††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Scalable nanomechanical logic gate

Erick Romero  Nicolas P. Mauranyapin  Timothy M. F. Hirsch  Rachpon Kalra
Christopher G. Baker  Glen I. Harris  Warwick P. Bowen<EMAIL_ADDRESS>The
Australian Research Council Centre of Excellence for Engineered Quantum
Systems, School of Mathematics and Physics, University of Queensland,
St.Lucia, Queensland 4072, Australia

###### Abstract

Nanomechanical computers promise robust, low energy information processing.
However, to date, electronics have generally been required to interconnect
gates, while no scalable, purely nanomechanical approach to computing has been
achieved. Here, we demonstrate a nanomechanical logic gate in a scalable
architecture. Our gate uses the bistability of a nonlinear mechanical
resonator to define logical states. These states are efficiently coupled into
and out of the gate via nanomechanical waveguides, which provide the
mechanical equivalent of electrical wires. Crucially, the input and output
states share the same spatiotemporal characteristics, so that the output of
one gate can serve as the input for the next. Our architecture is CMOS
compatible, while realistic miniaturisation could allow both gigahertz
frequencies and an energy cost that approaches the fundamental Landauer limit.
Together this presents a pathway towards large-scale nanomechanical computers,
as well as neuromorphic networks able to simulate computationally hard
problems and interacting many-body systems.

##  I Introduction

The miniaturisation of semiconductor electronics has fueled remarkable
progress in computer performance over more than six decades. However, this
progress has now markedly slowed [1, 2].

Thermal fluctuations of the energies of electrons and holes are a key barrier.
Known as Boltzmann’s tyranny [3], they enforce a minimum supply voltage for
error-free computation which constrains the energy cost of logic operations
above around 30 aJ [3, 4, 5]. This is a factor of $10^{4}$ higher than the
fundamental Landauer limit, which arises from the creation of entropy when
information is erased in irreversible computing [6, 2]. Semiconductor
electronics are also susceptible to degradation in harsh conditions, such as
high temperature and radiation environments [7, 8]. This imposes challenges
for applications in space, medical and nuclear facilities, and in areas such
as geothermal exploration and advanced propulsion systems [8].

In nanomechanical computing, information is encoded in mechanical motion
rather than electrical charge [9]. This evades Boltzmann’s tyranny, in
principle allowing the Landauer limit to be reached [10], and affords
intrinsic robustness to radiation and temperature [11, 12]. Nanomechanical
computers also offer new computing modalities, such as adaptive information
processing where the computer directly interacts with its environment via
forces that it exerts or that are exerted upon it [13]. These qualities,
together with rapid advances in nanofabrication, have provided the impetus for
much recent progress, including nanomechanical computers built from
metamaterials [14, 12] and DNA-origami [15, 16], emulation of emergent
phenomena such as ferromagnetism [17] and symmetry breaking [18], and
reversible mechanical computing [19]. Nanomechanical elements are now even
being applied as memories and interfaces for quantum computers [20, 21, 22].
However, scaling nanomechanical gates into complex circuits remains an
outstanding challenge.

Unlike nanoelectromechanical computing, where information is stored in a
combination of mechanical motion and electrical charge [23, 24], scalable
purely nanomechanical gates have yet to be developed. Purely mechanical gates
generally rely on parametric interactions between mechanical waves [25, 26,
27], but these shift the frequencies of the bits so that the output of one
gate cannot easily be used as the input of the next. Alternatively, direct
physical contact has been used to interconnect gates [16] but, lacking the
equivalent of nanomechanical wires, this is constrained to simple circuit
topographies.

Here, we report a highly efficient nanomechanical logic gate that is built
into a scalable architecture. Our gate has a mechanical quality factor two
orders of magnitude higher than previous nanomechanical gates [19, 26, 28,
10]. It is connected to nanomechanical waveguides which act as wires for the
input and output bits. We show that input bits can be coupled into the gate
with 99% efficiency, and can drive transitions between two bistable states
[29, 30]. This enables a new approach to nanomechanical computing [31], for
which we demonstrate a universal set of purely mechanical logic operations.
Importantly – and unlike previous approaches [25, 26, 27] – the output bits
have the same frequency as the inputs and are therefore compatible with
downstream gates. Our approach is CMOS compatible, facilitating both
miniaturisation and scaling to complex architectures, and our modelling
predicts that the Landauer limit is well within reach. This provides a pathway
towards highly efficient, robust computer processing.

##  II Architecture

Our nanomechanical computing architecture is constructed on a silicon chip by
combining single-mode acoustic waveguides [30] with evanescent tunnel barriers
(Fig. 1a). The tunnel barriers are created by locally narrowing the width of
the waveguide until it no longer supports any acoustic modes. They then act as
acoustic mirrors with reflectivity that can be finely tuned by controlling
their length in fabrication [32]. Here, we use pairs of closely-separated
tunnel barriers to create a new kind of acoustic resonator reminiscent of an
optical Fabry-Pérot cavity. These resonators form the nanomechanical logic
gates in our computing architecture, with the single-mode acoustic waveguides
used to input and output acoustic logical bits in the form of acoustic pulses.

Our acoustic resonators exhibit a strong hardening Duffing nonlinearity which
introduces a bi-stability in the oscillation amplitude [33]. This causes an
abrupt transition from low to high amplitude when the input acoustic wave
reaches a critical amplitude (Fig. 1 b). It has been suggested recently that
this transition can be exploited for nanomechanical logic [31], with the major
advantage that the input and output bits have near identical spatiotemporal
properties so that the output of one gate can be used as the input for the
next. We follow this approach here, defining the low and high amplitude
states, respectively, as the logical ‘0’ and ‘1’ states.

Figure 1: Concept for scalable nanomechanical logic. a, The nanomechanical
gate is based on a nonlinear mechanical resonator that is created by placing
two tunnel-barrier acoustic mirrors within a single mode acoustic waveguide.
The gate can be precisely coupled to input and output waveguides by
controlling the length of the tunnel-barriers. b, Gate operations are
implemented using the bistability of the resonator, which results in an abrupt
jump from a ‘0’ to a ‘1’ state at a critical input amplitude $x_{\rm crit}$.

##  III Experiment

###  III.1 Fabrication

The physical platform for our nanomechanical logical architecture uses a CMOS-
compatible mesh-phononic approach that we have developed previously [32]. As
described in the Methods, here we fabricate an array of $80$ nm thick, 80
$\mu$m-square mechanical resonators from a stressed silicon nitride membrane
[32]. Each resonator is connected through evanescent tunnel barriers to single
mode input and output waveguides. Figure 2, for example, shows a chip
containing eight waveguide-coupled resonators each with different tunnel-
barrier lengths. The factor-of-a-thousand aspect ratio between the size of a
resonator and the thickness of the film results in a strong geometry-induced
Duffing nonlinearity, reducing the energy required for logic operations (see
Supplementary Information). Gold electrodes are patterned on the input
waveguide. We use these to generate the acoustic inputs to each logic gate.

Figure 2: Fabricated nanomechanical computing architecture. (Top) Schematic of
the nanomechanical logic architecture, showing the on chip electrodes,
acoustic interference, and nonlinear mechanical resonator. (Bottom) Microscope
images of fabricated components. Left: Optical microscope image of device
chip, showing eight logic devices. Middle and right: Scanning electron
micrographs of the on-chip electrodes (false color) and nanomechanical
resonator. To minimize acoustic impedance mismatch, and therefore reflections,
gold strips are deposited both between and on either side of the electrodes
with the length of the strips on the sides gradually tapered to zero to create
an adiabatic transition to the bare acoustic waveguide (see Methods). Inset:
magnified image of the silicon nitride meshed structure.

###  III.2 Experimental Setup

The input acoustic logical bits to our nanomechanical logic gate, which we
refer to as $A$ and $B$, are introduced by electrostatically actuating two of
the electrodes situated on the input acoustic waveguide. The third electrode
is used to provide a pump acoustic wave $P$ that drives the nanomechanical
gate close to the nonlinear Duffing transition (see Fig. 3). This provides
gain for logic operations, amplifying the amplitude of the output bit relative
to the inputs and therefore offsetting any attenuation that might occur in
propagation between logic gates. A combination of a DC voltage and a voltage
oscillating at frequency $\Omega$, close to the natural resonance frequency of
the mechanical resonator $\Omega_{\textnormal{m}}$, is applied between each
electrode and a ground-plane under the chip (see Fig. 3). This results in
three spatially separated forces on the waveguide $F_{j}=a_{j}\cos(\Omega
t+\phi_{j})$ [30, 32], where $j\in\\{A,B,P\\}$ label the electrodes that each
force is applied to, and $a_{j}$ and $\phi_{j}$ are the amplitude and phase of
the force. These forces generate out-of-plane acoustic waves that propagate
along the input waveguide and interfere with each other to produce the
combined acoustic input to the nanomechanical logic gate.

The electrodes and ground-plane are separated by the roughly half-millimetre
thickness of the silicon chip. Since capacitive forces are roughly inversely
proportional to the square of separation [34], this configuration would
usually require kilovolts of potential difference to deflect the waveguide
enough to access nonlinear oscillation amplitudes (see Supplemental
Information). However, by choosing a boron-doped silicon substrate, which
exhibits extremely high permittivity [35, 36], we are able to effectively
displace the ground-plane potential to the top of the silicon substrate,
around one micron away from the on-chip gold electrodes. This increases the
capacitive forces by more than five orders of magnitude, reducing the required
potential difference to as little as tens of volts, as shown in the next
section.

Figure 3: Experiment setup for acoustically driven nanomechanical logic.
Devices are actuated with electrostatics capacitive forces between three
individual on chip electrodes A, B,P and a bottom electrode situated below the
chip. Motion of the silicon nitride membrane is detected with a custom-built
vibrometer combining a lensed fibre with heterodyne detection and recorded on
a spectrum analyser. AOM: acousto-optic modulator.

##  IV Results

###  IV.1 Nanomechanical resonator characterization

To detect the motion of the mechanical resonator, we use a custom-built laser
vibrometer based on a combination of lensed fibre and heterodyne measurement
[30, 32], as described in the Methods. As a first test, we observe the thermal
motion of seven nanomechanical resonators, each with different tunnel barrier
lengths and therefore different quality factors. The input and output tunnel
barriers lengths are chosen to be equal. In the absence of intrinsic
dissipation, this forms an impedance matched acoustic cavity such that a
resonant input acoustic wave is fully transmitted through the resonator and
into the output waveguide.

An example measurement of the power spectral density of a nanomechanical
resonator with 105 $\mu$m barrier length is shown in Fig. 4a. The mechanical
resonance frequency of $\Omega_{\textnormal{m}}/2\pi=4.13$ MHz agrees well
with our predictions based on finite-element modelling (see Supplementary
Information.).

The ability to engineer the waveguide-resonator coupling enables wide
tunability of both the resonator quality factor and transmission. Figure 4b
shows the loaded mechanical quality factors extracted from power spectra for
each of the seven measured resonators. For barrier lengths above $150~{}\mu$m,
the quality factor saturates at the intrinsic quality factor of the resonator,
which we find to be $Q_{\rm int}=275,000$, around two orders of magnitude
higher than has previously been reported for a nanomechanical gate [19, 26,
28, 10]. At barrier lengths below $50~{}\mu$m the quality factor instead
approaches $Q_{\rm WG}\sim 3,000$, bounded above zero due to reflections at
the ends of the waveguides. Apart from one resonator, for which a dust
particle was observed on the meshed-membrane (barrier length of 78 $\mu$m), we
find that the transition between the two regimes agrees well with modelling.

A key development in our work is the ability to drive a nanomechanical
resonator into the bi-stable regime through an acoustic waveguide. To explore
this we drive the pump electrode $P$ on the input waveguide to introduce an
acoustic wave. Sweeping the acoustic frequency through the nanomechanical
resonance frequency from below, we monitor the resonator amplitude in response
to the acoustic wave and observe the skewed Lorentzian response characteristic
of a hardening Duffing resonator [37]. Figure 4c shows the response of a
resonator with a measured loaded quality factor of $Q=28,000$ ($105~{}\mu$m
tunnel-barrier lengths) for a range of drive forces. We find that the bi-
stable regime, where the resonance curve is multivalued, is reached for a
combination of DC and root-mean-square AC drive voltages as low as $10$ V and
$0.5$ V, respectively, corresponding to a drive force on the resonator of 1.7
$\mu$N. The inset of Fig. 4c shows the normalised resonator amplitude as a
function of drive strength at a fixed resonator-drive detuning of 660 Hz. At
this detuning, we see a clear abrupt transition from low to high amplitude at
a drive strength of 1.7 $\mu$N. In our computing scheme, these low and high
amplitude states correspond to the logical ‘0’ and ‘1’ states.

Finite-element simulations provide a nonlinear Duffing coefficient of
$\alpha=2.6\times 10^{13}~{}\textnormal{N/m}^{3}$ for the resonator (see
Supplementary Information.). From this we calculate a critical energy to reach
the bistable regime required for computing of
$E_{\mathrm{crit}}=k^{2}/(3\alpha Q)=28$ fJ (see Supplementary Information),
where $k=253$ N/m is the linear spring constant of the resonator. This roughly
defines the minimum energy cost of a gate operation with our architecture, and
is comparable to the energy cost of a gate in a 1995 Pentium Pro semiconductor
microprocessor [38]. In practice, the energy cost depends on whether the
output is a ‘0’ or a ‘1’, since the lower amplitude of a ‘0’ means the gate
must reject more of the input energy in this case. It also depends on the
duration of the logical pulses and amplitudes of the logical bits, among other
factors. We model the energy cost for both ‘0’ and ‘1’ outputs in the
Supplementary Information using our experimental parameters (below). The
results are shown as a function of loaded quality factor in Fig. 4d, along
with their average. The minimum average energy cost of 140 fJ is predicted to
occur for a loaded quality factor of $Q\sim 0.41\,Q_{\rm int}$. For technical
reasons we choose to work with a device for which $Q\sim 0.1\,Q_{\rm int}$
($Q=28,000$). This results in a somewhat higher average energy cost of 270 fJ
(dashed red line in Fig. 4d), but improved impedance matching – on-resonant
input acoustic waves are transmitted into the gate with 99% efficiency and
into the output waveguide with 81% efficiency.

Figure 4:  Purely acoustic driving of nanomechanical logic gates. a,
Experimental measurements of the thermal noise power spectral density of a
resonator with $Q\sim 28000$. b, Loaded quality factor as a function of the
length of the acoustic mirrors. c, Normalized response of the nanomechanical
meshed resonator to a swept-frequency acoustic tone. As the driving force is
increased the response becomes increasing skewed, characteristic of a Duffing
nonlinearity. The inset shows the acoustic input amplitude for a 660 Hz
detuning of the acoustic tone from the resonance frequency, illustrating the
abrupt transition used for our logic operations. d, Predicted energy cost of
logical operations as a function of loaded quality factor. Dashed red line:
predicted energy cost for our loaded quality factor of $Q=28000$.

###  IV.2 Nanomechanical logic

To demonstrate nanomechanical logic, we drive the $A$, $B$ and pump $P$
electrodes on the input waveguide with logic pulses. This creates spatially
separated acoustic pulses that are synchronised to interfere with each other
as they propagate, prior to the nanomechanical resonator (see Fig. 2). The
pulses have a rectangular envelope, and the same carrier frequency $\Omega$.
Careful choice of their phases and amplitudes enables a full set of Boolean
logic gates to be implemented within the nanomechanical resonator. As an
example, in Fig. 5a we demonstrate a universal NAND gate using a
nanomechanical resonator with $Q\sim 28,000$. Here, the pump acoustic wave $P$
is chosen to have an amplitude that is high enough to drive the nanomechanical
logic gate beyond the critical amplitude
$x_{\mathrm{crit}}=\sqrt{2\,k/(3\,\alpha\,Q)}=15~{}$nm (See Supplementary
information), and into the ‘1’ state with an oscillation amplitude of $\sim$18
nm. The $A$ and $B$ logical inputs are arranged to be out-of-phase with the
pump, destructively interfering with it. Their amplitudes are chosen so that
the nanomechanical logic gate transitions to the ‘0’ state (with an
oscillation amplitude below 2 nm) only if both inputs are ‘1’s.

In Fig. 5a, we drive the gate with a sequence of all possible input
combinations (‘00’, ‘01’, ‘10’ & ‘11’), which allows us to record the truth
table of the NAND gate in a single measurement of the peak of the
nanomechanical power spectral density over time. As can be seen, the NAND gate
operates as expected: the ‘00’, ‘01’ and ‘10’ acoustic inputs all result in a
‘1’ output state while a ‘11’ input results in a ‘0’ output state. Similar
performance was achieved for XOR, AND and NOR gates. From the measured
amplitude of motion of the resonator, its quality factor and the resonator-
drive detuning, our model gives an average gate energy cost of 270 fJ as
described above. We have also shown faster gate operations, reducing the
operation time from the 120 ms shown in Fig. 5a to as short as 10 ms, slightly
longer than the decay rate of the nanomechanical resonator. This reduced the
average gate energy cost to 23 fJ.

###  IV.3 Error statistics and fatigue

To test for mechanical fatigue, we pumped a logic gate continuously for two
months, corresponding to more than $10^{13}$ cycles, and separately performed
around $10^{6}$ logic operations. We observed no statistically significant
changes in mechanical quality factor or nonlinearity in either experiment.

To investigate the reliability of logical operations, we make repeated
measurements of the NAND gate truth table of Fig. 5a. To do this, we automate
the control and data taking systems as described in the Methods, repeating the
sequence of four possible sets of input states (‘00’, ‘01’, ‘10’, & ‘11’) a
total of 1500 times. We record the nanomechanical oscillation amplitude for
each sets of input state, averaged over 10 ms. We observe no failures of gate
operations for the full 6000 operations tested.

To investigate the statistical probability of a failure, we correct for slow
thermal drifts in the output amplitude and compute histograms of this
normalised amplitude for each of the four sets of input states (see
Supplementary information). As shown in Fig. 5b, the amplitude fluctuations
are far smaller than the separation of the ‘0’ and ‘1’ output states,
consistent with our observation of no errors over the full sequence of
measurements. Indeed, we find that the separation between the ‘0’ bit value
(red trace) and the lowest ‘1’ bit value (blue trace) corresponds to $\sim
10^{7}$ times the characteristic thermal energy ($k_{B}T$, with $k_{B}$ the
Boltzmann constant and $T$ the temperature), so that thermally driven errors
can be expected to be exceptionally rare. Fig. 5c confirms this, extrapolating
the histograms for these bit values to predict that it would take an
349$\sigma$ event for the ‘0’ state to be mistaken for a ‘1’, or a 47$\sigma$
event for the reverse to occur. Thus, statistical errors in logical operations
using our nanomechanical computing architecture can be safely neglected, with
errors likely dominated in practice by external perturbations to the system.

Figure 5: NAND gate. a, NAND gate truth table. Time trace showing the measured
resonator amplitude for all four combinations of logical inputs when our
nanomechanical device is configured to implement a universal NAND gate. b,
Time evolution of the amplitudes of each of the four NAND gate outputs over
1500 measurements, corrected for slow time drift due to chip heating effects
(see Supplementary Information). Histograms are plotted on the right. These
are found to be consistent with Gaussian distributions (solid lines), as
expected from the central limit theorem. c, Histograms of the output bit
amplitudes for ‘11’ and ‘10’ inputs. The inset shows an interpolation of the
histogram fits, illustrating the low probability of a statistical error. All
results used a drive-resonator detuning of approximately 660 Hz.

##  V Discussion

The potential of nanomechanical technologies to allow robust, low energy
computing has long been recognised [9]. However, the lack of a purely
nanomechanical computing architecture that is scalable has been a major
barrier, with electronics instead generally used to interconnect gates [28].
The five essential criteria for scalable computing are [39, 40]: nonlinearity,
gain, feedback prevention, functional completeness, and concatenability. While
previous purely nanomechanical logic gates have, like ours, achieved
functional completeness [19], used nonlinearities [26, 41], and demonstrated
gain [42], none have been fully concatenatable due, either, to the reliance on
physical contact between gates [16], or to differences in the frequencies of
input and output bits [25, 26, 29]. A similar challenge exists for many
nanoelectromechanical gates, with reported part-in-ten-million conversion
efficiencies between electronic and mechanical domains preventing efficient
information transfer from one gate to the next [19].

Our nanomechanical logic architecture allows concatenation due to the use of
both the Duffing bistability which ensures that input and output bits have the
same spatiotemporal characteristics, and low-dissipation acoustic waveguides
which ensure near-lossless transfer of information between gates. Indeed, the
attenuation of our waveguides is well beneath a dB-per-cm [32], so that losses
on transmission can be safely neglected. Our architecture also allows feedback
prevention, so that acoustic reflections from one gate will not influence the
function of upstream gates. For instance, higher-order mechanical resonances
can be exploited to create a nanomechanical transistor for which the input
(‘gate’) is naturally isolated from both pump (‘source’) and output (‘drain’).
We validate that this prevents feedback via finite-difference time-domain
simulations in the Supplementary Information. Beyond conventional approaches
to nanomechanical computing, we expect that this capacity to concatenate
nonlinear mechanical resonators and control how they interact will open up new
approaches to neuromorphic computing that overcome energy and density
constraints of existing approaches using photonics, nanoelectromechanics, and
semiconductor electronics [43]. It could also enable nonlinear phononic
circuits that can simulate complex many-body phenomena such as symmetry
breaking, synchronisation, and strongly-interacting spin systems [18, 44, 17].

The energy cost of the nanomechanical logic operations we achieve is
competitive with the best previously reported purely nanomechanical gates
[26]. However, at around $10^{7}k_{B}T$ it remains far above the Landauer
limit. This is caused by the need to pump the mechanical resonator to energies
that exceed its critical energy of 28 fJ ($\sim 7\times 10^{6}\,k_{B}T$) in
order to drive transitions between ‘0’ and ‘1’ states. Our demonstration is
also slow, with each gate operation lasting at least 10 ms. Both speed and
critical energy can be greatly improved through miniaturisation and material
engineering. As shown in the Supplementary Information, reducing the width and
thickness of the resonator to 2 $\mu$m and 10 nm, respectively, and decreasing
the tensile stress by a factor of 100, would allow the Landauer limit to be
reached and the speed of gate operations to be increased to around 50 kHz.
Replacing the silicon nitride device layer with graphene would allow further
miniaturisation into the nanoscale [45, 46, 47], increasing the gate speed
into the gigahertz regime while maintaining Landauer limited operation.
Interestingly, in this regime, the mechanical resonance frequency is
sufficiently high to allow passive cooling of the mechanical resonator into
its quantum ground state using commercial cryogenic systems. Our computing
architecture could also then be used as a coherent Ising machine to solve non-
deterministic polynomial-time (NP) hard computing problems [48], or
potentially even as the basis for a purely nanomechanical quantum computer.

##  VI Methods

###  VI.1 Measurement setup

To detect the mechanical motion of the membrane, we use a home-built laser
vibrometer based on a combination of lensed fibre and heterodyne measurement
[30, 32]. Light from a laser is reflected off the membrane (see Fig. 3). The
periodic motion of the membrane at frequency $\Omega_{\textnormal{m}}$
modulates the phase of the reflected optical field. This modulation is then
detected through its interference with a frequency-shifted local oscillator
field $\Omega$ [30, 32]. After detection, a beat-note that has an amplitude
proportional to the amplitude of motion of the membrane is observed using a
spectrum analyser (see Supplementary information). We find that the
measurement is sufficiently precise to resolve both the thermal motion of the
nanomechanical resonators that form our logical gates, and the driven motion
of the input and output acoustic waveguides.

###  VI.2 Nanomechanical gate sequence

To create the truth table sequence in Fig. 5a, we use a series of transistor-
transistor-logic (TTL) pulses to trigger short bursts of coherent drive from
three signal generators. The first of these three signal generators is used to
create the electronic actuation for the pump, which consists of a sinusoidal
wave at $\Omega$ gated by four 120 ms square pulses separated by 300 ms. The
other two signal generators are used similarly to create the actuation for
logical inputs A and B. The phase of the three sinusoidal signals are adjusted
to create a NAND gate as explained in the the main text. The TTL pulses also
trigger a spectrum analyser to record the amplitude of motion of the membrane
at the drive frequency $\Omega$ with zero span and a resolution bandwidth of
510 Hz. In Fig. 5a we show a sequence that demonstrates a NAND gate, where the
pump and two logic inputs $A$ and $B$ have a carrier frequency $\Omega$, which
is slightly blue detuned from the bare resonant frequency
$\Omega_{\textnormal{m}}$. The ratio of the amplitudes of the 0 and 1 states
is close to a factor of ten, due to the nonlinear mechanical response of the
resonator, as shown in Fig. 4c.

###  VI.3 Fabrication details

The devices are fabricated on a chip that is diced from a commercially
available wafer which has a film of stoichiometric silicon-nitride (Si3N4)
deposited on a silicon substrate. The Si3N4 film is $\sim$80 nm thick with an
effective tensile stress of $\sigma_{T}$=0.67 GPa [32]. We pattern the
electrodes through a combination of electron beam lithography (EBL) and
evaporation of gold ($\sim$50 nm), followed by lift-off. The hole pattern in
the Si3N4 film is created using EBL and reactive-ion-etching. This pattern
consists of a grid of 1 $\mu$m square holes separated by 2 $\mu$m (centre-to-
centre). The Si3N4 membrane is then released from the silicon substrate via a
potassium hydroxide wet etch. On each chip we define arrays of devices, each
consisting of an 80 $\mu$m-square mechanical resonator connected with tunnel
barriers to single mode input and output acoustic waveguides. Together, this
procedure allows low-loss acoustic architectures to be built on a CMOS-
compatible platform.

###  VI.4 Error statistic setup

To record the data displayed in Fig. 5b and Fig. 5c, the pulse sequence shown
in Fig. 5a is repeated over 1500 times. However, for this measurement, we
operate with a resolution bandwidth of 100 Hz and only record the central
point of each pulse displayed on the spectrum analyser.

###  VI.5 Adiabatic impedance transitions

To avoid reflections and resonances between the three gold on-chip electrodes,
the acoustic impedance between the silicon nitride waveguide without gold and
the silicon nitride waveguide with gold needs to be matched. This is done by
patterning nine gold pads on the waveguide between the on-chip electrodes,
minimising the impedance difference. Similarly, to efficiently launch the
acoustic wave in the membrane, adiabatic tapering of the impedance is used
with ten decreasing size gold pads patterned on the left and right side of
electrodes A and P (see Fig. 2).

### Acknowledgments

This research was primarily funded by the Australian Research Council and the
Lockheed Martin Corporation through the Australian Research Council Linkage
Grant No. LP160101616. Support was also provided by the Australian Research
Council Centre of Excellence for Engineered Quantum Systems (Grant No.
CE170100009). This work was performed in part at the Queensland node of the
Australian National Fabrication Facility and the Australian Microscopy &
Microanalysis Research Facility at the Centre for Microscopy and
Microanalysis. G.I.H. and C.G.B. acknowledge their Australian Research Council
(Grants No. DE210100848 and No. DE190100318), respectively. The authors would
like to acknowledge HCPhotonics for providing a second harmonic module that
was used in our heterodyne probing system.

### Author contributions

W.P.B., R. K. and C.B. conceived the idea. E. R. fabricated the devices. E.R.,
N.M. and T.M.F.H. carried out the experiment and data analysis with support
from G.H. Theoretical modelling and simulations were done by N.M., E.R., and
C.B. The manuscript was written by E. R., N. M and W.P.B with contributions
from the rest of the authors. W.P.B. supervised the work.

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## Supplementary Information

##  S1 Derivation of linear and nonlinear spring coefficients

The origin of the non-linearity present in our system is known as geometric
non-linearity [1] with non-linear coefficient $\alpha$. This non-linearity
emerges from a high amplitude displacement of the mechanical resonator. The
calculation of $\alpha$ can be directly derived analysing the modeshape and
displacement of our resonators. Here we derive $\alpha$ for the simplest
system, a 1D string, extend the derivation for a 2D membrane using numerical
simulations and finally derive it for a meshed membrane just as the one in our
experiment.

###  S1.1 String

The simplest system to consider is a 1D string, with undeflected length

$L=\int_{0}^{L}\mathrm{d}x,$ (1)

while the length $L^{\prime}$ of the deformed string is given by

$L^{\prime}=\int_{0}^{L}(\Delta x^{2}+\Delta
u^{2})^{1/2}=\int_{0}^{L}\left(1+\left(\frac{\partial u}{\partial
x}\right)^{2}\right)^{1/2}\mathrm{d}x.$ (2)

In a deflected string, the local change in string length is

$\mathrm{d}x\left(1+\left(\frac{\partial u}{\partial
x}\right)^{2}\right)^{1/2}-dx\simeq\frac{1}{2}\left(\frac{\partial u}{\partial
x}\right)^{2}\mathrm{d}x,$ (3)

which corresponds to a local strain

$\varepsilon(x)=\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^{2}.$
(4)

The energy density $e(x)$ associated with this elongation is:

$e(x)=\frac{1}{2}YA\varepsilon^{2}=\frac{1}{8}AY\left(\frac{\partial
u}{\partial x}\right)^{4},$ (5)

where $A$ is the cross-sectional area of the string, and $Y$ the Young’s
modulus of the material. The total elongation energy is thus given by the
integral over the volume

$E_{\mathrm{elongation}}=\int_{0}^{L}e(x)\mathrm{d}x=\frac{1}{8}YA\int_{0}^{L}\left(\frac{\partial
u}{\partial x}\right)^{4}\mathrm{d}x.$ (6)

Identifying this with a quartic Duffing potential energy of the form
$E=\frac{1}{4}\alpha x^{4}$, we get $\alpha$ the non-linear coefficient

$\boxed{\alpha=\frac{1}{2}YA\int_{0}^{L}\left(\frac{\partial u_{0}}{\partial
x}\right)^{4}\mathrm{d}x},$ (7)

where $u_{0}$ is the unperturbed normalized displacement profile, normalized
such that $\mathrm{max}(u_{0})=1$.

Similarly, since the energy stored in the string’s kinetic energy/tension is
given by

$E=\frac{1}{2}\rho A\int_{0}^{L}\left(\frac{\partial u}{\partial
t}\right)^{2}\mathrm{d}x=\frac{1}{2}\sigma A\int_{0}^{L}\left(\frac{\partial
u}{\partial x}\right)^{2}\mathrm{d}x$ (8)

we can identify the linear spring $k$ and the effective mass
$m_{\mathrm{eff}}=k/\Omega^{2}$,

$k=\rho\Omega_{\textnormal{m}}^{2}A\int u_{0}^{2}(x)\mathrm{d}x=\sigma
A\int_{0}^{L}\left(\frac{\partial u_{0}}{\partial
x}\right)^{2}\mathrm{d}x,\hskip 14.22636pt\textnormal{and}\hskip
14.22636ptm_{\mathrm{eff}}=\rho A\int u_{0}^{2}(x)\mathrm{d}x,$ (9)

both in geometric terms.

###  S1.2 Membrane

For the 2D case, in the regime that the resonator is thin (i.e. the resonator
is thin enough to behave as a membrane and not a plate), the bending stiffness
is neglected, and the Young’s modulus is assumed isotropic. The membrane’s
Duffing nonlinear coefficient $\alpha$, by analogy to the string case (Eq.
(7)), takes the form:

$\boxed{\alpha=\frac{1}{2}Y\,h\iint\left(\left(\frac{\partial u_{0}}{\partial
x}\right)^{4}+\left(\frac{\partial u_{0}}{\partial
y}\right)^{4}\right)\mathrm{d}x\,\mathrm{d}y},$ (10)

where $h$ is the membrane thickness. Similarly, the spring constant $k$ and
effective mass $m_{\mathrm{eff}}=k/\Omega_{\textnormal{m}}^{2}$ take the form

$k=\sigma\,h\iint\left(\left(\frac{\partial u_{0}}{\partial
x}\right)^{2}+\left(\frac{\partial u_{0}}{\partial
y}\right)^{2}\right)\mathrm{d}x\,\mathrm{d}y,\hskip
14.22636pt\textnormal{and}\hskip 14.22636ptm_{\mathrm{eff}}=\rho\,h\iint
u_{0}^{2}(x,y)\,\mathrm{d}x\mathrm{d}y$ (11)

expressed in geometric terms.

Using the derived geometric terms, we can directly feed them into finite
element models (FEM) such as COMSOL. The numerical results are calculated for
simple systems such as strings or membranes and compared with the analytical
expressions. Once verified, we modify the geometry and introduce the meshed
membrane. The modeshapes obtained are plotted in Fig. S1b and S1c, from these
simulation we can extract the non-linear coefficient for the meshed membrane
$\alpha=2.64\times 10^{13}$ N.m-3.

Figure S1:  (a) Elongation energy in a deformed string. (b) FEM simulation of
a non-meshed acoustic fabry perot cavity. Estimated parameters: $k=418$ N.m-1;
$m_{\mathrm{eff}}=5.11\times 10^{-13}$ kg; $\alpha=3.65\times 10^{13}$ N.m-3;
$x_{\mathrm{crit}}=16.5$ nm; $\Omega_{\textnormal{m}}=4.5$ MHz. (c) FEM
simulation of a meshed acoustic Fabry Perot cavity. Estimated parameters:
$k=253$ N.m-1; $m_{\mathrm{eff}}=3.8\times 10^{-13}$ kg; $\alpha=2.64\times
10^{13}$ N.m-3; $x_{\mathrm{crit}}=15$ nm; $\Omega_{\textnormal{m}}=4.1$ MHz.
Physical parameters: $Y=250$ GPa; $\sigma=1$ GPa; $\rho=3100$ kg.m-3,
$Q=28000$. Resonator width=length=80 $\mu$m, tunnel width 44 $\mu$m.

##  S2 Electrostatic actuation and electric power requirements

Sections S3 and S4 focussed on the acoustic energy required for nonlinear
nanomechanical logic operation, considering acoustic inputs and outputs.
Before the acoustic wave is present in the circuit, it is generated by
electrostatic actuation. Here we discuss the parameters of this process, and
discuss the overall (electric+acoustic) efficiency.

###  S2.1 Electrostatic actuation

A finite element simulation of the electrostatic actuation process is shown in
Fig. S2. It shows a cross-section of the 500 $\mu$m thick silicon wafer,
covered by the 80 nm thick silicon nitride membrane, and 40 $\mu$m wide gold
electrode. As these are $\sim$10 000 times thinner than the wafer, these are
only visible in the zoomed-in sub-plots. Actuation of the membrane is provided
by the electrostatic interaction between the gold electrode atop the phononic
waveguide and an electrode on the underside of the wafer (see also Figs. 2 & 3
of the main text). The silicon handle wafer is P-doped (Boron), with a
resistivity of 1-10 Ohm cm. As such, its low frequency permittivity
$\varepsilon$ is greatly enhanced over its optical frequency permittivity
($\varepsilon_{\mathrm{Si,THz}}=n^{2}=3.5^{2}\simeq 12$), due to charge
transfer through the silicon, with a MHz-range permittivity of $\sim 10^{4}$
(so-called colossal permittivity [2, 3]). The effect of this is shown in Fig.
S2, which plots the electric field in response to a 1V bias applied between
both electrodes. Because of the extremely large permittivity of doped silicon
at MHz frequencies, the electric energy density is well localized in the
undercut region between the silicon nitride and the silicon substrate (see
Fig. S2(c)). This means the capacitance can be well approximated by that of a
parallel plate capacitor with an air gap ($C=\frac{\varepsilon_{0}A}{d}$),
with a plate separation distance $d$ equal to the depth of the undercut. The
error from this approximation is under 4% here, with a simulated capacitance
per unit length (along $\hat{y}$) of 367 pF/m from the finite element
simulation and 354 pF/m from the parallel-plate approximation.

Figure S2: (a) 2D electrostatic simulation of the electric potential in
response to a 1V DC bias applied between a gold electrode located above the
released silicon nitride film and a ground plane below the silicon substrate.
Because of the extremely large permittivity of doped silicon at MHz
frequencies, the electric energy density is well localized in the undercut
region between the silicon nitride and the silicon substrate. This means the
capacitance can be well approximated by that of a parallel plate capacitor
($C=\frac{\varepsilon_{0}\varepsilon_{r}A}{d}$) with a plate separation
distance $d$ equal to the depth of the undercut—with an error under 4% here.
Simulation parameters: substrate thickness 500 $\mu$m; undercut depth 1
$\mu$m; silicon permittivity $1\times 10^{4}$; electrode width: 40 $\mu$m;
simulated capacitance per unit length 367 pF/m; parallel plate approximation:
354 pF/m. (b) Schematic showing the relevant parameters for the estimation of
stored mechanical and electrostatic energies.

It is instructive to relate the mechanical energy stored in the membrane
deflection, to the stored energy in the capacitor. As justified above, we
employ the parallel plate approximation. The mechanical energy stored in the
deflection of amplitude $u$ is:

$E_{M}=\frac{1}{2}ku^{2}=\frac{1}{2}k\left(\frac{F_{\mathrm{es}}}{k}\right)^{2}=\frac{1}{2}\frac{F_{\mathrm{es}}^{2}}{k}$
(12)

where $F_{\mathrm{es}}=\frac{1}{2}\frac{\varepsilon_{0}A}{d^{2}}V^{2}$ is the
electrostatic force applied on the membrane, in the limit of small
displacements $u\ll d$. Similarly, the stored electrostatic energy in the
capacitor is

$E_{e}=\frac{1}{2}CV^{2}=\frac{1}{2}\frac{\varepsilon_{0}A}{d}V^{2}=F_{\mathrm{es}}d$
(13)

The ratio $\eta$ of these two energies, valid in the limit of small
displacements, takes the form:

$\eta=\frac{E_{M}}{E_{e}}=\frac{F_{\mathrm{es}}}{2kd}=\frac{\varepsilon_{0}AV^{2}}{4kd^{3}}=\frac{1}{2}\frac{u}{d}.$
(14)

Note that this energy analysis can be readily converted to a power analysis
when the waveguide is operated sufficiently high above its cutoff frequency,
such that the group velocity of the acoustic wave is close to its phase
velocity: $v_{g}\sim vp\sim\sqrt{\frac{\sigma}{\rho}}$, in which case the
acoustic energy has left the capacitor region before the start of a new cycle.
In the current experiments, where the capacitor is charged with an AC voltage
source, there is no fundamental lower bound on power loss in the electric
circuit (ohmic losses may in principle be made arbitrarily small). On the
other hand, if operated in a regime where a given fraction of the capacitor
energy is lost during the charge & discharge process—as when the capacitor is
charged from a constant voltage power supply, which is the case for CPUs,
where this fraction reaches one half [38]—it is beneficial to operate in a
regime of small capacitor gap $d$, which maximizes the electrostatic force for
a given capacitor energy, see Eq.(13). Using an oxide sacrificial layer under
the nitride as in Ref. [4], this gap can reliably be made sub-100 nm.

##  S3 Critical amplitude and critical energy

Considering the total energy of the resonator $E$ as

$\displaystyle E$
$\displaystyle=\frac{1}{2}k\,x^{2}+\frac{1}{4}\alpha\,x^{4}=\frac{1}{2}m_{\mathrm{eff}}\Omega_{\textnormal{m}}^{2}\left(1+\frac{\alpha}{2k}x^{2}\right)x^{2},$
(15)

where the resonance frequency becomes amplitude dependent

$\Omega=\Omega_{\textnormal{m}}\left(1+\frac{\alpha}{2k}x^{2}\right)^{\frac{1}{2}}.$
(16)

The frequency shift $\delta\Omega(x)$ is given by

$\delta\Omega(x)=\frac{\alpha}{4k}\Omega_{\textnormal{m}}x^{2},$ (17)

where $\alpha$ is the non-linear coefficient. The critical amplitude is
reached when this amplitude-dependent frequency shift
$\delta\Omega\sim\Gamma=\Omega_{\textnormal{m}}/Q$, that is:

$x_{\mathrm{crit}}=\sqrt{\frac{2\,k\,\Gamma}{3\,\alpha\,\Omega_{\textnormal{m}}}}=\sqrt{\frac{2\,\Gamma\,\Omega_{\textnormal{m}}\,m_{\mathrm{eff}}}{3\,\alpha}}=\sqrt{\frac{2\,k}{3\,\alpha\,Q}}.$
(18)

Using these derivations, we calculate the linear elongation energy for a
string S1(a), for an evanescently coupled membrane S1(b) and an evanescently
coupled meshed membrane, as shown in S1(c). The meshed case has lower mass and
frequency compared to the unperturbed membrane. The critical amplitude is
however not significantly reduced, since both $k$ and $\alpha$ drop by
comparable amounts (divided by 1.65 vs 1.4). This observation is expected, as
the mode profile is essentially unperturbed by the subwavelength hole pattern.

Finally, the last relevant parameter is the critical energy
$E_{\mathrm{crit}}$, which is the energy cost associated to take the resonator
amplitude to the critical amplitude, defined as

$E_{\mathrm{crit}}=\frac{1}{2}k\,x_{\mathrm{crit}}^{2}=\frac{k^{2}}{3\,\alpha\,Q}=\frac{m_{\mathrm{eff}}^{2}\,\Omega_{\textnormal{m}}^{4}}{3\,\alpha\,Q}.$
(19)

It is important to notice that most of the energy stored in the resonator
corresponds to the linear elastic energy. Since
$\frac{1}{2}kx^{2}\gg\frac{1}{4}\alpha x^{4}$, the energy arguments presented
in (26) refer to the energy in the linear regime, but apply to the non-linear
case.

##  S4 Energy cost derivation

In practice, the energy cost depends on other parameters, specifically the
desired contrast between the ‘0’ and ‘1’ amplitudes, the duration of the logic
pulses, and how much of the energy in the Duffing oscillator is transmitted
into the output waveguide (and therefore not lost). We develop a simple model
of the total energy cost, in the limit that the pulse duration is long
compared to the decay time of the nanomechanical resonator and that the
majority of energy in the gate comes from the pump. The model predicts
different energy costs and functional dependencies for ‘0’ and ‘1’ output
states. This can be understood directly from the difference in amplitudes of
the two outputs: to achieve a lower amplitude for a ‘0’ the gate must reject
more of the input energy. The results of the model are shown as a function of
the level of impedance matching of the resonator in Fig. 4d of the main text
using the calibrated Duffing coefficient and spring constant of our
resonators.

To estimate the energy cost for a single operation in our device, we consider
the double-sided non-linear mechanical cavity of Figure S3 which oscillates
with an amplitude proportional to $|b|$ with an energy $E=|b|^{2}$, when
excited by an input acoustic wave of power $|b_{\rm{in}}|^{2}$. The power of
the wave exiting the right side of the cavity is $|b_{out}|^{2}$ and the power
of the wave reflected by the cavity is $|b_{r}|^{2}$. The equation of motion
of such cavity is given by [5]:

$\dot{b}=-\frac{\gamma_{\rm{tot}}}{2}b-i\Delta b+\sqrt{\gamma}b_{\rm{in}}$
(20)

where $\gamma_{\rm{tot}}=2\gamma+\gamma_{\rm{int}}$ is the total energy loss
rate, $\gamma$ the coupling rate of both sides of the cavity,
$\gamma_{\rm{int}}$ the intrinsic loss rate of the cavity, $\Omega$ the
angular frequency of the exciting wave, $\Delta=\Omega-\Omega_{\rm{m}}$ the
drive frequency detuning from the cavity resonance $\Omega_{\rm{m}}$. This
equation can be solved for the steady state when the duration of the acoustic
wave pulse $T=N/\gamma_{\rm{tot}}$ is larger than the inverse of the decay
rate. This is the case here since the quality factor of the resonators is
around $10^{4}$ leading to a decay rate of hundreds of Hertz which inverse is
two orders of magnitude smaller than the pulse duration used. In this
situation, $N\gg 1$ and $\dot{b}\sim 0$ which leads to:

$b=\frac{\sqrt{\gamma}}{\gamma_{\rm{tot}}/2+i\Delta}b_{\rm{in}}.$ (21)

As shown by figure S3, $b_{\rm{out}}=\sqrt{\gamma}b$, therefore the steady-
state efficiency defined as $\eta=|b_{\rm{out}}|^{2}/|b_{\rm{in}}|^{2}$ can be
expressed as

$\eta=\frac{\gamma^{2}}{\gamma_{\rm{tot}}^{2}/4+\Delta^{2}}$ (22)

Using eq. 21, the mechanical energy of the cavity $E$ is then:

$E=\frac{\gamma}{\gamma_{\rm{tot}}^{2}/4+\Delta^{2}}|b_{\rm{in}}|^{2}.$ (23)

The steady-state energy can be expressed as a fraction $A$ of the critical
energy $E_{\mathrm{crit}}$, $E=AE_{\mathrm{crit}}$, where $A=A^{(1)}>1$ when
the duffing oscillator is in the high amplitude state corresponding to a logic
operation output “1” and $A=A^{(0)}<1$ when the oscillator is in the low
amplitude state, corresponding to a logic operation output “0”. Therefore we
have:

$|b_{\rm{in}}|^{2}=\frac{AE_{\mathrm{crit}}(\gamma_{\rm{tot}}^{2}/4+\Delta^{2})}{\gamma}.$
(24)

Since the right output of the cavity guides the result of the logic operation,
only the wave propagating away from the cavity in the left direction and the
energy lost to the environment at a rate $\gamma_{\rm{int}}$ will contribute
to the losses (see figure S3). Therefore the energy lost $E_{\mathrm{lost}}$
during the logic operation can be expressed as [5]

$E_{\mathrm{lost}}=\int_{0}^{T}(|b_{\rm{in}}|^{2}-|b_{\rm{out}}|^{2})\,dt.$
(25)

Since $b_{\rm{out}}=\sqrt{\gamma}b$ and, for the steady state, $b_{\rm{in}}$
can be assumed to be constant as function of time. Therefore using eq. 21 and
24,

$E_{\mathrm{lost}}=\frac{NAE_{\mathrm{crit}}}{\gamma_{\rm{tot}}\gamma}\left(\frac{\gamma_{\rm{tot}}^{2}}{4}+\Delta^{2}-\gamma^{2}\right)$
(26)

When the output of the logic operation is a binary “1”, $A=A^{(1)}>1$ and
$\Delta=\Delta^{(1)}=0$ since the cavity has shifted on resonance. Using eq.
26, the energy lost for an output “1” (Output 1 in Fig. 4d in the manin text),
$E_{\mathrm{lost}}^{(1)}$ is therefore:

$\boxed{E_{\mathrm{lost}}^{(1)}=\frac{NA^{(1)}E_{\rm
crit}}{4}\left(\frac{\gamma_{\rm
int}}{\gamma}\right)\left(2-\frac{\gamma_{\rm{int}}}{\gamma_{\rm
tot}}\right)}$ (27)

where we have used the relation $\gamma_{\rm tot}=2\gamma+\gamma_{\rm int}$.

When the output of the logic is a binary “0”, $A=A^{(0)}<1$. However, the
detuning is not negligible since the resonance frequency of the duffing
oscillator has not shifted to the driving frequency. Knowing that the acoustic
pump signal excites the oscillator close to the critical energy independently
if the logic operation result is a “0” or a “1”, the contribution of the
acoustic signal A and B to the input acoustic power will be negligible and we
can assume that $b_{\rm{in}}^{(1)}\sim b_{\rm{in}}^{(0)}$. The detuning can
then be expressed using equation 24 as :

$\Delta^{(0)}=\sqrt{\frac{\gamma_{\rm{tot}}^{2}}{4}\left(\frac{A^{(1)}}{A^{(0)}}-1\right)}.$
(28)

Using equation 26 it is straightforward to see that the energy lost when the
output of the logic operation is a ”0“ is (Output 0 in Fig. 4d in the main
text):

$\boxed{E_{\mathrm{lost}}^{(0)}=\frac{NA^{(1)}E_{\rm
crit}}{4}\left(\frac{\gamma_{\rm{tot}}}{\gamma}-\frac{4\gamma}{\gamma_{\rm{tot}}}\left(\frac{A^{(0)}}{A^{(1)}}\right)\right)}$
(29)

The evolution of $E_{\mathrm{lost}}^{(0)}$ and $E_{\mathrm{lost}}^{(1)}$ as
function of $Q/Q_{\rm{int}}=\gamma_{\rm{int}}/\gamma_{\rm{tot}}$ is displayed
in the main text figure 4d). For this figure, we use the experimental values
of $\Omega_{\textnormal{m}}$, $\gamma_{\rm{int}}$, $\gamma$, $\alpha$ and $N$.
$m_{\mathrm{eff}}$ is calculated from finite element simulation using the
equations of section S1.

Parameter |  Energy value (fJ) |  Energy value (fJ) | Expression
---|---|---|---
| (Gate duration 10 ms) | (Gate duration 120 ms) |
Energy lost (0) | 38 | 455 | $\displaystyle E_{\mathrm{lost}}^{(0)}=\frac{NA^{(1)}E_{\rm crit}}{4}\left(\frac{\gamma_{\rm{tot}}}{\gamma}-\frac{4\gamma}{\gamma_{\rm{tot}}}\left(\frac{A^{(0)}}{A^{(1)}}\right)\right)$
Energy lost (1) | 7 | 89 | $\displaystyle E_{\mathrm{lost}}^{(1)}=\frac{NA^{(1)}E_{\rm crit}}{4}\left(\frac{\gamma_{\rm int}}{\gamma}\right)\left(2-\frac{\gamma_{\rm{int}}}{\gamma_{\rm tot}}\right)$
Critical energy | 28 | 28 | $E_{\mathrm{crit}}=\displaystyle\frac{1}{2}k\,x_{\mathrm{crit}}^{2}=\frac{k^{2}}{3\,\alpha\,Q}=\frac{m_{\mathrm{eff}}^{2}\,\Omega_{\textnormal{m}}^{4}}{3\,\alpha\,Q}$
Average energy cost | 22 | 272 | $\bar{E}=\displaystyle\frac{1}{2}\left(E_{\mathrm{lost}}^{(1)}+E_{\mathrm{lost}}^{(0)}\right)$
Minimum average energy cost | 12 | 140 | $\displaystyle 0=\frac{d\bar{E}(Q)}{dQ}$
Table S1: Values of the energies involved in the gate operation considering a
$Q=28000$.

As can be seen, the energy cost for an output ‘1’ state increases
monotonically as the resonator becomes increasingly undercoupled, while going
to a constant minimum value of $E_{\rm lost,min}^{(1)}=NAE_{\rm
crit}(\gamma_{\rm int}/\gamma_{\rm tot})$ in the impedance-matched regime
where the loaded $Q$ is much smaller than $Q_{\rm int}$. By contrast, the
energy cost for an output ‘0’ state is optimal when $Q/Q_{\rm int}=0.49$, in
between the well- and poorly-impedance matched regimes. The increased energy
cost at low $Q$ in this case can be understood because as the $Q$ decreases
the decay rate of the resonator increases, so that a larger nonlinear
frequency shift is required to operate the gate. Achieving this larger shift
requires a higher amplitude of drive, and therefore increases the energy cost.

Our experiments operate in the near-impedance-matched regime, with
$\gamma_{\rm int}/\gamma\sim 0.2$, $\gamma_{\rm int}/\gamma_{\rm tot}\sim
0.1$, and $\gamma_{\rm tot}/\gamma\sim 1$. The measured oscillation amplitudes
of 2 nm and 18 nm, respectively, for a ‘0’ and a ‘1’ give a ratio
$A^{(0)}/A^{(1)}=(2/18)^{2}\sim 0.01$. The oscillation energy of a ‘1’ is
around forty percent higher than the critical energy, with
$A^{(1)}=(x^{(1)}/x_{\rm crit})^{2}=(18\,{\rm nm}/15\,{\rm nm})^{2}=1.4$. For
the data shown in the main text, $N\sim 18$, while for our shorter 10 ms
duration gates, this reduces to $N\sim 1.4$. Using these values in Eqs. (29)
and (27), we find that the energy lost during an operation of our longer gates
is roughly $2.4E_{\rm crit}$ and $14E_{\rm crit}$ for an output ‘1’ and ‘0’,
respectively; while the equivalent energy losses for our shorter gates are
roughly $0.25E_{\rm crit}$ and $1.35E_{\rm crit}$.

Figure S3: Schematic of the model used to calculate the energy cost of each
logic operation. Double sided mechanical resonator excited by an input wave of
amplitude $x_{\rm{in}}$ with a coupling rate $\gamma$. The waves exiting the
cavity on the right ($x_{\rm{out}}$) and left ($x_{\rm{r}}$) are also coupled
out to the cavity at a rate $\gamma$. The intrinsic cavity loss rate is
$\gamma_{\rm{int}}$.

##  S5 Drift correction

Each point $i$ of the four traces shown in Fig. 5 b represents the mean value
around the center (withing the resolution bandwidth window) of each NAND gate.
During the data acquisition of error statistics (main text figure 5 b)), a
slow drift of the amplitude of each window was observed, due to thermal
effects. The maximum drift was observed for the red trace which dropped by 2.7
dBm after 0.6 hours of continuous measurements. To remove this systematic
error from our measurement the drift was compensated by applying the same
simple linear transformation to the four traces. That is to say, each point
$i$ of the four traces is transformed $i\rightarrow i^{\prime}$ as
$i^{\prime}=m\times i+p$. The blue trace (named $A$) and red trace (named $B$)
of figure 5 b) were used as a reference. We define the variables $m$ and $p$
as:

$m=\frac{\overline{A}-p}{\langle A\rangle},\hskip
28.45274ptp=\frac{\overline{B}\langle A\rangle-\overline{A}\langle
B\rangle}{\langle A\rangle-\langle B\rangle},$ (30)

where $\overline{X}$ represents the average of $X$ over the 1500 measurements
and $\langle X\rangle$ represents the moving average of $X$ over a window of
100 measurements centered on point $i$ but excluding point $i$. The resulting
corrected data is presented in Fig. 5 b showing that the systematic error
caused by drift is corrected efficiently. Histograms are plotted on the right
of Fig. 5 b. These are found to be consistent with Gaussian distributions
(solid lines), as expected from the central limit theorem.

##  S6 Five requirements of a logic device

Five requirements have been identified as necessary for a viable logic device
[39, 40]: nonlinearity, gain, cascadability, feedback prevention and
functional completeness. A device that meets these requirements is viable in
the sense that it is effective and scalable. Previous nanomechanical devices
have achieved strict subsets of these requirements, but not met all five
simultaneously. We posit that our proposed device is the first nanomechanical
system that does meet meets all five criteria simultaneously.

Figure S4: FIVE-essentials: a) Diagram of a nanomechanical transistor used for
feedback prevention. The source, gate and drain are formed by input, and
output waveguides. The time domain simulations are shown in Fig. S5(a). b)
Configuration of a gate for gain $\geq 1$, where the pump (left waveguide)
provides majority of the oscillation energy. The logic input, carrying a small
fraction of the energy is delivered from the top waveguide. The logic output
of the gate is sent into the next gate through the right waveguide. c) In the
transistor shown in a) the feedback is eliminated using using orthogonal but
degenerate modes between the between the logic input and the logic output.
Further simulations are shown in Figure S5 (b)-(c). The complete set of
Boolean operations is composed the gates AND, OR, NOT, XOR and NAND. e)
Schematic of cascadability architecture for our devices where the ultrasonic
outputs waves are carrier in the mode as the input, allowing multiple devices
to concatenate in the same circuit.

  1. 1.

Non-linear characteristics (for signal-to-noise)

Nonlinearity is necessary to achieve a good signal-to-noise ratio [6]. A
linear logic gate would poorly distinguish between ‘1’ and ‘0’ responses.
Ideally, a gate should show a marked change in output when the input crosses
from the ‘1’ to the ‘0’ preimage, and vice versa. The more easily a ‘1’ can be
told apart from a ‘0’, the less noise is introduced into the logical circuit
by the gate. Figure 4 (c) in the main text shows the clear difference in the
two amplitudes of oscillation introduced by Duffing bistability in our
resonators. The probability of mislabelling the amplitudes is negliglbe, as
seen in Figure 5 (c).

  2. 2.

Power amplification

To string multiple logic gates together without gradually diminishing the
signal, the gain of a single gate must be $\geq 1$. In our gate this is
achieved by the pump signal, which provides the majority of the oscillation
energy in each logic operation. The pump allows an input signal with only a
fraction (for example, 10%) of the jump up energy to induce a jump up. The
output signal from the pump-driven resonator is more energetic than this
fraction—i.e. gain is $>1$—more than making up for any propagation losses that
will occur before the next resonator.

  3. 3.

Cascadability

Logic circuits with complex truth tables typically require multiple logic
devices. Additionally, inputs and outputs of gates may come from and go into
other, non-logic components. Therefore a useful logic device should produce an
output compatible with other components in the circuit, including other gates.
This is a requirement that has not been met by previous NEMS work, usually for
one of two reasons: (i) the gate is bookended by a pair of electrical
actuators, which have unscalable efficiencies on the order of $10^{-7}$ [19];
or (ii) the gate outputs a mechanical signal, but in a different mode which
cannot serve as input for another gate [26]. In contrast our device outputs
ultrasonic waves in the same mode as the input, allowing multiple devices to
concatenate in the same circuit.

  4. 4.

Feedback prevention (output does not affect input)

To sequence multiple logic gates, the output channel of a gate should not
affect either of its input channels. Our logic gate in its simplest form does
not have this property, because energy from the resonator leaks into both the
input and output waveguides. However, feedback prevention can be achieved by
pairing the gate with an ‘acoustic transistor’ device that we have designed.
We propose using transistors immediately logic gates, as illustrated in Figure
6 e), prevent unwanted feedback in a logic circuit.

The transistor device has been simulated with a finite difference time domain
code [32]. Figure S5 (a) provides a diagram of the device and the pump, input,
and output waveguides. The pump is blue detuned and therefore by default
transmits inefficiently across the resonator to the output waveguide.
Efficient transmission is enabled by the gate signal, which increases the
resonance frequency of the resonator through Duffing stiffening. The gate
signal thereby determines the output signal. Feedback in the opposite
direction is prevented because the gate and pump signals induce orthogonal
modes in the resonator, which do not couple. This was tested in simulation by
allowing and disallowing reflections in the output waveguide: Figure S5
(b)-(c) show that the reflections did not change the amplitude in the input
waveguide.

  5. 5.

Complete set of Boolean operations

A logic device should be functionally complete, that is, able to construct any
desired truth table. Functional completeness is certainly provided by the set
of all standard logic gates (AND, NOT, OR, and so on), however it can also be
provided by smaller subsets. In particular, sequences of NAND gates can
reproduce the truth tables of any other gate [7]. Therefore by achieving a
NAND gate we have shown that Boolean completeness may be achieved within our
nanomechanical architecture. We also note that other logic gates can be
achieved directly without sequencing NAND gates—for example, we experimentally
achieved OR and AND gates.

Figure S5: (a) Diagram of the transistor mechanism, where the source, drain
and gate signals are provided by the bottom, top, and left waveguides
respectively. Each waveguide contains absorbing regions that were toggled on
and off in the simulation. (b) Amplitude in gate waveguide when pumping the
resonator and inputting a ‘1’ (high amplitude) gate signal. The plot contains
two time series drawn as red and blue lines, which correspond to running the
simulation with and without absorption in the drain waveguide. The series are
identical which shows reflections are not affecting the gate waveguide. (c) As
previous but with a ‘0’ (low amplitude) gate signal.

##  S7 Photocurrent derivation

To measure the displacement of the meshed membrane, we use an optical
heterodyne detection system. In this measuring system, the optical probe field
$E_{p}$ is reflected off the oscillating membrane (see main text figure 3).
For a given polarisation, the probe field can be expressed as:

$E_{\textnormal{p}}=A_{\textnormal{p}}\exp(i\omega t+\phi),$ (31)

where $A_{\textnormal{p}}$ is the amplitude of the field, $\Omega$ its
frequency, $\phi$ is the phase modulation created by the change of the light
path length due to the membrane motion. Here for simplicity, we have neglected
any noise in amplitude and phase of the field and we have chosen a fixed
observation position at the position of the detector. Similarly, the local
oscillator field $E_{LO}$ can be expressed as:

$E_{\textnormal{LO}}=A_{\textnormal{LO}}\exp\left[i(\omega+\omega_{\textnormal{AOM}})t\right]$
(32)

where $A_{\textnormal{LO}}$ is the amplitude of the field and
$\omega_{\textnormal{AOM}}$ is the frequency shift created by the acousto
optical modulator (AOM). These two fields are interfered and once detected,
the photocurrent $i$ transduced by the detector is proportional to:

$i\propto
A_{\textnormal{p}}^{2}+A_{\textnormal{LO}}^{2}+2A_{\textnormal{p}}A_{\textnormal{LO}}\cos(\omega_{\textnormal{AOM}}t-\phi).$
(33)

Since the membrane is driven with a sinusoidal excitation at frequency
$\omega$, $\phi$ can be expressed as $\phi=A_{\phi}\cos(\Omega t)$ with
$A_{\phi}$ the amplitude of the phase modulation in radian. Using the Jacobi-
Anger relation, to first order, the response of the detector at the frequency
$\omega_{\textnormal{AOM}}\pm\Omega$ is:

$i(\omega_{\textnormal{AOM}}\pm\Omega)\propto-\frac{1}{2}A_{\textnormal{p}}A_{\textnormal{LO}}A_{\phi}\left[\sin(\omega_{\textnormal{AOM}}-\Omega)t\right)-\sin\left((\omega_{\textnormal{AOM}}+\Omega_{\textnormal{m}})t)\right],$
(34)

with the amplitude of the photocurrent at the frequencies
$\omega_{\textnormal{AOM}}\pm\Omega$ is proportional to the amplitude of the
membrane motion. Meanwhile the photocurrent at frequency
$\omega_{\textnormal{AOM}}$ is given by

$i(\omega_{\textnormal{AOM}})\propto
A_{\textnormal{p}}A_{\textnormal{LO}}\cos(\omega_{\textnormal{AOM}}t).$ (35)

Furthermore, to first order the ratio
$i(\omega_{\textnormal{AOM}}\pm\Omega)/i(\omega_{\textnormal{AOM}})=A_{\phi}/2$.
Knowing that $A_{\phi}=2\pi$ corresponds to 780 nm, the free space wavelength
of the laser, we can directly retrieve the amplitude of motion of the membrane
in nanometers. Typically, in our experiment, when the membrane is driven to
high amplitude, the amplitude of the photocurrent at
$\omega_{\textnormal{AOM}}$ and at $\omega_{\textnormal{AOM}}\pm\Omega$ is
0.14 VRMS (-4 dBm) and 0.022 VRMS (-20 dBm) respectively which give an
amplitude of motion of $\sim$19 nm.

##  S8 Scaling with resonator size

Here we quantify the gate energy cost scaling with size of a resonator. We
consider an out-of-plane flexural motion of a square membrane of side $L$,
thickness $h$, density $\rho$, Youngs’s modulus $Y$, under tensile stress
$\sigma$. The mode profile $u$ takes the form:

$u(x,y)=\sin\left(\frac{n\pi x}{L}\right)\sin\left(\frac{m\pi y}{L}\right).$
(36)

We consider hereafter the fundamental $n=1;m=1$ mode, with effective mass
given by Eq.(37):

$m_{\mathrm{eff}}=\frac{1}{4}L^{2}h\rho,$ (37)

its spring constant, given by Eq.(38), is:

$k=\frac{1}{2}\pi^{2}\,h\,\sigma,$ (38)

and the nonlinear Duffing coefficient given by Eq.(10) is:

$\alpha=\frac{9\pi^{4}\,h\,Y}{64\,L^{2}}.$ (39)

In the linear regime, the oscillation has a small-amplitude, the resonance
frequency $\Omega_{\textnormal{m}}$ is:

$\Omega_{\textnormal{m}}=\sqrt{\frac{k}{m_{\mathrm{eff}}}}=\frac{\sqrt{2}\,\pi}{L}\sqrt{\frac{\sigma}{\rho}},$
(40)

where we recognize the speed of sound $\sqrt{\frac{\sigma}{\rho}}$. The
critical amplitude $x_{\mathrm{crit}}$ given by Eq.(18) is:

$x_{\mathrm{crit}}=\frac{8L}{3\sqrt{3}\pi}\sqrt{\frac{\sigma}{QY}},$ (41)

with $Q$ the mechanical quality factor. The critical amplitude is linearly
proportional to the resonator size $L$. The critical energy
$E_{\mathrm{crit}}$ associated with reaching this critical amplitude is given
by:

$\boxed{E_{\mathrm{crit}}=\frac{1}{2}k\,x_{\mathrm{crit}}^{2}=\frac{16L^{2}h\,\sigma^{2}}{27QY}.}$
(42)

This energy scales with the resonator area $L^{2}$ and thickness $h$,
underscoring the merits of miniaturization. A large Young’s modulus is also
beneficial, as it increases the Duffing nonlinearity (which arises in response
to an elongation of the material - see Eq. (10)) and therefore reduces the
critical energy. Reducing the tensile stress $\sigma$ also reduces the
critical energy, at a cost of a reduced operational frequency and reduced
impedance mismatch between the suspended membrane and the substrate, leading
to increased acoustic radiative losses.

As an example, here we estimate the prospects for miniaturization of our
platform considering a resonator made of a membrane with thickness as low as
10 nm and tensile stress of $\sigma\sim 10$ MPa.

We use Eq. (42) to estimate the potential energy cost reduction for nonlinear
nanomechanical logic. A significant step forward towards the can be achieved
by using a resonator made of a Si3N4 membrane with thickness of 10 nm and
tensile stress of $\sigma\sim 10$ MPa and a $Q$ factor of $\sim 4000$. With
these feasible parameters, our platform would operate at the Landauer limit,
having a resonance frequency of $\sim 180$ MHz and speed of gate operations
$\sim 50$ kHz. The ultimate reduction in thickness can be achieved through the
use of a purely two-dimensional nanomechanical resonator, made from a 2D
material such as graphene [8, 9, 10, 11, 45, 12, 13] or molybdenum disulfide
(MoS2) [14]. Such materials have been employed to build high-Q suspended
membrane resonators exhibiting hardening Duffing nonlinearity [12, 14], and
are therefore compatible with our scalable approach to nanomechanical
computing.

We consider a square single-layer graphene drum resonator operating at the
Landauer limit with critical energy $E_{\mathrm{crit}}=\log(2)k_{B}T$. The
required parameters for this resonator are shaded in Table S2 and compared
with our current platform.

Parameter |  Symbol |  Meshed Si4N3 Membrane |  Graphene Membrane |  Unit
---|---|---|---|---
Young’s modulus | Y | 250 | 1000 | GPa
Size | $L$ | 80 | 0.1 |  $\mu$m
Quality factor | $Q$ | 28000 | 60 | NA
Resonance frequency | $\Omega_{\textnormal{m}}/2\pi$ | 4.1 | 20000 | MHz
Duffing coefficient | $\alpha$ | 2.6 $\times 10^{13}$ | $4.1\times 10^{17}$ | N/m3
Critical amplitude | $x_{\mathrm{crit}}$ | $15\times 10^{-9}$ | $1\times 10^{-10}$ | m
Critical energy | $E_{\mathrm{crit}}$ | $2.6\times 10^{-14}$ | $1.23\times 10^{-21}$ | J
Critical energy | $E_{\mathrm{crit}}$ | $7.2\times 10^{6}$ | $\log(2)$ | $k_{\textnormal{B}}\,T$
Table S2: The parameters in the shaded region for Si4N3 are determined from
our current experimental system. The parameters in the shaded region for
graphene membrane are chosen as feasible experimental values for platform
miniaturization. The rest of the parameters are derived here from Eqs. (39),
(41) and (42). These parameters are represented as dashed lines in Fig. S6
calculated assuming a square membrane resonator. These values are in good
agreement with those derived from FEM simulations. The small differences arise
due to the presence of the tunnel barriers which modify the mode shape, which
is not taken into account here.

In Fig.S6 we plot the predicted operating frequency (Fig.S6a), speed of gate
operations given by the dissipation rate (Fig.S6b) and the minimum required
quality factor (Fig.S6c), all for a nanomechanical logic gate based on a
single layer graphene membrane. These predictions are calculated for three
different intrinsic stresses and as a function of the size of the graphene
drum. For the case where the drum size is $L=100~{}$nm, Eqs. ((37), (40),
(42)) predict a resonator with a resonance frequency of $\sim 20$ GHz and gate
speed in the range $\sim$100 MHz. These calculations were done using a Young’s
modulus for graphene on the order of 1 TPa [8, 12, 9, 10], a density
$\rho_{\mathrm{2D}}=8\times 10^{-19}$ kg/$\mu$m2 [11, 45] for single-layer
graphene, a thickness $h=0.3$ nm and tensile stress up to $\sigma=250\times
10^{6}$ Pa (consistent with the literature [15, 16]).

Figure S6: A graphene drum resonator operating at the Landauer limit for three
cases of intrinsic stress $\sigma=2.5$, 25 and 250 MPa. a) Operating frequency
of a graphene drum as a function of the drum size. b) Dissipation rate of the
drum resonator, therefore maximum gate speed. c) Lower bound to operate at the
Landauer limit for the quality factor of the graphene drum as a function of
the drum size.

All parameters are summarized in Table S2. This change in resonator design
leads to a significant reduction in $E_{\mathrm{crit}}$ of the order of 7
million, down to a value of only $\ln 2k_{B}T$ reaching the Landauer limit for
irreversible computing [6].

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