. This field changes the equilibrium curvature $C(y)\sim\beta
T(y)$, which in turns creates a stress distribution. Using the Euler-Bernoulli
description of the resonator, this stress is equivalent to an external
distributed back-action force $f_{\mathrm{ba}}(y)$, which we project on
$\phi_{n}(y)$ to compute the lumped force $F_{\mathrm{ba}}$ acting on the
resonant mode. Since this force is proportional to the original mode amplitude
$\Xi_{n}$, we can define the associated back-action stiffness
$k_{\mathrm{ba}}$, that embeds how the resonance is affected by the probe
beam.

We use the methodology described in Fig. 1. We start by imposing an
oscillation amplitude $\Xi_{n}$ at the angular frequency $\omega$, close to
the resonance $\omega_{n}$ corresponding to the eigenmode $\phi_{n}(y)$. The
corresponding deflection is given by $\xi(y,t)=\Xi_{n}\phi_{n}(y)e^{i\omega
t}$. The distance $\xi(y_{p},t)-x_{p}$ between the probe beam and the
oscillator is therefore modulated, as is the absorbed power $aP$. This motion
thus creates a dynamic temperature field $T(y)$, that in turns creates a
distributed force $f_{\mathrm{ba}}(y)$ along the NW, which we project on the
eigenmode shape $\phi_{n}(y)$ to infer the force $F_{\mathrm{ba}}$ acting on
the resonant mode. This back-action force is proportional to the imposed
deflection amplitude $\Xi_{n}$, with the proportionality constant
$k_{\mathrm{ba}}$ embedding its effect on the resonance. The signature of this
interaction is the modification of the resonance frequency (real part of the
stiffness $k_{\mathrm{ba}}$), and of the damping (imaginary part of
$k_{\mathrm{ba}}$, in quadrature with the oscillation). The methodology is
conceptually similar to Refs. 13, 14, which are each dedicated to a very
specific sample and detection. Here we develop a very broad framework that
explores generic resonators and different classes of optomechanical coupling,
and will demonstrate the dramatic effect of the probe position.

The following sections propose an analytical framework to implement this
methodology. In Section III, we compute the dynamic temperature field due to
the absorption of the probe beam. In Section IV, we translate this temperature
distribution into the back-action force acting on the oscillation mode. In
section V, we present several hypotheses for the stress generated by the
temperature field, and their implication for the back-action stiffness
$k_{\mathrm{ba}}$, and thus for the resonance. In the next two sections, we
explore the effect of the back-action force noise associated with the quantum
fluctuations of the probe beam (section VI), and the additional, classical
mechanical thermal noise component acting on the system (VII). We then compare
the relative contribution of these two sources of noise in a precision force
measurement scheme in section VIII. Finally, we present a numerical
application to a typical mechanical nanowire, highlighting the orders of
magnitude of the various effects in IX, before the conclusions and
perspectives of this work.

##  III Heat equation

We assume that the transverse directions of the resonator are much smaller
than any relevant length scale, so that a one-dimensional description of the
fields is sufficient. The nanowire radius $R$ is therefore much smaller that
its length $L$. Since vacuum is assumed, any heat exchange with the
surroundings (including thermal radiation) is neglected. The heat equation for
the temperature field $T(y,t)$ along the nanowire is:

$\partial_{y}(\kappa\partial_{y}T)=\rho c\partial_{t}T$ (1)

with $\kappa$ the thermal conductivity, $\rho$ the density and $c$ the heat
capacity of the oscillator. The boundary conditions (BC) correspond to a fixed
temperature $T_{0}$ at the origin (that is the clamping position $y=0$), no
heat flux at the nanowire free end ($y=L$), and a power influx $aP$ at
position $y=y_{p}$, with the hypothesis that the probe beam spot is small and
can be described as punctual for the thermal problem:

$\displaystyle T(0,t)$ $\displaystyle=T_{0},$ (2a)
$\displaystyle\kappa\partial_{y}T(L,t)$ $\displaystyle=0,$ (2b)
$\displaystyle\kappa\partial_{y}T(y_{p}^{-},t)-\kappa\partial_{y}T(y_{p}^{+},t)$
$\displaystyle=\frac{aP}{\pi R^{2}}.$ (2c)

Assuming $\kappa$ is uniform, the static temperature profile $\overline{T}$
solution of these equations is piecewise linear:

$\overline{T}(y)=T_{0}+\bar{\theta}\frac{1}{L}\min(y,y_{p}),$ (3)

where $\bar{\theta}=\overline{aP}L/(\pi R^{2}\kappa)$ is the absolute maximum
static temperature elevation (corresponding to $y=y_{p}=L$), and
$\overline{aP}$ is the mean absorbed power from the probe beam. We now define
$\widetilde{T}=T-\overline{T}$ as the temperature deviation from this
stationary state, and study the dynamics of the temperature field due to the
non-stationary part of the absorbed power $\widetilde{aP}=aP-\overline{aP}$.
In the Fourier space, the heat equation writes:

$\partial_{y}^{2}\widetilde{T}=\alpha^{2}\widetilde{T},\ \text{with}\
\alpha=\sqrt{i\omega\frac{\rho c}{\kappa}},$ (4)

with the dynamic BCs:

$\displaystyle\widetilde{T}(0,\omega)$ $\displaystyle=0,$ (5a)
$\displaystyle\partial_{y}\widetilde{T}(L,\omega)$ $\displaystyle=0,$ (5b)
$\displaystyle\partial_{y}\widetilde{T}(y_{p}^{-},\omega)-\partial_{y}\widetilde{T}(y_{p}^{+},\omega)$
$\displaystyle=\frac{\widetilde{aP}(\omega)}{\pi R^{2}\kappa}.$ (5c)

We solve this equation with the ansatz $\widetilde{T}=A\sinh\alpha y$ for
$y<y_{p}$ and $\widetilde{T}=B\cosh\alpha(y-L)$ for $y>y_{p}$ (satisfying the
heat equation and the initial and final BC). The continuity of the temperature
field in $y=y_{p}$ imposes:

$A\sinh\alpha y_{p}=B\cosh\alpha(y_{p}-L),$ (6)

while the discontinuity of the heat flux writes:

$A\alpha\cosh\alpha
y_{p}-B\alpha\sinh\alpha(y_{p}-L)=\frac{\widetilde{aP}}{\pi R^{2}\kappa}.$ (7)

We solve these two equations for $A$ and $B$ and conclude that the dynamic
temperature field is given by:

$\displaystyle\widetilde{T}(y,\omega)$
$\displaystyle=\widetilde{\theta}(\omega)h\left(y,\sqrt{i\omega\rho
c/\kappa}\right),$ (8) $\displaystyle\widetilde{\theta}(\omega)$
$\displaystyle=\frac{L}{\pi R^{2}\kappa}\widetilde{aP}(\omega),$ (9)
$\displaystyle\begin{split}h(y,\alpha)&=H(y_{p}-y)\frac{\sinh\alpha
y\cosh\alpha(y_{p}-L)}{\alpha L\cosh\alpha L}\\\
&\quad+H(y-y_{p})\frac{\sinh\alpha y_{p}\cosh\alpha(y-L)}{\alpha L\cosh\alpha
L}.\end{split}$ (10)

The dynamic temperature field is expressed here as the product of 2 terms,
$\widetilde{\theta}(\omega)$ with quantifies the amplitude of the forced
temperature variation, and $h(y,\alpha)$ (where $\alpha$ is the complex number
proportional to $\sqrt{i\omega}$ defined in Eq. 4) which accounts for the
propagation of the damped thermal wave according to the heat equation. It
should be noted that all three quantities expressed in Eqs. 8 to 10 are
implicit variables of the position $(x_{p},y_{p})$ of the probe beam.

Figure 2: Temperature field along the 1D resonator. The top panel reports the
static profile $\bar{T}$ assuming a constant thermal conductivity, normalized
to the maximum temperature one would get by choosing $y_{p}=L$, for 3 values
of $y_{p}$ reported above the plots. The bottom panels present the dynamic
profile $\tilde{T}$ for three values of thermal diffusion length $1/|\alpha|$.
$|\alpha|L=1$ is the low frequency / high conductivity limit. The thick line
corresponds to the envelope of the thermal waves (thin lines at different
times) inside the NW, and the colors match the static profiles corresponding
to the 3 values of $y_{p}$. As expected, this low frequency limit is
equivalent to the static field. $|\alpha|L=10$ is the intermediate thermal
diffusivity regime. The thermal waves travel with a complex shape across the
nanowire, with an asymmetric profile reflecting that of the clamped-free
boundary conditions. $|\alpha|L=100$ is the high frequency / low conductivity
limit. The thermal waves are localized around the heating point $y_{p}$. The
amplitude of all dynamic fields is computed with a $100\text{\,}\%$ modulation
depth of the static profile ($\widetilde{aP}=\overline{aP}$). The lower the
thermal length, the lower the temperature wave amplitude (note the
multiplicative factors on the vertical scales).

Those temperature fields, both static and dynamics, are reported in Fig. 2.
The value of $\alpha L$ sets the regime to be considered: the dynamic profile
is equivalent to the static one in the low frequency / high conductivity limit
($\alpha L\ll 1$), and tends to be very localized and of low amplitude in the
opposite limit ($\alpha L\gg 1$).

##  IV Temperature mediated force back-action mechanism

As the temperature of the nanowire varies, it can induce stresses that result
in a change in the local curvature of equilibrium $C=\partial_{y}^{2}\xi$. For
example, let us assume the local curvature at temperature $T_{0}$ is
$C_{0}\neq 0$. When the temperature changes to $T$, the equilibrium curvature
changes to $C=C_{0}/[1+\varepsilon(T-T_{0})]\sim
C_{0}[1-\varepsilon(T-T_{0})]$, with $\varepsilon$ the linear thermal
expansion coefficient. In other cases, for instance when several materials are
involved in the oscillator composition, a bimetal effect can be present [13],
which also results in an equilibrium curvature depending on the local
temperature. Yet in other cases, some defects or geometric asymmetries may be
present and trigger similar effects. To describe the generic case, we assume
that these curvature effects are small and define a linear thermal bending
coefficient $\beta$, so that in equilibrium:

$C=C_{0}+\beta(T-T_{0}).$ (11)

The equation of motion of the nanowire is a variant of Euler-Bernoulli’s, and
writes in Fourier space:

$\displaystyle\partial_{y}^{2}[EI(\partial_{y}^{2}\xi-\beta\widetilde{T})]-\rho\pi
R^{2}\omega^{2}\xi$ $\displaystyle=f_{\mathrm{ext}},$ (12)
$\displaystyle\partial_{y}^{2}[EI\partial_{y}^{2}\xi]-\rho\pi
R^{2}\omega^{2}\xi$
$\displaystyle=\partial_{y}^{2}(EI\beta\widetilde{T})+f_{\mathrm{ext}},$ (13)

where static terms have been removed, $EI$ is the flexural rigidity of the
nanowire, and $f_{\mathrm{ext}}$ a potential external distributed force. The
temperature field can thus be seen as such an external force field,

$\displaystyle f_{\mathrm{ba}}(y,\omega)$
$\displaystyle=\partial_{y}^{2}[EI\beta(y)\widetilde{T}(y,\omega)],$ (14)
$\displaystyle=EI\widetilde{\theta}(\omega)\partial_{y}^{2}[\beta(y)h(y,\sqrt{i\omega\rho
c/\kappa})].$ (15)

We suppose that its effect on the mechanical response of the nanowire is
small, so that the resonant mode shape $\phi_{n}(y)$ is unaffected by
$f_{\mathrm{ba}}$. We can then project Eq. 13 on the solution of the
unperturbed Euler-Bernoulli equation $\phi_{n}(y)$, and derive in the Fourier
space the equation of the driven simple harmonic oscillator corresponding to
the resonant mode:

$(k^{r}_{n}+ik^{i}_{n}-m\omega^{2})\Xi_{n}(\omega)=F_{\mathrm{ba}}(x_{p},y_{p},\omega)+F_{\mathrm{ext}}.$
(16)

Here $m=\rho R^{2}L$ is the mass of the oscillator, $k^{r}_{n}$ is the lumped
stiffness of mode $n$, and $k^{i}_{n}$ describes the damping, accounted for in
Eq. 13 by the imaginary part $E^{i}$ of Young’s Modulus $E=E^{r}+iE^{i}$:

$k^{r}_{n}+ik^{i}_{n}=\int_{0}^{L}(E^{r}+iE^{i})I\phi_{n}^{\prime\prime}(y)^{2}\differential
y.$ (17)

Finally, the temperature mediated force back-action is:

$\displaystyle F_{\mathrm{ba}}(x_{p},y_{p},\omega)$
$\displaystyle=\int_{0}^{L}f_{\mathrm{ba}}(y,\omega)\phi_{n}(y)\differential
y,$ (18) $\displaystyle=\widetilde{aP}(x_{p},y_{p},\omega)\frac{L}{\pi
R^{2}\kappa}g(y_{p},\omega),$ (19)

with

$g(y_{p},\omega)=\int_{0}^{L}\partial_{y}^{2}[EI\beta(y)h(y,\sqrt{i\frac{\omega\rho
c}{\kappa}},y_{p})]\phi_{n}(y)\differential y.$ (20)

In those last equations, we explicitly expressed the dependence of the back-
action force with respect to the position $(x_{p},y_{p})$ of the probe beam.

There are two origins for the variations of $\widetilde{aP}$ that drive the
dynamic temperature field. On the one hand, the probe beam power intrinsically
fluctuates in time, yielding to first order to $\widetilde{P}(\omega)\neq 0$
and $\widetilde{aP}=\overline{a}\widetilde{P}$ (with $\overline{a}$ the static
absorption). On the other hand, the nanowire is laterally deflected with
respect to the probe beam, changing the dynamic absorption $\widetilde{a}$:

$\displaystyle\widetilde{a}(x_{p},y_{p},\omega)$
$\displaystyle=\frac{\differential a}{\differential
x_{p}}(x_{p})\xi(y_{p},\omega)$ (21a) $\displaystyle=\frac{\differential
a}{\differential x_{p}}(x_{p})\phi_{n}(y_{p})\Xi_{n}(\omega).$ (21b)

In the following sections, we independently treat each of these two
mechanisms.

##  V Back-action effects on the oscillator mechanical response

Figure 3: Back-action stiffness spatial profile for different spatial
distributions of $\beta$. The top row presents the real part
$k_{\mathrm{ba}}^{r}$ of the stiffness (modifying the resonance frequency),
the middle one the imaginary part $k_{\mathrm{ba}}^{i}$ (modifying the
damping), and the bottom one the modulus $|k_{\mathrm{ba}}|$. The three
columns correspond to the thermal diffusivity length $|\alpha|L$ reported in
the upper left corner of the plots. The curves are normalized to the maximum
of $|k_{\mathrm{ba}}(y_{p})|$ on all positions $y_{p}$. The dashed curve
corresponds to a uniform $\beta$ (Eq. 25), the plain ones to a single defect
in $y_{d}/L=0.1,0.5$ or $0.9$ (Eq. 26, assuming $\tau_{d}=0$). All curves are
computed for the first oscillation mode of the nanowire, when the position
$y_{p}$ of the probe beam is scanned along the resonator length. The signature
of the temperature mediated back-action is very different for the various
hypotheses probed here, and displays a non-trivial shape, including sign
changes in some cases.

In this section, we focus on the driving effect resulting from the temperature
field induced by the vibration of the nanowire:
$\widetilde{aP}=\widetilde{a}\overline{P}\propto\Xi_{n}(\omega)$. This
vibration creates a thermal wave, which in turn drives the motion of the
nanowire via the above-described thermomechanical coupling. The fields of
deformation and of temperature are thus coupled, and the induced feedback
modifies the effective mechanical response of the system due to the presence
of the measurement probe beam. This coupling can cause self-oscillation or
back-action cooling of the oscillator [10, 8, 13, 30, 9, 29].

Injecting the expression of $\widetilde{a}$ from Eq. 21b into Eq. 19, we can
define a back-action stiffness $k_{\mathrm{ba}}$ with

$\displaystyle F_{\mathrm{ba}}(x_{p},y_{p},\omega)$
$\displaystyle=-k_{\mathrm{ba}}(x_{p},y_{p},\omega)\Xi_{n}(\omega),$ (22)
$\displaystyle k_{\mathrm{ba}}(x_{p},y_{p},\omega)$
$\displaystyle=-\frac{\differential a}{\differential
x_{p}}(x_{p})\overline{P}\frac{L}{\pi
R^{2}\kappa}\phi_{n}(y_{p})g(y_{p},\omega).$ (23)

This back-action mechanical stiffness is a complex quantity
$k_{\mathrm{ba}}=k_{\mathrm{ba}}^{r}+ik_{\mathrm{ba}}^{i}$. To describe the
dynamics of the equivalent resonator, Eq. 16 therefore has to be modified in:

$[k^{r}_{n}+k_{\mathrm{ba}}^{r}+i(k^{i}_{n}+k_{\mathrm{ba}}^{i})-m\omega^{2}]\Xi_{n}(\omega)=F_{\mathrm{ext}}.$
(24)

The real part $k_{\mathrm{ba}}^{r}$ changes the effective stiffness of the
oscillator, thus the resonance frequency, whereas the imaginary part
$k_{\mathrm{ba}}^{i}$ modifies the damping.

A first feature of $k_{\mathrm{ba}}$ in this framework is that it is linear in
the probe power $\overline{P}$. A noticeable second feature is that
$k_{\mathrm{ba}}$ is proportional to $\differential a/\differential x_{p}$,
which is antisymmetric with respect to the lateral position $x_{p}$. The
frequency shift and the damping modification resulting from the back-action
should be opposite on the left and right sides of the nanowire. This
essentially stems from the assumption that the heat-induced mechanical
deformation is purely dependent on the absorbed power. In particular, by
choosing two symmetric points $\pm x_{p}$, the average of the two complex
stiffnesses cancels the back-action effect, such that both the intrinsic
stiffness and the damping of the resonator can be recovered. The two effects
(power dependency and antisymmetry) have been experimentally reported, where
the back-action typically results in amplification or cooling of mechanical
modes depending on the side sensed by the probe, proportionally to the power
implied [9, 11].

To go further, additional hypotheses are needed, regarding the spatial
distribution of the thermo-elastic deformation $EI\beta$. For example, for a
nanowire with a uniform natural curvature $C_{0}$, the linear thermal bending
coefficient is uniform: $\beta=-C_{0}\varepsilon$. The same uniformity would
also apply to a nanowire coated on one side only [13] and thus subject to the
“bimetal effect” (that is, the differential expansion of its layers). In such
cases, we extract $EI\beta$ from the integral in Eq. 20 and get:

$g(y_{p},\omega)=iEI\beta\omega\frac{\rho
c}{\kappa}\int_{0}^{L}h(y,\sqrt{i\frac{\omega\rho
c}{\kappa}},y_{p})\phi_{n}(y)\differential y.$ (25)

The dependency on the longitudinal coordinate $y_{p}$ of the back-action force
can be estimated from this formula since the functions $h$ and $\phi_{n}$ are
known.

Another relevant hypothesis on $\beta$ is that it corresponds to a localized,
point-like defect:
$\beta(y,\omega)=\beta_{d}e^{i\omega\tau_{d}}\delta^{D}(y-y_{d})$, with
$\delta^{D}$ the Dirac distribution centered at the position $y_{d}$ of the
defect, $\beta_{d}$ the amplitude of the punctual thermal bending coefficient,
and $\tau_{d}$ a microscopic response time of the mechanical stress to the
temperature change. After a double integration by parts, Eq. 20 leads to:

$g(y_{p},\omega)=EI\beta_{d}e^{i\omega\tau_{d}}h(y_{d},\sqrt{i\frac{\omega\rho
c}{\kappa}},y_{p})\phi_{n}^{\prime\prime}(y_{d}).$ (26)

Here again, the spatial dependency of the back-action effect on $y_{p}$ can be
predicted and compared to the experimental data. If several defects are
present, all that is required is to sum the functions $g$.

The effect of the spatial distribution of $\beta$ on the spatial evolution of
the back-action stiffness $k_{\mathrm{ba}}$ is illustrated in Fig. 3, using
for the mechanical mode shape $\phi_{1}(y)$, the first normal mode of a
clamped-free nanowire. Other modes could be computed using the very same
recipe. The different hypotheses yield very different signatures both for the
real and imaginary part. Experimental observations on the resonance frequency
shift and the damping variation, both along and across the nanowire, will thus
point towards the best model to describe a specific thermal bending mechanism.
These quantities can be extracted either from measuring the response function
to an external drive (_i.e._ by applying an external force $F_{\mathrm{ext}}$
and measuring the deflection while sweeping the driving frequency), or
directly from a thermal noise measurement. Indeed, providing a high enough
motion readout sensitivity, the spontaneously observed Brownian fluctuations
directly lead to the resonance characteristics, with a predictable,
antisymmetric behavior from one lateral position of the probe beam to the
opposite: By simply changing the sign of $x_{p}$, the effect of the thermal
back-action can be switched from an upward to a downward frequency shift and
from a cooling to an amplification of the noise amplitude. In particular, the
case described in Fig. 3 corresponds to a reduced damping, stemming from the
arbitrary convention $(\differential a/\differential x_{p})_{x_{p}}>0$ which
yields a negative imaginary part $k_{\mathrm{ba}}^{i}$, associated with motion
amplification, and which can even lead to self-oscillations if the negative
back-action damping overcomes the intrinsic dissipation [30, 9, 29]. This
effect can be reversed, by symmetrically moving the measurement probe,
resulting in motion cooling [9, 11]. Note that such antisymmetries essentially
reflect those displayed by the mechanical structure in response to heat, by
nature, or by design. Their presence therefore enables to unequivocally
conclude as to the nature of the observed measurement back-action force,
specifically when the expected fundamental mechanism has distinct symmetries
(e.g., scattering forces [31, 32], gradient forces [33]…). Likewise, our
approach can be straightforwardly extended to other, defect-sensitive
measurement processes [34, 35, 36].

##  VI Quantum back-action from power fluctuations

In this section, we focus on the variations of $\widetilde{aP}$ due to the
power fluctuations: $\widetilde{aP}=\overline{a}\widetilde{P}$. Even for a
stabilized input beam, such fluctuations are unavoidably present due to the
quantum nature of the measurement probe e.g., photon-shot noise, electron-shot
noise, etc. These quantum fluctuations exhibit a white power spectrum:

$S_{P}=\frac{\langle\widetilde{P}\rangle}{\Delta\omega}=2\epsilon\overline{P},$
(27)

with $\Delta\omega$ the spectral bandwidth, $\epsilon$ the quantum of energy
of the particle involved ($\epsilon=h\nu$ for a photon of frequency $\nu$,
with $h$ the Plank constant, $\epsilon=eV$ for an electron of charge $-e$ and
acceleration voltage $V$).

Eq. 19 simply leads to:

$\displaystyle F_{\mathrm{ba}}^{\mathrm{SN}}(x_{p},y_{p},\omega)$
$\displaystyle=\overline{a}(x_{p})\widetilde{P}(\omega)\frac{L}{\pi
R^{2}\kappa}g(y_{p},\omega),$ (28)

where the superscript SN stand for “shot noise” driven. The measured
deflection $\xi(y_{p},\omega)$ is deduced from the transfer function of the
harmonic oscillator, yielding

$\xi^{\mathrm{SN}}(y_{p},\omega)=\frac{F_{\mathrm{ba}}(x_{p},y_{p},\omega)}{k^{r}_{n}+k_{\mathrm{ba}}^{r}-m\omega^{2}+i(k^{i}_{n}+k_{\mathrm{ba}}^{i})}\phi_{n}(y_{p})$
(29)

The power spectrum density of $\xi^{\mathrm{SN}}(y_{p},\omega)$ is strongly
peaked around the resonance angular frequency
$\omega_{n}=\sqrt{(k^{r}_{n}+k_{\mathrm{ba}}^{r})/m}$. We can therefore
neglect the frequency dependence of other terms when computing the mean square
deflection $\langle\xi_{n}^{\mathrm{SN}}(y_{p})^{2}\rangle$ of mode $n$
measured in $y_{p}$ by integrating the power spectrum on all frequencies
around $\omega_{n}$, which leads to:

$\displaystyle\\!\\!\langle\xi_{n}^{\mathrm{SN}}(y_{p})^{2}\rangle$
$\displaystyle=\frac{\pi\omega_{n}\phi_{n}^{2}(y_{p})}{2(k^{r}_{n}+k_{\mathrm{ba}}^{r})(k^{i}_{r}+k_{\mathrm{ba}}^{i})}S_{F_{\mathrm{ba}}}(x_{p},y_{p},\omega_{n})\\!$
(30)
$\displaystyle=\frac{\pi\omega_{n}}{(k^{r}_{n}+k_{\mathrm{ba}}^{r})(k^{i}_{n}+k_{\mathrm{ba}}^{i})}|k_{\mathrm{ba}}|^{2}\frac{\overline{a}^{2}}{a^{\prime}(x_{p})^{2}}\frac{\epsilon}{\overline{P}}.$
(31)

The variance of the quantum back-action-induced motion detected at the probe
position $y_{p}$ is therefore essentially determined by the profile of the
complex stiffness back-action $|k_{\mathrm{ba}}|$. In particular, if the
modification of the mechanical resonance is weak (no sizeable frequency shift
or damping alteration), the shot noise driven deflection noise follows the
square modulus of $k_{\mathrm{ba}}$, presented in Fig. 3. In this case, the
motion standard deviation is proportional to $\overline{P}$ (since
$k_{\mathrm{ba}}\propto\overline{P}$). Conversely, if
$(k^{i}_{n}+k_{\mathrm{ba}}^{i})$ significantly varies, the power scaling of
the measured fluctuations may become challenging to analyze, with a net motion
variance that may either increase or stagnate, depending on the thermal back-
action gradient sign. Thereby, it is legitimate to question whether the
observation of such back-action-driven vibrations enables to conclude as to
the quantum origin of the underlying driving processes, as opposed to
fluctuations resulting from thermal agitation, to which the next section is
dedicated.

##  VII Thermal back-action noise

In this section, we tackle the question of the thermal noise of the resonator,
a nontrivial problem since the hypotheses of the fluctuation-dissipation
theorem are not fulfilled. Indeed, the system is in a non equilibrium steady
state (NESS), as it bears a constant heat flow from the probe beam position
$y_{p}$ to the thermostat, here the clamp assumed to be at a fixed temperature
$T_{0}$. The temperature itself is thus a field, and the fluctuation-
dissipation theorem needs to be extended to this NESS. We follow Refs. 15, 37,
38, 39, which demonstrate that the amplitude of the thermal noise in this
instance is

$\langle\xi_{n}^{\mathrm{TN}}(y_{p})^{2}\rangle=\frac{k_{B}T^{*}}{k^{r}_{n}+k_{\mathrm{ba}}^{r}}\frac{k_{n}^{i}}{k^{i}_{n}+k_{\mathrm{ba}}^{i}}\phi_{n}(y_{p})^{2},$
(32)

where the superscript TN stands for thermal noise driven, and $T^{*}$ is the
equivalent temperature of the oscillator, defined by

$\displaystyle T^{*}$
$\displaystyle=\frac{1}{k_{n}^{i}}\int_{0}^{L}\overline{T}(y)E^{i}I\phi_{n}^{\prime\prime}(y)^{2}\differential
y$ (33)
$\displaystyle=T_{0}+\bar{\theta}\frac{1}{Lk_{n}^{i}}\int_{0}^{L}\min(y,y_{p})E^{i}I\phi_{n}^{\prime\prime}(y)^{2}\differential
y.$ (34)

$T^{*}$ describes the amplitude of the thermal noise driven fluctuations, and
corresponds to the average temperature of the system with a weighing that
matches the normalized local dissipation of mechanical energy
$E^{i}I\phi_{n}^{\prime\prime}(y)^{2}/k_{n}^{i}$. If $\bar{\theta}\ll T_{0}$
(small heating effects), then $T^{*}\sim T_{0}$ and Eq. 32 is very similar to
the equipartition at equilibrium, except that the effective temperature is
$T_{0}k^{i}_{n}/(k^{i}_{n}+k_{\mathrm{ba}}^{i})$. Once again, we see the
effect of the back-action that effectively cools or warms the resonant mode
depending on the sign of $k_{\mathrm{ba}}^{i}$.

Figure 4: Thermal noise driven fluctuations. (a) The NESS contribution to the
thermal noise is summarized in $T^{*}$, the effective temperature that should
be taken into account when extending the fluctuation-dissipation theorem to
this non-equilibrium steady state. The overhead to $T_{0}$ is proportional to
$\bar{\theta}$, and increases with the probe position $y_{p}$. Four hypotheses
on the mechanical dissipation distribution are illustrated here: uniform
damping (purple dashed line, computed for the first normal mode), localized
damping in $y_{d}=0.1,0.5$ or $0.9$. (b) Appart from the amplification/damping
refactor coming from the back-action stiffness, the thermal noise profile when
scanning the probe along the resonator is proportional to
$T^{*}\phi_{n}(y_{p})^{2}$. It is illustrated here for $\bar{\theta}=T_{0}$
and $n=1$, and mainly follows the square of the mode shape since the effective
temperature is a pre-factor bounded between 1 and 2 in this case.

The NESS contribution to $T^{*}$ comes from the non-uniform temperature field,
$\bar{\theta}\min(y,y_{p})/L$, proportional to $\overline{P}$. It depends on
the probe position and power, due to the interplay between the fields of
temperature and dissipation. The latter is expressed here with the imaginary
part of the Young modulus and can have several origins, such as distributed
thermoelastic or anelastic damping, internal friction [5, 15, 39], or be
point-like and localized at the defects’ positions or at the clamp [15, 38]. A
reasonable assumption is that the spatial distributions of this dissipation
and of the linear thermal bending coefficient are correlated: a uniform
$\beta$ corresponds to a uniform $E^{i}$, a peaked $\beta$ in $y_{d}$ leads to
a localized damping $E^{i}\propto E^{i}_{d}\delta^{D}(y-y_{d})$. We report in
Fig. 4 the dependency of $T^{*}$ as a function of $y_{p}$ under those various
hypotheses, and the corresponding thermal noise driven fluctuations profile.
The latter is increasing when the probe gets closer to the free end of the
resonator, and the dissipation distribution only has a mild influence on this
qualitative shape.

At this stage, we can conclude that the shot and thermal noises are
distinguished by their evolution as a function of $y_{p}$, while showing the
same linear power dependence. It is however not possible to draw general
conclusions so as to the relative weights of these two effects, which stems
from the thermal bending parameter $\beta$, which essentially determines the
quantum back-action. This parameter may acquire sizeable values (corresponding
to large bending under small temperature variations), with a correspondingly
large shot noise therefore having to be accounted for, together with the NESS
contribution. In the next section, we further consider the consequences of the
combination of these two noises, in particular in the context of force
sensing.

##  VIII Back-action noises in force sensing

Here we assume that force sensing is performed from a local measurement of the
deformation of the interacting resonator. The estimated force is therefore
deduced using Eq. 24, which we rewrite here as:

$F_{n}=\chi(x_{p},y_{p},\omega_{n})\frac{\xi(y_{p},\omega_{n})}{\phi_{n}(y_{p})},$
(35)

where $F_{n}$ is the projection of the external force field to be measured on
the considered mode $\phi_{n}$, and
$\chi=k^{r}_{n}+k_{\mathrm{ba}}^{r}-m\omega^{2}+i(k^{i}_{n}+k_{\mathrm{ba}}^{i})$
is the harmonic oscillator response function. On top of the thermal
($\xi^{\mathrm{TN}}$) and shot ($\xi^{\mathrm{SN}}$) noises described
previously, we add the detection noise ($\xi^{\mathrm{DN}}$) for which we
assume a white noise spectrum. The total equivalent force noise detected in
presence of the back-action processes is then:

$S_{F}=S_{F}^{\mathrm{TN_{0}}}+S_{F}^{\mathrm{TN^{*}}}+S_{F}^{\mathrm{SN}}+S_{F}^{\mathrm{DN}},$
(36)

with $S_{F}^{\mathrm{TN_{0}}}(\omega)=2k_{n}^{i}k_{B}T_{0}/\omega_{n}$ the
thermal noise associated with the thermal bath (that is, in absence of
measurement-induced heating),
$S_{F}^{\mathrm{TN^{*}}}(x_{p},y_{p},\omega)=2k_{n}^{i}k_{B}(T^{*}-T_{0})/\omega_{n}$
the NESS contribution associated with the heat flow along the NW,
$S_{F}^{\mathrm{SN}}(x_{p},y_{p},\omega)$ the force spectral density
associated with the shot noise back-action, and
$S_{F}^{\mathrm{DN}}(y_{p},\omega))\propto 1/\phi_{n}(y_{p})^{2}$ the
detection noise floor. The latter term diverges on nodes of the mode shape,
and in particular at the origin: though the NESS and shot noise contributions
are simultaneously minimal for $y_{p}=0$, it make no sense measuring there as
the sensitivity also vanishes. $S_{F}$ is plotted in Fig. 5 for equivalent
contributions from all three driving sources (that is,
$\mathrm{max}(S_{F}^{\mathrm{TN^{*}}})=\mathrm{max}(S_{F}^{\mathrm{SN}})=S_{F}^{\mathrm{TN_{0}}}$,
top panel) and for dominating quantum back-action effects
($\mathrm{max}(S_{F}^{\mathrm{TN^{*}}})=3\times S_{F}^{\mathrm{TN_{0}}}$,
$\mathrm{max}(S_{F}^{\mathrm{SN}})=100\times S_{F}^{\mathrm{TN_{0}}}$, bottom
panel). It shows that measuring the mechanical motion at the tip of the NW
yields a close to minimal force noise in presence of defects implanted far
from the tip ($y_{d}/L=0.1$ and $y_{d}/L=0.5$). However and interestingly,
this is no longer true for a defect close to the apex ($y_{d}=0.9$): in this
case, the measurement becomes substantially noisier, due to the contribution
of the quantum back-action. By moving closer to the clamp, one sees that
significant reductions of the force noise may be achieved. This may be
particularly noteworthy in the context of nanomechanical scanning probes [40,
41], which generally functionalize the tip of a free-standing
micro-/nanomechanical resonator, and measure its mechanical response close to
it, therefore with potential important quantum back-action noise influence.

Figure 5: Total force noise in a photothermal back-action-driven mechanical
resonator computed from Eq. 36, expressed in unit of equilibrium thermal noise
$S_{F}^{\mathrm{TN_{0}}}=2k_{n}^{i}k_{B}T_{0}/\omega_{n}$. The plain lines are
computed for $|\alpha|L=10$ and a single defect contribution to $\beta$ and
$E^{i}$, localized in $y_{d}=0.1,0.5$ or $0.9$. The dashed-dotted line stands
for the equilibrium thermal limit $S_{F}^{\mathrm{TN_{0}}}$, and the dotted
line to the detection noise contribution. (a) Case where the three first
contributions are assumed to have the same, maximal magnitude
($\mathrm{max}(S_{F}^{\mathrm{TN^{*}}})=\mathrm{max}(S_{F}^{\mathrm{SN}})=S_{F}^{\mathrm{TN_{0}}}$).
(b) Strongly driven, quantum dominated case: the maximal magnitudes of the
NESS term is $\mathrm{max}(S_{F}^{\mathrm{TN^{*}}})=3\times
S_{F}^{\mathrm{TN_{0}}}$ and the one of the shot noise contribution is
$\mathrm{max}(S_{F}^{\mathrm{SN}})=100\times S_{F}^{\mathrm{TN_{0}}}$.

##  IX Application cases

The experimental relevance of our approach can be confirmed by considering the
case of cylindrical SiC or Si nano-optomechanical scanning probes, which have
recently been praised for their high sensitivity and reliability [25, 34, 42,
43]. Assuming typical length $L=$100\text{\,}\mu\mathrm{m}$$ and diameter
$2R=$100\text{\,}\mathrm{n}\mathrm{m}$$, and with the following approximate
material values $E\simeq$300\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$,
$\rho\simeq$3000\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$
(yielding a mechanical resonance frequency
$\omega_{1}/2\pi\simeq$14\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$),
$c\simeq$750\text{\,}\mathrm{J}\,\mathrm{K}^{-1}\,\mathrm{k}\mathrm{g}^{-1}$$,
$\kappa\simeq$10\text{\,}\mathrm{W}\,\mathrm{K}^{-1}\,\mathrm{m}^{-1}$$ [9,
44], one gets $|\alpha|L\simeq 14$, similar to the intermediate case treated
here (middle column of Fig. 3, Fig. 5). Assuming that the optomechanical
readout is achieved with an optical power $P=$100\text{\,}\mu\mathrm{W}$$, in
presence of an average absorption rate of $\overline{a}=0.1\%$, and that the
absorption gradient is essentially set by the optical waist
$w_{0}=$1\text{\,}\mu\mathrm{m}$$, i.e. $\differential a/\differential
x_{p}\simeq a/w_{0}=${10}^{3}\text{\,}\mathrm{m}^{-1}$$, we find
$\mathrm{max}\left(|k_{\mathrm{ba}}|\right)\simeq$2.5\text{\times}{10}^{-9}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$$
for a point-like, centered defect with $y_{\mathrm{d}}/L=0.5$ and
$\beta_{d}=${10}^{3}\text{\,}\mathrm{K}^{-1}$$. Note that this latter value
means that, when hitting the defect with the laser probe, the nanowire bends
so that its tip laterally moves approximately from
$2\text{\,}\mathrm{n}\mathrm{m}$, that is, only $2\%$ of the NW diameter,
while experimental observations routinely range within several diameters,
suggesting that standard values of $\beta_{d}$ are likely to be orders of
magnitude larger. Gauging upon the efficiency of the back-action stiffness can
be done by comparing it to the dissipation, $|k_{\mathrm{ba}}/k_{n}^{i}|\simeq
Q|k_{\mathrm{ba}}/m_{\mathrm{eff}}\omega_{1}^{2}|\simeq 6$
($m_{\mathrm{eff}}\simeq m/4$ the effective mass, and $Q\simeq${10}^{4}$$ the
quality factor): this means that such measurement would operate with an
absorbed probe power that is more than five times the parametric instability
threshold. The prominence of the dynamical back-action would only be
reinforced at higher Q-factor, an issue that is ubiquitous to the development
of ultra-low dissipation mechanical systems [45, 46].

Another type of systems specifically prone to this phenomenology are ultra-low
diameter, high aspect-ratio nanomechanical systems,including carbon nanotubes
and semiconducting NWs [47, 48, 11, 23]. On top of a generally enhanced
sensitivity to defects, stemming from their extremely reduced size, these
systems generally are amongst those with the lowest thermal noise, which
results in a relative reinforcement of their responsivity towards external
fluctuation processes, including measurement quantum back-action. Since those
effects are parameterized by $\beta_{d}^{2}$, they may acquire very large
spectral densities, to the point that they overwhelmingly dominate other noise
sources. This, together with a full, quantitative verification of the hereby
reported theoretical model, is the object of a recent work from our
collaboration [49], and matches the choice of parameters used in Fig. 5(b).

##  X Conclusions and perspectives

In conclusion, measuring the vibrations of micro- and nanoresonators using a
high-energy probe beam will unavoidably result in heating those tiny devices.
The induced temperature field feeds back to the motion of the resonator,
yielding changes in its vibrational response (frequency, amplitude, and
quality factor). This back-action effect strongly depends on the probe
position, the thermal diffusion length, and the spatial distribution of the
thermomechanical coupling mechanism. Assuming a strong correlation between
this thermomechanical coupling ($\beta$) and the mechanical losses ($E^{i}$),
we are able to quantitatively describe the fluctuational processes associated
with this back-action, including the thermal noise of the system in the
nonequilibrium steady state generated by the heat flow, and the quantum noise
due to the shot noise of the probe. The signatures of all these processes
_when scanning the probe_ along and across the resonator are found to be very
specific, especially when the losses / thermo-mechanical coupling are strongly
localized, which is a commonly expected case. For functionalized sensors for
example, the modification of the resonator by e.g. grafted molecules [18, 19],
strain-coupled single quantum emitters [50, 51, 35], magnetic nanoparticles
[52, 34, 41], purposely alters the mechanical response, with predictable side
effects in terms of backaction and associated noise. For defect-free
resonators, the clamp itself is a singular area where broken symmetries or
peculiarities [36], together with clamping losses [53, 54, 15, 38, 39], are
rather the rule than the exception. The analysis of the real (frequency shift)
and imaginary (damping) part of the response function and of the noise profile
provides a rich tool to characterize the nanoresonator: a single model should
encompass all three observations, making the conclusions very robust. Our
approach is certainly applicable in many instances and should be of general
and wide interest.

The analytical framework presented in this article is exemplified with the
study of a 1D resonator, with clamped-free boundary conditions, all figures
being generated considering the first resonant mode. Exploring the signatures
with other modes simply requires changing the strain profile to the
corresponding eigenfunction $\phi_{n}(y)$. Experimentally, this should prove
to be very informative, as the thermal diffusion length changes significantly
with frequency, while the spatial distribution of the thermomechanical
coupling remains essentially unaltered. In particular, such a multimode
approach may provide additional accuracy in addressing the nature and location
of mechanical losses.

Extending our approach to various boundary conditions is straightforward for
1D oscillators. For example, for double clamp bridges, Eq. 2b changes to
fixing the temperature in $L$: $T(L)=T_{0}$. The static and dynamic thermal
profiles can subsequently be solved using the same methodology as that
outlined in Section III, with the mechanical modes being also analytically
computable. From there, hypotheses on $\beta$ (and $E^{i}$) can be made to
test different scenarios, and compared to experimental data, thus providing
quantitative insights about the dominant loss mechanisms, their spatial
location and possible mitigation strategies. For 2D or 3D resonators, the
methodology remains valid but can turn very complex, as the analytical
description of the temperature and deformation fields would most likely be
limited to specific geometries and boundary conditions. In such cases, turning
to numerical simulations may become unavoidable [14], making it a less
efficient tool to probe a vast range of thermo-mechanical coupling hypotheses
or probe positions.

Another possible extension of this work is towards pump-probe approaches.
Indeed, one key learning of this study is that a wealth of information can be
unraveled by scanning the probe beam along the resonator. In our framework,
this single beam acts both as a probe (measuring the deflection) and as a pump
(creating the temperature field that interacts with the motion). An even
richer behavior could be explored by decoupling the two functions, i.e. using
a high-power pump to create the thermomechanical effects, and a much weaker
one for probing the response at a different location [55, 56], for instance
where the sensitivity is maximized.

Lastly, our analytical derivation shows that, by controlling the distribution
of $\beta$, both the thermomechanical coupling and the back-action dynamics
can actually be tuned with the position of the probe along the resonator. It
should then be possible to engineer the back-action so as to selectively
address the damping/effective temperature with no associated frequency
detuning, or vice versa, to control the frequency with no corresponding
modification of the dissipation. Such a perspective could even be expanded
with a pump-probe configuration, opening the perspective of spatially
resolved, non-equilibrium thermal state engineering.

###### Acknowledgements.

This work has been partially funded by projects ANR-16-CE09-0010,
ANR-22-CE42-0022 and ERC StG 758794. We thank J. Claudon, M. Hocevar, A.
Fontana, and O. Bourgeois for constructive scientific discussions.

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# Dynamic twisting and imaging of moiré crystals

Qixuan Zhang  Program in Material Science and Engineering, University of
California, San Diego, CA 92093, USA  Trevor Senaha  Ruolun Zhang  Chen Wu
Lingyuan Lyu  Leonard W. Cao  Department of Physics, University of California,
San Diego, CA 92093, USA  Jason Tresback  Center for Nanoscale Systems,
Harvard University, Cambridge, MA 02138, USA  Andrew Dai  Department of
Physics, University of California, San Diego, CA 92093, USA  Kenji Watanabe
Takashi Taniguchi  National Institute for Materials Science, Namiki 1-1,
Tsukuba, Ibaraki 305-0044, Japan  Monica T. Allen  Department of Physics,
University of California, San Diego, CA 92093, USA

††preprint: APS/123-QED

The electronic band structure is an intrinsic property of solid-state
materials that is intimately connected to the crystalline arrangement of
atoms. Moiré crystals, which emerge in twisted stacks of atomic layers,
feature a band structure that can be continuously tuned by changing the twist
angle between adjacent layers Li  _et al._ (2010); Bistritzer and MacDonald
(2011); Yankowitz  _et al._ (2012); Carr  _et al._ (2017). This class of
artificial materials blends the discrete nature of the moiré superlattice with
intrinsic symmetries of the constituent materials, providing a versatile
platform for investigation of correlated phenomena whose origins are rooted in
the geometry of the superlattice, from insulating states at “magic angles” Kim
_et al._ (2017); Iglovikov  _et al._ (2014); Cao  _et al._ (2018a, b) to flat
bands in quasicrystals Moon  _et al._ (2019). Here we present a route to
mechanically tune the twist angle of individual atomic layers with a precision
of a fraction of a degree inside a scanning probe microscope, which enables
continuous control of the electronic band structure in situ. Using
nanostructured rotor devices, we achieve the collective rotation of a single
layer of atoms with minimal deformation of the crystalline lattice. In twisted
bilayer graphene, we demonstrate nanoscale control of the moiré superlattice
periodicity via external rotations, as revealed using piezoresponse force
microscopy McGilly  _et al._ (2020). We also extend this methodology to create
twistable boron nitride devices, which could enable dynamic control of the
domain structure of moiré ferroelectrics Zheng  _et al._ (2020); Woods  _et
al._ (2021); Yasuda  _et al._ (2021). This approach provides a route for real-
time manipulation of moiré materials, allowing for systematic exploration of
the phase diagrams at multiple twist angles in a single device.

Figure 1:  Real-time mechanical manipulation of a moiré superlattice, enabling
in situ control over the twist angle. (A) Schematic illustration of a twisted
bilayer graphene rotor device and dynamic manipulation of the twist angle
using a scanning probe tip. The circular opening in the middle of the rotor
frame provides space for high resolution imaging and measurements of the moiré
superlattice. (B) Cross-sectional view of the tip-induced mechanical
manipulation of the rotor frame during a contact-mode AFM line scan. (C)
Schematic of the bilayer graphene moiré superlattice in the rotor device
before (left panel) and after (right panel) the twisting process, revealing
the change in the moiré unit cell size $\lambda_{1}\to\lambda_{2}$ due to the
change in the twist angle $\theta\to\theta+\Delta\theta$.

In conventional solid-state materials, the band structure is a fixed property
determined by the composition and arrangement of atoms in a crystalline
lattice. Many recent breakthroughs have been enabled by synthesis of crystals
with novel band structures, ranging from the development of flat band
materials that host correlated phases to the discovery of quantum-relativistic
crystals whose low energy electronic band structures give rise to charges that
mimic relativistic particlesWehling  _et al._ (2014); Derzhko  _et al._
(2015). A contrasting “bottom-up” approach is the realization of artificial
quantum matter using platforms such as ultracold atoms and trapped ions, which
benefit from exquisite control and tunability Goldman  _et al._ (2016);
Spielman  _et al._ (2010, 2007). However, the weakness of many-body
interactions in these artificial systems has hampered investigation of certain
classes of correlated states, while access to low temperature regimes has
proven challenging. Two-dimensional moiré materials merge the best of both
approaches by pairing strong correlations (and access to non-trivial band
topology) with the tunability afforded by twisted heterostructures, for
example enabling simulation of the Hubbard model in triangular superlattices
Hubbard (1963); Li  _et al._ (2010); Bistritzer and MacDonald (2011);
Yankowitz  _et al._ (2012); Carr  _et al._ (2017); Tang  _et al._ (2020); Mak
and Shan (2022); Pan  _et al._ (2020).

Twisted moiré superlattices – consisting of two atomic crystals rotated by an
angle – exhibit behavior that departs dramatically from that of the
constituent monolayers and form a highly tunable platform for exploration of
emergent phenomena driven by interactions. Such artificially layered systems
can be created in the lab by isolating single atomic layers from a bulk
crystal and stacking them in a precise configuration using micromanipulators,
a technique known as “van der Waals epitaxy” Novoselov  _et al._ (2004); Geim
and Grigorieva (2013); Dean  _et al._ (2010). The moiré interference pattern
features a modulation of the relative alignment between sublattice atoms, thus
yielding spatial variations in the tunneling amplitude and interlayer
potential difference across the superlattice. At small rotations that yield
flat bands, electronic interactions dominate over the kinetic energy, giving
rise to a rich phase diagram that can be tuned with twist angle Bistritzer and
MacDonald (2011); Yankowitz  _et al._ (2019); Kim  _et al._ (2017); Wu  _et
al._ (2019). For example, twisted bilayer graphene exhibits a constellation of
correlated states at a “magic angle” of $1.1\degree$, including
superconductivity, Mott insulators, and a quantum anomalous Hall effect
Bistritzer and MacDonald (2011); Cao  _et al._ (2018a, b). In semiconductor
Group IV transition metal dichalcogenides (TMDs), a richer picture emerges
because broken inversion symmetry and spin-orbit coupling lift the spin
degeneracy in the +/-K valence bands, which can lead to the formation of many-
body topological phases in twisted heterostructures Wu  _et al._ (2019); Pan
_et al._ (2020); Zeng  _et al._ (2023); Cai  _et al._ (2023).

The periodic structure of the superlattice can also be leveraged for a broad
range of practical applications, from the creation of photonic crystals for
nano-light Sunku  _et al._ (2018) to tunable angle-dependent platforms for
electrochemistry Yu  _et al._ (2022). However, existing methods for the
construction of moiré heterostructures lock the atomic arrangement of the
superlattice in place after device fabrication, so investigation of multiple
twist angles requires measurements of ensembles of devices with different
disorder profiles, strain, electrostatic environments. This motivates our
development of a new technology to dynamically rotate individual layers of an
active device structure, which should allow for continuous investigation of
phase diagrams as a function of angle in a highly controlled setting.

In this work, we present a method for real-time rotation of individual atomic
layers with a precision of a fraction of a degree, which should enable
unprecedented control over the topology of the band structure and electronic
interactions in moiré superlattices. We demonstrate manipulation of the
superlattice periodicity with nanoscale resolution in response to this angular
rotation, achieved through direct imaging of the moire superlattice using
piezoresponse force microscopy (PFM) McGilly  _et al._ (2020). Furthermore,
the proposed nanomechanical approach offers a precise method of tuning the
device to targeted angles that may host interesting phenomena, such as “magic
angles” in twisted bilayer graphene or specific angles that host topological
insulator states Bistritzer and MacDonald (2011); Pan  _et al._ (2020); Cao
_et al._ (2018b, a); Wu  _et al._ (2019); Zeng  _et al._ (2023); Cai  _et al._
(2023).

Figure 2:  Dynamic rotation and translation of twisted bilayer graphene driven
by an atomic force microscope. (A) and (B) Tapping-mode AFM image of rotor
device I before and after rotation, respectively. The rotor frame is
lithographically patterned on a twisted bilayer graphene heterostructure,
allowing the upper monolayer to rotate freely with respect to the lower layer.
Because the rotor is several hundred times thicker than a graphene monolayer,
the large height contrast makes the graphene flake almost invisible. (C) A
superimposed image of panels (A) and (B) reveals the clockwise twisting of the
rotor induced by in situ manipulation. The red arrow indicates the direction
of the horizontal force applied by the AFM probe. (D) and (E) Tapping-mode AFM
image of rotor device I before and after translation, respectively. (F) A
superimposed image of panels (D) and (E) reveals the lateral translation of
the upper graphene monolayer with respect to the lower layer.  Figure 3:
Nanoscale tunability of the moiré superlattice geometry, achieved via in situ
rotation of twisted bilayer graphene. (A) Image of the moiré superlattice that
forms at a twist angle of $\theta=0.06\degree$, obtained using piezoresponse
force microscopy (PFM). Lower panel: Cross-sectional line cut that shows
modulations in the PFM signal associated with the superlattice. (B) After a
tip-driven rotation of this twisted bilayer graphene device, which introduces
a twist angle change of $\Delta\theta=0.01\degree$, a modified moiré
superlattice with a smaller period emerges at the new twist angle of
$\theta+\Delta\theta=0.07\degree$. Both the in situ rotation and PFM imaging
are performed within the same microscope. (C) and (D) PFM images of the moiré
superlattice in the same rotor device after additional rotations, revealing
the real-space evolution of the superlattice geometry that is induced by each
successive rotation of the twist angle. (C) and (D) show the moiré patterns at
$\theta=0.37\degree$ and $\theta=1.2\degree$, respectively. The insets are
higher resolution
$$150\text{\,}\mathrm{n}\mathrm{m}$\times$150\text{\,}\mathrm{n}\mathrm{m}$$
PFM scans.  Figure 4:  Tuning the ferroelectric domain structure of twisted
double-layer hBN. (A-C) A sequence of AFM images of an hBN rotor device before
and after two successive rotations driven by the scanning probe tip. The white
and red dashed lines mark the initial and final orientations of rotor,
respectively. The green arrows in (B) and (C) indicate the push point location
and direction of the applied in-plane force, while the red squares mark the
locations of the PFM imaging windows. (D-F) Sequence of PFM images showing the
evolution of the moiré domain structure in the twisted double-layer hBN
induced by the mechanical rotation process. The imaging and in situ
manipulation steps were performed within the same microscope.

A schematic illustration of the rotor geometry and manipulation methodology is
presented in Figure 1. We employ mechanical exfoliation and standard dry
transfer methods to assemble twisted bilayer graphene and twisted double-layer
hexagonal boron nitride (hBN) heterostructures, which are deposited onto a
thicker hBN base layer Dean  _et al._ (2010). As shown in Fig. 1 A, a
nanostructured metal rotor is patterned on the top monolayer of the twisted
heterostructure, allowing it to rotate freely with respect to the bottom
flake. The rotor frame provides structural support that enables rigid rotation
of the entire top atomic layer, minimizing the likelihood of stretching or
tearing during the mechanical manipulation process. The rotors are patterned
with electron beam (e-beam) lithography, followed by e-beam evaporation of 15
nm Cr / 100 nm Au (Fig. S1 in the Supplementary Information). The rotor is
designed to cover most of the exposed area of the top monolayer, which allows
the entire flake to move freely without being pinned onto the lower layer.

A Park NX20 scanning probe microscope with combined atomic force microscopy
(AFM) and piezoresponse force microscopy (PFM) capabilities is used to both
image and manipulate the twisted heterostructures in situ. As demonstrated in
prior work, an AFM cantilever probe can be used to controllably apply lateral
forces and move nanostructures (including multilayer hBN and graphite flakes)
on a flat surface Koren  _et al._ (2016); Chari  _et al._ (2016); Ribeiro-
Palau  _et al._ (2018); Hu  _et al._ (2022); Kapfer  _et al._ (2022); Barabas
_et al._ (2023); Yang  _et al._ (2020); Inbar  _et al._ (2023). Our in situ
rotation process for twisted bilayer graphene is conceptually illustrated in
Fig.1 B. When the moving AFM probe comes into contact with the metal rotor, it
will exert a horizontal force on the structure. The adhesive force between the
gold rotor frame and the top flake is considerably larger than the inter-layer
friction and van der Waals forces between the graphene layers; therefore, the
rotor-supported top flake will rigidly rotate on the surface of the bottom
flake once the force exerted by the AFM probe exceeds the static friction
limit between the two layers. Thus, the rotation process can be precisely
controlled by programming the trace of the contact-mode AFM line scans. The
rotation of the top layer with respect to the lower layer directly changes the
moiré periodicity, as illustrated in Fig. 1 C.

This in situ manipulation procedure allows for highly controlled translational
and rotational inter-layer movement in twisted heterostructures, as shown in
Fig. 2. The first step involves acquisition of a tapping-mode AFM image
encompassing the entire metal frame and the surrounding area, which serves as
a reference for later manipulation steps (Fig. 2 A, D). To obtain precise
information on the initial orientation of the metal frame, high resolution AFM
scans of the targeted pushing point on the lever arm are also obtained (Fig.
S2 in the Supplementary Information). The manipulation process is performed
with the AFM tip in contact mode, and the scanning direction of the tip should
be oriented perpendicular to the lever arm of the rotor. The white arrow in
Fig. 2A indicates the point of contact between the scanning tip and the rotor,
and the resulting direction of the induced rotation is shown in Fig. 2B. When
the AFM tip is scanned from the substrate towards the lever arm, the tip will
push on the side wall of the rotor and a large longitudinal force exceeding
$500\text{\,}\mathrm{n}\mathrm{N}$ is applied to the metal frame. (The forces
required to achieve in situ twisting of double-layer graphene typically fall
within the range of $600\text{\,}\mathrm{n}\mathrm{N}$ to
$1400\text{\,}\mathrm{n}\mathrm{N}$.) Note that the contact-mode scan window
should be defined to have a high aspect ratio or even be set to a single line
scan, which pinpoints the specific location of the point of contact between
the tip and rotor. The rotation angle can be tuned by changing the distance
pushed by the AFM tip and the location of the point of contact on the rotor.
Once the manipulation step is completed, another tapping-mode AFM image of the
entire rotor frame and the surrounding area is obtained to quantify the
induced rotation (Fig. 2B). To track the change in orientation of the rotor,
Fig. 2 C plots the “before” and “after” images superimposed on top of each
other. By comparing the two images, it is possible to resolve the twisting of
the heterostructure (see Fig. S3 in the Supplementary Information). To more
accurately identify the twist angle of the bilayer before and after the
manipulation, PFM can be used to image the moiré pattern and extract its
periodicity.

Fig. 2 D-F shows the second type of manipulation: lateral translation of the
rotor. While tip-controlled manipulation can be used to induce both rotation
and translation, their relative proportions can be controlled by adjusting the
position of the push point. Nearly pure translation could be achieved by
aligning the tip contact point with the center of mass of the metal rotor
structure. The sliding resistance between the two graphene sheets arises
mainly from interlayer friction and resistive forces caused by uneven
topography and organic residues from nanofabrication. We find that contact-
mode cleaning, a standard technique for improving the surface cleanliness of
van der Waals heterostructuresGoossens  _et al._ (2012); Lindvall  _et al._
(2012), effectively reduces the sliding resistance between two layers by
removing organic residues from the e-beam lithography process. Additionally,
heating the sample stage decreases the interlayer friction, which facilitates
the nanomechanical manipulation process.

The main goal of this manipulation technique is to exert precise, controllable
changes to the structure and unit cell size of a moiré crystal. In Fig. 3, we
demonstrate dynamic rotation of a single monolayer within a twisted bilayer
graphene heterostructure and observe a series of controlled changes in the
superlattice period. During the rotation process, the device was tuned to four
distinct twist angles: 0.06°, 0.07°, 0.37°, and 1.20°. The sequence of PFM
images in Fig.3A-D show the evolution of the moiré pattern with twist angle;
all scans were captured from the same location on the heterostructure. These
images reveal nearly isotropic changes in the period of the superlattice for
each successive twist angle change $\Delta\theta$. The twist angle $\theta$
for each configuration can be extracted experimentally from the measured
superlattice period, as discussed in the Supplementary Information. From the
PFM data, we observe no obvious stretching or deformation introduced to the
superlattice during the rotation process. This is one difference between the
proposed approach and other methods for mechanical manipulation of van der
Waals heterostructures, which can simultaneously introduce strain in the
process. To show reproducibility, we also demonstrate the twisting-induced
manipulation of the moiré superlattice in a separate twisted bilayer graphene
device, which exhibits similar behavior (see Fig.S6 in the Supplementary
Information).

To achieve the sequence of rotations featured in Fig.3A-D, the sample stage
was heated up to $190$ to lower the interlayer friction between the graphene
flakes, and the longitudinal forces used in the AFM manipulation process
ranged from $1000\text{\,}\mathrm{n}\mathrm{N}$ to
$1500\text{\,}\mathrm{n}\mathrm{N}$. The smallest twist angle change that we
could demonstrate experimentally is roughly $\Delta\theta=0.01\degree$, as
illustrated in Fig. 3A-B, which was achieved by inducing a
$30\text{\,}\mathrm{n}\mathrm{m}$ displacement of the rotor arm. The
differences between the expected rotation angle ($\Delta\alpha=0.012\degree$,
estimated based on the lateral displacement of the lever arm) and the actual
twist angle change ($\Delta\theta=0.01\degree$, extracted experimentally from
the measured superlattice period) can be attributed to the natural scale
errors in the AFM scanning program and the lateral deflection of the tip
during the pushing process, which adds a small offset to the tip travel
distance (see Supplementary Information for details). AFM images of the rotor
device used for these measurements are provided in Supplementary Figs. S4-S5.

In order to demonstrate the broader applicability of this method for other
classes of 2D materials, we extend this technique to twisted double-layer hBN,
which for example could enable in situ tuning of the ferroelectric domain
structure by changing the twist angle Zheng  _et al._ (2020); Woods  _et al._
(2021); Yasuda  _et al._ (2021). The fabrication procedure and heterostructure
geometry are conceptually similar that illustrated schematically in Fig. 1A
(see Fig.4 and Fig. S1 in the Supplementary Information). Fig. 4A-C show
tapping-mode AFM images of the rotor configurations prior to manipulation,
after the first rotation, and after the second rotation, respectively. By
manipulating the rotor, we are able to induce corresponding changes in the
moiré domain structure, as revealed in the PFM images displayed in Fig. 4 D-F.
The longitudinal forces required to manipulate hBN rotors, which fall in the
range of $600\text{\,}\mathrm{n}\mathrm{N}$ to
$1200\text{\,}\mathrm{n}\mathrm{N}$, are typically smaller than those required
to rotate twisted bilayer graphene.

Conclusion and Outlook. In contrast to many existing state-of-the-art twisted
bilayer heterostructures, in which the atoms are locked into a specific
configuration after fabrication, the devices presented here offer a new route
to dynamically tune and image moiré superlattices. This is achieved via
controlled rotation of a single atomic layer with respect to another, which
can be performed in situ within a microscope. The high level of angular
control provides access to multiple twist angles and superlattice
configurations, which enables systematic investigation of a wide variety of
phenomena within a single device. This method can also be extended to
monolayer van der Waals materials beyond graphene, opening the door to dynamic
control over the ferroelectric domain structure in angle-aligned hBN and TMDs
Wang  _et al._ (2022), or manipulation of the periodicity of topological spin
textures in moiré magnetsXie  _et al._ (2023).

The twisted heterostructure remains exposed throughout the entire rotation
process to allow for easy interrogation with optical or scanning probes, which
could be useful for a broad range of experiments across many scientific
disciplines. The in situ twisting technique presented here does not require
any specialized scanning probe microscopy instrumentation to perform; it can
be implemented using a commercial AFM and nanofabrication equipment available
in a standard cleanroom. We hope that this accessibility will allow for broad
adoption of this methodology across the community.

##  Acknowledgements

We gratefully acknowledge funding support from the UC Office of the President,
specifically the UC Laboratory Fees Research Program (award LFR-20-653926) and
the AFOSR Young Investigator Program (award FA9550-20-1-0035). This work was
performed, in part, at the San Diego Nanotechnology Infrastructure (SDNI) of
UCSD, a member of the National Nanotechnology Coordinated Infrastructure,
which is supported by the National Science Foundation (Grant ECCS-2025752).

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Supplementary Information

Figure S1:  Optical microscope images of the twisted bilayer graphene rotor
featured in Fig.S6 (left panel) and the double-layer hBN rotor featured in
Fig.4 (right panel). The nanopatterned rotor devices are defined using
electron beam lithography, followed by deposition of 15nm Cr / 100nm Au using
electron beam evaporation. The hBN and graphene flakes used in these devices
are mechanically exfoliated from bulk crystals, and heterostructures are
assembled using standard dry flake transfer methods.  Figure S2:  Example
tapping-mode AFM scans of the rotor edge before and after the tip-driven
manipulation step. Before and after the pushing process, pairs of 1D line
scans obtained in the same position are used to confirm the displacement of
the rotor.  Figure S3:  Superimposed AFM images of the rotor device from Fig.
2 before and after rotation. In the two smaller panels on the right, the
brightest feature on the right side is the edge of the rotor frame. Figure S4:
A tapping-mode AFM image of the rotor device featured in Fig. 3 in the main
text. The green line marks the perimeter of the top graphene monolayer
attached to the rotor frame.  Figure S5:  Superimposed AFM images of the rotor
device featured in Fig. 3 prior to manipulation and after the final rotation.
The red and blue dotted lines mark the orientations of the same edge before
and after the rotation, respectively. Figure S6:  Nanoscale control over the
period of the moiré superlattice by tip-driven rotations. (A, C) PFM images of
a twisted bilayer graphene rotor device before (A) and after (C) rotation,
revealing the change of moiré superlattice period. The twist angle change is
$\Delta\theta=1\degree$. (B, D) 1D line cuts extracted from the PFM images in
panels (A) and (C), respectively, showing the change in periodicity of the
superlattice induced by the rotation. (E) AFM image of the rotor device,
illustrating the tip-induced manipulation responsible for the rotation of the
twist angle. The orange arrow represents the location of the AFM tip push,
which introduced a displacement of 100 nm. (F, H) PFM images of a twisted
bilayer graphene rotor device before (F) and after (H) a separate rotation
process, showing the change in the moiré period that is caused by a twist
angle change of $\Delta\theta=-0.24\degree$. (G, I) 1D line cuts extracted
from the PFM images in panels (F) and (H), respectively, showing the change in
periodicity of the superlattice induced by the rotation. (H) The AFM image of
the rotor device, illustrating the manipulation responsible for the change in
twist angle. The orange arrow represents the location of the AFM tip push,
which introduced a displacement of 50 nm.

The estimated rotation angle ($\Delta\alpha$) extracted from the tapping-mode
AFM images is calculated based on the push distance of the rotor and rotation
radius. The pushing distance can be extracted from the 1D line scan curve
before and after the push (Fig. S2), while the rotation radius is determined
from superimposed tapping-mode AFM images (Fig. 2; Fig. S3). The push distance
can be finely controlled by pausing the tip in the middle of the contact mode
line scan. The rotation radius is determined by drawing extension lines along
one straight edge of the rotor before and after the rotation; the cross point
can be considered as the rotation center.

The actual twist angles extracted from the measured moiré patterns are
calculated using the following equation:

$\theta=2\arcsin\frac{a}{2\lambda}$

where $\lambda$ is the moiré periodicity, $a$ is the lattice constant, and
$\theta$ is the twist angle. For a graphene lattice,
$a=$0.246\text{\,}\mathrm{n}\mathrm{m}$$. The moiré periodicity $\lambda$ can
be determined experimentally from the piezoresponse force microscopy images
(Fig. 3; Fig. S6).