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Bistable metamaterial for switching and cascadingelastic vibrationsOsama R. Bilala,b,c,1, Andre Foehrb,c, and Chiara Daraioc,1

aInstitute of Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland; bDepartment of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich,Switzerland; and cDivision of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125

Edited by Rajeev Ram, Massachusetts Institute of Technology, Cambridge, MA, and accepted by Editorial Board Member Evelyn L. Hu March 17, 2017(received for review November 3, 2016)

The realization of acoustic devices analogous to electronic sys-tems, like diodes, transistors, and logic elements, suggests thepotential use of elastic vibrations (i.e., phonons) in informationprocessing, for example, in advanced computational systems,smart actuators, and programmable materials. Previous experi-mental realizations of acoustic diodes and mechanical switcheshave used nonlinearities to break transmission symmetry. How-ever, existing solutions require operation at different frequenciesor involve signal conversion in the electronic or optical domains.Here, we show an experimental realization of a phononictransistor-like device using geometric nonlinearities to switch andamplify elastic vibrations, via magnetic coupling, operating at asingle frequency. By cascading this device in a tunable mechan-ical circuit board, we realize the complete set of mechanicallogic elements and interconnect selected ones to execute simplecalculations.

phononic metamaterials | tunable materials | phonon switching andcascading | phononic computing | acoustic transistor

The idea of realizing a mechanical computer has a long estab-lished history (1). The first known calculators and comput-

ers were both mechanical, as in Charles Babbage’s concept of aprogrammable computer and Ada Lovelace’s first description ofprograming (2). However, the discovery of electronic transistorsrapidly replaced the idea of mechanical computing. Phononicmetamaterials, used to control the propagation of lattice vibra-tions, are systems composed of basic building blocks, (i.e., unitcells) that repeat spatially. These materials exhibit distinct fre-quency characteristics, such as band gaps, where elastic/acousticwaves are prohibited from propagation. Potential applications ofphononic metamaterials in computing can range from thermalcomputing (3–5) (at small scales) to ultrasound and acoustic-based computing (6, 7) (at larger scales). Phononic devices anal-ogous to electronic or optical systems have already been demon-strated. For example, acoustic switches (8, 9), rectifiers (10, 11),diodes (12–15), and lasers (16, 17) have been demonstratedboth numerically and experimentally. Recently, phononic com-puting has been suggested as a possible strategy to augment elec-tronic and optical computers (18) or even facilitate phononic-based quantum computing (19, 20). All-phononic circuits havebeen theoretically proposed (21, 22) and phononic metamateri-als (23–26) have been identified as tools to perform basic logicoperations (6, 7). Electromechanical logic (27) and transistors(28) operating using multiple frequencies have also been demon-strated. Most of these devices operate using electronic signals(26) and/or operate at mixed frequencies. When different fre-quencies are needed for information to propagate, it becomesdifficult, if not impossible, to connect multiple devices in acircuit.

Electronic transistors used in today’s electronic devices arecharacterized by their ability to switch and amplify electronicsignals. Conventional field-effect transistors (FETs) consist of atleast three terminals: a source, a drain, and a gate. The switchingfunctionality takes place by applying a small current at the gate tocontrol the flow of electrons from the source to the drain. Due to

the big difference between the low-amplitude controlling signal(in the gate) and the higher amplitude controlled signal (flowingfrom the source to the drain), one can cascade an electronic sig-nal by connecting multiple transistors in series to perform com-putations. In this work, we experimentally realize a phononictransistor-like device that can switch and amplify vibrations withvibrations (operating in the phononic domain). Our phononicdevice uses elastic vibrations at a gating terminal to control trans-mission of elastic waves between a source and a gate, operatingat a single frequency (f0 = 70 Hz; Materials and Methods andSupporting Information). It consists of a one-dimensional arrayof geometrically nonlinear unit cells (representing the metama-terial), connecting the source to the drain (Fig. 1A). Each unitcell is composed of a spiral spring containing a magnetic massat its center. The magnetic mass allows the stiffness of the spi-ral spring to be tuned by an external magnetic field. The meta-material is designed to support a resonant, subwavelength bandgap, whose band edges vary with the stiffness of the spiral springs(Fig. 1B). To tune the mechanical response of the metamaterial,we place an array of permanent magnets under the unit cells. Wecouple the permanent magnets to a driven, magnetic cantilever(herein referred to as the gate, Fig. S1), designed to resonateat a frequency f0 = 70 Hz. When the gate is excited by a rel-atively small mechanical signal, the resonance of the cantilevershifts the array of magnets (Movie S1) and tunes the transmis-sion spectrum of the metamaterial (Fig. 1 B and C). The energypotential of the gating system is bistable (Fig. S2) and providescontrol of the transmission of a signal from the source to thedrain (Fig. S3). We characterize the transistor-like device exper-imentally (Materials and Methods and Supporting Information)by fixing one end of the metamaterial structure to the source

Significance

One of the challenges in materials physics today is the real-ization of phononic transistors, devices able to manipulatephonons (elastic vibrations) with phonons. We present a real-ization of a transistor-like device, controlled by magnetic cou-pling, which can gate, switch, and cascade phonons withoutresorting to frequency conversion. We demonstrate all logicoperations for information processing. Because our systemoperates at a single frequency, we use it to realize experi-mentally a phonon-based mechanical calculator. Our device canhave a broad impact in advanced acoustic devices, sensors, andprogrammable materials.

Author contributions: O.R.B. and C.D. designed research; O.R.B. and A.F. performedresearch; O.R.B., A.F., and C.D. analyzed data; and O.R.B., A.F., and C.D. wrote thepaper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.R. is a guest editor invited by the EditorialBoard.1To whom correspondence may be addressed. Email: daraio@caltech.edu or bilalo@

ethz.ch.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1618314114/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1618314114 PNAS | May 2, 2017 | vol. 114 | no. 18 | 4603–4606

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Fig. 1. Phononic switching and cascading. (A) Schematic diagram of ourdevice, including a zoom-in view of a unit cell. A, Inset shows the anal-ogy with a typical FET electronic device. (B) Conceptual visualization ofthe switching principle. As the unit cell shape changes, the transmissionspectrum of the signal through the metamaterial shifts between ON/OFFstates. The shaded area highlights the band gap in the OFF configura-tion. (C) Experimental transmission spectrum of the transistor in the ON(solid black line) and OFF (brown dotted line) states. C, Inset shows a timeseries of the mechanical signal at the operational frequency (marked bythe green dashed line). (D) Transistor as a switch: experimental measure-ment of the transmission at the drain as a function of gate input, for dif-ferent source input. (E) Transistor as an amplifier: two connected transis-tors with Inset showing the relative measured signal amplitudes when gate1 is OFF (gray) and ON (green). (F) Experimental time series of the signalmeasured at gate 1 and drain 2. F, Inset shows the circuit in analogy toelectronics.

(i.e., a mechanical shaker) and the other end to the drain (a fixed,rigid support). The signal at the drain is measured using a laserDoppler vibrometer. The operation frequency of the transistoris indicated by the dashed-green line in Fig. 1C. The gate tunesthe transmission signal from the source to the drain, acting as aswitch for a harmonic, elastic wave with frequency f0. In Fig. 1C,the black and brown transmission lines represent 0 and 1 statesand the highlighted orange region is the band gap obtained fromthe numerical analysis of the infinite periodic medium theory, asin Fig. S4. The ratio between transmission coefficients in the ONand OFF states is 12.5 in Fig. 1C. Whereas most transistors retainasymmetric transmission, transistor models (e.g., junction gatefield-effect transistor) exist that require one to choose, beforeusing them, which terminal to assign to a drain and a source (i.e.,it is symmetric with respect to transmission from the source andthe drain). Moreover, if an asymmetric transmission is required,a defect can be simply incorporated into the metamaterial array(15); however, this might come at the expense of operatingat a unified single frequency (because of the nonlinear modeconversion).

To demonstrate the switching functionality of the transistor,we measure the transmission at the drain as a function of gateinput (Fig. 1D). As we increase the vibration amplitude at thesource, we observe an increase in the efficiency of the transis-tor as a switch. For instance, when the source input is relativelysmall, e.g., the velocity amplitude of the driving signal is set to15 mm/s, the gate can be opened (switching the system ON) witha gate excitation of 5 mm/s, with a source-to-gate input ratio of3. If the source input is larger, e.g., 45 mm/s, a smaller excitationof the gate, 3 mm/s, is sufficient to switch the system ON, with asource-to-gate input ratio of 15. Stronger source signals translateto larger displacements of the metamaterials that, in turn, affectthe coupling strength between the array of control magnets andthe magnets in the metamaterial. The strengthening of this cou-pling leads to an increase in efficiency in the gating functionality,defined as source-to-gate input, by at least a factor of 5 in theconfigurations tested experimentally.

To demonstrate signal cascading, we connect two transistorsin a mechanical circuit (Fig. 1E). During the experiments, thesources of both transistors (sources 1 and 2) are always turnedON, and the transmission at gate 1, drains 1 and 2 is measuredwith a laser vibrometer (Fig. 1E). We normalize the signal atboth drain terminals to its maximum amplitude of transmission,to visualize the difference between the ON and OFF states. Theleakage at the two drain terminals is less than 10% of the trans-mission amplitude when the gate is closed (Fig. 1E, Inset). Thisvalue is close to the background noise level captured by the laservibrometer in absence of excitation. To show the cascading effectbetween the two transistors, we consider the time signal of gate1 as the circuit input and the measured signal at drain 2 as thecircuit output (Fig. 1F). Once gate 1 is open, both drain 1 anddrain 2 gain transmission, with a delay of less than 1 s. The twosignals (drain 2 output)/(gate 1 input) demonstrate an amplifica-tion factor bigger than 5. This value can be increased by changingthe excitation amplitude of source 2.

Modern electronic devices use connected stacks of switches toperform logic operations. For example, in an AND logic opera-tion, when two switches are connected in series, no signal passesthrough unless both switches are open. In other words, the out-put of the circuit is 1 (true) when both inputs are 1 (true)and 0 (false) otherwise. If two switches are connected in par-allel, when either one of them is open (true), the circuit pro-duces an output signal (true). This circuit represents an OR gate.Following the same approach, one can create all of the basicelectronic logic gates, using switches as basic building blocks(Fig. S5). In our work, we use a configuration of four intercon-nected switches to experimentally realize phononic logic gatesanalogous to all existing electronic ones. We refer to this con-figuration as the universal phononic logic gate (UPLG) (Fig.2A). One terminal (Fig. 2A, Left) of the UPLG is connectedto a vibration source that excites harmonic elastic waves. Wemeasure the signal transmitted to the other terminal (Fig. 2A,Right) in the location indicated by the blue and orange dots.To implement all of the logic operations, we change the cou-pling between the magnetic control stages (Mi , with i = 1,...,4)for each gate in a systematic manner (Fig. 2B). For instance,to have an AND gate, we connect M1 and M3 into one mag-netic stage that could be driven by a magnetic cantilever as a sin-gle control unit, representing a binary state, A. Stages M2 andM4 are coupled in a similar manner representing another binarystate, B. The positions of the magnetic stages, resembling theAND gate functionality, represent the logic operation 0 AND 0resulting in 0 (i.e., all switches are black/OFF, Fig. 2B). Anotherexample is the XNOR gate; both pairs M1,3 and M2,4 are cou-pled, but with a different coupling distance than the AND gate.The XNOR represents the operation 0 XNOR 0 that resultsin 1 (two green/ON switches along the same transmission line,Fig. 2B). The numerical simulations of the operations 1 UPLG

4604 | www.pnas.org/cgi/doi/10.1073/pnas.1618314114 Bilal et al.

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Fig. 2. Phononic logic gates. (A) Schematic of four interconnected switches.(B) Schematic of the relative positions of the magnetic stages in referenceto the metamaterial for the logic gates representing the first row in thetable (A = 0 and B = 0). The NOT gate operates on a single input (A = 0).(C) Numerical simulations of the different logic gates representing the sec-ond row in the table in B. (D) Experimental time series for different gates.Orange and blue curves correlate with the dots in A and C. The red, dashedlines in A–C are the reference frames for the magnetic control stages M1−4.The green/ON and black/OFF represent the position of the stage, relative tothe metamaterial. The magnetic stage is either underneath the spiral springs(ON) or displaced laterally by 6 mm (OFF). The cyan arrows represent the log-ical inputs A and B. The vibration source powering the logic gates with theharmonic excitation F is always on.

0 (Fig. 2C) agree with the corresponding experimental measure-ments (Fig. 2D). In experiments, we measure the output at thetwo end points of the UPLG, to characterize each transmissionline and define the output of the gate as the summation of thetwo signals. We define logical 1 as signals with a transmissionamplitude bigger than two-thirds of the maximum signal trans-mitted through 14 resonators (one line of resonators in Fig. 2A)and logical 0 as signals with less than one-third of that maximum.For example, for the logic operation 1 OR 0 (Fig. 2D, Center Left)one transmission line is open (blue) whereas the other is closed(orange). In this case, we measure about 10% transmission leak-age from the ON to the OFF line.

The realization of all logic gates is the basis for performingcomputations in various devices. In a binary calculator for exam-ple, the addition operation is executed in the calculator circuitboard by a group of contiguous full adders (Fig. 3A). The numberof binary bits a calculator can add is equivalent to the number of

full adders it contains. Each of these adders can be split into twohalf adders, each composed of interconnected XOR and ANDgates (Fig. 3B). The circuit board for a mechanical full addercomposed of two half adders (Fig. 3C) is similar to an electroniccalculator, in which the two half adders are linked together byan OR gate. We experimentally construct the phononic circuitboard for a simple binary calculator. We demonstrate, withoutany loss of generality, the calculator carrying out the operation1 + 1. To add two bits, only a single half adder is needed (thereis no carry-in from any previous operation). The experimentallyobtained time-series envelope of the operation 1 + 1 is presentedin Fig. 3G. Fig. 3D shows the mechanical signal as it propa-gates in the circuit in red lines, whereas the attenuated signalis plotted in black lines. The difference between the transmittedvs. the attenuated signal is more than one order of magnitude,drawing a distinct line between the 0 and 1. The finite-elementmode shapes (Fig. 3E) of the same operation agree with the mea-sured transmission pathway. The realization of a circuit boardfor a full-fledged mechanical calculator (addition, subtraction,

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Fig. 3. Phononic calculator. (A) Schematic of the input/output signals ina full adder with its truth table. (B) Detailed schematic of the full adderbroken down into its basic logic elements. (C) Schematic of the mechani-cal analogues full adder composed of two phononic half adders. In A–C,green lines indicate a signal between two different interconnected fulladders. (D) Schematic of a half adder with logical inputs (A = B = 1) indi-cated by red lines. (E) Numerical simulations of a mechanical half adder.(F) Schematic of the relative position of the magnetic stages to the meta-materials. (G) Experimental time series for the half adder. The transmittedsignal is normalized by the maximum transmission amplitude at the mea-surement (right) end of the phononic half adder. D–G represent the execu-tion of the operation 1 + 1.

Bilal et al. PNAS | May 2, 2017 | vol. 114 | no. 18 | 4605

multiplication, and division) using the same design principle caneasily follow. Although our proof-of-principle experiments wereperformed at the macroscopic scale, the planar geometry of ourdevices makes them scalable to micro/nanoscales. Miniaturiza-tion of the concept would allow the realization of devices thatoperate at higher-frequency ranges and speeds. For instance, ifwe scale the unit cell size to 25 µm, a dimension easily achievablewith conventional microfabrication tools, the device will operatein the ultrasonic frequency range ∼70 kHz. The speed of switch-ing will scale by a power of 4 [mass scales approximately O(3) andthe distance scales linearly]. This translates to a possible switch-ing speed much higher than the frequency of operation.

This work presents a conceptual design of a dynamicallytunable phononic switch and provides experimental realiza-tion for its functionality. The connection of multiple elementsin a circuit allows it to perform computing operations. Thisprovides an experimental demonstration of switching and cas-cading of phonons at a single frequency and serves as thebasis for mechanical information processing, which can have animpact in soft robotics, programmable materials, and advancedacoustic devices.

Materials and MethodsFor the experimental realization of the phononic transistor-like device, thetunable metamaterial, connecting source and drain, was fabricated frommedium-density fiberboard (MDF), using a Trotec Speedy 300 laser cutter.The mechanical properties of the MDF were characterized using a standardtensile test with an Instron E3000. The measured Young’s modulus is 3 GPaand the density is 816 Kg/m3. The quality factor of the metamaterial is ≈20

(Fig. S6). The unit cells in the metamaterial consist of four concentric andequally spaced Archimedean spirals, with lattice spacing of 25 mm. Cylindri-cal neodymium nickel-plated magnets, with a 3-mm diameter and a 3-mmthickness, were embedded in the center of each unit cell. An array of con-trol magnets with a 20-mm diameter and 5-mm thickness was placed on amovable stage below the array of unit cells. To control the movable stage,we coupled it to a vibrating cantilever (the gate) attached to a separatevibration source (Fig. 1A and Supporting Information). The gate is a rectan-gular cantilever of dimensions 125 mm × 15 mm, connected to the drain(right end) of the metamaterial. The gate-control magnets, attached to theside of the control stage, are squares, with 10-mm side length and 3-mmthickness, whereas the ones attached to the cantilever are rectangular with6-mm × 4-mm × 2-mm dimensions.

Mechanical oscillations in the source terminal were excited using a Mini-Shaker (Bruel & Kjær Type 4810) with two identical audio amplifiers (Top-ping TP22) and were then optically detected at the drain by a laser Dopplervibrometer (Polytec OFV-505 with a OFV-5000 decoder, using a VD-06 decodercard). A lock-in amplifier from Zurich Instruments (HF2LI) was used to filterthe signal. The frequency response functions plotted in Fig. 1C are the velocityamplitude at the output (drain) normalized to the signal at the input (source).The time series in Figs. 1 C and F, 2D, and 3G are obtained using an oscillo-scope (Tektronix DPO3014). The time signals in Figs. 2D and 3G are filteredusing a passband filter between 50 Hz and 90 Hz. The excitation amplitude atthe source is 15 mm/s in Fig. 1C. All of the numerical simulations are done inCOMSOL 5.1, using the structural mechanics module.

ACKNOWLEDGMENTS. The authors thank Sandeep Deshpande for assist-ing in the construction of the setup of the signal amplification experiment.The authors acknowledge the fruitful discussions with V. Costanza, R. DiGiacomo, and M. S. Garcia. This work was supported by ETH PostdoctoralFellowship FEL-26 15-2 (to O.R.B.) and ETH Grant ETH-24 15-2.

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Supporting InformationBilal et al. 10.1073/pnas.1618314114Gating SystemThe phononic transistor shown in Movie S1 can be representedby the schematic diagram in Fig. S1. A metamaterial, consist-ing of an array of seven spiral-spring resonators with embeddedmagnets (not shown in the schematic), connects the source (i.e.,a mechanical shaker) to the drain. A control stage, which con-tains seven magnets corresponding to the number of unit cellsin the metamaterial, is positioned 7 mm below the metamaterial.The control stage also contains two other magnets, one on eachside (left and right in the schematic diagram), which couple totwo magnetic, resonant cantilevers.

The ON/OFF switching occurs when a low-amplitude mechan-ical signal of frequency (f0) excites the magnetic cantilever, caus-ing it to resonate. This resonance disturbs the energy potential ofthe control stage due to its magnetic coupling with the cantileverand induces a shift of the control stage from left to right (Fig.S2). When the stage moves, the magnetic field between the con-trol stage magnets and the metamaterial magnets causes a shapechange in the spiral-spring geometry from a flat (2D) shape to ahelical (3D) shape (Fig. 1B). The shape change, in turn, shiftsthe band-gap frequency. This allows a mechanical signal withthe same frequency (f0) and higher amplitude to flow from thesource to the drain through the metamaterial.

Bistable PotentialThe system in Fig. S1 can be described analytically with a dis-crete mass-spring model. We assume the two cantilevers to beidentical, having a stiffness k1 and mass m1. Each cantilever iscoupled with the control stage by a magnet, which is representedby a nonlinear spring with stiffness k2. The control stage has amass m2 and has two hard-end stops represented by the nonlin-ear spring k3 in both direction (Fig. S2). The side-to-side distancebetween the two resting positions of the control stage is 6 mm,which is the required displacement to switch the transistor ONand OFF.

The forces acting on the control stage are emerging only fromthe magnetic cantilevers (k2) and the end stops of the stage (k3).Their potential can be approximated as

VTot = VS (x − xend) + VS (−x − xend)

+

∫(FMag(rS,L, pL, pS ) + FMag(rS,R, pR, pS ))∂x ,

where VS represents the displacement of the control stage, withxend = 3 mm. The magnetic potential is obtained by integrat-ing FMag over ∂x , where p is the polarization vector and ri,jis the vector connecting the two magnets i and j . The hard-end stops are represented by the nonlinear springs k3 that areone sided:

VS (∆x ) =1

2k [∆x ]2+.

The magnetic dipole–dipole interaction force is calculated usingref. 29,

FMag(r , pL, pS ) :=3µ

4π ‖ r ‖5 ((r × pL)× pS + (r × pS )× pL

−2r(pL · pS ) + 5r

‖ r ‖2 ((r × pL) · (r × pS ))),

where µ is the magnetic permeability of air, ≈ 4∗π10−7(N/A2).An approximation of the magnetic momenta pL, pR, and pS is ob-tained using the forces indicated by the manufacturer (www.

supermagnete.ch), using FMag for touching magnets. The vectorsrS,L and rS,R connect the stage to the cantilever magnets.

pL = pR = [2.24× 10−2, 0, 0]ᵀ

A∗m2,

pS = [−8.22× 10−2, 0, 0]ᵀ

A∗m2,

rS,L = [−x0,mags − x , d0 + dpL , 0]ᵀ m,

and rS,R = [x0,mags − x , d0 + dpR , 0]ᵀ m,

where

x0,mags = xend + 1.7× 10−3,

and d0 = 1.5× 10−3 m,

are the x and y components of the equilibrium position of thecantilever magnets relative to the centered stage. The param-eters dpL and dpR represent the deflection of the gate (i.e., ofthe cantilever) having a spring (k1). For a given deflection ofthe left- or right-hand side magnets that happens as a resultof the harmonic excitation at the gate in the physical system,the potential becomes asymmetric and unstable, which inducesa shift in the control stage position (Fig. S2). The energy poten-tial of the control stage when it is exactly in the center betweenthe two cantilevers is defined as the reference (i.e., equilib-rium) state. The potential of the control stage moving from leftto right or from right to left is plotted against the referencecase (Fig. S2). The amount of energy required to switch a sin-gle transistor from one state to the other is estimated to be≈70 µJ, corresponding to the required shift to modify the energypotential.

To demonstrate the stability of the proposed concept, wepresent the measured output signal in a NOT gate to alternatinginputs of ones and zeros (Fig. S3). The output signal is consistentbetween multiple cycles.

Band-Structure Analysis for the MetamaterialWe calculate the band structure of an infinite array of the spiral-spring unit cells, using the finite-element method. We solve theelastic wave equations in three dimensions and apply the Blochwave formulation in plane (i.e., Bloch boundary conditions) (30).We consider the wave propagation along the direction of peri-odicity, in the Γ-X direction (Fig. S4, Left). We then constructa finite system, corresponding to the experimental setup, con-sisting of an array of seven unit cells. We fix one end of thestructure and apply a dynamic load to the other end. We obtainthe frequency response function of the metamaterial in Fig. S4,Right. Both finite and infinite predictions agree well, particu-larly in the band gap frequency range, highlighted in gray. Theband-gap range also agrees well with the experimental results(Fig. 1C).

Different Realizations of Logic GatesThe unified logic platform (Fig. 2), designed to perform all ofthe mechanical logic operations, is composed of four intercon-nected transistors. However, some of these operations could alsobe performed with a lower number of transistors, in differentconfigurations. For example, the NOT operation requires onlya single transistor (Fig. S5A). The OR and NAND operationscan be achieved using only two transistors connected in parallel(Fig. S5B). Similarly, the AND and NOR operations can be real-ized with two transistors connected in series (Fig. S5C). The only

Bilal et al. www.pnas.org/cgi/content/short/1618314114 1 of 5

two gates that require four interconnected transistors are theXOR and XNOR (Fig. S5D).

Metamaterial Quality FactorWe characterize the quality factor, Q, of the spiral-spring res-onators in the metamaterial by measuring the resonance ampli-

fMagnetic Cantilever

Magnetic Cantilever

Cont

rol S

tage

+ -+ - ++ - + -

+

+

+

+

+

+

+

+

+

+

+

+

+

Source

Drain

f

f

Met

amat

eria

l

( )

Fig. S1. Schematic of the phononic transistor-like device and its components.

Magnetic-stage m Displacement (mm)

ReferenceLeft-to-RightReference

Right-to-Left

Pot

entia

l Ene

rgy

(J)

m m mk k

k kk k

3-3 -2 -1

10 -3

1

2

3

4

5

6

7

-3 -2 -1

10 -3

1

2

3

4

5

6

7

1 2 30 0 1 2

x

y

Magnetic Cantilever Control Stage

Magnetic Cantilever

Fig. S2. Schematic of the dynamically bistable switching mechanism approximated into masses and springs. Shown is the energy potential of the controlstage when it moved from one of its hard-end stops (k3) to the other end (blue and green lines). The red line represents the equilibrium state of the controlstage when at equal distance from the magnetic cantilevers.

tude of a single spring, using a mechanical shaker as a harmonicexcitation source and a laser Doppler vibrometer to detect thevelocity of the central mass of the spring. The calculated Q =f /∆f is ≈20 (Fig. S6), where f is the resonance frequency and∆f is the width of the frequency range for which the amplitudeis 1/√

2 of its peak value.

Bilal et al. www.pnas.org/cgi/content/short/1618314114 2 of 5

Time (s)10 12 14 16 18 20

Tra

nsm

issi

on

-1

-0.5

0

0.5

1

0 2 4 6 8

Fig. S3. Experimental time signal acquired for a NOT gate with 12 cycles of alternating input starting with a logical 1 at time t = 0.F

requ

ency

(H

z)

0

20

40

60

80

100

120

140

160

180

200

Transmission10-4 10-2 100

Fre

quen

cy (

Hz)

0

20

40

60

80

100

120

140

160

180

200

XWavenumber (1/m)

. . .

. . . . . .

. . . F

Fig. S4. (Left) Infinite dispersion curves in the Γ-X direction of the metamaterial, based on Bloch theorem. (Right) Finite frequency response function of ametamaterial consisting of seven unit cells. The band-gap region is highlighted in gray in both panels.

Bilal et al. www.pnas.org/cgi/content/short/1618314114 3 of 5

NOR

M1 M2 M1 M2

AND

OR

M2

M1

M3 M4

M1 M2

XOR

NAND

M2

M1

M3 M4

M1 M2

XNOR

M1

NOT

M1M1

OFFON

A B

C

D

(One transistor) (Two transistors in parallel)

(Two transistors in series)

(Four transistors)

Fig. S5. Schematic representation of the minimum number of transistors necessary to realize the various logic gates. (A) the NOT gate with one transistor.(B) Both OR and NAND gates with two transistors in parallel. (C) Both AND and NOR gates with two transistors in series. (D) Both XNOR and XOR gateswith four interconnected transistors. Similar to Fig. 2, the red, dashed lines are the reference frames for the magnetic control stages M1−4. The green/ONand black/OFF represent the position of the stage, relative to the metamaterial. The magnetic stage is either underneath the spiral springs (ON) or displacedlaterally by 6 mm (OFF). The cyan arrows represent the input signal.

Frequency (Hz)50 60 70 80 90 100

Am

plitu

de (

mm

/s)

100

101

Fig. S6. Experimental characterization of the quality factor of a single spiral-spring resonator.

Bilal et al. www.pnas.org/cgi/content/short/1618314114 4 of 5

Movie S1. The experimental realization of the transistor’s bistable gating system. By exciting the magnetic cantilever at one side of the metamaterial theenergy potential becomes asymmetric, causing the magnetic control stage to flip to the opposite side and vice versa.

Movie S1

Bilal et al. www.pnas.org/cgi/content/short/1618314114 5 of 5