An on-chip platform for multi-degree-of-freedom control of two-dimensional quantum and nonlinear materials

Abstract

In one aspect, a MEMS system is disclosed, which includes a first platform, a first two-dimensional material (2DM) structure coupled to the first platform, a second platform separated from the first platform, a second two-dimensional material coupled to the second platform, and a MEMS actuator mechanically coupled to said first platform and configured to provide displacement and rotation of the first platform relative to the second platform.

Description

[0001]Attorney Docket No. 4424-0063WO01 AN ON-CHIP PLATFORM FOR MULTI-DEGREE-OF-FREEDOM CONTROL OF TWO-DIMENSIONAL QUANTUM AND NONLINEAR MATERIALS Cross-Reference to Related Applications This application claims priority to U.S. Provisional Application No. 63 / 548,314 filed on November 13, 2023, the content of which is incorporated herein in their entirety. This application is also related to PCT Application No. PCT / US2023 / 037206, which is also herein incorporated by reference in its entirety. Statement Regarding Federally Sponsored Research or Development This invention was made with government support under W911NF21-2-0147 awarded by U.S. Army Research Office (ARO). The government has certain rights in the invention. Background The present teachings are generally related to MEMS systems that allow tuning separation and relative rotational positions of a pair of two-dimensional material structures relative to one another. Two-dimensional materials (2DMs) and their derived heterostructures have electrical and optical properties that are widely tunable via several approaches, most notably electrostatic gating and interfacial engineering such as twisting. While electrostatic gating is inherently simple and has thus been ubiquitously employed on 2DMs, the ability to tailor the interfacial properties in a similar real-time manner represents the next leap in modulating the underlying physics and build exotic quantum devices with 2DMs. However, all existing approaches rely on external machinery such as scanning microscopes, which often limit their scope of applications, and there is currently no systems for tuning a 2D interface with similar accessibility and scalability as device-level electrostatic gating. As such, there is a need for enhanced MEMS systems that can provide precise movement of 2DM structures relative to one another. Attorney Docket No. 4424-0063WO01 Summary In one aspect, a MEMS system is disclosed, which includes a first platform, a first two- dimensional material (2DM) structure (composition) coupled to the first platform, a second platform separated from the first platform, a second two-dimensional material structure coupled to the second platform, and a MEMS actuator mechanically coupled to said first platform and configured to provide displacement and rotation of the first platform relative to the second platform. In some embodiments, one or both of the 2D material structures can be in direct contact with the respective platforms, while in other embodiments, one or both of the 2D material structures (compositions) can be coupled to the respective platform via one or more intermediate material structures. In various embodiments, at least one of the first and the second platform can include a metal film that is configured to allow establishing an electrical connection with a respective two- dimensional material structure coupled to that platform. By way of example, and without limitation, the metal film can be fabricated via a thermal or an e-beam evaporation on a respective platform so as to be in electrical contact with a respective two-dimensional material structure coupled to that platform. In various embodiments, one of the platforms can be stationary and the other platform can move relative to the stationary platform. In various embodiments, the movable platform can be displaced relative to the stationary platform so as to change a separation distance between the two platforms and / or rotate the movable platform relative to the stationary platform so as to adjust a twist angle between the two platforms. By way of example, the movable platform can be displaced relative to the fixed platform along an axis orthogonal to a surface of the fixed platform and can also be rotated about that axis relative to the fixed platform. In some embodiments, the MEMS system can be configured to allow optical access to at least one of the 2DMs, and preferably to both. For example, one or both of the platforms can be formed of a material that is substantially transparent to visible and / or infrared radiation so as to allow passage of such radiation therethrough for interrogating one or both of the 2DMs. By way Attorney Docket No. 4424-0063WO01 of example, and without limitation, some examples of suitable materials include fused silica, silicon, diamond, lithium niobate, and glass. In various embodiments, the MEMS system can include at least one MEMS displacement motor mechanically coupled to the first platform for causing the displacement of the first platform relative to the second platform, and a MEMS rotational motor mechanically coupled to the first platform for causing the rotation of the first platform relative to the second platform. In some such embodiments, the MEMS displacement motor can include a first electrode, a second electrode, and a lever coupled to said second electrode via at least one fixed hinge and coupled at a distal end thereof to the first platform via at least one flexible hinge. Further, the lever can be coupled at its proximal end via a flexible hinge to an anchor, which can be, e.g., a part of the device layer of the MEMS system. In some embodiments, each electrode can include a semiconductor, a metal or a combination thereof. By way of example, each of the electrodes can include a semiconductor portion on which an electrically conductive material, e.g., a metal layer, is deposited. By way of example, and without limitation, the semiconductor can be any of silicon, germanium, gallium arsenide, and gallium phosphide and the metal layer can be any of nickel, copper, gold, and aluminum. By way of another example, each of the electrodes can be formed of a heavily doped semiconductor, such as doped silicon. Further, each electrode can have a length in a range of about 10 microns to about 1000 microns, a width in a range of about 10 microns to about 5000 microns, and a thickness in a range of about 10 microns to about 100 microns, all by way of example. A plurality of insulating spacers can separate the first and the second electrodes from one another. By way of example, and without limitation, the insulating spacers can include an oxide material, e.g., silicon oxide. The application of a voltage difference across the first and the second electrodes can cause a deflection of the lever, thereby causing a displacement of the first platform relative to the second platform. Further, in some embodiments, a ratio of a total length of the lever relative to a separation between the at least one fixed hinge coupling the lever to the second electrode and the at least one flexible hinge coupling the lever to the anchor is in a range of about 1.5 to about 10, Attorney Docket No. 4424-0063WO01 e.g., in a range of about 2 to about 5, thereby allowing the lever to amplify the displacement of the first platform relative to the second platform. In some embodiments, the voltage difference applied across the two electrodes can be in a range of about 10 V to about 200 V, though other voltages can also be employed. By way of example, a DC voltage source operating under control of a controller can be utilized to apply the voltage across the two electrodes. In some embodiments, the first and the second electrodes can be separated by a distance in a range of about 1 micron to about 5 microns in absence of application of the voltage across said first and second electrodes. In some such embodiments, the MEMS displacement motor can be configured to cause a maximum displacement of the first platform of about 5 microns. As noted above, in some embodiments, the lever is coupled at a proximal end thereof via a flexible hinge to an anchor, which can form a portion of a device layer of the MEMS system in which the MEMS displacement motor is incorporated. In some such embodiments, the lever is coupled at a distal end thereof via another flexible hinge to the movable platform. Further, the second electrode can include two segments, where each segment is coupled to the lever via a fixed hinge. The fixed hinges coupling the lever to the second electrode can be positioned between the flexible hinges coupling the lever to the anchor and the movable platform. In some embodiments, the lever can further include a mechanical limiter for constraining the deflection of the lever. Such a mechanical limiter can be integrally formed with the rest of the lever and can be formed of the same, or a different material, relative to the rest of the lever. By way of example, upon the deflection of the lever through a certain angle, the mechanical limiter can come into contact with the edge of the first electrode and hence inhibit further deflection of the lever. Some suitable materials for forming the mechanical limiter include, without limitation, any of semiconductors or metals mentioned above. In various embodiments, the MEMS rotational motor can include at least two electrodes (which are herein referred to as a third and a fourth electrode to distinguish them from the electrodes of the MEMS displacement motor) each of which includes a plurality of poles, and a rotor that is mechanically coupled to the first platform and includes a plurality of poles and is Attorney Docket No. 4424-0063WO01 positioned relative to said at least third and fourth electrodes to allow independent electrical interaction between the poles of the rotor and the poles of each of said at least third and fourth electrodes in response to application of a DC potential difference between the poles of the rotor and the poles of each of said at least two electrodes to cause rotation of the rotor, thereby causing rotation of the first platform relative to the second platform. The rotation of the first platform relative to the second platform in turn results in a change in a twist angle between the two 2DMs coupled to the two platforms. While in some embodiments the two 2DMs have the same compositions, in other embodiments, they can have different compositions. By way of example, and without limitation, any of the 2DMs can be any of graphene, hexagonal boron nitride, transitional metal dichalcogenides, high-temperature superconductors, and topological insulators. Some examples of suitable transitional metal dichalcogenides include MoS2 and WSe2. The high-temperature superconductor can be, for example, Bi2Sr2CaCu2O8+x. In various embodiments, any of the first and the second platform can be substantially transparent to radiation with wavelengths in the visible and / or infrared portion of the electromagnetic spectrum so as to allow accessing the 2DMs via radiation. In some embodiments, a pillar is coupled to the first platform and provides a flat surface on which the first 2DM is deposited. The coupling between the pillar and the first platform is such that a displacement and / or a rotation of the first platform via any of the MEMS displacement and / or rotational motors can result in a respective displacement and / or rotation of the pillar and consequently that of the 2DM structure deposited on the flat surface of the pillar. By way of example, the pillar can be formed of a semiconductor, such as silicon. In some embodiments, a pyramidal protrusion is fabricated at the flat surface of the pillar to provide extra height to the 2DM. The tip of the pyramid can be sharp or can be flat. By way of example, the pyramid can be formed from the same or different material as the pillar, such as silicon. Attorney Docket No. 4424-0063WO01 In some embodiments, each of the electrodes of the MEMS rotational motor can include an inner surface including the plurality of electrode’s poles and the rotor includes an outer surface including the plurality of the rotor’s poles. The MEMS system can further include at least one voltage source operating under control of a controller that is configured for establishing the DC potential difference between the poles of the rotor and the poles of each of the electrodes of the MEMS rotational motor. The controller can be programmed so as to cause the voltage source to activate each of the electrodes in a different temporal interval so as to provide a desired rotation of the rotor. By way of example, the MEMS rotational motor can include three independent electrodes, where each electrode can be independently activated in a different temporal interval such that each activated electrode can cause some rotation of the rotor such that the electrodes can collectively cause a total desired rotation of the rotor. In some embodiments, the MEMS system can include at least one anchor, which can be different from the anchor to which the proximal end of the lever is coupled, that is mechanically coupled to the first platform, where the anchor is isolated from the rotor thereby remaining stationary as the rotor rotates. In some such embodiments, the anchor is mechanically coupled to the first platform via one or more suspension beams. By way of example, the suspension beam can be formed of a semiconductor, such as silicon. Further, in various embodiments, the suspension beams can have a width in a range of about 1 micron to about 10 microns. Further understanding of various aspects of the present teachings can be obtained by reference to the following detailed description in conjunction with the associated drawings, which are described briefly below. Brief Description of the Drawings FIG. 1A schematically depicts a MEMS system according to an embodiment of the present teachings, Attorney Docket No. 4424-0063WO01 FIG. 1B schematically depicts an embodiment of a MEMS system according to the prsent teachings in which a thin metal film is deposited on each of the 2D materials to allow establishing electrical contact therewith, FIGS. 2A schematically depicts a cross-sectional view of an implementation of the MEMS device illustrated in FIG.1, FIG. 2B is a schematic exploded view of the MEMS system depicted in FIG. 2A, FIG. 3 is a schematic partially top view of the MEMS system depicted in FIGS. 2A and 2B, illustrating four displacement MEMS motors and a rotational MEMS motor of the MEMS system, FIG. 4A is a schematic partial cross-sectional view of a MEMS displacement motor suitable for use in a MEMS system according to embodiments of the present teachings in an inactivated state, FIG. 4B is a schematic view of the MEMS displacement motor of FIG. 4A in an activated state, FIG. 4C depicts a perspective view of a lever and an upper electrode of the MEMS displacement motor shown in FIGS. 4A and 4B, where the lever is positioned between two segments of that electrode, where the lever is in inactivated state, FIG. 4D depicts a perspective view of the lever and the upper electrode shown in FIG. 5A, where the lever is in an activated state, FIG. 5A is a top schematic view of a MEMS rotational actuator according to an embodiment of the present teachings, FIG. 5B schematically depicts the establishment of a voltage differential across a pole of a rotor of the MEMS rotational actuator depicted in FIG. 2A and a respective pole of one of the electrodes, FIG. 5C presents an example of actuation of three electrodes of a rotational actuator illustrated in FIG. 5A in three phases, Attorney Docket No. 4424-0063WO01 FIG. 6A shows a top schematic view of the MEMS rotational motor in one rotational state, FIG. 6B is another view of the MEMS rotational motor depicted in FIG. 6A is another rotational state, FIG. 7A is a diagram schematically depicting activation of the three electrodes of the MEMS rotational actuator illustrated in FIG. 2A via three voltage sources operating under control of a controller, FIG. 7B is an example of an implementation of a controller suitable for use in the practice of various embodiments, FIGS. 8A – 8F illustrate various stages of a fabrication method according to an embodiment as applied to a wafer, FIG. 9A schematically depicts a measurement setup utilized for performing SHG measurements, FIG. 9B shows a quarter-wave plate polarimetry setup for arbitrary polarization generation, FIG. 10A schematically depicts a simplified version of the experimental set-up that was employed for making the measurements, FIG. 10B shows measured SHG signals as a function of axial separation between the 2DMs, FIG. 11A shows a typical map of the SHG signal in a free-standing state of the device, FIG. 11B shows a map of SHG signal when the 2DMs were in contact with one another with a twist angle zero degree between them, FIG. 11C shows a map of SHG signal when the 2DMs were in contact with one another with a twist angle of 9 degrees between them, FIG. 11D an optical micrograph of the state shown in FIG. 11C, Attorney Docket No. 4424-0063WO01 FIG. 12 shows the SHG signal measured during forward and backward scanning of the vertical control voltage as the two-dimensional flakes were brought into and out of contact with one another, FIG. 13 shows the SHG circular dichroism, as function of θ and h, FIG.14A shows a typical polarization-dependent SHG measurement in twisted h-BN, FIG. 14B shows a pseudospin ψ representing a direction on the Bloch (Poincar é) sphere,FIG. 14C shows the calculated values of the polarization angle α of the maximumSHG power in a synthetic space generated by separation (h) and the twist angle (θ) betweentwo h-BN two-dimensional materials, FIGS. 14D – 14G show theoretical calculation for four types of synthetic merons (Q = +1 / 2) and anti-merons (Q = −1 / 2), FIGS.15A – 15D show the measured and calculated polarization angle α and SHG power ratio Pmin / Pmax in two MEMS devices according to the present teachings, FIG.16A shows schematically that in a surface SHG process, two fundamental photons of energy ℏω combine into a single photon at 2ℏω, FIG.16B schematically depicts the use of a MEMS system according to various embodiments for performing spontaneous parametric down-conversion (SPDC), FIG. 16C schematically depicts the measurement of SHG polarization, FIG. 16D shows major polarizations that can be accessed using an example of a device according to the present teachings, and FIG. 16E shows coverage of the Poincar é sphere associated with variouspolarization of SHG accessible via an example of a MEMS system according to the present teachings. Detailed Description The present disclosure relates generally to MEMS systems (devices) that can be utilized to move a pair of two-dimensional materials relative to another. By way of example, such a Attorney Docket No. 4424-0063WO01 MEMS device can allow changing a separation between a pair of two-dimensional materials and / or rotate the two-dimensional materials relative to one another. As discussed in more detail below, a MEMS device according to some embodiments can both cause displacement as well as rotation of a pair of two-dimensional materials relative to one another. The terms “two-dimensional material,” its abbreviation “2DM,” and “two-dimensional material structure” and similar phrases are used herein to refer to a material composition (structure) that is formed of one atomic layer or multiple stacked atomic layers such that the thickness of the layers is not more than a few hundred nanometers, e.g., less than about 500 nanometers, or less than about 300 nanometers, or less than about 200 nanometers. The term “visible range of the electromagnetic spectrum,” and similar terms, as used herein, refer to a wavelength range of about 450 nm to about 750 nm. The term “infrared portion of the electromagnetic spectrum,” and similar terms, as used herein, refer to a wavelength band in a range of about 0.7 micrometer (micron) to about 25 microns. The term “substantially transparent” is used to describe a material structure that allows passage of at least about 80%, or at least about 90%, of visible and / or infrared radiation incident thereon. The term “about” as used herein denotes at most 10%, or at most 5%, variation about a numerical value. The term “substantial” denotes a deviation, if any, of at most 20%, or at most 10%, from a complete condition and / or state. Two-dimensional materials (2DMs) have emerged as a transformative class of materials with the potential to uncover novel physics and new device applications, in particular due to their remarkable tunability by electrostatic gating and extensibility through van der Waals stacking. The introduction of twist(s) in stacks of 2DMs, in conjunction with the intrinsic lattice mismatch in heterostructures, further offers control over their band structures and many-body correlationsvia moir é effects.For nearly two decades, the assembly of 2D heterostructures has predominantly relied on dry and wet transfer methods. These methods have proven reliable, relatively simple to implement, and amenable to subsequent processing steps, such as etching and electrode Attorney Docket No. 4424-0063WO01 evaporation, utilizing standard semiconductor processes. However, as the complexity of stacked 2D material structures continues to evolve, a limitation of the transfer methods becomes increasingly evident. Each stacked configuration produced through transfer methods is inherently unique and non-reconfigurable. The associated non-reproducibility essentially precludes convenient exploration of various interfacial degrees of freedom (IDoFs), such as a twist angle of two layers relative to one another, which in turn limits the statistical inference of properties of such materials. Recent developments have led to the rise of scanning microscope-based platforms, which can control the twist angle between successive layers and further allow performing tunneling spectroscopy at the same time. While such platforms represent a significant step forward, they are inherently limited in the scope of their applicability. For example, such platforms do not allow direct optical access to the 2DMs without obstruction. Moreover, the frontier in the study of 2DMs often requires exposing the 2DMs to ultra-low temperatures in combination with magnetic fields. Reaching cryogenic temperatures and high magnetic fields in a scanning microscope is possible but challenging, as highly specialized setups are needed. Such setups are inherently costly, highly complex, and are often susceptible to vibrations from the low- temperature apparatus. Other available methods for altering a twist angle between a pair of 2DMs typically are not suitable for achieving real-time variation of the twist angle and / or are restricted to near-ambient conditions. The present disclosure provides an on-chip platform based on MEMS for generic manipulation of 2DMs with unprecedented flexibility and accuracy. Such a platform (which is herein also referred to as MEMS-based Generic Actuation Platform for 2DM (MEGA2D)) not only addresses the long-felt need for in situ control over 2D material interfacial degrees of freedom (IDoFs) but also opens a gateway to a wealth of new possibilities in condensed-matter physics, optics and beyond. The MEMS devices according to various embodiments of the present teachings can be employed as a ‘knob’ for tuning a 2D heterostructure, e.g., via changing the separation and / or the twist angle between one or more pairs of 2DMs. In various embodiments, such tuning of 2D heterostructures can be on par with electrostatic gating, in terms of convenience, scalability, and accessibility to the broader research and industrial community. Attorney Docket No. 4424-0063WO01 Various embodiments of the present teachings provide MEMS actuators that allow independent adjustment of two critical IDoFs, namely, the twist angle, θ, and the separation distance, h, between two layers of 2DMs. Two primary challenges arise when utilizing MEMS for the manipulation of 2DMs: (1) MEMS actuators typically exhibit limited travel ranges, necessitating that the 2DM layers be in close proximity, typically within a few micrometers, at their rest positions, (2) the mating interfaces between the 2DM layers must possess a high degree of flatness and parallelism to ensure consistent and uniform interaction. The former challenge is exacerbated when working under conditions involving low temperatures and / or high magnetic fields, where thermal expansion and magnetostriction can distort the MEMS device by more than several micrometers. To maintain platform scalability, it is imperative to maintain this initial gap between the two 2DMs without relying on an external positioner, such as a mechanical or piezo stage. Further, ensuring a high degree of parallelism over a region spanning even a few micrometers is challenging without the use of a two-axis goniometer. The introduction of such a goniometer, however, can significantly increase system complexity while compromising mechanical stiffness and overall robustness of the system. In various embodiments, a MEMS according to the present disclosure solves the above shortcomings of conventional systems. For example, as discussed in more detail below, in various embodiments, a MEMS device according to the present disclosure includes integrated MEMS displacement and rotational motors that allow adjusting the separation as well as the twist angle between a pair of two-dimensional materials. FIG. 1A schematically depicts a MEMS system (device) 100 according to an embodiment, which includes a platform 102 (herein also referred to as a substrate 102) on a top surface of which a two-dimensional material 104 is disposed. By way of example, substrate 102 can be formed of a suitable semiconductor material, such as silicon and is fixedly positioned (i.e., it is not movable). A variety of techniques can be employed for depositing the 2D material 104 on the surface of the substrate 102. Some examples of such techniques include, without limitation, dry transfer, wet transfer, and chemical vapor deposition. The system 100 can further include a MEMS actuator 106 to which another 2D material 108 is coupled. In various embodiments, the composition of the 2D materials coupled to the substrate 102 and MEMS actuator 106 are the same. In other embodiments, the pair of 2D Attorney Docket No. 4424-0063WO01 materials 104 and 108 have different compositions. In this embodiment, the MEMS actuator 106 can change the axial separation between the two 2D materials 104 and 108 along a direction perpendicular to the substrate’s surface (designated as the z-dimension), which is also herein referred to as the vertical direction. In addition, MEMS actuator 106 can rotate 2D material 108 relative to 2D material 104, thereby adjusting the twist angle between the pair of 2D materials. In this embodiment, MEMS actuator 106 is configured so as to allow the adjustment of the vertical separation between the two 2D materials independently of the adjustment of the twist angle between the two materials. In some embodiments, a thin metal film can be deposited on at least one of the 2DMs to allow establishing an electrical connection to one or both of the 2DMs. By way of example, FIG. 1B shows two thin metal films 104a and 108a, which are deposited, respectively, on the 2D materials 104 and allow establishing electrical contact with the 2D materials. FIGS. 2A and 2B schematically depict a system 200, which is an example of implementation of system 100 discussed above. The system 200 includes a fixed substrate (platform) 202 on a top surface of which the 2D material 104 is disposed. Further, the system 200 includes a MEMS actuator 204 that includes a MEMS handle layer 209 and a MEMS device layer 212. The handle layer 209 is separated from the fixed platform 202 via a plurality of spacers 210. In this embodiment, the spacers 210 are formed of SU-8 resin and can have a thickness in a range of about 0.5 µm to about 10 µm, by way of example. The MEMS device layer 212 is in turn separated from the handle layer 209 via an SiO2separation layer 214. The handle layer further includes a silicon pillar 206 that is coupled to a movable platform via a portion of the SiO2 separation layer 214 and is suspended over substrate 202. The other 2D material 108 is coupled to the silicon pillar 206. More specifically, the silicon pillar 206 includes a base portion 206a from which a pyramidal-shaped tip portion 206b extends in a downward direction toward the substrate 202 such that a shoulder 206c surrounds the tip portion. The tip portion terminates at a substantially flat surface 207. In this embodiment, the 2D material 108 is coupled to the pillar 206 such that a circumferential portion of the 2D material 108 is fixed to the peripheral portion of the shoulder 206c and a central portion of the 2D material 108 is in contact with the substantially flat surface 207 of the tip portion such that the Attorney Docket No. 4424-0063WO01 central portion of the 2D material is maintained in a substantially flat configuration and is separated from a respective portion of the 2D material 104. Referring to FIG. 3, the MEMS actuator 212 includes four MEMS displacement motors 216a, 216b, 216c, and 216d (herein collectively referred to as MEMS displacement motors 216) that allow causing axial displacement of the movable platform 213 to which the pillar 206 is coupled relative to the fixed platform 202. The axial displacement of the movable platform 213 relative to the fixed platform in turn adjusts the axial separation between the 2D material 104 that is coupled to the fixed platform 202 and the 2D material 108 that is coupled to the movable platform 213 via the silicon pillar 206. In various embodiments, the displacement motors 216 allow adjusting the separation between the two 2D materials 104 and 108 by a maximum separation of about 5 microns. As discussed in more detail below, in some embodiments, a mechanical limiter can be utilized to constrain the vertical motion of a lever associated with a displacement motor. MEMS actuator 212 further includes a MEMS rotational motor 218 that allows the rotation of the movable platform 213, and consequently the rotation of the silicon pillar 206, relative to the fixed platform 202, which in turn causes the adjustment of the twist angle between the two 2D materials. The displacement motors 216 function as electrostatic actuators that can be modeled as parallel plate capacitors. As is known in the art, the energy stored in a parallel plate capacitor is given by the following relation: ^^^ ^= ^ ^ ^^^^ ^ = ^^Eq. (1) where, E denotes the energy of the capacitor, C denotes the capacitance of the capacitor, V denotes the voltage differential across the capacitor’s plates, A denotes the surface area of the overlap between the two plates, d denotes the separation between the two plates, and ε0 denotes the vacuum permittivity. Attorney Docket No. 4424-0063WO01 For a capacitor in which one plate is fixed and the other plate is movable and can be displaced relative to the fixed plate (e.g., moving toward the fixed plate along a dimension perpendicular to the plates that can be referred to without loss of generality as the z-dimension), the electrostatic force applied to the movable plate can be defined as follows: ^^^^^ = ^^^^ = −^^^^^^^^Eq. (2) MEMS electrostatic actuators essentially utilize this force to balance a force from MEMS spring(which in some cases can be modeled by Hook’s law ^ = ^^^) to achieve a displacement ^^ =^ / ^. As discussed in more detail below, the displacement motors according to various embodiments utilize a lever-based mechanism to amplify motion of one of the 2D materials relative to the other. FIGS. 4A and 4B schematically depict two of the displacement motors 216, which are disposed on opposed sides of the movable platform 213, e.g., displacement motors 216a and 216b. For ease of description, in the following discussion, reference will be made only to displacement motor 216a. The other displacement motors are structurally and functionally identical to the displacement motor 216a. The displacement motor 216a includes a movable upper electrode 400 and a fixed lower electrode 404 and a lever 406 that is fixedly coupled to the upper electrode, as discussed in more detail below. In this embodiment, the fixed lower electrode 404 is part of the handle layer 209 and is formed of silicon portion 404a on which a thin metal layer 404b is deposited. With continued reference to FIGS. 4A and 4B, as well as FIGS. 4C and 4D, the movable upper electrode 400 includes two arcuate electrically conductive segments 400a and 400b. In some embodiments, the electrically conductive segments can be formed of a doped semiconductor, e.g., doped silicon. The lever 406 is positioned between the two conductive segments 400a / 400b and is coupled to those segments via a pair of fixed hinges (joints) 403a / 403b (herein collectively referred to as fixed hinges 403). At its proximal end, the lever 406 is coupled to an anchor 408, which forms part of the device layer, via a fixed hinge 405. At Attorney Docket No. 4424-0063WO01 its distal end, the lever 406 is coupled to the movable platform 213 via a pair of flexible hinges 407a / 407b (herein collectively referred to as hinges 407). A voltage source 410 allows application of a voltage difference between the upper 400 and the lower 404 electrodes so as to cause deflection of the upper electrode toward the lower electrode, thereby causing a deflection of the lever 406, which in turn causes a displacement of the movable electrode. The lever advantageously amplifies the deflection of the platform. By way of example, the amplification can be characterized, e.g., by a ratio between the total length of the lever relative to the distance between the fixed hinges 403 coupling the lever 406 to the upper electrode 400 and the flexible hinge 405 coupling the lever to the anchor 408, can be in a range of about 1.5 to about 10. By way of illustration, FIGS. 4A and 4B show the depicted MEMS displacement motors 216a and 216b in an inactive state and FIGS. 4B and 4C show the MEMS displacement motors 216a and 216b in an activated state in which the lever of each of the motors is deflected towards the lower fixed electrodes 404. In this embodiment, the voltages applied to the four displacement motors are configured such that the motors collectively translate the movable platform 213 along the vertical direction (i.e., along a direction perpendicular to the lower fixed platform). By way of example, and without limitation, the voltage applied across the upper and the lower electrodes can be in a range of about 10 V to about 100 V for causing a full range of displacement of the movable platform relative to the fixed platform. In addition, in this embodiment, the upper electrode 400 includes a mechanical range limiter 412 that constrains the motion of the lever 406. In particular, in this embodiment, a contact between the mechanical range limiter 412 and the lower electrode, due to the deflection of the lever, will constrain further deflection of the lever, e.g., to prevent shock damage and / or electrostatic pull-in. For example, in this embodiment, the mechanical range limiter limits the maximum vertical displacement of the movable platform to about 4 microns. FIGS. 5A, 5B, and 5C schematically depict a MEMS rotational motor 2000 as an example of the implementation of a rotatable carriage 2020 (herein also referred to as a rotor), which includes arcuate segments 2040 and 2060, where each arcuate segment 2040 and 2060 includes a plurality of poles 2050 and 2070, respectively, disposed on an outer perimeter thereof. Attorney Docket No. 4424-0063WO01 In this embodiment, the poles 2050 and 2070 are in the form of a plurality of teeth, though in other embodiments, other structures may be employed. The rotor 2020 includes two connecting segments 2050a / 2050b that mechanically connect the arcuate segments 2040 and 2060 to the movable platform 213 that is positioned at the center of the rotor. The rotational actuator 2000 further includes a plurality of suspension beams 2220a, 2220b, 2220c, and 2220d that connect the carriage 2020 to platforms 2250 and 2260. Further, the rotational actuator 2000 includes two anchors 2280a / 2280b (herein referred to collectively as anchors 2280). The anchor 2280a is mechanically connected to the platform 2260 via a pair of suspension beams 2300a / 2300b. The anchor 2280b is, in turn, mechanically connected to the platform 2250 via another pair of suspension beams 2320a / 2320b. The anchors 2280a / 2280b are not directly mechanically connected to the rotatable carriage. Rather, they are separated from the interior perimeter of the carriage by a pair of gaps. The rotational MEMS actuator 2000 also includes two sets of electrodes 2080 and 2100, which are positioned in proximity to the outer perimeters of the arcuate segments 2040 and 2060, respectively, to engage electrically with those arcuate segments to cause their rotation, as discussed in more detail below. More specifically, the electrode set 2080 includes three arcuate-shaped electrodes 2080a, 2080b, and 2080c (herein also referred to as stators A, B, and C, respectively), which are electrically isolated from one another, with each surrounding a portion of the outer perimeter of the arcuate segment 2040 of the carriage. The electrodes 2080a, 2080b, and 2080c include, respectively, a plurality of poles 2090a, 2090b, and 2090c, which are herein in the form of a plurality of teeth, that are positioned in close proximity to the poles of the arcuate segment 2040 to be able to interact via electrostatic forces with those teeth so as to cause the rotation of the carriage, as discussed below. Similarly, the electrode set 2100 includes three arcuate-shaped electrodes 2100a, 2100b, and 2100c (herein referred to as stators A, B, and C), which are electrically isolated from one another, with each surrounding a portion of the outer perimeter of the arcuate segment 2060 of the carriage. The electrodes 2100a, 2100b, and 2100c include, respectively, a plurality of poles Attorney Docket No. 4424-0063WO01 2110a, 2110b, and 2110c, in the form of teeth in this embodiment, that are positioned in proximity to the teeth of the arcuate segment 2060 to be able to interact via electrostatic forces with those teeth so as to cause rotation of the carriage, as discussed below. In some embodiments, the poles can be in the form of a comb. By way of illustration, as shown schematically in FIG. 5B, a voltage differential between a tooth A of the rotatable carriage and a respective tooth B of one of the electrodes, which are rotationally offset relative to one another, results in the generation of an electric field E that causes the movement of the tooth B of the carriage relative to the tooth A of the electrode, thereby causing the rotation of the carriage relative to the electrode. In other words, tooth A and tooth B function as the two plates of a small capacitor, where the energy stored in the capacitor is given by above Eq. (1). The tangential force between the two electrodes as a function of the distance (x) between them is given by: ^^^^^ ^^^^ = − ^^Eq. (2) where, d is the dimension of the electrode in the direction perpendicular to the plane of the actuator. In this embodiment, the force between tooth A and tooth B is substantially parallel to the plane of the carriage. Hence, for rotation of the carriage, a plurality of smaller capacitors is employed, which can be actuated, with a proper phase offset as discussed below, to cause the rotation of the carriage relative to the electrodes. In this example, in phase B, the poles of an electrode align precisely with the carriage poles, while in phases A and C, they are misaligned by 1 / 3 and -1 / 3 of the pitch of the poles (i.e., the spacing between the poles). With reference to FIG. 5C, in this embodiment, the electrodes of the MEMS rotational actuator can be activated in three phases, which are herein referred to as phase A, phase B, and phase C. Each electrode can be individually energized via a voltage waveform to attain the desired rotation. Energizing the three phases induces an electrostatic force between the poles of the rotor and those of the electrodes (stators), causing the rotor’s poles to re-align with the stator poles, neutralizing the generated electric field. Subsequently, the adjacent phase is activated, Attorney Docket No. 4424-0063WO01 causing the carriage to once again align its poles with the relevant stator poles. Thus, in this example, at any given moment, only one-third of the rotor poles are synchronized with the stator poles. More specifically, when a MEMS rotor is actuated in phase B, its poles align precisely with those of the electrode B, while the poles are misaligned with electrode A and C by 1 / 3 and - 1 / 3 of the pitch of the poles (i.e., the spacing between the poles). In some embodiments, micro-stepping of the applied voltage can be employed for a finer rotational tuning. In the simplest scenario, actuating phase A and phase B at the same time with the same voltage will result in a position halfway between aligning with stator A poles and stator B poles. Adjusting the ratio between these two voltages can in principle continuously interpolate any angular position between phase A alignment and phase B alignment. The same principle can be applied to any pair of phases. The result is the “micro-stepping” waveform shown in FIG. 5C, in which any angular position on the rotor can be in principle reached by smoothly interpolating using all three phases. By way of further illustration, FIGS. 6A and 6B show the rotatable carriage of the MEMS rotational motor at two rotational states. As the rotatable carriage rotates, it causes a respective rotation of the movable platform to which the pillar supporting one of the 2D materials is coupled, thereby changing the twist angle between the two 2D materials. In various embodiments, the precision of the rotational angle control can be in a range of about 0.1 to about 1 degree, e.g., about 1 / 6 degrees, without the usage of micro-stepping. In some embodiments, the rotatable carriage can have a diameter in a range of about 500 microns to about 5000 microns and a thickness in a range of about 10 microns to about 100 microns. Further, in various embodiments, the suspension beams can have a length in a range of about 100 microns to about 2000 microns and a width in a range of about 1 micron to about 10 microns. Further the electrodes A, B, and C can have a width in a range of about 10 microns to about 200 microns. With reference to FIG. 7A, in use, a desired rotation angle is inputted into a controller 601 that is in communication with a waveform generator 602, which is in turn in communication with voltage sources 604a / 604b / 604c (collectively referred to as voltage sources 604) to cause the voltage sources to actuate the electrodes via application of desired three-phase voltage waveforms to the electrodes. More specifically, in this embodiment, the three voltage sources Attorney Docket No. 4424-0063WO01 are utilized to establish independent potential differences between the teeth of each electrode A, B, and C and the respective teeth of the rotor. FIG. 7B schematically depicts an example of an implementation of the controller 601 that includes, among other components known in the art, a processor 612, a random-access- memory (RAM) module 614, a permanent memory module 616, a communications module 615, and a communications bus 613 that allows various components to communicate with one another. Instructions for generating control signals for application to the waveform generator for causing the activation of the MEMS actuators can be stored in the permanent memory and can be transferred to the RAM module 614 during runtime, via the processor, to be executed. With reference to FIGS. 8A – 8F, one example of a method for fabricating a MEMS device according to the present teachings includes the following steps: (1) coating a commercial silicon-on-insulator (SOI) wafer with thermal silicon oxide (FIG. 8A), (2) oxide etching (FIG. 8B), (3) tip KOH etching and oxide removal (FIG. 8C), (4) TSV etching and coating (FIG. 8D), (5) MEMS structure etching and vapor HF release (FIG. 8E), and (6) top layer fabrication and bonding (FIG. 8F). In the following description of the fabrication process, it is assumed that the 2DMs are h-BN flakes, but the same processes can be employed with other 2DMs. By way of example, the fabrication process can start with a 6-inch silicon-on-insulator (SOI) wafer, comprising a 450 µm silicon (Si) handle layer, a 2 µm buried oxide (BoX) layer, and a 60 µm Si device layer. In some cases, each wafer can contain 332 dies. The wafer can be first coated with a silicon oxide hard mask on both sides (FIG. 8A). One side of the hard mask can be etched into a small square (FIG. 8B). The underlying silicon layer can be etched into a pyramid using, e.g., a 45% KOH solution at 75 ⁰C. In some implementations, the height of the pyramid can be controlled via the duration of the etching to be around 600 nm. The hard mask can be subsequently removed in diluted HF solution (FIG. 8C). TSVs (through silicon vias) are employed as electrical connections between the top and bottom p-doped silicon layers. The TSV holes can be first patterned on the device layer using photolithography. The device layer and the BoX layer can be subsequently removed via reactive-ion etching (RIE). A conductive layer can be conformally deposited on the wafer, ensuring full coverage of the TSV holes. Before the deposition, the backside of the wafer can be protected by plasma-enhanced chemical vapor deposition (PECVD) grown silicon oxide. This Attorney Docket No. 4424-0063WO01 oxide layer and the conductive layer grown on it can be removed by RIE before proceeding to the next fabrication steps (FIG. 8D). The MEMS structure fabrication can include the following three steps: backside patterning, front side patterning, and HF release (FIG. 8E). A tetraethylorthosilicate (TEOS) oxide hard mask can be first grown on both sides of the wafer. The Si handle layer can be first patterned and etched from the back side up to the BoX layer, using deep RIE. After plasma cleaning, the front side can then be patterned and etched from the front side down to the BoX layer. At this stage, the wafer may become fragile in which case it would require handling with extreme care. The final stage of the fabrication process is HF release. The wafer can be thoroughly cleaned to remove fluorocarbon residues. The BoX layer, as well as the remnant TEOS silicon oxide hard mask, can be undercut using an HF vapor etcher at 35 ⁰C for 45 minutes. This will fully suspend the MEMS vertical actuators and the MEMS rotator. Subsequently, the individual dies can be removed from the wafer, e.g., by a tweezer without the need for dicing. In some implementations, the bottom layer can include a 1×1 cm2fused silica chip pre- patterned with alignment markers. After 2D material flakes are transferred to a fused silica wafer, they are cleaned and annealed at 300◦C∼500◦C overnight, and a 2.2 µm thick SU-8 layer is patterned on the wafer. The wafer can then be diced using a commercial dicing saw. Each MEMS die can be aligned to a fused silica chip and bonded using heat and pressure (See, FIG. 8F). Finally, the device can be electrically connected to a measurement PCB by wire bonds. For transfer of the flakes of 2DM material onto the MEMS pyramid without damaging the suspended MEMS structures, which can be susceptible to shock and cannot be exposed to any liquid, a modified dry transfer method using poly bisphenol-A carbonate (PC) film has been developed. A thin PC and polypropylene carbonate (PPC) film can be spin coated on a clean Si wafer and stacked on a pre-molded polydimethylsiloxane (PDMS) flat-top pyramid. The PPC lies between PC and PDMS to provide a thermal releasing mechanism at 200◦C. The PC surface of the stack can be used to pick up h-BN flakes at 70◦C∼90◦C. A sharp Si knife (e.g., a knife formed by 40 µm- tall ridges etched on a Si wafer) can be used to cut the PC / PPC stack at 180◦C. This step makes it much easier to release the PC / PPC stack without resorting to tearing. The Attorney Docket No. 4424-0063WO01 segregated PC / PPC square with h-BN flake can then be released on the MEMS pyramid tip at 200◦C. The PC and PPC can be removed in chloroform in a special holder that prevents the solvent from entering the active MEMS structure, since any liquid in the MEMS will cause stiction. The MEMS die can be subsequently vacuum annealed at 300◦C∼500◦C overnight to remove PC and PPC residue. A simpler procedure can be performed on the fused silica substrate side. A PC film can be used to pick up the h-BN at similar temperature as above and directly release it onto clean fused silica substrate at 150◦C. SU-8 can be subsequently spun and patterned on the substrate. The substrate can then be cut with dicing saw into 1 × 1 cm2chips. The assembly of the MEMS chip and fused silica substrate can take place at 200◦C, which remelts the SU-8 and forms a strong bond between the two chips. Cleanliness of the fused silica and MEMS chips can be important to the success of the fabrication process, which means any dust that is larger than a few microns should be removed. In all the steps described above, note should be taken of the orientation of the h-BN flakes that was pre-determined via SHG measurement and the information be utilized to align h-BN flakes to within 2◦. The following examples are provided for further elucidation of various aspects of the present teachings and are not presented to provide necessarily optimal ways of practicing the present teachings and / or optimal results that may be achieved. Examples Example 1 A prototype MEMS device was fabricated using two layers of hexagonal boron nitride (h-BN) single crystals as the 2DMs to measure second-harmonic generation (SHG). FIG. 9A schematically depicts a measurement setup utilized for performing SHG measurements. The light source was a 780-nm femtosecond fiber laser with a minimal pulse width of about 100 fs. After rotating the polarization using a λ / 2 plate and optionally a λ / 4 plate (for SHG CD measurements), the light beam was focused onto the fused silica side of the sample using a 40x objective with a correction collar to compensate for the fused silica thickness. The reflected SHG light was collected with the same objective, passed through the same waveplates, and was Attorney Docket No. 4424-0063WO01 separated from the incident beam using a dichroic mirror. The reflected beam was further filtered using a bandpass filter to remove the fundamental light, and then it passed through a linear polarizer, and was measured using a homemade camera spectrometer. FIG. 9B shows a quarter-wave plate polarimetry setup for arbitrary polarization generation. A quarter-wave plate (QWP) polarimetry was used to directly measure the polarization of the reflected SHG wave. The polarizer was also fixed to allow passage of vertical polarization. The SHG signal was measured as a function of QWP angle Θ as the twist angle between the two 2DM layers was rotated 0 to 180⁰. This information was used to extract the Stokes parameters S0, S1, S2, S3of the SHG light. 2DMs with broken C2symmetry, such as h-BN, have strong second-order nonlinear response (χ(2)) that is highly sensitive to lattice orientation, as well as the layer arrangement along the optical path. Enhancement of the maximum SHG signal can be achieved via several mechanisms, including cavity enhancement, plasmonic enhancement, and quasi-phase matching. In this example, the SHG signal enhancement was utilized as a sensitive probe for the interlayer distance h between two h-BN flakes, which were employed as the 2DMs. The MEMS device was assembled with a h-BN flake on the pyramid associated with the MEMS actuator and another h-BN flake on a fused silica fixed substrate. A 40x objective was used to a focus a λ = 780 nm femtosecond laser beam through the fused silica substrate onto the h-BN flakes and collect the reflected SHG signal at λ / 2 = 390 nm, using the same objective. As the size of the pyramid’s flat tip was 4 x 4 µm, the light spot, which had a diameter less than 1 µm, could be fully contained within the h-BN flake coupled to the pyramid’s tip. FIG. 10A schematically depicts a simplified version of the experimental set-up that was employed for making the measurements. The SHG signal was measured as a function of a voltage applied to the MEMS displacement motor for tuning the separation (h) between the two h-BN flakes along the z- dimension (the voltage is herein referred to as Vz), as shown in FIG. 10B. It was discovered that the SHG signal was tunable by more than an order of magnitude via tuning h, a manifestation of the enhancement of the signal due to a cavity formed by the two h-BN flakes. Remarkably, the SHG data clearly shows four periods before it settles to a constant value, which is an indication Attorney Docket No. 4424-0063WO01 of contact between the two h-BN flakes. The cavity enhancement is periodic in second harmonic wavelength, such that each period corresponds to a motion of approximately 390 nm, and the observed four periods therefore match well with the designed free-standing gap of about 1.6 µm. The reduction of the peak height as the separation between the flakes is increased may be attributed to decoherence of the femtosecond laser radiation, which had a coherence length of about 10 µm. FIG. 11A shows a typical map of the SHG signal in a free-standing state of the device (h = 1.6 µm). When the vertical drive of the MEMS displacement motor was activated the two flakes were brought into contact, the observed SHG signal initially became non-uniform, which may be attributed to formation of bubbles between the flakes, as it is typically found during stacking of 2DMs, as shown in FIG. 11B. However, upon several rotations between ±10⁰ while the flakes remained in contact, the uniformity of the SHG signal significantly improved, as shown in FIG. 11C, demonstrating the self-cleansing property of the 2D interface. FIG. 11D is an optical micrograph of the state shown in FIG. 2C. The data shows that in various embodiments, it is important to have the mating surface between the 2DM and the underlying surface be flat. A MEMS device according to various embodiments can allow directly measuring mechanical properties of a 2D interface, such as the van der Waals attraction between 2DMs. These quantities are traditionally measured also by scanning probe techniques. FIG. 12 shows the SHG signal measured during forward and backward scanning of the vertical control voltage as the h-BN flakes were brought into and out of contact with one another. A clear hysteresis was observed upon removing contact between the flakes (i.e., the flakes were disengaged from one another), where the SHG signal abruptly jumped from the contact value to the engagement curve, at a separation of h = 220 nm. Using the calculated MEMS vertical stiffness of kz = 300 Nm-1, the maximum van der Waals attraction was estimated as kzd = 66 µN, which corresponds to a van der Waals tensile strength of 4 MPa (contact area was assumed to be 16 µm2). This value is comparable to, but smaller than, the intrinsic cleavage strength of h-BN reported in literature, presumably due to the misalignment between the two h-BN flakes used in the measurements. h-BN crystals feature a mirror plane aligned with its armchair direction. This mirror symmetry remains intact when two h-BNs are twisted by an angle θ that is a multiple of 60⁰. Attorney Docket No. 4424-0063WO01However, for other values of θ, this mirror symmetry is lifted by the emergent moir é pattern,which results in observable effects. It was possible to directly detect this breakdown of chiral symmetry. Chiral materials respond differently to left (LCP) and right (RCP) circularly polarized light. To probe the chirality of twisted h-BN via SHG, a quarter-wave plate (QWP) was inserted before the objective to convert linear polarized light to circular polarized light (and vice versa for collected SHG light) and the SHG circular dichroism (CD) was computed, where CD was defined as (P^^ ) / ^^^^ SHG− PSHG ( PSHG+ PSHG), wherein PSHG andPSHG denote the SHG powers using LCP respectively. The MEMS device was fabricated with a free-standing h-BN twist angle near 0 degrees. FIG. 13 shows the SHG CD, as function of θ and h. It is clearly observed that while the measured SHG CD varies strongly with h, it is always antisymmetric with respect to θ and vanishes at θ = 0. The observed SHG CD signal was on the order of unity, much stronger than the CD measured in the transmittance of twisted 2DMs. Thus, it has been demonstrated that the symmetry breaking properties of 2DM can be continuously varied ‘on-the-fly’ using the MEMS device. Example 2 The tuning of h and θ provided by MEMS devices according to the present teachings makes it possible to study and engineer physics phenomena with new dimensions. The IDoFs provided by various MEMS devices according to the present teachings can be treated by the MEMS device as synthetic dimensions. These additional continuous parameters allow higher- dimensional physics, such as topology, to be realized in a system with fewer physical dimensions. In nonlinear optics involving 2DMs, the rank-3 nonlinear susceptibility tensor χ(2)has eight components that are of importance. Materials with C3or higher crystallographic symmetry, such as bulk h-BN (which have additional mirror symmetry, as mentioned above) or arbitrarily twisted h-BN (no additional symmetry), have two independent groups of coefficients: χ1= χ(2)xxx= −χ(2)yyx=−χ(2) yxy= −χ(2) xyyand χ2= −χ(2) yyy= χ(2) xyx= χ(2) xxy= χ(2) yxx. These two (D3h symmetry), ψ is proportional to [1, 0] if x-direction is parallel to its armchair direction. For the twisted h-BN, the effective nonlinear susceptibility can be complex and the effective ψrepresents a direction on the Bloch (Poincar é) sphere (See, FIG. 14B). This direction can beviewed as the orientation of a pseudospin vector. Attorney Docket No. 4424-0063WO01 For an arbitrary stack of twisted h-BN flakes (or any 2DM with three-fold symmetry), ψ can be derived to be: $%ℎ, () = *∑, ^,%ℎ)-. / 01,%1)∑, ^,%ℎ) / 2^01,%1) 3 Eq. (3)where At(h) is an h- from t-th h-BN monolayer, and θt(θ) describes the angle between x axis and the armchair direction of the t-th monolayer, which changes when the relative twist angle θ is varied. Ψ is a function of synthetic variables h and θ. Importantly, ψ is periodic in both h (period is λ / 2) and θ (period is 120◦), and the h-θ synthetic space has the same torus-like topology as the first Brillouin zone in a 2D crystal lattice. The normalized pseudospin ψ / |ψ| also defines a polarization state (generally elliptical), at which maximum SHG power is reached. The second-order nonlinear optics in twisted C3- symmetric 2DMs realizes an effective two-band system in the h-θ synthetic space. The spectra and eigenstates of this two-band system represents SHG power and the polarization state mentioned above, respectively. To experimentally probe the nonlinear pseudospin ψ, the SHG power as a function of linear polarization was measured in a parallel configuration and with the same polarization of the incident and reflected SHG waves. FIG.14A shows a typical polarization-dependent SHG measurement in twisted h-BN. From each curve like this, the SHG power can be fitted to a cosine function with respect to the polarization to obtain three independent parameters: SHG power maxima Pmax, minima Pmin, and polarization α of the maxima. As shown in FIG. 14A, this type of measurements can yield a characteristic six-fold symmetry pattern with respect to the incident polarization. The following three quantities can be identified for each measurement: the maxima Pmax, the minima Pmin, and the polarization angle of the maxima α, which is in the range of 0◦≤ α < 60◦. With reference to FIG. 14B, these quantities are directly tied to the geometricalrepresentation of ψ on the Poincar é sphere. The azimuthal angle of ψ on the sphere equals 6α, andits altitude is given by 2 tan−1(Pmin / Pmax)1 / 2. FIG. 14C shows the calculated values of α in the synthetic space, for ntop = 20 layers and nbottom= 30 layers of h-BN on the fused silica (top, in the measurement configuration, See FIG.8A) and MEMS die (bottom) respectively. It was surprisingly found that rich features exist at certain locations of the synthetic space, where α acquires a rotation from 0◦to 60◦(or vice versa) Attorney Docket No. 4424-0063WO01 around singularities. With the vie1w of ψ as a pseudospin, these singularities are analogous to topological singularities found in magnetic materials, known as half-skyrmions or merons. Themerons are characterized by their topological charge Q = 2 p · v = ±1 / 2, which is determined by thecore polarity p = ±1 and the vorticity v = ±1. p = +1 or −1 means that ψ points to +z or −z on thePoincar é sphere at the core of the meron, whereas v = +1 or −1 means that ψ forms a vortex oranti-vortex around that core. There are therefore four possible types of merons / anti-merons in total. In these examples, all four types of synthetic merons (Q = +1 / 2) and anti-merons (Q = −1 / 2) were found, as illustrated in FIGS. 14D, 14E, 14F and 14G. Using MEMS devices according to the present teachings, such topological features can be accessed experimentally. FIGS.15A – 15D show the measured and calculated polarization angle α and SHG power ratio Pmin / Pmaxin two MEMS devices according to the present teachings. In both devices the experiments can be well-modeled by simulation. In one device (FIGS. 15A and 15B), an anti-meron (Q = −1 / 2) labeled by (p, v) = (+1, −1) can be clearly identified. At the core of the anti-meron, Pmin / Pmaxapproaches unity, consistent with the expectation that ψ points towardspoles of the Poincar é sphere. In another device (FIGS. 15C and 15D), a meron of the type (+1,+1) was found. It is noted that the polarity p of the meron cannot be directly determined by linear polarization SHG, but can be identified via SHG CD measurement and compared with simulation. The pseudospin texture of an isolated meron covers half of the Poincar é sphere, and thatof a meron-meron pair, or bimeron, in principle provides a full coverage on the Poincar ésphere. A practical implication of the existence of these synthetic topological quasiparticles is that the second-order susceptibility χ(2)of an active optical stack can be engineered to cover the full available space permitted by symmetry, while tuning h and θ over a small range. Example 3 In some applications, the tunability of the nonlinear susceptibility near a synthetic meron can be utilized to design full-stokes tunable classical and quantum light sources. To facilitate an intuitive understanding of the process, the pseudospin ψ can be reparametrized using a single complex number γ: ψ ∝ [cos γ, sin γ]. In the surface SHG process, two fundamental photons of energy ℏω combine into a single photon at 2ℏω (See, FIG.16A). If the polarization of the two fundamental photons αω, α′ωand the second- harmonic polarization α2ωare all different, their Attorney Docket No. 4424-0063WO01 polarization states obey a simple polarization sum rule that relates these three angles to γ, as follows: αω+ α′ ω + α2ω= γ + kπ, k = 0, ±1, ±2, ... α can be complex too, with the imaginary parts by a Mercator conformal projection onto the Poincar ésphere. With γ being tunable via the IDoFs h and θ, a light source at 2ω frequency can be realized with a tunable polarization state α2ω. In conventional approaches using 2DMs, active tuning of SHG polarization can only be achieved via intricate control of the arrival times of the two fundamental photons, and the tuning range is limited to a subset of polarizations. Incontrast, in various embodiments, the tunability of ψ can in principle span the entire Poincar ésphere, and a full-Stokes tunable SHG source can thus be implemented, with h and θ being the only necessary tuning knobs. The fundamental laser beam was fixed to vertical polarization α ′ ω = αω= 0 and the SHG polarization α2ωwas measured using quarter-wave plate polarimetry in a MEMS device according to the present teachings, as shown in FIG. 16C. The polarization vortex in α2ω at a synthetic meron, around which α2ω rotates by 180◦can be directly observed. Exactly at the center of the vortex, the SHG wave becomes left circularly polarized; this can be described by α2ω= γ = +i∞ in Eq. 3. By tuning h and θ, nearly any polarization state can be achieved. FIG. 16D shows that all sixprincipal polarizations on the Poincar é sphere can be approximately reached within theaccessible synthetic space, and FIG.16E shows the coverage of the sphere. With reference to FIG.16B, in another application, a MEMS device according to various embodiments of the present teachings can be employed in performing spontaneous parametric down-conversion (SPDC) for converting a single photon at 2ω to two photons at ω. SPDC is inherently a quantum mechanical process and can be utilized to form pairs of entangled photons, the entanglement of which is further tunable by varying γ(h, θ) (See, FIG. 16B). For α ′ ω + α ω = 0, a Bell’s state (|↔↔^ − |↕↕^) / √2. If αω+ α′ω = π / 2, we obtain another Bell’s state (|↔↕^ + |↕↔^) / √2. For γ = +i∞ (i.e. ψ ∝ [1, i]), however, αωand α′ω are both forced resulting photons have zero entanglement (|^^^ state). It is believed that such flexibility of tuning the polarization and the extent of entanglement is not available in Attorney Docket No. 4424-0063WO01 other SPDC platforms, because it takes advantage of the fact that SHG / SPDC in 2DMs does not rely on double refraction and are not restricted by phase-matching requirements. A SPDC system, when implemented with MEMS device according to the present teachings, could enable further compactification of quantum optics instruments. In some cases, a 2DM with a stronger nonlinear response than h-BN may be needed for experimental realization of such a tunable quantum light source. Some suitable material may include 3-R TMDCs and other noncentrosymmetric 2DMs. Those having ordinary skill in the art will appreciate that various changes can be made to the above embodiments without departing from the scope of the present teachings.

Claims

Attorney Docket No. 4424-0063WO01 What is claimed is:

1. A MEMS system, comprising: a first platform, a first two-dimensional material coupled to the first platform, a second platform separated from the first platform, a second two-dimensional material coupled to the second platform, and a MEMS actuator mechanically coupled to said first platform and configured to provide displacement and rotation of the first platform relative to the second platform.

2. The MEMS system of Claim 1, wherein said MEMS actuator is configured to allow optical access to at least one of said first and said second two-dimensional materials.

3. The MEMS system of Claim 1, wherein said MEMS actuator is configured to allow optical access to both of said first and said second two-dimensional materials.

4. The MEMS system of Claim 1, wherein the MEMS actuator comprises: at least one MEMS displacement motor mechanically coupled to the first platform for causing the displacement of the first platform relative to the second platform, and a MEMS rotational motor mechanically coupled to the first platform for causing the rotation of the first platform relative to the second platform.

5. The MEMS system of Claim 4, wherein the at least one MEMS displacement motor comprises: a first electrode, a second electrode, a lever coupled to said second electrode via at least one fixed hinge and coupled at a distal end thereof to the first platform via a flexible hinge,Attorney Docket No. 4424-0063WO01 wherein an application of a voltage difference across the first and the second platform causes a deflection of the lever, thereby causing a displacement of the first platform relative to the second platform.

6. The MEMS system of Claim 5, wherein said voltage difference is in a range of about 10 V to about 200 V.

7. The MEMS system of Claim 5, wherein said lever is coupled at a proximal end thereof via a flexible hinge to an anchor.

8. The MEMS system of Claim 5, wherein ratio of a total length of the lever relative to a separation between said at least one fixed hinge coupling the lever to the second electrode and said at least one flexible hinge coupling the lever to the anchor is in a range of about 1.5 to about 10.

9. The MEMS system of Claim 8, wherein said second electrode comprises two segments and said at least one fixed hinge coupling the lever to the second electrode comprises a pair of fixed hinges each coupling the lever to one of said two segments of the second electrode.

10. The MEMS system of Claim 9, wherein said pair of fixed hinges coupling the lever to the second electrode are positioned between said at least a fixed hinge coupling the lever to the anchor and said at least a flexible hinge coupling the lever to the first platform.

11. The MEMS system of Claim 5, wherein said lever comprises a mechanical limiter for constraining deflection of the lever.

12. The MEMS system of Claim 5, wherein said first and said second electrodes are separated by a distance in a range of about 1 micron to about 5 microns in absence of application of the voltage across said first and second electrodes.

13. The MEMS system of Claim 4, wherein said at least one MEMS displacement motor is configured to cause a maximum displacement of the first platform of about 5 microns.Attorney Docket No. 4424-0063WO01 14. The MEMS system of Claim 5, wherein each of said first and second electrodes has a length in a range of about 10 microns to about 1000 microns, a width in a range of about 10 microns to about 5000 microns, and a thickness in a range of about 10 microns to about 100 microns.

15. The MEMS system of Claim 5, wherein any of said first and second electrodes comprises a semiconductor, a metal, or a combination thereof.

16. The MEMS system of Claim 15, wherein any of said first and second electrodes comprises a semiconductor portion with a metal film deposited on a surface thereof.

17. The MEMS system of Claim 16, wherein said metal film has a thickness in a range of about 10 nm to about 1 µm.

18. The MEMS system of Claim 15, wherein said semiconductor comprises any of silicon, germanium, gallium arsenide, and gallium phosphide.

19. The MEMS system of Claim 15, wherein the metal comprises any of nickel, copper, gold, and aluminum.

20. The MEMS system of Claim 5, further comprising a plurality of insulating spacers for separating the first and the second electrodes.

21. The MEMS system of Claim 20, wherein said insulating spacers comprise an oxide material, and wherein optionally the oxide material comprises silicon oxide.

22. The MEMS system of Claim 4, wherein the MEMS rotational motor comprises: at least a third and a fourth electrode each of which includes a plurality of poles, and a rotor mechanically coupled to the first platform and having a plurality of poles and positioned relative to said at least third and fourth electrodes to allow independent electrical interaction between the poles of the rotor and the poles of each of said at least third and fourth electrodes in response to application of a DC potential difference between the poles of the rotorAttorney Docket No. 4424-0063WO01 and the poles of each of said at least two electrodes to cause rotation of the rotor, thereby causing rotation of the first platform relative to the second platform.

23. The MEMS system of Claim 1, wherein said first and said second two-dimensional materials comprise a same material.

24. The MEMS system of Claim 1, wherein said first and said second two-dimensional materials comprise different materials.

25. The MEMS system of Claim 1, wherein any of said first and said second two- dimensional materials comprises any of graphene, hexagonal boron nitride, transitional metal dichalcogenides, high-temperature superconductors, and topological insulators.

26. The MEMS system of Claim 25, wherein the transitional metal dichalcogenides comprise any of MoS2 and WSe2.

27. The MEMS system of Claim 25, wherein the high-temperature superconductors comprise Bi2Sr2CaCu2O8+x.

28. The MEMS system of Claim 1, further comprising a plurality of insulating spacers for separating the first and the second electrodes.

29. The MEMS system of Claim 28, wherein said plurality of insulating spacers comprises an oxide, and wherein optionally said oxide comprises silicon oxide.

30. The MEMS system of Claim 1, wherein any of said first and second platform is substantially transparent to radiation in the visible and the infrared region of the electromagnetic spectrum.

31. The MEMS system of Claim 30, wherein said second platform comprises any of fused silica, silicon, diamond, lithium niobate, or glass.Attorney Docket No. 4424-0063WO01 32. The MEMS system of Claim 1, further comprising a pillar coupled to said first platform such that a displacement of the first platform results in a displacement of the pillar, said pillar providing a flat surface on which said first two-dimensional material is deposited.

33. The MEMS system of Claim 32, wherein said pillar comprises a semiconductor, and wherein optionally said semiconductor comprises silicon.

34. The MEMS system of Claim 22, wherein each of said third and fourth electrodes comprises an inner surface including the plurality of electrode’s poles and said rotor includes an outer surface including the plurality of poles of the rotor.

35. The MEMS system of Claim 34, further comprising at least one voltage source configured for establishing the DC potential difference between the poles of the rotor and the poles of each of said third and fourth electrodes.

36. The MEMS system of Claim 35, further comprising a fifth electrode configured for independent actuation for causing rotation of the first platform.

37. The MEMS system of Claim 36, further comprising a controller configured to send control signals to the at least one DC voltage source to cause the at least one DC voltage source to establish said DC potential difference between the poles of the rotor and the poles of said electrodes by a desired rotational angle.

38. The MEMS system of Claim 24, further comprising at least one anchor mechanically coupled to said first platform, wherein said anchor is isolated from said rotor thereby remaining stationary as the rotor rotates.

39. The MEMS system of Claim 38, wherein the at least one anchor is mechanically coupled to said first platform via one or more suspension beams.

40. The MEMS system of Claim 39, wherein said one or more suspension beams comprise a semiconductor.Attorney Docket No. 4424-0063WO01 41. The MEMS system of Claim 40, wherein any of said at least first and second suspension beam has a width in a range of about 1 micron to about 10 microns.

42. The MEMS system of Claim 1, wherein any of said first and said second platform comprises at least a metal film configured to provide electrical connection to any of the said first and said second two-dimensional materials, respectively.

43. The MEMS system of Claim 42, wherein said at least one metal film is fabricated via any of a thermal and e-beam evaporation in vacuum.

Images