Electrical Control of Surface Acoustic Waves
Linbo Shao, Di Zhu, Marco Colangelo, Dae Hun Lee, Neil Sinclair, Yaowen Hu, Peter T. Rakich, Keji Lai, Karl K. Berggren, Marko Loncar
EElectrical Control of Surface Acoustic Waves
Linbo Shao, ∗ Di Zhu, Marco Colangelo, Dae Hun Lee, Neil Sinclair,
1, 4
YaowenHu,
1, 5
Peter T. Rakich, Keji Lai, Karl K. Berggren, and Marko Lonˇcar † John A. Paulson School of Engineering and Applied Sciences,Harvard University, Cambridge, Massachusetts 02138, USA Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 01239, USA Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA Division of Physics, Mathematics and Astronomy, and Alliance for Quantum Technologies (AQT),California Institute of Technology, Pasadena, California 91125, USA Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA (Dated: January 6, 2021)Acoustic waves at microwave frequencies have been widely used in signal processing applications.They are also emerging as a promising approach for communication and manipulation of quantuminformation. However, dynamic control of acoustic waves in a low-loss and scalable manner remainsan outstanding challenge, which hinders the development of phononic integrated circuits. Here wedemonstrate electrical control of traveling acoustic waves on an integrated lithium niobate platformat both room and cryogenic temperatures. We electrically tune the material elasticity to modulatethe phase and amplitude of the acoustic waves and demonstrate an acoustic frequency shifter byserrodyne phase modulation. Furthermore, we show reconfigurable nonreciprocal modulation bytailoring the phase matching between acoustic and quasi-traveling electric fields. Our scalableelectro-acoustic platform comprises the fundamental elements for arbitrary acoustic signal processingand manipulation of phononic quantum information.
Acoustic waves in solids are the basis for numeroussignal processing applications [1] including microwave fil-ters, delay lines, and sensors. Furthermore, they can beused to provide an interface between quantum systemssuch as superconducting circuits [2–8], defect centers[9, 10], and optical devices [11–15]. Compared to elec-tromagnetic waves, acoustic waves feature five-orders-of-magnitude shorter wavelength and do not radiate intofree-space. This therefore allows coherent informationprocessing and manipulation in an ultra-compact foot-print with negligible crosstalk between devices and withthe environment. For these reasons, on-chip phononicsystems have emerged as a promising candidate for quan-tum computing and storage [15–17]. A phononic inte-grated circuit requires a few essential functionalities, in-cluding efficient transduction from and to electromag-netic waves at microwave frequencies, low-loss waveg-uiding and routing of acoustic waves, and, importantly,active control of the phase and amplitude of travelingacoustic waves with low energy consumption. The latteris especially important for low-temperature applications.Further, nonreciprocal acoustic isolators and circulatorsare needed for source protection, bi-directional commu-nication, and noise mitigation. While significant effortshave been made to realize passive phononic integratedcircuits and transducers, efficient modulation and isola-tion of acoustic waves have not been demonstrated, hin-dering the development of scalable phononic integratedcircuits. ∗ [email protected] † [email protected] Acoustic waveguides on chip have been realized usingsuspended structures [18, 19], including two-dimensionalphononic crystals [20], and high acoustic velocity sub-strates [21, 22]. Efficient transduction between travelingacoustic waves and electromagnetic waves have also beendemonstrated, leveraging piezoelectric coupling with mi-crowaves [18, 19, 21, 22] and optomechanical couplingwith light [20]. Controlling the amplitude and phase of anon-chip acoustic wave, on the other hand, has proven tobe more challenging since it requires changes in the mate-rial elasticity. Previously explored approaches based on,for example, acoustic four-wave mixing [21] and nonlin-ear mechanical cavities [19], were inefficient and requiredlarge acoustic powers due to the weak nonlinearity in theelastic response of most materials. On the other hand,strategies used to achieve acoustic nonreciprocity basedon nonlinear materials [23], circulating fluids [24], water-submerged phononic crystals [25], deformed water-air in-terfaces [26], and optomechanics [27], were all limited toacoustic frequencies below a few megahertz, which ren-ders them unsuitable for applications that require mi-crowave acoustic frequencies. Recent demonstrations ofnonreciprocal microwave phonon transmission based onelectric amplifiers [28] and semiconductor acoustoelectriceffects [29] are not suitable for applications in quantuminformation processing. Furthermore, approaches usingferromagnetic materials [30] require a magnetic field andtherefore are not compatible with quantum technologiesthat rely on superconducting circuits or spins in solids.Here we demonstrate electrical control of the funda-mental degrees of freedom of acoustic waves at microwavefrequencies – their phase and amplitude – using an in-tegrated lithium niobate (LN) electro-acoustic platform. a r X i v : . [ phy s i c s . a pp - ph ] J a n B LN SiNAl
500 μm
IDT IDTTaper TaperElectro-acoustic modulator A XY Z30ºDisplacement0 10.5 D P r opaga t i on l o ss ( d B / c m ) Temperature (K)
10 μm LNAlSiN C Modulation lengthModulationelectrodesPort 1 Port 2
AlLN
Bias electric field
Acoustic mode
FIG. 1.
Lithium niobate (LN) electro-acoustic platform. ( A ) Optical micrograph of a fabricated device. Bright regionsare aluminum (Al). The etched silicon nitride (SiN) layer showing dark boundaries is used to define the acoustic waveguide andregions where the interdigital transducers (IDT) are fabricated. IDTs are used to excite and detect the acoustic waves. Thedevice is on an X-cut LN substrate, and the acoustic waveguide is at 30 ° angle with respect to the crystal Z-axis (coordinatesindicated). The out-of-focus dark lines are scratches on the back side of the chip, which do not affect the device. ( B )Cross-section of the acoustic waveguide in the modulation region. The normalized displacement field intensity (blue shading)shows the simulated fundamental acoustic mode. The displacement field is exaggerated for visualization. Arrows indicate thesimulated electric field direction and magnitude due to a bias voltage on the modulation electrodes. ( C ) False-colored scanningelectron microscope image of the acoustic waveguide. ( D ) Measured propagation loss of the acoustic waveguide as a functionof temperature. Furthermore, we utilize these devices to shift the fre-quency of propagating microwave phonons, and to real-ize nonreciprocal phase modulation using quasi-travelingelectric fields. Modulation of the acoustic waves are en-abled by the electro-acoustic effect [31], which is equiv-alent to electro-optic effect widely utilized to controlthe phase and amplitude of optical signals. Electro-acoustic effect describes the change in the elasticity ofa solid due to an applied electric field, which resultsin a change of the phase velocity of traveling acousticwaves. The electro-acoustic effect is characterized by thethird-order piezoelectric tensor d . The change of elas-tic constants ∆ c due to the applied electric field E isgiven by ∆ c ij = d kij E k , where i , j , k can take val-ues of 1 to 3, corresponding to crystal X, Y and Z di-rections, and d kij is subject to the material symmetry[32]. However, since this electro-acoustic effect is relativeweak, bulk components show small phase changes [31]and are unsuitable for practical applications. Here weovercome this limitation by confining the acoustic waveto a wavelength-scale acoustic waveguide and placing themodulation electrodes closely across the waveguide, al-lowing π -phase shift to be achieved.Our electro-acoustic modulators are fabricated on anX-cut LN substrate (Fig. 1A), where we employ the third- order piezoelectric coefficients of LN by using guidedacoustic modes and applying electric fields mainly in theY direction. The acoustic waveguide is formed by cre-ating a 10 µ m slot inside thin silicon nitride (SiN) filmdeposited on top of LN (Figs. 1B and 1C). Since theacoustic velocity (index) of SiN is greater (smaller) thanthat of LN, an acoustic mode is confined (Fig. 1B). Wenote that our waveguide supports a Rayleigh-type acous-tic wave [1] with most of its strain field is in the XZcomponent. Inter-digital transducers (IDT) are used toelectrically excite and detect microwave acoustic waves.The pitch of the IDT finger electrodes is 650 nm andequal to the half wavelength of the acoustic waves at2.5 GHz. To optimize the transduction efficiency, thewidth of the IDT (75 µ m) is designed to be larger thanthe acoustic waveguide (10 µ m), and tapered waveguidestructures are used to couple the wave into the acousticwaveguide. Importantly, the waveguide is oriented along30 ° angle with respect to the crystal Z axis, as this direc-tion features the smallest acoustic velocity on the X-cutsurface and thus provides the best acoustic wave confine-ment (Fig. S1). Finally, aluminum (Al) electrodes aredeposited on the SiN layer and used to apply the electricfield needed for acoustic index modulation.Low loss is critical for realizing large-scale phononic in- E N o r m a li z ed po w e r ( d B c ) V = 2.3 V π pp -8 -6 -4 -2 0 2 4 6 8-40-200 N o r m a li z ed po w e r ( d B c ) Acoustic frequency shifting f - f (kHz) c Frequency-shifted Unmodulated d B -1 09095100 E ffi c i en cy ( % ) A B μ mBiasedUnbiased N o r m a li z ed a m p li t ude A c ou s t i c pha s e ( deg r ee ) Time (ms) / f mod RF signalgenerator Modulating signal Acoustic waveguide Real-timespectrumanalyzer200 μm f mod f = GHz c I npu t O u t pu t Scanning probe for TMIM L = 1 cm DC -V π V π Time
Serrodyne
FIG. 2.
Electro-acoustic phase modulation. ( A ) Experimental setup for characterizing the electro-acoustic phasemodulator. A signal generator is used to excite the acoustic wave via an IDT at carrier frequency f c = 2 .
483 GHz. Amodulating signal with frequency f mod is applied to the modulation electrodes, and a real-time spectrum analyzer is usedto detect the phase and amplitude of transmitted acoustic signal detected by an identical IDT. In the transmission-modemicrowave impedance microscopy (TMIM), a scanning probe is used to detect the acoustic field. ( B ) TMIM images showphase shift of the acoustic wave due to an applied bias voltage on the modulation electrodes. The scanning region is located atthe center of the waveguide near the output of the modulator. ( C ) Phase of the transmitted acoustic wave measured by thereal-time spectrum analyzer when the modulator is driven by a f mod = 10 kHz sine wave with peak-to-peak voltage V pp = 53V, showing full 180 ° phase oscillation. This indicates a half-wave voltage V π = 53 V. ( D ) By driving the phase modulatorharder, additional acoustic sidebands can be generated, resulting in formation of an electro-acoustic frequency comb. Here, an f c = 2 .
483 GHz acoustic wave is modulated by a 10 kHz sine wave with V pp = 2 . V π . ( E ) Serrodyne frequency shift of anacoustic wave by 1 kHz. This is achieved by applying a repeating linear voltage ramp (serrodyne) with V pp = 2 V π (Left inset).Right inset plots the acoustic powers in a linear scale, showing an efficiency of 92%. The spectral powers in (D) and (E) arenormalized to the unmodulated signal received by the spectrum analyzer. The results shown in this figure are measured fromthe same device with a modulation length of 1 cm. tegrated circuits. We measure the propagation loss of theacoustic waveguide [32] at different temperatures down to1 K (Fig. 1D). At room temperature (300 K) and undervacuum, the acoustic propagation loss is α = 17 dB/cm.It decreases with lowered temperature to α = 5 dB/cmat liquid nitrogen temperature (77 K) and < f c = 2 .
483 GHz and detect-ing phase and amplitude of the modulated acoustic wave(Fig. 2A). The insertion loss of the device, measured froma microwave signal applied to one IDT and detected afterthe other, is 10 dB at 1.3 K. This loss is dominated by thesymmetric IDTs that excite and collect acoustic wavesbidirectionally, which result in a 3 dB loss at each IDT, and the tapers that guide acoustic waves to the waveg-uide. At room temperature, the insertion loss increasesby 25 dB due to higher propagation loss and a lower ef-ficiency of IDTs (Fig. S2). We used transmission-modemicrowave impedance microscopy (TMIM), a scanningprobe technique that coherently measure the profiles oftraveling acoustic waves near the output of the modula-tor waveguide [32]. We observe a π/ V π of the modula-tor, we apply a sinusoidal signal at f mod = 10 kHz onthe phase modulator electrode and analyze the outputby a real-time spectrum analyzer. The phase change in-creases linearly with of the amplitude of modulating sig-nal (Fig. S3). When the peak-to-peak voltage ( V pp ) ofthe modulating signal reaches 53 V, a π -phase change N o r m a li z ed a m p li t ude Time (ms) 1012141618 M odu l a t i on v o l t age ( V ) C -30 -20 -10 0 10 20 300.00.20.40.60.81.0 N o r m a li z ed a m p li t ude Modulation voltage (V) -20 0 20-15-10-50 P o w e r ( d B c ) V = 29 V π B Mach-Zehnder modulatorModulating signal I npu t O u t pu t A Acoustic waveguide 200 μm
FIG. 3.
Electro-acoustic amplitude modulation. ( A ) Schematic of the electro-acoustic Mach-Zehnder modulator. Themodulating signal is applied at the middle electrode while two outer electrodes are ground, and thus two acoustic waveguidesexperience electric fields with opposite polarities. ( B ) Measured output acoustic amplitude (main figure) and power (Inset)with slowly varying modulating voltage to determine the V π of the modulator. ( C ) Measured acoustic amplitude with a weakmodulating signal at f mod = 10 kHz and biased at 0 . V π . of the received acoustic wave is observed at room tem-perature (Fig. 2C), inferring a V π = 53 V. At 1.3 K, thepeak transmission frequency shifts to 2.532 GHz, and the V π increases to 133 V (Fig. S3). An important figure ofmerit for the modulator, the product between its half-wave voltage V π , length L , and propagation loss α , i.e., V π Lα , is reduced by a factor of 7 (from 900 V · dB to120 V · dB) at 1.3 K owing to reduced propagation losscompared to room temperature. The V π of the phasemodulator could be further reduced by a factor of 2 byusing narrower acoustic waveguides, which is possible us-ing materials with higher acoustic contrast. In addition,we measure the 3-dB bandwidth to be 110 kHz and ob-serve zero modulation at f m od = 336 kHz when the pe-riod of the modulating signal matches the traveling timeof the acoustic wave (Fig.S4).We demonstrate two proof-of-concept applications –electro-acoustic frequency comb and acoustic frequencyshifting. By driving the phase modulator with a 10kHz sinusoidal signal of V pp = 2 . V π , we generate 19equidistant frequency comb lines centered at f c = 2 . π phase allows us to demonstrate acoustic fre-quency shifting using a serrodyne approach. Specifically,by applying a repeating linear voltage ramp signal atthe frequency of 1 kHz with V pp of 2 V π , the modulatedacoustic wave experiences an approximately linear phaseramp in time, which results in change (shift) in frequencyof the acoustic wave. We measure shift efficiency of 92% (Fig. 2E), defined as the ratio of detected acoustic powerat the shifted frequency and the power of unmodulatedacoustic wave. The carrier suppression of our frequencyshifter is 21 dB.We realize amplitude modulation of acoustic wave byconstructing an acoustic Mach-Zehnder interferometer(MZI) in the push-pull configuration (Fig. 3A). The in-put acoustic wave is split equally between two MZI arms,and the two waves experience different phase shifts asthey propagate in each arm. The phase difference is con-trolled by the voltage applied to the electrodes. Whenthe two waves recombine, the acoustic interference yieldsan amplitude modulation. Maximum output amplitudeoccurs when the phase difference between the two pathsis zero (no voltage applied) or an even integer number of π , while the minimum amplitude occurs when the phasedifference is an odd integer number of π . We measure V π = 29 V at quasi-DC frequency for an 8-mm-longacoustic Mach-Zehnder modulator. The extinction ra-tio between the maximum and minimum output power isover 15 dB (Fig. 3B), which could be limited by the fab-rication imperfection of the Y-splitter or the existing ofhigher order modes of the waveguide. When the modula-tor is biased to the quadrature point (i.e., 50% transmis-sion), the amplitude of the output acoustic waves followsthe small input signal accordingly (Fig. 3C).To achieve nonreciprocal transmission of acousticwaves, we employ a quasi-travelling electric field to breakthe symmetry of counter-propagating acoustic waves. Byseparating modulation electrode into three segments, wecan control the wave number (momentum) of the quasi-traveling electric field by adjusting the relative phase of A c ou s t i c pha s e ( deg r ee ) Time (ms)
Forward
Backward 180 120 60 0 -60 -120 -180-60-50-40-30-20-10 S i deband po w e r ( d B c ) Electrode phase delay (degree)
Forward
Backward
C D
Electrodes V GNDGNDGND P o r t Acoustic waveguide A TimeForward Backward B Forward Backward P o r t V V V V V t t t t t t t t t
200 μm
FIG. 4.
Nonreciprocal phase modulation of acoustic waves. ( A ) Schematic of the device used for nonreciprocalphase modulation. The modulation electrode is separated into three segments with independently controlled voltage V to V . The acoustic traveling time through each segment is t . ( B ) Illustration of nonreciprocal modulation of forward andbackward propagating acoustic waves. The voltage applied on each segment of the electrodes is progressively delayed by 120 ° .For forward traveling acoustic wave, the phase accumulated in each segment is the same and results in net modulation. Forbackward traveling acoustic wave, the accumulated phase in each segment is different and can be designed to result in a net zerophase shift. ( C ) Measured acoustic phases acquired for both forward and backward propagating acoustic waves. ( D ) Measuredmodulation sideband power of the forward and backward propagating acoustic waves for varying phase delays between thevoltages applied to the electrodes. The dashed line indicates the operating condition in (C) and a >
40 dB nonreciprocity. Themodulation frequency is f mod = 1 / (3 t ) = 336 kHz in (C) and (D). modulating signals applied to each electrode (Fig. 4A).This approach enables nonreciprocal acoustic phase mod-ulation when the quasi-traveling modulating signal isphase matched with the traveling acoustic wave in onedirection but mismatched with that in the opposite direc-tion (Fig. 4B and Movie S1). Maximum nonreciprocityoccurs when the signals applied to each succeeding mod-ulation electrode segments are phase delayed by 120 ° andwhen the modulation frequency matches the total trav-eling time of the acoustic wave, i.e., 1 /f mod = 3 t , where t is the time for acoustic wave to travel through oneelectrode segment. In this case, the forward propagatingacoustic wave always experiences the same phase modu-lation when it traverses the electrodes and thus resultsin maximum modulation. On the other hand, the back-ward propagating wave effectively experiences a full (2 π )modulation cycle, which results in no net phase change.To implement this concept, we fabricate such a nonre-ciprocal acoustic modulator with 1 cm length and applythe required modulation frequency f mod = 1 / (3 t ) = 336kHz for maximum nonreciprocity. At this condition, weobserve the presence (absence) of acoustic phase modu-lations in the forward (backward) propagation direction(Fig. 4C).The acoustic nonreciprocity can be adjusted by the rel-ative phase of the signals applied to the modulation elec-trodes (Fig. 4D). We observe a maximum nonreciprocity of over 40 dB of the modulation sideband power. Further-more, we sweep the modulation frequency and the phasedelay between the applied voltages on the three elec-trodes, and clearly observe maximum modulation side-band powers when the traveling acoustic wave and themodulating signal are phase-matched, and zero modula-tion when they are phase mismatched by any positiveinteger number of 2 π phases (Fig. S5). With additionalfilters and couplers, we could build acoustic isolators andcirculators based on our frequency shifter and nonrecip-rocal phase modulator.In conclusion, we demonstrate an integrated electro-acoustic platform that provides arbitrary control of on-chip traveling acoustic waves, namely their phase andamplitude. Compared with approaches based on op-tomechanics and nonlinear acoustics, our electro-acousticmodulators show significant advantages in terms of mod-ulation efficiency, simplicity in fabrication, and scalabil-ity. Taken together, these advantages may enable re-alization of large scale integrated acoustic informationprocessing systems. Using advanced nanofabrication, wecould push the operation frequency of our device to tensof GHz, covering the 5G millimeter wave bands. We ex-pect the half-wave voltage V π to decreases quadraticallywith higher frequency, as both the width of the acousticwaveguide and the acoustic wave number scale linearlywith frequency. Our devices may find applications inemerging acoustically mediated quantum networks thatallow connecting different solid-state systems and thusmay enable hybrid quantum networks that leverage thedistinct functionalities of each system for quantum com-puting, communication and sensing [35]. In particular,phase modulation is necessary for control of coherent in-teractions and entanglement between solid state systems,dynamical routing and synchronization for addressabil-ity and error mitigation, compensating environmentalchanges such as local temperature drifts, or for com-pensating unavoidable detuning between quantum sys-tems. Finally, our devices offer an opportunity to buildversatile acoustic signal processing components, such as,tunable filters and reconfigurable duplexers, which wouldminimize the number of acoustic components for next-generation telecommunications. ACKNOWLEDGMENTS
We thank Prof. Cheng Wang and Cleaven Chia forfruitful discussion.
Funding : This work is supportedby ONR QOMAND grant no. N00014-15-1-2761, DOE HEADS-QON grant no. DE-SC0020376, NSF grantno. DMR-2004536, the Welch Foundation Grant F-1814, and NSF RAISE/TAQS grant no. ECCS-1839197.N.S. is supported by the Natural Sciences and En-gineering Research Council of Canada (NSERC), theAQT Intelligent Quantum Networks and Technologies(INQNET) research programme and the DOE/HEPQuantISED programme grant and DOE award no. DE-SC0019219.
Author contributions : L.S.: Conceptu-alization, Methodology, Investigation, Formal analysis,Visualization, Writing - Original Draft. D.Z.: Method-ology, Investigation, Writing - Original Draft. M.C.: In-vestigation, Writing - Review & Editing. D.H.L.: In-vestigation, Writing - Review & Editing. N.S.: Writing- Original Draft. Y.H.: Writing - Review & Editing.P.T.R.: Writing - Review & Editing. K.L.: Resources,Methodology, Writing - Review & Editing, Supervision.K.K.B.: Resources, Writing - Review & Editing, Super-vision. M.L.: Resources, Writing - Review & Editing,Supervision.
Competing interests : M.L. is involved indeveloping lithium niobate technologies at HyperLightCorporation. The other authors declare no competinginterests. [1] C. Campbell,
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I. DESIGN OF THE ELECTRO-ACOUSTIC MODULATORS
The interdigital transducers (IDTs) are optimized for maximum transduction between acoustic and electrical wavesat 2.5 GHz. The aperture of the IDT is 75 µ m, the pitch of the finger electrode is 650 nm, and the number offinger electrode pairs per IDT is 25. The maximum transmission between two IDTs at 2.5 GHz is -8 dB at roomtemperature, with -6 dB resulting from the symmetric design of the IDTs (-3 dB per IDT). The adiabatic taper thatconnects an IDT to an acoustic waveguide is 400 µ m long. The insertion loss per acoustic taper is about 5 dB atroom temperature, as extracted from the measured transmission of devices with and without the tapered structure.An X-cut lithium niobate (LN) substrate is used for all devices. The direction of the acoustic waveguide is at anangle of 30 ° with respect to the crystal Z-axis (Fig. 1A). This direction features the slowest surface acoustic wavephase velocity using X-cut LN and thus leads to a well-confined acoustic mode for the waveguide (Fig. S1). Themodulating electric field applied across the waveguide is mainly in the crystal Y-direction. As most strain field of theguided acoustic mode is in the XZ component (corresponding to index 5 in the Voigt notation), our device employsa non-zero electro-acoustic modulation coefficient d . II. DEVICE FABRICATION
A 400 nm-thick silicon nitride (SiN) layer is deposited by plasma enhanced chemical vapor deposition on the X-cutLN substrate. The SiN layer is patterned by a direct write lithography tool (Heidelberg Instruments MLA150) andetched by reactive ion etching using carbon tetrafluoride (CF ), sulfur hexafluoride (SF ) and hydrogen (H ) gases.The metal layer is patterned by an electron lithography tool (Elionix ELS-F125) using polymethyl methacrylate(PMMA) resist. A 115 nm-thick layer of aluminum is deposited by an electron beam evaporation tool, and lift-off ina 1-methyl-2-pyrrolidone (NMP) solvent for more than 3 hours at 80 ° C. III. DEVICE MEASUREMENTS
The devices are mounted and wire-bonded to a printed circuit board (PCB). The transmission spectra of the devicesare measured using a vector network analyzer (Keysight N5224A). For the modulation experiments, a microwave signalgenerator drives one IDT using a single-frequency tone around 2.5 GHz, and a real-time spectrum analyzer (RSA)is connected to the other IDT. The RSA not only measures the power spectrum of the acoustic wave received bythe IDT, but also demodulates the signal to provide real-time in-phase and quadrature data, which are converted tothe phase and amplitude of the received signal. The microwave signal generator and the RSA are synchronized by a10 MHz clock. An arbitrary waveform generator is used to provide any low-frequency modulation signals, and a 20kHz-bandwidth voltage amplifier (Falco Systems, WMA-005) is used to provide a 20 times amplification in voltage upto ±
75 V when necessary. For nonreciprocal phase modulation, a four-channel arbitrary waveform generator (TaborWS8104A-DST) is used to generate the three synchronized modulation signals with various phase delays.
IV. LOW-TEMPERATURE SETUP
The low-temperature performance of the device is measured in a closed-cycle cyrostat (ICE Oxford) that reaches abase temperature of ∼ ), and half-wave voltage of the devices are monitoredcontinuously as the cryostat cools from room temperature. These measurements are repeated as the cryostat warmsup. Cable losses are independently calibrated during a separate cooldown. To characterize the propagation loss ofthe SAWs, the transmission of two acoustic waveguides with different lengths is measured and compared. Thesewaveguides are fabricated on the same chip, packaged on the same PCB, and connected using identical cables. V. TRANSMISSION-MODE MICROWAVE IMPEDANCE MICROSCOPY
The acoustic-wave profiles in the main text are directly imaged using the transmission-mode microwave impedancemicroscopy (TMIM) [36, 37], which is implemented on a commercial atomic-force microscopy (AFM) platformParkAFM XE-70. Here the IDT is driven by a continuous microwave input signal (Anritsu MG 3692A), whichlaunches the propagating surface acoustic wave. During the AFM scanning, the customized cantilever probe fromPrimeNano Inc. picks up the GHz piezoelectric potential accompanying the acoustic wave. By using the same ex-citation frequency as the reference, the TMIM electronics demodulate the tip signal into a time-independent spatialpattern that is shown in Fig. 2B. Note that the TMIM image contains information on the phase of the propagatingwave [37]. As a result, a lateral shift of the wave pattern indicates that the acoustic wave is modulated by the DCbias electric field. Due to the charging effects at interfaces between layers, a higher DC bias voltage is required toachieve same phase shift than that of a modulating signal at f mod = 10 kHz. VI. ELECTRO-ACOUSTIC EFFECT
Hooke’s law says that the force on a spring is proportional to its displacement. The square of the resonant frequencyof a mass-spring system is equal to the ratio of the spring proportionality constant to the mass. Thus, tuning thespring constant also varies the resonance frequency of the spring.Weakly-excited acoustic waves in solids follow a generalized Hooke’s law that relates stress σ and strain (cid:15) by anelasticity (stiffness) matrix C , which is a 6-by-6 matrix in Voigt notation. LN is of point group 3m, which has athree-fold rotation symmetry about its Z axis and mirror symmetry on its X axis. With vanishing components of C due to the symmetry of LN, the relation is σ σ σ σ σ σ = c c c c c c c − c c c c c − c c c c c
14 12 ( c − c ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) . The elasticity matrix of LN can be varied by an applied electric field, ∆ c ij = d kij E k , where i , j =1 · k = 1 , , , E is the applied electric field, and D ( d kij ) is the third-order piezoelectric tensor. Subject to the symmetry of LN, D has the following form, d ij = d d d d d d d
145 12 ( d − d ) d d d d d d d
136 12 ( d − d ) 0 0 d ij = ( d + 3 d ) ( d − d ) d d ( d − d ) − (3 d + d ) − d d d − d d d d d − d d
145 12 ( d − d )0 0 0 0 ( d − d ) ( d − d ) d ij = d d d d d d d − d d d d d − d d d d d
314 12 ( d − d ) . VII. COUPLING BETWEEN A TRAVELING ACOUSTIC WAVE AND A BIAS ELECTRIC FIELD
An applied electric field affects the traveling acoustic wave by tuning the elasticity of the material. When theapplied electric field is small, such tuning in elasticity can be treated by the perturbation theory. The wave equationfor a guided acoustic mode is − ρω u = ∇ • ( C (cid:15) ) , where ρ is the mass density of the material, u is the displacement field, ω is the angular frequency of the eigenmodeat given wavenumber k .For a guided acoustic mode, the first-order shift in the eigenfrequency at the given wavenumber k due to theperturbation of elasticity ∆ C is given by∆ ωω = (cid:82) dr ∆ c ij (cid:15) ∗ i (cid:15) j (cid:82) drc ij (cid:15) ∗ i (cid:15) j = (cid:82) dr d kij E k (cid:15) ∗ i (cid:15) j (cid:82) drc ij (cid:15) ∗ i (cid:15) j . The integral is over the whole cross section that perpendicular to the acoustic propagation direction. Further, thechange of wavenumber ∆ k at certain mode frequency ω is calculated by the dispersion relation of the acoustic mode,∆ kk = ∆ ωω v p v g , where v p and v g are the phase and group velocity of the guided acoustic mode. The overall acoustic phase changeover length L due to the applied electric field is ∆ kL . VIII. MEASUREMENT OF THE MODULATION BANDWIDTH
First, we measure the modulation bandwidth of the 1-cm-long phase modulator (Fig. 2). We apply a weak signalmodulation ( V pp = 0 . V π ) with varied frequency and measure the modulation efficiency. The modulation efficiency isindicated by the first sideband power generated by the phase modulation. Due to the phase mismatch between theslowly propagating acoustic wave and the fast varying electrical modulation signals, we measure the 3-dB bandwidthto be 110 kHz for the 1-cm phase modulator and observe periodic variations of the sideband power as a function ofmodulation frequency (Fig. S4). The modulation efficiency approaches zero every time when the modulation frequencyis an integer multiple (N) of f mod = 336 kHz. At these frequencies, the electric field oscillates exactly N full cyclesas the acoustic wave travels through the modulator, resulting in a vanishing cumulative modulation effect. The zero-modulation frequency is related to the modulation length L and the acoustic group velocity v g by f mod = v g L . Usingthis relationship, we infer an acoustic group velocity of v g = 3 .
36 km/s, which is consistent with simulated velocity of3.38 km/s (Fig. S1).
CAPTION FOR MOVIE S1
Principle of nonreciprocal acoustic phase modulation. First (last) 30s depicts a wave-front of the acoustic waveas a blue (red) dot traveling in the forward (backward) direction. Top window illustrates three electrodes of themodulator, labelled as V , V and V , that have sinusoidal signals applied which are 120 ° out of phase with eachother. Bottom window depicts the signals in the time domain, depicting the electrode regions for each signal. Middlewindow indicates the phase accumulation experienced by the wave-front as it propagates through the modulator. -90 -60 -30 0 30 60 9033003400350036003700 A c ou s t i c pha s e v e l o c i t y ( m / s ) Direction (degree) Shorted surface Free surface E l ec t r o m ec h a n i ca l c oup li ng k ( % ) FIG. S1.
Simulated acoustic phase velocities for varying directions on X-cut LN.
The direction is defined by theangle respective to the crystal Z axis. The electromechanical coupling coefficient k = 2 ( v o − v m ) /v o , where v o and v m are thephase velocities when the top surface is free and electrically shorted, respectively. The direction of the waveguide used in ourdevice is 30 ° , as indicated by the dash line. T r an s m i ss i on ( d B ) Frequency (GHz)
25 dB
FIG. S2.
Measured transmission spectra of the acoustic modulator at temperatures of 300 and 1.3 K.
Theresults indicate a 25 dB improvement in peak transmission at low temperature. The frequency shift of the spectrum is due tothe temperature dependent elasticity of LN. A c ou s t i c pha s e c hange ( deg r ee ) Modulation amplitude (V ) pp FIG. S3.
Peak-to-peak acoustic phase changes due to sinusoidal modulating signals of varying peak-to-peakvoltage ( V pp ) at room and cryogenic temperatures. The sinusoidal modulation signals are of the frequency f mod = 10kHz. Linear fits show V π of 53 V at room temperature (300 K) and 135 V at 1.3 K, respectively.
500 1000 1500 2000-70-60-50-40-30-20 S i deband po w e r ( d B c ) Modulation frequency (kHz) f mod f =
336 kHz mod
FIG. S4.
Modulation bandwidth of the 1-cm-long electro-acoustic phase modulator.
The modulation efficiency isindicated by the first sideband power due to the phase modulation. The measured 3-dB bandwidth is 110 kHz. The modulationapproaches zero at f mod = 336 kHz when the acoustic traveling time through the modulator equals 1 /f mod . The same deviceis measured as that in Fig. 2.
200 400 600 800 1000 1200-150-100-50050100150 Modulation frequency (kHz) E l e c t r ode pha s e de l a y ( deg r ee ) -20-40-60-80 M odu l a t i on s i deband po w e r ( d B c ) P h a s e m a t c h i n g w i t h a c o u s t i c w a v e
336 kHz
FIG. S5.