WO2023102349A1 - Couplage paramétrique interfréquence et transfert d'énergie de mouvement mécanique - Google Patents

Couplage paramétrique interfréquence et transfert d'énergie de mouvement mécanique Download PDF

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WO2023102349A1
WO2023102349A1 PCT/US2022/080498 US2022080498W WO2023102349A1 WO 2023102349 A1 WO2023102349 A1 WO 2023102349A1 US 2022080498 W US2022080498 W US 2022080498W WO 2023102349 A1 WO2023102349 A1 WO 2023102349A1
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frequency
mechanical
electromechanical device
mechanical mode
nonlinearities
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PCT/US2022/080498
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English (en)
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Andrew N. CLELAND
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The University Of Chicago
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H9/00—Networks comprising electromechanical or electro-acoustic devices; Electromechanical resonators
    • H03H9/25—Constructional features of resonators using surface acoustic waves
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/40—Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H9/00—Networks comprising electromechanical or electro-acoustic devices; Electromechanical resonators
    • H03H9/02—Details
    • H03H9/02228—Guided bulk acoustic wave devices or Lamb wave devices having interdigital transducers situated in parallel planes on either side of a piezoelectric layer
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H9/00—Networks comprising electromechanical or electro-acoustic devices; Electromechanical resonators
    • H03H9/02—Details
    • H03H9/02244—Details of microelectro-mechanical resonators
    • H03H9/02338—Suspension means
    • H03H9/02362—Folded-flexure

Definitions

  • the present disclosure relates to methods, devices, systems, and applications that utilize an energy transfer in a solid-state micromechanical structure between mechanical motions at different frequencies that are coupled via various types of parametric nonlinearities.
  • Mechanical modes of motion in micromechanical devices may be relied upon in various sensing, transducing, and actuation applications. These mechanical modes may be characterized by frequencies at distinct ranges. In some applications, it may be beneficial to enhance mechanical motion at a particular frequency while suppressing mechanical motion in other frequencies, thereby prompting a need for a mechanism to induce and control energy transfer between these mechanical modes of motion at different frequencies.
  • FIG. 1 illustrates an example solid-state structure that provides interfrequency nonlinear coupling of mechanical modes of motion.
  • FIG. 2 illustrates another example solid-state structure that provides interfrequency nonlinear coupling of mechanical modes of motion.
  • FIG. 3 illustrates an example mechanically-suspended solid-state structure that provides inter-frequency nonlinear coupling of mechanical modes of motion based on a nonlinear piezoelectric effect.
  • FIG. 4 illustrates a system that incorporates the example solid-state structure in FIGs. 1 -3 into an electromechanical device for sensing, transducing, actuation, or quantum information processing applications.
  • FIG. 5 illustrates an example set of conductive traces for an interdigitated transducer.
  • FIG. 6 illustrates a mechanically-suspended solid-state structure that incorporates a piezoelectric material layer and the interdigitated transducer of FIG. 5.
  • FIG. 7 illustrates an example silicon-based mechanically-suspended solid- state structure that incorporates an interdigitated transducer, a piezoelectric material layer, and phononic crystal structures.
  • FIG. 8 shows an SEM micrograph of an example mechanically-suspended solid-state micro-electromechanical structure.
  • FIG. 9 shows a top-view SEM micrograph of an example mechanically- suspended solid-state micro-electromechanical structure.
  • FIG. 10 shows a top-view SEM micrograph of another example mechanically-suspended solid-state micro-electromechanical structure.
  • FIG. 1 1 shows an SEM micrograph of another example mechanically- suspended solid-state micro-electromechanical structure.
  • FIG. 12 shows a simulated surface acoustic wave profile along a cross section of the mechanically-suspended solid-state structure in FIG. 7.
  • FIG. 13 shows an intermediate structure during a fabrication process for the mechanically-suspended solid-state structure of FIG. 7.
  • FIG. 14 shows another intermediate structure during a fabrication process for the mechanically-suspended solid-state structure of FIG. 7.
  • FIG. 15 shows an SEM micrograph an example phononic crystal structure.
  • FIG. 16 shows a unit cell structure of the example phononic crystal structure of FIG. 15.
  • FIG. 17 illustrates a calculated band structure of the example phononic crystal structure of FIG. 15 showing an acoustic transmission bandgap.
  • the present disclosure generally relates to methods, devices, and systems that utilize an energy transfer in a solid-state micromechanical structure between mechanical motions at different frequencies that are coupled via various types of parametric nonlinearities.
  • the methods, devices, and systems described herein take advantage of certain types of parametric nonlinearities in some solid-state structures, the parametric nonlinearities being either engineered or intrinsically present, to achieve an energy transfer between mechanical modes of motion of different frequency ranges.
  • Example parametric nonlinearities that may be utilized include but are not limited to any combination of geometric nonlinearities and various types of constitutive nonlinearities. Specific non-limiting examples of parametric nonlinearities are described in more detail below.
  • These and other types of parametric nonlinearities provide nonlinear coupling between the mechanical motions at different frequencies. Because of such nonlinear coupling, these normally independent and non-interacting mechanical motions that may be far apart in frequency may interact with one another. Such coupling by the parametric nonlinearities thus allows for an energy transfer, in either direction, between these mechanical motions of different frequencies.
  • nonlinearity-based interactions and energy transfer between mechanical motions at different frequency ranges may be utilized to enable a variety of applications.
  • low-frequency mechanical vibrations sensed by a large number of resonant sensors may be read out at a single much higher frequency, e.g., in the microwave range, via electric means, when such nonlinearities are present and utilized.
  • damping of low-frequency mechanical modes in systems or devices possessing such nonlinearities may be achieved using electrically-based methods to increase transfer of energy in these low-frequency mechanical vibrations into higher frequencies, thereby reducing low-frequency mechanical noise.
  • electrically-based methods may be implemented to reduce rather than increase the damping and amplify the motion of the low-frequency mechanical modes by transferring energy from high frequency modes into the low- frequency vibrations.
  • nonlinear coupling may enable a use of quantum bits (qubits) to measure the motion of the low-frequency mechanical modes, thereby providing quantum sensing of the motion in these modes for quantum information processing applications.
  • low-frequency mechanical devices form a basis for many sensors designed for force sensing, accelerometry, mass sensing and the like.
  • These sensors typically use a mechanically-resonant element such as a cantilevered beam subject to mechanical induction by the subject being sensed.
  • a force being sensed may induce a motion of the cantilevered beam, and the act of sensing involves measuring and monitoring the resulting on-resonance motion of that cantilever.
  • Methods that reduce the noise inherent in the mechanical motion, improve the sensitivity of the measurement of that motion, or improve the dynamic range of the measurement, may all enhance the performance of such a sensor.
  • the parametric nonlinear coupling and energy transfer described in this disclosure may enable a lowering in the motional energy of the lower-frequency mechanical mode, and thus prepare or maintain the sensor in a low-noise state to provide more sensitive measurements.
  • the induced low-frequency motion in the sensor may be amplified by energy transfer from a higher frequency excitation motion as a result of the parametric nonlinear coupling, thereby providing low frequency actuation or improving detection signal level and signal to noise ratio at the low frequency.
  • energy of an induced low-frequency motion in a sensor may be transferred to motion in a higher frequency range via the parametric nonlinear coupling, and may then be measured electrically rather than mechanically to provide improved measurement precision and higher signal-to-noise ratio.
  • the high-frequency mechanical motions may be phononic, e.g., in the form of an acoustic wave.
  • phononic is used interchangeably with “acoustic” to refer to motions due to propagating or standing vibrational waves through the lattice of the mechanical system rather than the low-frequency collective bulk vibrations (such as structural deformational and vibrational motions, including but not limited to flexural vibrations, torsional vibrations, extensional vibrations, and the like).
  • the phononic motion or modes in addition to being coupled to low-frequency mechanical motion via one or more parametric nonlinear interactions, may be directly excitable electromagnetically via, for example, a piezoelectric effect in the mechanical system.
  • an electromagnetic excitation may be used to induce strain modulations in a piezoelectric material layer in the mechanical system, thereby generating phononic motion or acoustic waves.
  • the piezoelectric effect may be nonlinearly coupled to the low-frequency mechanical modes, providing at least part of, and in some cases, the main contribution to the parametric nonlinearities for the mode coupling and energy transfer between mechanical motions at phononic/acoustic frequency and at low frequencies.
  • Devices relying on piezoelectric nonlinearities provide the advantage of being able to excite the phononic or acoustic motion electromagnetically.
  • the piezoelectric effect also provides a capability of sensing the energy transferred from a low-frequency motion to the phononic or acoustic motions electrically, as such motions would inversely induce electromagnetic fields that may be detected via electrodes attached to the piezoelectric materials.
  • Devices based on piezoelectric materials thus may only need to rely on electric control and/or actuation signals for their operations.
  • Devices relying on piezoelectric nonlinearities for mode coupling and energy transfer may be completely solid-state. They may be fabricated as micromechanical devices, or micro electromechanical devices. Fabrication of these devices thus would leverage the mature solid-state processing and integration technologies including but not limited to controlled material deposition, evaporation, impurity implantation, annealing, patterning, etching, lithography, packaging, and the like, to yield high quality and compact devices.
  • FIG. 1 illustrates an example basic solid-state structure 100 that utilizes the inter-frequency nonlinear coupling of mechanical modes of motion as the underlying operating principle.
  • the solid-state structure 100 may be fabricated in the form of a micromechanical component or micro-electromechanical component incorporated into various sensing, transducing, actuation, and quantum information processing devices.
  • FIG. 1 illustrates that the solid-state structure 100 may support motions in various mechanical modes at distinct frequencies.
  • the solid-state structure 100 may support low-frequency mechanical modes 102 and high-frequency mechanical modes 104.
  • the low-frequency mechanical modes 102 may represent resonant mechanical bulk deformation and structural vibrational modes (such as flexural, torsional, extensional or other types of structural vibration modes) and are thus related to the geometry, size, stiffness of the solid-state structure 100 and their support structure.
  • the high-frequency mechanical modes 104 may represent acoustic modes (e.g., phononic modes, or mechanical lattice wave motions) in the solid-state structure 100.
  • the mechanical modes such as the low-frequency mechanical modes 102 and high-frequency mechanical modes 104, may be coupled via a variety of interfrequency nonlinearities, as shown by 1 10 in FIG. 1.
  • Such nonlinear coupling may be geometric in nature, originating from modification, by mechanical motion in one frequency range, of geometric attributes of the solid-state structure 100 that affects the mechanical modes in another frequency range.
  • These types of nonlinearities may be referred to as geometric nonlinearities or nonlinear coupling.
  • such nonlinear coupling may originate from modification of one or more mechanical properties or responsivities (such as the mechanical stiffness or compliance of the constituent material(s) of the solid-state structure 100) in one frequency due to mechanical motion at another frequency.
  • constitutive nonlinearities may be intrinsic to the constituent materials of the solid-state structure 100. Alternatively, in other implementations, they may be achieved through material engineering. Examples for both geometric and constitutive nonlinearities are provided in various sections of this disclosure below. The manner in which they couple the motions in different frequencies are also described in further detail below.
  • FIG. 1 further illustrates that the mechanical motions, particularly the mechanical motions at high frequency mechanical modes 104, may be directly induced by or may be generated with assistance from external excitation via an excitation component 120.
  • such high-frequency motion may be in the radio frequency range and may be directly or indirectly induced via electromagnetic means.
  • radio-frequency electromagnetic waves, or microwaves may be applied to assist in the generation of the high-frequency mechanical motion 104, thereby providing an electric control of the mechanical motions in the solid-state structure 100.
  • the high-frequency mechanical motion 104 may be induced as an excitation of one or more acoustic modes with the assistance of the excitation component 120 by external electromagnetic waves via the piezoelectric effect in the solid-state structure 100.
  • the mechanical motions in the solid-state structure 100 may be detected or sensed via a detection component, as shown by 130.
  • such high-frequency motion may be detected or sensed electrically; in other examples it may be detected or sensed optically; in other examples it may be detected through magnetic means.
  • the high-frequency mechanical motion 104 in the solid-state structure 100 may generate electromagnetic waves via the piezoelectric effect, and the electromagnetic waves so generated may then be detected.
  • Such high- frequency mechanical motion for example, may be induced from a low-frequency mechanical motion as a result of the inter-frequency nonlinear coupling.
  • the high frequency motion may be externally induced (such as via the excitation component 120).
  • the nonlinear coupling between the low frequency motion and the high frequency motion may generate sidebands in the high frequency motion, which may be converted into electric signals by the detection component 130 for detection.
  • the excitation component 120 and detection component 130 may be implemented separately or as a single component.
  • a single component such as the interdigitated transducer described below, may be used for both excitation and detection of the high frequency motion.
  • FIG. 2 illustrates a solid-state structure 200 that may include elements 202 and 204 in addition to the solid-state structure 100 of FIG. 1 .
  • the elements 202 and 204 may be incorporated into the solid-state structure 100 of FIG. 1 to support or improve the resonant modes of the mechanical motion.
  • the elements 202 and 204 may be provided to support resonant modes for the high-frequency mechanical motion.
  • Each resonant mode may be associated with a resonant frequency.
  • On-resonance mechanical motion in the solid-state structure 200 may thus be enhanced due to the presence of the elements 202 and 204.
  • the elements 202 and 204 may be constructed as mirrors or other equivalent structures for forming a resonant cavity of the acoustic wave (e.g., a Fabry-Perot resonator). Examples of elements 202 and 204 are provided in further detail in the disclosure below.
  • FIGs. 1 and 2 The shape, geometry, and relative positions of various components of the solid-state structure 100 and 200 in FIGs. 1 and 2 are only shown for illustrative purposes.
  • the material composition, geometric shape, layer scheme, patterning of the various components of the solid-state structures 100 and 200 are not limited by the specific illustrations of FIG. 1 and FIG. 2, and other examples further given below.
  • These solid-state structures may form basic components in mechanical or electromechanical devices that may be used in various sensing, transducing, actuation, and quantum information processing devices.
  • FIG. 3 shows a layered solid-state structure 300 as an example of the general solid-state structure 200 of FIG. 2 that may be used to harness mechanical inter-frequency nonlinear coupling for achieving various sensing, transducing, actuation, and quantum information processing applications.
  • the layered solid-state structure 300 of FIG. 3 may include a support or base layer 302 for the solid-state structure 300.
  • the support or base layer 302 may be implemented as a single-layer or a multi-layer structure.
  • the solid-state structure 300 may further include an excitation/detection layer 310 for inducing or sensing mechanical motions in the base layer 302.
  • the excitation/detection layer 310 for example, may be particularly configured for excitation or sensing of the high-frequency mechanical motion in the solid-state structure 300.
  • the excitation/detection layer 310 may include a layer of piezoelectric material associated with a set of nonlinear piezoelectric coefficients that relate an external applied electric field to stress induced in the piezoelectric material layer by the external electric field.
  • the excitation/detection layer 310 may be conformably attached to the base layer 302 such that the mechanical motion, such as that induced in the excitation/detection layer 310, is directly transferred to the base layer 302 for excitation of the mechanical motion in the solid-state structure as a whole, and likewise, mechanical motion that could be present in the base layer 302 is transferred to the excitation/detection layer 310 for detection of the mechanical motion.
  • base layer 302 and the excitation/detection layer 310 are shown as separate layers in FIG. 3, in some other implementations, these two layers may be integrally formed as a single layer structure.
  • a single layer of piezoelectric material can be used as both the base layer and the excitation/detection layer of the solid-state structure 300.
  • the base layer and the excitation/detection layer may be separate layers of different piezoelectric materials. Mechanical modes of motion of the combined base layer and the excitation/detection layer 310 at different frequencies may not be separable and may be considered as a whole and determined by various electric, mechanical, and electro-mechanical attributes of the combined layers.
  • the solid-state structure 300 of FIG. 3 may further include an electrode layer 320 for applying electromagnetic excitations to the excitation/detection layer 310 or for outcoupling electromagnetic fields included in the excitation/detection layer 310 for detection.
  • the electrode layer 320 may be connected to an external electromagnetic source or an electromagnetic detector via electric connections 330. While the electrode layer 320 is illustrated in FIG. 3 as a general layer component, it may include any type of pattern of conductive traces having polarities for applying a spatial pattern of static or time-varying electric fields, or for outcoupling electric fields induced in the excitation/detection layer 310.
  • the solid-state structure 300 may further include components 340 as, e.g., mirrors for the mechanical motion in the base layer 302 such that a resonant cavity is formed, for example, for some high-frequency mechanical modes.
  • the components 340 may be fabricated as an extension of the base layer 302 with patterns formed for reflecting the mechanical motions at certain frequencies in the base layer 302.
  • the component 340 may not be limited to the base layer 302 in the direction normal to the in-plane direction of the base layer 302. It may extend further, for example, to the excitation/detection layer 310 in the direction normal to the in-plane direction.
  • the solid-state structure 300 of FIG. 3 may also include mechanical support elements 304.
  • the solid-state structure 300 may be constructed as a mechanically-suspended structure (e.g., a thin-film structure) anchored to a host device by the support elements 304.
  • the support elements 304 may be implemented as micro-beam, pillar, or micro-spring structures.
  • the support elements 304 may be fabricated as part of the base layer 302.
  • the support elements 304 may be designed and fabricated to achieve a particular desired stiffness or spring constant.
  • the support elements 304 together with the solid-state structure 300 may be characterized by one or more low-frequency mechanical deformation or structural vibrational modes of mechanical motion, including but not limited to flexural vibration modes, torsional vibration modes, extensional vibrational modes, and the like. The characteristic frequencies of these vibrational modes may be determined by the stiffness, geometry, and sizes of the support elements 304 and the solid-state structure 300.
  • the support elements 304 may be fabricated in a spring-like or meandered manner (see component 1 1 10 of SEM micrograph of FIG. 1 1 ) for releasing internal stress in the solid-state structure 300.
  • the solid-state structures 100, 200, and 300 are essentially configured to support both low-frequency modes of mechanical motion (such as the various structural vibrational modes described above) and high-frequency modes of mechanical motion (such as the acoustic or phononic modes described above). Under normal conditions, these mechanical modes at distinct frequencies may not be coupled (or coupled only very weakly) such that the motions in these frequencies are independent or nearly independent, and motion in one frequency range would minimally affect the motion in the other distinct frequency range. As such, there would be no or little energy transfer between the mechanical modes at these distinct frequencies.
  • the solid-state structures 100, 200, 300 may be designed and configured to provide a nonlinear coupling between these frequencies based on various types of nonlinear inter-frequency interactions. Because of such nonlinear interactions, the mechanical motions in the distinct frequencies may be mixed. A motion in the low-frequency range may induce motion(s) in the high-frequency range or generate sidebands in the high-frequency motion(s). Conversely, a motion in high frequency may be mixed into motion(s) at low frequency. In other words, the nonlinear coupling or interaction may lead to energy transfer between these mechanical modes of motion at very distinct frequencies.
  • the solid-state structures 100 and 200 may be intrinsically nonlinear in that the mechanical modes at different frequencies may be intrinsically coupled in a geometrical manner: a mechanical distortion in one frequency may intrinsically change geometric or structural factors that affect dynamic parameters associated with a mechanical mode at another frequency.
  • Such inter-frequency coupling that is geometric or structural in nature may be referred to as geometric nonlinearities.
  • Such geometric nonlinearities may be more appreciable within vibrational modes of low frequencies than between low frequency vibrations and the high-frequency phononic or acoustic modes. But in general, such nonlinearities may be weak for practical purposes.
  • the solid-state structure 100, 200, and 300 may be designed to additionally provide stronger nonlinear interfrequency coupling relying on constitutive nonlinearities.
  • the constitutive nonlinearity may be provided via a nonlinear piezoelectric coefficient; a nonlinear stiffness coefficient; a nonlinear density; and other nonlinearities not normally found in such structures.
  • the constitutive nonlinearity may be provided by a nonlinear piezoelectric coefficient.
  • a high-frequency mechanical motion (phononic or acoustic mechanical motion) that may be excited by a high-frequency electric field applied via the electrode layer 320 of FIG. 3 may be coupled to low- frequency vibrations.
  • energy in the high-frequency mechanical mode may be transferred into and thereby amplify the low-frequency motion(s).
  • energy in a low-frequency motion may be transferred into the high-frequency mechanical mode(s) for cooling/damping the low-frequency motion.
  • the low-frequency mode (e.g., in the form of vibrational noises) may be cooled to its vibrational ground state.
  • the energy transferred into the high- frequency mechanical modes from the low-frequency modes may in turn induce an electric field in the extraction/detection layer 310 that may be used for sensing the original low-frequency motions.
  • FIG. 4 shows an example system 400 for using a device incorporating the solid-state structure 100, 200, or 300 of FIGs. 1 -3 above for various sensing, transducing, actuation, or quantum information processing applications.
  • the system 400 may include an electro-mechanical device 430 that incorporates one or more solid-state structures above (or in alternate implementations, device 430 may additionally include quantum bits (qubits) for controlling or sensing or measuring the electromechanical device that also makes up part of device 430), a controller 410, electromagnetic source 420, DC electrical and/or magnetic field source 426, a readout/measurement circuitry 440 for sensing.
  • the system 400 may include the device 430 and optionally include one or more of the other components of FIG. 4.
  • the controller 410 may be of any form of dedicated circuits, general-purpose instruments, and mobile or fixed electronic devices including but not limited to desktop computers, laptop computers, tablets, mobile phones, personal digital assistants, and the like.
  • the controller 410 may be configured to provide a user interface for controlling the electromagnetic source 420 (for controlling its timing, frequency, phase, and amplitude), the electric/magnetic field sources 426, and the readout/measurement circuitry 440 for excitation and detection of mechanical modes of motion at various frequencies in the device 430.
  • the controller 410 may include communication interfaces, one or more processors, input/output (I/O) interfaces, storages, and display circuitry.
  • the controller 410 may communicate with one or more of the other components of the system 400 to send/receive commands, data, or the like.
  • the electromagnetic source 420 may include one or more devices to generate an electromagnetic wave in, for example, the microwave-frequency range that may be applied to the device 430.
  • the electromagnetic sources 420 may include a microwave generator including but not limited to an inductor, capacitor, microwave resonator, and may be connected to a microwave-frequency alternating current (AC) power supply.
  • the electromagnetic signal generated by the electromagnetic source 420 may be applied to, for example, the electrode layer 320 of the solid-state structure 300 that may be incorporated in the device 430.
  • the frequency of the electromagnetic signal may be tunable and its intensity and/or phase (or electric field amplitude) may be adjusted by the controller 410.
  • the DC electric or magnetic sources 426 may be optionally used to generate and apply DC electric or magnetic fields to the device 430.
  • the DC electric or magnetic fields may be applied for various purposes. For example, a DC electric field may be applied for shifting or compensating mechanical properties (e.g., to apply some baseline mechanical strain).
  • the readout/measurement circuitry 440 may be configured to detect the electromagnetic field generated in the device 430.
  • the readout/measurement circuitry 440 may include inductors, capacitors, electromagnetic resonators, electromagnetic or microwave amplifiers, and may be connected to, for example, the electrode layer of 320 of the solid-state structure 300 as incorporated into the device 430 for readout.
  • the readout/’measurement circuitry 440 may further include other components, including but not limited to analog-to-digital converters, other analog and digital data processing components, and memories for analyzing the signal obtained from the device 430 and for storing the various raw and processed information.
  • mechanical systems are sometimes intrinsically nonlinear, in the sense that motion in one mode may affect motion in a nominally completely different mode.
  • One source of this nonlinearity is one or more geometric nonlinearities: the distortion of the solid in one mode may modify the dynamical parameters of the other mode.
  • a displacement in a flexural mode of a cantilevered beam involving an extension of the beam on one side of a neutral surface (the midplane of the beam) and a compression on the other side of the beam distorts the shape of the beam, and may change rotational moments of inertia for other torsional modes of the same beam.
  • Such changes in the rotational moments represents a parametric change in the dynamical torsion response, and thus may affect the frequency and amplitude of motion in the torsional modes.
  • the torsional motion may change the flexural parameters and thus change the frequency and amplitude of the flexure of the beam.
  • these geometric effects may be small and can be ignored to first order, yielding nearly independent motion in the different modes.
  • the mechanical modes at different frequencies may no longer be independent, and energy exchange may be realized between the modes even when the modes are associated with motions at very different frequencies.
  • Such geometric nonlinearities when non-negligible (particularly between low frequency mechanical modes, less between a low frequency mode and an acoustic mode), may be utilized as a mechanism that couples the mechanical modes at distinct frequencies to achieve the energy transfer for the sensing, transducing, actuation, and quantum information processing applications described above.
  • cn represents the relevant material stiffness.
  • the uniaxial notation herein is used merely as an example and to simplify the description. The underlying principles described below apply equally to the non-uniaxial situations.
  • cn represents the relevant material stiffness
  • en represents the relevant piezoelectric constant
  • e 1 represents the relevant dielectric constant. More generally, this may be written as a multi-component tensor relation between stress, strain, electric displacement, and electric field, usually written using, for example, six-component vectors representing stress and strain, a six-by-six component matrix representing the stiffness, three-component vectors for the displacement and electric fields, and a six-by-three matrix representing the relevant piezoelectric constants.
  • This constitutive relation can be modified to include stress-strain nonlinearities.
  • the additional first-order correction for the dependence of the piezoelectric constant e on strain r) may be considered, yielding a modified relation (again only representing an example uniaxial situation for simplicity)
  • nonlinearities could include derivatives of the piezoelectric tensor element en with other components of the strain vector rj, as well as the other piezoelectric tensor elements e y that depend on strain.
  • the parametric piezoelectric nonlinearity may be used to, for example, generate coupling between mechanical strains and electric fields. Measurements in quartz and lithium niobite, for example, have been made for the piezoelectric constant en and its nonlinearity 3en/3/)i with fractional nonlinearity values de 11 /e 11 dr] 1 of the same order in both materials of around 10-25. The maximum strain r/ (before material failure) is typically a few percent, so the maximum achievable change in the piezoelectric constant may be of order 10-100 percent. The observed values are thus relatively significant, of the same order or larger than the geometric nonlinearities above. Theoretical calculations for the piezoelectric nonlinearity in some materials have also been made, mostly supporting the various measurements, albeit in different piezoelectric materials.
  • Equation (1 ) or (3) may give rise to acceleration in the strain , as expressed by a typical strain-stress dynamic equation:
  • Equation (1 ) The linear constitutive relation in Equation (1 ), combined with Equation (4) and the dynamical Equation (6), has the property that the dynamic response is completely linear and thus yields completely independent responses at each frequency f of interest, when ignoring the geometric nonlinearities that couple the different mechanical modes mentioned above.
  • Equation (3) For a material with the linear response of Equation (1 ), or with a linear piezoelectric material as in Equation (3), given, for example, an external driving force (or in the piezoelectric case, a driving electric field) at frequency fd, the stress, strain, self-consistent electric field, etc., may all respond uniquely at the frequency fd, and in the absence of driving terms at other frequencies, this frequency may be the only frequency in a steady-state solution (that is, after any transients have decayed away).
  • This means that the dynamical response at each frequency is completely independent, thus yielding eigenfrequencies (self-resonant frequencies) that can be found mathematically using e.g. matrix diagonalization, and the steady-state response solved using matrix inversion.
  • This eigenmode-eigenfrequency analysis is a characteristic of systems of linear dynamic equations (ignoring the geometric parametric nonlinearities described above, which appear even in the absence of any constitutive nonlinearities).
  • Equations (2) and (5) The effect of a nonlinearity such as that described in Equations (2) and (5) may be understood as a perturbation on the linear system, where the completely linear system represents a reasonable approximation to the nonlinear system and has a set of completely independent modes.
  • the linear response may thus include a family of frequencies, typically not simply related to one another. By including the geometric or constitutive nonlinearity, additional modes may appear in the response due to the nonlinear coupling.
  • additional harmonics of a particular mode may appear.
  • modes at sum and/or difference frequencies of other modes may appear.
  • the nonlinearity may generate a response at the sum and at the difference frequencies, Fj + F k , ⁇ Fj - F k ⁇ .
  • the linear response at two (or more) frequencies may generate higher-order intermodulation products, e.g., at the sum and differences of integer multiples of these frequencies, ⁇ nFj ⁇ mFk ⁇ , where n, m are integers 1 , 2, 3 This extends naturally to more frequencies Fj, Fk, Fe, . . ..
  • the effect and utility of the parametric coupling may be the largest when the frequencies involved are close to the natural mechanical resonance frequencies of the system.
  • This system may be driven at both frequencies, e.g., mechanically at Fb and also using a drive signal at fd, the latter close to f a (close here meaning detuned from f a by a frequency of order Fb).
  • the system dynamics may display a mechanical response at both Fb and fd, but in addition, due to the nonlinearity, at fd ⁇ Fb as well as at the higher-order intermodulation products nfd ⁇ mFb.
  • By changing (tuning) the drive fd, the lower sideband fd - Fb, the drive fd, or the upper sideband fd + Fb, can be brought equal to f a , with a significantly increased response when one of these frequencies matches f a .
  • a quantum bit operating with its transition frequency near f a can be used to quantum-sense the amplitude or phase of the response in either one ofor both sidebands, yielding potentially quantum- limited sensing of the low-frequency motion.
  • the solid-state structure 100, 200, and 300 of FIGS. 1 -3 may be implemented as surface acoustic wave transducers utilizing piezoelectric materials. Such transducers may be used in various types of delay lines, resonators and filters.
  • SAWs surface acoustic waves
  • Rayleigh waves which travel along the surface of a solid at a speed slightly less than the shear wave velocity in the bulk solid.
  • an interdigitated transducer IDT is placed on a smooth surface of a piezoelectric material, as shown by 500 of FIG. 5, comprising two sets of parallel interleaved electrically-conducting fingers 502 and 504 that are separately driven by radiofrequency or microwave electrical signals.
  • IDT interdigitated transducer
  • An electrical signal at the frequency f applied to alternating interdigitated fingers may, through the piezoelectric response of the underlying material, generate a SAW with wavelength that travels on the surface in the two directions perpendicular to the line of the transducer fingers, as indicated by arrow 510 in FIG. 5.
  • Alternative implementations may use double-finger geometries or chirped spacing variants or other specific geometries, which are all understood to be included as variants of FIG. 5.
  • FIG. 6 further shows a view of the example interdigitated figures being disposed on the layered structure of the example solid-state structure 300 of FIG. 3.
  • the interdigitated conductive fingers 500 may be patterned over the piezoelectric layer 310 which may further be disposed on the base layer 302.
  • unidirectional transducers that emit SAWs in only one direction
  • these other elements may include acoustic mirrors (that reflect acoustic waves back in the direction they came from), acoustic resonators, delay lines, pulseshaping devices, and the like.
  • a surface acoustic wave resonator may be constructed.
  • the surface acoustic wave resonator may be constructed by placing one or two transducers of the type described above on the surface of the piezoelectric material, and then placing a set of acoustic mirrors in the path of the surface acoustic waves generated by the transducer(s).
  • These mirrors can be made with electrically conducting fingers similar in geometry to the IDT fingers, with a spacing close to d, or by cutting shallow grooves with a spacing close to d, in such a way that the surface acoustic wave is reflected back towards the transducer(s).
  • the two mirrors act together to form an acoustic Fabry-Perot resonator, trapping sound waves with a frequency close to f or its integer multiples, as the Fabry-Perot resonances of the mirror pair.
  • These sound-trapping Fabry-Perot resonance frequencies may display larger electrical response and larger amplitude SAW waves between the pair of mirrors than at other frequencies.
  • the nonlinearities in the piezoelectric effect may operate to couple motions at different frequencies as follows.
  • a mechanical motion at a first frequency may cause distortions of the solid-state structure, introducing strain in the structure modulated at the first frequency.
  • Such strain would cause modulation in the piezoelectric coefficient via the piezoelectric nonlinearity, which further causes modulation of the speed vof sound in the solid-state structure.
  • This type of device can be used as the solid-state structure 100, 200, and 300 of FIGs. 1 -3, where a SAW Fabry-Perot resonance maybe used as the high frequency resonance f a , interacting with a lower-frequency mechanical mode for the device at frequency Fb, such as a flexural, torsional, or extensional mode.
  • Fb frequency
  • parametric effects would occur when signals at f ⁇ ⁇ Fbor Malign closely with the Fabry-Perot resonance at f a , yielding the resonance effects described above.
  • the solid-state structure 100, 200, and 300 of FIGS. 1 -3 may be implemented as a Lamb resonator.
  • a mechanically-suspended structure in which the surface acoustic waves are generated and reflected on a mechanically-suspended thin film comprising only the piezoelectric material, or the piezoelectric material on top of a non-piezoelectric material (the “support”), or some more complex arrangement may be used.
  • the thickness of the structure may be less than, or of the order of, the wavelength of the acoustic waves, and such acoustic waves are thus termed Lamb waves, as their form and velocity now depend on the thickness of the structure.
  • Such a device may be referred to as a Lamb wave resonator.
  • the transducer that generates and detects the acoustic excitations may have a design similar as for a SAW resonator, with a finger spacing designed using the Lamb wave velocity in the suspended structure.
  • the length of the suspended structure in the direction transverse to the parallel lines of the transducer IDT fingers may be chosen to be an integer multiple of the acoustic wavelength A, such that the edges of the structure may act as mirrors for the acoustic waves. In some implementations, the edges can be made slightly convex, to refocus the Lamb waves to the central part of the structure.
  • the acoustic energy in the Fabry-Perot resonances may be better trapped in the suspended structure, yielding a sharper resonant response, i.e. a high quality (Q) factor.
  • the edges of the structure can also be made using additional structures, as shown by 340 of Figure 3.
  • additional structure may be fabricated as a phononic crystal, where if the frequency of the Lamb waves is in the bandgap of the phononic crystal, the sounds waves will be reflected by the phononic crystal and the phononic crystal thus serves as a mirror.
  • a Lamb wave structure may be designed to have low frequency modes with small effective spring constants (by, using, for example, thinner and larger structures), yielding a better sensor design, and the interaction between the high frequency Lamb wave resonance and the low frequency mode may thus be designed to be stronger, yielding stronger parametric effects.
  • a variation on the surface acoustic wave resonator above using a mechanically-suspended structure may be made.
  • waves traveling in the interior (bulk) of the resonator volume may be used. This may be fabricated using either a bulk piezoelectric material or a thin film of piezoelectric on the surface of a thicker film of non-piezoelectric material.
  • the acoustic waves may be made to resonate with the thickness of the overall structure, so that an integer number of half-wavelengths would match the thickness of the device.
  • the thickness may be much larger than the wavelength, so these structures may be referred to as high-overtone bulk acoustic resonators or “HBAR” structures.
  • these structures may be designed so that the thickness is a half wavelength, or a small multiple of the halfwavelength, in which case the resonators may be referred to as film bulk acoustic resonators, or “FBAR” structures (the “bulk” referring to the nature of the acoustic wave).
  • the mechanical supports for the mechanically suspended structure can be made to trap the acoustic energy in the volume of the resonator, using e.g. a phononic crystal, achieving higher quality factors, and again the structure can be designed to have low-frequency resonances that interact with the higher frequency acoustic waves to achieve the desired nonlinear piezoelectric effects described above.
  • the solid-state structures/devices may be used in quantum information processing applications for coupling quantum bits (qubits) through the nonlinear coupling mechanism to low-frequency mechanical motion, thereby enabling, for example, further quantum-limited readout.
  • the qubits operate in the microwave frequency range.
  • Such qubits may include and are not limited to superconducting qubits.
  • These qubits may be integrated with the solid-state structures/devices described above.
  • These qubits may be coupled to the low-frequency mechanical motion of the solid-state structure/device via the parametric nonlinear coupling described above.
  • Such coupling may enable transfer and control of quantum information and thus allows for coupling between qubits at different microwave frequencies via low-frequency mechanical modes.
  • one qubit operating at a higher frequency may induce higher frequency mechanical mode via piezoelectric effect, which in turn may be coupled to another qubit via lower mechanical modes.
  • the firs qubit may thus be adapted to control and/or measure another qubit via the lower mechanical modes.
  • FIG. 7 shows an example solid-state structure 700 that may be fabricated according to the principles underlying FIG. 3 and other portions of the disclosure above.
  • the solid-state structure may be configured to support various low-frequency mechanical modes as well as resonant SAW modes, and to implement the inter-mode energy transfer for various applications described above.
  • the example solid-state structure 700 may include a piezoelectric material layer 702 with IDT fingers and electrodes 704 deposited thereover.
  • the piezoelectric material layer 702 with the IDT fingers and electrodes 704 may be disposed over the base layer or support layer 710.
  • Part of the support layer at the two transverse ends of the IDT finger region may be patterned to form phononic crystal structures 720, which are characterized by particular phononic bandgap for reflecting SAWs in the piezoelectric material layer and the support layer at a frequency within the bandgap of the phononic crystal to form a SAW resonator at that frequency.
  • the IDT fingers are spaced at 1 micron and are deposited as 30 nm thick aluminum conductive traces. Such spacing and thickness are merely examples. In practice, the IDT finger spacing may be designed according to a desired SAW wavelength and may range from, for example, a hundred nanometers to many microns. Likewise, the thickness of the conductive traces may be deposited at thickness other than the 30 nm value as illustrated.
  • FIG. 8 shows an SEM micrograph of one example design 802 of the solid- state structure.
  • FIG. 8 further shows a blow-up of portion 810 of the solid-state structure illustrating the IDT fingers 820 and the phononic crystal structures 830.
  • FIG. 8 also shows that the solid-state structure 802 may be implemented as a mechanically suspended thin-film structure supported or anchored by elements 840 (also shown in FIG. 7 as 740).
  • the suspended structure including the IDT traces, the piezoelectric layer and the support layer may be much thinner than the relevant SAW acoustic wavelength. As such, the SAW may experience all the layers as a whole.
  • a top view of the solid-state structure 802 is further shown in the SEM micrograph 900 of FIG. 9.
  • FIG. 10 shows a top-view SEM micrograph of another example solid-state structure 1000.
  • the example solid-state structure 1000 may include IDT fingers 1002 over a piezoelectric material layer, phononic crystal structures 1004, and the support anchors 1010 for the solid-state structure 1000 as a mechanically suspended structure.
  • FIG. 1 1 shows an SEM micrograph of another example solid-state structure 1 100.
  • the example solid-state structure 1 100 may include IDT fingers 1 102 over a piezoelectric material layer, and support micro-mechanical spring anchors 1 1 10 for the solid-state structure 1 100 in thin film.
  • the transverse edges 1120 of the mechanically suspended thin-film structure 1 100 may function as a reflector for the SAW generated inside the thin film.
  • the anchor elements 1 1 10 may be particularly designed with a soft spring that may help release strain in the solid-state structure 1 100.
  • FIG. 12 illustrates an example SAW mode in the thin-film structure of the solid-state structure 700 shown in FIG. 7.
  • an example resonant SAW profile 1210 (acoustic oscillation amplitude) along the cross section 1202 of the mechanically suspended solid-state structure 700 is shown in 1210, having a wavelength determined by the IDT finger spacing.
  • FIGs. 13-14 and 7 illustrate intermediate and final structures during an example fabrication process for the example solid-structure 700 of FIG. 7.
  • FIG. 13 shows an intermediate structure 1300 after depositing a layer of piezoelectric material 1302 over a support substrate layer 1304.
  • the support layer 1304 may be a mechanically suspended thin-film silicon fabricated in various manners.
  • the thin film silicon layer 1304, for example, may be fabricated with a thickness between tens of nanometers to hundreds of nanometers.
  • Other substrates material, such as gallium arsenide, silicon nitride, and silicon dioxide fabricated in mechanically- suspended thin films may be used as the supporting layer.
  • the piezoelectric material layer 1302 may include, for example, an aluminum nitride layer.
  • the aluminum nitride piezoelectric material layer 1302 may be deposited with a thickness in the range from tens of nanometers to hundreds of nanometers.
  • Other piezoelectric material such as zinc oxide, lead zirconate titanate, gallium arsenide, lithium niobate, and quartz as examples deposited on the thin-film support layer may be used.
  • FIG. 14, for example, further illustrates patterning of portions of the support layer to form phononic crystal structures 1402 as mirrors for the Fabry-Perot resonator for the SAW.
  • the IDT traces including the fingers and corresponding electrodes 704 may be deposited and patterned over the piezoelectric material layer 1302 of FIGs. 13 and 14 to form the final solid-state structure 700.
  • FIG. 15 further illustrates an example top view SEM micrograph of the phononic crystal structure 720 of FIG. 7 and 1402 of FIG. 14.
  • the phononic crystal structure pattern includes void portions (e.g., 1502) and remaining portions (e.g., 1504) of the support layer. The shapes of these portions may be designed such that the resulting phononic crystal structure supports a desired phononic bandgap.
  • the example design in FIG. 15 shows cross-shaped voids with a blow-up view in 1510.
  • the cross shaped voids for example, may be characterized by an overall size dimension H, an arm width W, and turning curvatures Ri and Ffc, as indicated in FIG. 15.
  • Figure 16 further shows a unit cell of the non-void portion 1504 of the support film in the phononic crystal structure.
  • a unit cell may be designed to support the desired phononic band gaps.
  • the unit cell containing the non-void portion 1504 is characterized by unit cell length a, thickness T, and an angle 6 for one of the vertical edges of the connecting arm (between unit cells) with the vertical axis (normal to the thin-film plane).
  • the photonic band structure of the example phononic crystal structure of FIG. 15 may be obtained by numerical simulation, as shown in FIG. 17.
  • FIG. 17 shows the propagation acoustic frequency as a function of propagation wavevector using standard Brillouin zone notation, indicating a bandgap 1702.
  • the solid-state structures described above may be adapted for parametric displacement sensing, where monitoring motion at a high frequency mode at f a reveals motion at a lower frequency Fb, yielding improved signal-to-noise and providing a multiplexing mechanism for reading out multiple mechanical modes at different frequencies.
  • the motion at low frequency F ⁇ may induce motion at high frequency via the nonlinear coupling, which may be detected by the detection component such as the interdigitated electrodes described above.
  • a high-frequency motion may be induced, and the presence of the low frequency motion and the inter-frequency coupling may lead to generation of sidebands of the high-frequency motion at one or a multiple of the low frequency apart from the high-frequency.
  • Such sidebands may be detected by the detection component such as the interdigitated electrodes described above.
  • the solid-state structures above may be adapted for sideband cooling, where a drive signal may be set to f a - Fb and the mode at Fb may be automatically cooled to a mode temperature limited by the mode’s coupling to the environment (inversely proportional to its mechanical quality factor) compared to the cooling rate ng, where n represents the number of quanta (phonons) in the driven mode at f a , and can reach 10 5 -10 6 or larger.
  • This may be used to greatly improve the signal-to-noise for measurement of the low-frequency mechanical mode, which may be used to detectweak signals.
  • the electromagnetic excitation may be tuned to f a - Fb, and as a result of the inter-frequency nonlinear coupling, a mechanical motion quantum at the low frequency Fb may be converted into a motion quantum at frequency f a , assisted by the drive at f a - Fb, leading to cooling of the low- frequency motion.
  • the solid-state structures above may be adapted for sideband damping reduction and amplification, achieved by tuning the drive instead to fa + Fb, reducing the intrinsic loss in the mechanical mode and, when driving sufficiently hard, amplifying the motion of the mechanical mode. Both of these effects can significantly improve the signal-to-noise for detecting motion in the mechanical mode.
  • the electromagnetic excitation may be tuned to f a + Fb, and as a result of the inter-frequency nonlinear coupling, a mechanical motion quantum at the low frequency Fb as well as a mechanical motion quantum at the high frequency f a may be generated from the electromagnetic drive at f a + Fb.
  • the solid-state structures above may be adapted for inducing a phase shift of the higher frequency mode response due to the low frequency motion, giving a dispersive readout method.
  • the solid-state structures above may be adapted for quantum non-demolition measurements, where a single variable (e.g. position) of the mechanical mode may be measured with arbitrary position, again with applicability to sensing.
  • the solid-state structures above may be adapted for coupling quantum bits (qubits) through this nonlinear coupling mechanism to low- frequency mechanical motion, thereby enabling further quantum-limited readout and control of audio frequency mechanics.
  • quantum bits quantum bits
  • many types of qubits operate in the microwave frequency domain, and the parametric coupling described and controlled herein provides a coupling mechanism of these qubits to audio frequency mechanics.
  • This is especially relevant for superconducting qubits, which are solid- state devices that could be easily integrated with mechanical sensors.
  • the solid-state device above may be connectable to a quantum-limited or other amplifier and being adapted to amplify electrical signals generated from the interdigitated electrodes therein.
  • the descriptions of the applications above focus on the lowest order nonlinear coupling.
  • the same underlying principles are applicable to higher order nonlinear coupling.
  • the higher order sidebands involving frequencies that are a multiple of the low-frequency apart from the high frequency maybe involved via the higher order nonlinearities.
  • higher sidebands may be used for cooling or amplification of the low-frequency mechanical motion.
  • the electromagnetic excitation may be accordingly tuned away from the high frequency by a multiple of the low-frequency.
  • terms, such as “and”, “or”, or “and/or,” as used herein may include a variety of meanings that may depend at least in part upon the context in which such terms are used.
  • the term “one or more” or “at least one” as used herein, depending at least in part upon context may be used to describe any feature, structure, or characteristic in a singular sense or may be used to describe combinations of features, structures or characteristics in a plural sense.
  • terms, such as “a”, “an”, or “the”, again, may be understood to convey a singular usage or to convey a plural usage, depending at least in part upon context.
  • the term “based on” or “determined by” may be understood as not necessarily intended to convey an exclusive set of factors and may, instead, allow for existence of additional factors not necessarily expressly described, again, depending at least in part on context.

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Abstract

Sont divulgués des procédés, des dispositifs et des systèmes permettant d'utiliser certains types de non-linéarités paramétriques dans certaines structures à l'état solide pour obtenir un transfert d'énergie entre des modes mécaniques de mouvement de plages de fréquences distinctes pour divers types d'applications. Les non-linéarités paramétriques peuvent être soit modifiées, soit intrinsèquement présentes. Des exemples de non-linéarités paramétriques qui peuvent être utilisées comprennent, sans y être limités, n'importe quelle combinaison de non-linéarités géométriques et divers types de non-linéarités constitutives. Ces non-linéarités paramétriques et d'autres types de non-linéarités paramétriques permettent un couplage non linéaire entre les mouvements mécaniques à différentes fréquences. Un tel couplage permet un transfert d'énergie, dans l'une ou l'autre direction, entre ces mouvements mécaniques de fréquences distinctes, ce qui permet d'utiliser des dispositifs nouveaux ou améliorés dans le cadre de diverses applications de détection, de transduction, d'actionnement et de traitement d'informations quantique.
PCT/US2022/080498 2021-11-30 2022-11-28 Couplage paramétrique interfréquence et transfert d'énergie de mouvement mécanique WO2023102349A1 (fr)

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Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6335667B1 (en) * 1998-08-28 2002-01-01 Seiko Epson Corporation Multi-longitudinal mode coupled saw filter
US20160109258A1 (en) * 2013-04-16 2016-04-21 The Regents Of The University Of California Continuous mode reversal for rejecting drift in gyroscopes
US20190270635A1 (en) * 2018-03-05 2019-09-05 California Institute Of Technology Techniques for bidirectionaltransduction of quantum level signals between optical and microwave frequencies using a common acoustic intermediary
US20200304093A1 (en) * 2018-02-08 2020-09-24 Kyocera Tikitin Oy Coupled mems resonator

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6335667B1 (en) * 1998-08-28 2002-01-01 Seiko Epson Corporation Multi-longitudinal mode coupled saw filter
US20160109258A1 (en) * 2013-04-16 2016-04-21 The Regents Of The University Of California Continuous mode reversal for rejecting drift in gyroscopes
US20200304093A1 (en) * 2018-02-08 2020-09-24 Kyocera Tikitin Oy Coupled mems resonator
US20190270635A1 (en) * 2018-03-05 2019-09-05 California Institute Of Technology Techniques for bidirectionaltransduction of quantum level signals between optical and microwave frequencies using a common acoustic intermediary

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