Solid-state tunable ionic oscillators, dielectric materials and resonator devices.

Abstract

An article comprising a ferroelectric material in its ferroelectric phase, the article configured to enable low signal propagation loss with ultra-low dielectric loss at one or more selected frequencies.

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 62 / 713,301, filed Aug. 1, 2018, which is incorporated herein in its entirety.

[0002] The technical field relates generally to dielectric, piezoelectric, and ferroelectric materials, and more specifically to solid-state tunable materials and oscillator devices.

[0003] Government Rights This invention was made with Government support under Grant No. W911NF-14-1-0500 awarded by the U.S. Army Research Laboratory. The Government has certain rights in this invention. [Background technology]

[0004] The ordering of ferroelectric polarization in domains [WJMerz,Domain formation and domain wall motions in ferroelectric BaTi03 single crystals,Phys.Rev.95,690-698(1954)] and its response trajectory under a field [Ruijuan Xu,Shi Liu,Ilya Grinberg,J Karthik,Anoop R Damodaran,Andrew M Rappe,and Lane W Martin,Ferroelectric polarization reversal via successive ferroelastic transitions,Nature Mater.14,79-86(2015)] has been proposed as a novel method for the development of non-volatile memories [A.Chanthbouala,A.Crassous,V.Garcia,K.Bouzehouane,S.Fusil,X.Moya,J.Allibe,B.Dlubak,J.Grollier,S.Xavier,et al,Solid-state memories based on ferroelectric tunnel junctions,Nature Nanotechnology 7, 101-104 (2012)], transducers, [P. Muralt, Ferroelectric thin films for micro-sensors and actuators: a review, Journal of Micromechanics and Microengineering 10, 136 (2000)], and electro-optical devices [BW Essels, Ferroelectric epitaxial thin films for integrated optics, Annu. Rev. Mater. Res. 37, 659-679 (2007)].

[0005] However, for voltage control of capacitance and frequency agility in telecommunication devices, domain walls have long been considered to be an impediment and obstacle, leading to high dielectric losses and hysteresis in response to applied electric fields. [G. Arlt, U. Bottger, and S. Witte, Dielectric dispersion of ferroelectric ceramics and single crystals at microwave frequencies, Ann. Phys. 506, 578 (1994)]. To avoid these adverse effects, tunable dielectrics are often designed to have a ferroelectric Curie T C Since tunable dielectrics rely on operation well beyond this range to operate under piezoelectric resonant conditions [Bob York, Tunable dielectrics for RF circuits, Ch. 6 of "Multifunctional Adaptive Microwave Circuits and Systems", M. Steer, ed. (2006)], it appears that there is an unavoidable trade-off between the two requirements of high tunability and low loss, leading to severe limitations on the figure of merit of tunable dielectric devices.

[0006] Modern telecom devices rely on the ability to tune device frequencies in the radio frequency (RF) spectrum, and current bulk and thin film based acoustic wave filters, resonators, and other devices have enabled cell phone miniaturization, antenna tuning, and the development of current mobile telecom technologies over the past few decades. Further telecom developments (5G, IoT) may require even more efficient use of the spectrum, necessitating the development of thin film dielectrics with higher dielectric tunability n, quality factor Q, and figure of merit, and stimulating intense research and development efforts.In particular, external influences such as defect, strain, interface, and polar ordering have been intensively investigated and have benefited from recent progress in understanding how functional properties can be significantly tuned, evolved from symmetry breaking, or even artificially induced [C.-H. Lee, ND Orloff, T. Birol, Y. Zhu, V. Goian, E. Rocas, R. Haislmaier, E. Vlahos, JAMundy, LF Kourkoutis, et al., Exploiting dimensionality and defect mitigation to create tunable microwave dielectrics, Nature 502, 532-536(2013); A. D. Damodaran, E. Breckenfeld, Z. Chen, S. Lee, and L. W. Martin, Enhancement of ferroelectric Curie temperature in BaTiO3 films via strain-induced defect dipole alignment,Adv.Mater.26,6341-6347(2014);ARDamodaran,C.-W.Liang,Q.He,C.-Y.Peng,L.Chang,Y.-H.Chu,and LWMartin,Nanoscale and mechanism for enhanced electromechanical response of highly strained BiFeO3 thin films, Adv. Mater. 23, 3170-3175 (2011).

[0007] Extrinsic enhancement of magnetic susceptibility from ferroelectric (FE) domain walls [YL Wang, AK Tagantsev, D. Damjanovic, and N. Setter, Giant domain wall contribution to the dielectric susceptibility in BaTiO3, Appl. Phys. Lett. 91, 062905 (2007)] can be achieved by pressure engineering through the creation of domain wall-rich films, whose extrinsic nature allows for dielectric properties that are not limited by the inherent limitations of defect-free bulk. Nevertheless, for tunable dielectrics, polar domains are not considered useful and are generally equivalent to other crystal imperfections (e.g., oxygen vacancies) that must be suppressed to achieve higher material quality in order to obtain lower dielectric losses and higher figure of merit values ​​[AK Tagantsev, VO Sherman, KF Astafiev, J. Venkatesh, and N. Setter, Ferroelectric materials for microwave tunable applications, J. of Electroceramics 11, 5-66 (2003)]. Therefore, domain engineering has not been explored for tunable, low-loss dielectrics.

[0008] Complex ferroelectric domain structures are predicted and have been realized in perovskite ABO3 thin films, such as PbTiO3, Pb(Zr,Ti)O3, BaTiO3, (Ba,Sr)TiO3, and BiFeO3. Domain patterns can be engineered by strain tuning, substrate termination orientation, uniform film composition or graded film thickness, and / or electrical boundary conditions. Such domain patterns can be configured to achieve substantial enhancements in dielectric, pyroelectric, and piezoelectric properties. Films composed of simple c-domains (e.g., out-of-plane polarization) are most common, as this domain organization is most effective for use in actuators, memory elements, and novel gates. However, the associated depolarization fields, especially in thin films, impair function and performance and even limit the written state on time scales of hours.

[0009] The ordering of ferroelectric polarization and its trajectory in response to an electric field is essential for ferroelectric and multiferroic (e.g., ferroelectric-ferromagnet) based non-volatile memories, transducers, electro-optical devices, analog voltage tunable filters, antennas and phase shifters, as well as voltage controlled oscillators and ferroelectric piezoelectric resonators, including but not limited to surface and bulk acoustic wave resonators.

[0010] Currently, for voltage control of capacitance and frequency agility of resonance in telecommunication devices utilizing ferroelectric materials, hysteresis in the ferroelectric polarization-voltage (or electric field) response is problematic in establishing accurate and facile (e.g., single-valued) control of capacitance with voltage (electric field), and it is necessary to design and / or operate ferroelectric materials in their higher temperature paraelectric phase to suppress the undesirable ferroelectric hysteresis. This is known as the ferroelectric phase transition or Curie temperature T c must be significantly lower than the temperature or temperature window of intended operation.

[0011] The maximum value of the dielectric constant, and therefore the tunability of the capacitance, of a given ferroelectric material is determined by the T cIt is well known that the dielectric constant of ferroelectric materials is found at or near T > T c decreases rapidly with temperature and the dielectric tunability depends on the field (or voltage) induced reorientation of the ferroelectric polarization as opposed to the remanent polarization. Thus, the amount of achievable dielectric or capacitance tunability is sacrificed to satisfy the need for suppression of ferroelectric hysteresis.

[0012] Dielectric loss, expressed as tan δ or its reciprocal, the quality factor Q, is a key measure of the performance of materials intended for use in many of the devices and applications mentioned above, and has important implications for their ability to transmit certain frequencies and reject others (frequency selectivity). It is known that the radio and microwave frequency dielectric loss (quality factor) of a material increases (decreases) with increasing frequency ω, and in general the scaling is less weakly linear with ω, according to fundamental principles elaborated for ferroelectrics by Akhiezer [Akhiezer, A. On the absorption of sound in solids, J. Phys. USSR 1, 277 (1939)] and Tagantsev et al.

[0013] Currently, ferroelectric and non-ferroelectric piezoelectric materials are utilized in RF and microwave applications such as filters, antennas, phase shifters, voltage controlled oscillators, and bulk and surface acoustic wave resonators. Ferroelectric materials are typically utilized in their non-ferroelectric phase. Where cost is not prohibitive and availability exists, bulk form single crystal piezoelectric materials such as sapphire or quartz are utilized to provide the lowest loss or highest quality factor. However, where cost is prohibitive and / or the material is not available in single crystal form, polycrystalline bulk ceramic materials are utilized, which have higher loss and reduced quality factor but at a lower cost.

[0014] Non-ferroelectric piezoelectric materials, single crystal or polycrystalline, do not have inherent tunability. That is, the application of a voltage or electric field compresses or expands the material, but there is no ability for the material to inherently change its dielectric constant. Thus, apart from small, slow changes with temperature or pressure, there is essentially no substantial ability for non-ferroelectric piezoelectric materials to change their frequency. Thus, non-ferroelectric piezoelectric materials resonate at a specific set of frequencies (modes) that depend on their material properties and the geometry of the material, and cannot actually be changed. Frequency change has been achieved with non-ferroelectric piezoelectric materials and has been widely employed in communications and information technology, for example, by designing and incorporating multiple sets of fixed-frequency resonators and then using techniques such as spread spectrum frequency hopping to switch between the different piezoelectric resonators when transmitting and receiving signals. This approach is popular as new spectrum is available, accounted for, published, and licensed; however, each time a new set of frequencies is offered, the piezoelectric resonators require the design and fabrication of new chipsets, potentially rendering currently designed and manufactured piezoelectric resonator chipsets obsolete.

[0015] Ferroelectric piezoelectric materials, either single crystal or polycrystalline, possess inherent tunability even when the material is in its paraelectric phase; voltage or electric fields can enable rotation of the remnant or induced ferroelectric polarization, thereby enabling larger changes in the dielectric constant, which can be realized as voltage-tuned capacitances. Voltage-tunable capacitors, when introduced into analog circuits, allow tuning of the resonant frequency, which depends in part on the adjustable capacitance. Thus, ferroelectric piezoelectrics remain attractive for voltage-tunable capacitors, resonators, and analog circuits, where a continuous frequency, as opposed to discrete fixed frequencies, may be desirable. Currently, the best ferroelectric piezoelectric resonators exhibit frequencies that can be voltage-tuned by at most a few percent (<5%). However, current ferroelectric piezoelectric materials, whether in single crystal or polycrystalline form, exhibit significantly higher (lower) dielectric losses (quality factors) than those associated with non-ferroelectric piezoelectric materials, making them less desirable despite this frequency agility.

[0016] The design and manufacture of RF and microwave components is driven in large part by the trend towards further miniaturization of devices and device components, particularly through the use of film and thin film ferroelectric and non-ferroelectric piezoelectric technologies for the efficient and scalable manufacture of resonators, filters, antennas, etc., and integrated circuits. This scaling of technology allows for operation at lower voltages and powers, ease of packaging, and even integration with other technologies. Even the manufacture of discrete capacitor elements benefits from miniaturization through scalable high-yield processing of membrane devices. Thus, the use of films and thin films of piezoelectric materials is highly advantageous over the technologies mentioned above.

[0017] As explained above, the preferred non-ferroelectric piezoelectric materials in film form have a ferroelectric constant of about 10 in the L, S, and C bands (<10 GHz). 3They can be engineered into device structures that exhibit Q values ​​at room temperature that can be as high as 1000 kHz. Well-developed microfabrication of piezoelectric resonators allows the transmission and reception of RF and microwave frequency signals at discrete frequencies with losses low enough to enable applications in mobile phones and other wireless devices, Internet of Things technologies, and biological and chemical sensors based on chemically intrinsic binding and frequency shifting due to mass accumulation. However, the frequencies are fixed according to the resonator material and geometry, requiring different sets of resonators for different frequencies.

[0018] Films of ferroelectric and non-ferroelectric piezoelectric materials, like their bulk counterparts, can be formed in single crystal or polycrystalline form. Polycrystalline films are advantageous for some applications and are easier to produce, while single crystal films have superior dielectric loss properties since they do not have grain boundaries to impede the flow of RF and microwave power. Thus, single crystal ferroelectric and non-ferroelectric piezoelectric films (in their paraelectric phase) are currently superior to polycrystalline films in this regard. Furthermore, ferroelectric or non-ferroelectric single crystals with higher concentrations of point defects and / or dislocations typically exhibit higher losses than those with fewer point defects and dislocations. Thus, when not preferred, polycrystalline films are preferred over the growth of high quality single crystal epitaxial films.

[0019] Since dielectric losses are associated with the dissipation of normal modes in a crystal (phonons), which have a thermal occupancy factor, a decrease in temperature results in lower dielectric losses due to a reduction in the contribution from phonons, all else being equal. Thus, currently, a perfect crystal of a given compound at constant temperature has a well-defined frequency scaling of its dielectric loss or quality factor. In real crystals with a non-zero concentration of finite thermodynamically limited defects, the so-called intrinsic limit on the temperature-dependent dielectric loss or quality factor is not achieved. This has facilitated RF and microwave applications that utilize bulk single crystals and low temperature operation, and also utilize superconducting electrodes to reduce the resistive contribution to losses.

[0020] Currently, because of favorable energetic considerations, the T of a material is limited because domain walls, like point defects, dislocations, and grain boundaries, can be a significant source of energy diffusion and therefore a significant source of dielectric loss. c Ferroelectric domain walls that form spontaneously in ferroelectric materials below the ferroelectric Curie temperature T are problematic. In particular, dielectric losses are now highly dependent on domain wall density, regardless of the type of domain wall. Therefore, to avoid these adverse effects, tunable dielectrics are often designed to have a ferroelectric Curie temperature T c Relying on operation well beyond piezoelectric resonance conditions, it appears that there is an unavoidable trade-off between the two requirements of high tunability and low loss, leading to severe limitations on the figure of merit of tunable dielectric devices. Summary of the Invention

[0021] Described herein are materials with high density of domain walls and engineered multiple domain wall variants, such as dielectric metamaterials. Such engineering can include polarization ordering. Spontaneous ferroelectric polarization ordering is important for, for example, non-volatile memory, piezoelectric transduction, and electro-optical devices, as well as hysteresis-free frequency agile filters and antennas that function via voltage tuning of capacitance.

[0022] Increased engineering of dielectric, piezoelectric, and ferroelectric properties has been achieved through interface-driven phenomena, achieved by arranging phases in superlattices, by engineering nanocomposites, by engineering films where point defects are aggregated in a way that reduces dielectric loss compared to randomly placed point defects, or by engineering solid solutions designed to operate at or near morphotropic phase boundaries that separate well-defined phases. In ferroelectric thin films, extrinsic enhancement of magnetic susceptibility from domain wall pinning can be achieved by strain engineering at room temperature through the creation of domain-wall rich films. The design and realization of heterogeneous dielectric metamaterials featuring phase proximity and accessibility between two or more thermodynamically predicted ferroelectric domain wall variant types is presented.

[0023] One of the most pressing unsolved scientific and technological challenges for communications, navigation, telemetry, etc., concerns new materials and device concepts that could provide disruptive advances in frequency selectivity, spectral management, and reduced power requirements through the nonlinear response properties of materials. To identify and enhance new sources of nonlinear reactions of media to external DC and EM fields, and ultimately to exploit these interactions, it is crucial to expand the range of possible interactions of RF microwave energy with dielectric solids that manage propagation and allow tuning of thermodynamic and physical properties.

[0024] Microwave oscillators play a key role, and the combination of low loss / high quality factor, narrow bandwidth or high tunability, low phase noise, low power consumption / high output power, temperature stability, and frequency settling time are all important considerations. Current solid-state microwave oscillators, whether implemented using dielectric resonator oscillators, surface acoustic wave (SAW) or bulk acoustic wave (BAW) resonators, membrane bulk acoustic wave resonators (FBARs), or in monolithic microwave integrated circuits (MMICs), suffer from a trade-off between the requirement of a high quality factor juxtaposed with inherent limitations on tunability in simple device elements, requiring complex circuitry to meet the design requirements. Furthermore, dielectric losses are understood to be inherently limited in theory, but this is rarely achieved. Parametric power amplifiers (PAs) remain an attractive alternative to transconductance PAs because of their near perfect RF-to-RF conversion efficiency. In addition, parametric systems are not expected to suffer from the gain-bandwidth product that limits the usable bandwidth of transconductance PAs. A parametric amplifier typically comprises at least two basic components: a variable ac capacitor (variac) and an oscillator.

[0025] The present disclosure relates to a new microwave oscillation medium based on novel atomic-scale ionic oscillators that can be realized in ferroelectric solids, and allows for ultra-low losses at selected frequencies that can be designed and controlled based on defect density, type, and properties rather than material density and acoustic wave velocity.

[0026] The appendices and appendices to this application are incorporated herein and are considered to be part of this application as if separately incorporated. The specific compositions and other features disclosed in these appendices and appendices, and obvious extensions thereof, are considered to be additional embodiments of the invention as if specifically set forth in the body of this specification. [Brief description of the drawings]

[0027] The file of this patent or application includes drawings / photographs made in at least one color. Copies of this patent or patent application publication with color drawing(s) / photograph(s) will be provided by the authority upon request and payment of the required fees. [Figure 1] It is a design of a domain wall variant-rich material and its microwave dielectric tunability. a. For Ba0.8Sr0.2TiO3, a thermodynamical landscape of in-plane polarization favorable for domain wall variants, showing the calculated average value |P1| of in-plane polarization as a function of temperature and in-plane strain. Arabic numerals represent various thermodynamically predicted domain wall variant structures. [Diagram 2] Domain structure phase field simulations regarding equivalent strain in BaTiO3 at ≒10K < TC under zero field (b, d, f) and corresponding non-degenerate (vertical dashed lines I and III) and degenerate (II) domain variant phase states persist even under a moderate in-plane field of 0.1 MV / cm applied along

[0100] (c, e, g), revealing high out-of-plane (b) and in-plane (d, f) domain wall densities. The plots of b and c are the normal-to-plane polarization P3 with a color map range of 30 μC / cm2 ±. In the plots (d - g), the angle of in-plane polarization θ is ±90°, where θ is the angle between

[0100] and the sum of the two in-plane components P1 + P2. [Diagram 3] It shows the theoretically predicted in-plane quasistatic field tunability of the relative permittivity ε11 / ε0 in Ba0.8Sr0.2TiO3 films on SrTiO3, on SmScO3, and on BaTiO3, corresponding to strain states I, II, and III, respectively, calculated using the GLD model, looking at the influence of proximity to this domain phase variant degeneracy point. [Figure 4] It is a diagram of the geometry of an experimental two-port comb-shaped electrode capacitor. [Diagram 5]Epitaxial paraelectric (Ba,Sr)TiO3 (BST) [C.J.G.Meyers, CRFreeze, S.Stemmer, and RAYork, (Ba,Sr)TiO3 tunable capacitors with RF commutation quality factors exceeding 6000, Appl.Phys.Lett.109,112902(2016)] and Sr7Ti6O19 [C.-H.Lee, ND Orloff, T.Birol, Y.Zhu, V.Goian, E.Rocas, R.Haislmaier, E.Vlahos, J.A.Mundy, LFKourkoutis, et al., Exploiting dimensionality and defect mitigation to create tunable microwave dielectrics, Nature 502, 532-536 (2013)] compared to the in-plane field tuning of the in-plane normalized capacitance Cnorm(E) measured at selected frequencies for a 400 nm thick manifold domain wall-variant material film (or metamaterial) sample (Ba0.8Sr0.2TiO3 / SmScO3(110)). [Figure 6]6A and 6B: Microwave spectroscopy revealing ultra-low loss field-dependent resonant domain wall spectral signatures, and tunable resonant performance. Experimentally determined Q plotted as a function of frequency and DC in-plane bias field for a thickness of 100 nm, with 10 and b 6 IDC electrode pairs, characterized by period 2W defined by electrode finger width w and interelectrode spacing d shown in optical micrographs as insets (scale bar: 10 μm). C and D: Extracted values ​​of frequency and field at which Q peaks were obtained for devices in A and B, showing that the spectrum of potential-dependent frequencies at which resonant Q peaks occur is essentially the same, and that the resonant frequency can be bias-tuned by ≈400% from ≈2 GHz at ≈0.1 MV / cm to ≈10 GHz at ≈0.67 MV / cm, with ≈103 ≦Q ≦≈106. Representative traces of DC field dependent Q for selected frequencies indicated in the legend, corresponding to the device shown in Figures 6a and b. [Figure 7] Peak Q collected at 100 frequencies for four different devices showing an increase of more than an order of magnitude over about 10 times the frequency, a significant departure from the usual 1 / f scaling law. For comparison, we show the highest values ​​for bulk single crystal quartz, sapphire, and AlN membrane piezoelectric resonators, none of which are inherently tunable (each dot represents an individual device), nor have any inherently tunable BST membrane including membrane bulk acoustic wave solid-mounted resonators (BAW-SMR) been reported. [Figure 8] MD simulation of Q. a, MD supercell and domain fluctuations at Ex = 0.6 MV / cm are shown, with domains with Py < 0 shown in black and domains with Py > 0 shown in gray. [Figure 9] Experimentally obtained Q (left panel) and MD simulated Q (right panel) for the aa1 / aa2 domain structure. Experimental data are shown for E = 0.09, E = 0.25 MV / cm, and E = 0.5 MV / cm. MD data are shown for E = 0, E = 0.3 MV / cm (green), and E = 0.6 MV / cm. [Figure 10]Q(f) of bulk-like layers 100–113 and DW layers 73–84 from MD simulations at E = 0.6 MV / cm. [Figure 11] Hopping speed of individual layers of a 120x10x10 supercell. [Figure 12] Autocorrelation functions of the total P time obtained from MD simulations for E = 0, E = 0.3 MV / cm, and E = 0.6 MV / cm. [Figure 13] Phase-field simulations of Case II with a cell containing 64 × 64 × 64 grid points using three different initial conditions. [Figure 14] Phase shield simulation for Case II with a cell containing 128 × 128 × 128 grid points using different values ​​for G11, BTO as shown at the bottom of each figure panel. [Figure 15] X-ray diffraction collected on 100 nm and 400 nm thick BST epitaxial films deposited on SmScO3(110). The small peaks are due to the Umweg peak in addition to the film and substrate. [Figure 16] Reciprocal space maps (RSM) showing, from right to left, the (103) film and (332) SSO substrate reflections for 100 nm and 400 nm thick, x=0.8 film samples, confirming that the films are epitaxial and strain coherent. [Figure 17] Rutherford backscattering spectroscopy analysis results obtained on BST films confirming the absence of A-site defects, within a known 1% error. [Figure 18] Dual Amplitude Resonant Tracking (DART™) lateral force piezoresponse force microscopy (PFM) images collected for a ≈100-120 nm thick Ba0.8Sr0.2TiO3 film on SSO showing the crystal orientation (amplitude on the left, phase on the right). The diagonal pattern confirms that the film has a dense aa1 / aa2 / aa1 / aa2 type superdomain structure, as described in the text. Scale bar corresponds to 1 μm. [Figure 19]1 is an optical micrograph of a representative two-port, ten-interdigitated finger capacitor (IDC) device; the device shown above was fabricated on a 100 nm thick film and has an inter-electrode gap spacing of 3 μm. [Figure 20] Representative measured S-parameter data collected from a 400 nm thick film device. Figure 21 shows the frequency dependence of tunability, n(f), for the in-plane |E|=0.67 MV / cm of film devices with thicknesses from 100 to 400 nm. [Figure 22] Quality factor Q measured at 0.5MV / cm. [Figure 23] Zero bias temperature dependent capacitance and loss measured at 1MHz. [Figure 24] FIG. 1 is a diagram of the voltage-time measurement sequence used for the voltage sweep. [Diagram 25] At the same frequency, the measured field dependence of Q corresponding to the sample presented in Fig. 1h of the main text. [Figure 26] We show that the commutation quality factor CQF(f)=(n(f)-1)2 Q(0,f) Q(E,f) / n(f) [IB Vendik, OG Vendik, and EL Kollberg, Commutation quality factor of two-state switchable devices, IEEE Transactions on Microwave Theory and Techniques 48, 802-808(2000)], a key metric incorporating n(E) and Q(E), is greater than that of the best films reported to date in this range [CJG Meyers, CR Freeze, S. Stemmer, and RAY Ork, (Ba,Sr)TiO3 tunable capacitors with RF commutation quality factors exceeding 6000, Appl. Phys. Lett. 109, 112902(2016)]. [Figure 27]Data plotting quality factor-frequency product as a function of frequency for four broadly tunable MDVM membrane devices, as well as values ​​for other BST and non-inherently tunable devices. References correspond to those within the text. [Figure 28] Single-domain Q calculated by bond-valence molecular dynamics (BVMD) for bulk BaTiO3 supercells at selected temperatures above and below TC, allowing to identify the intrinsic limits of the frequency- and temperature-dependent dielectric losses. [Figure 29] Q calculated by single-domain BVMD with and without clamping by the substrate, and with and without application of an E-field. [Diagram 30] Normalized capacitance from MD simulations. The normalized capacitance (static dielectric constant ε) is obtained from the total polarization fluctuations in MD simulations of a 120 × 10 × 10 aa1 / aa2 domain supercell. The peak capacitance is obtained at ≈170 K, which is 30 K lower than the TC of 200 K obtained from the single-domain calculations. [Diagram 31] Polarization trajectories of MD simulations. Global supercell polarization in the y direction as a function of time for the two simulations used for the calculated Q(f). Raw Py trajectories for the two simulations are shown in black and green. Trajectories smoothed using a Gaussian window function with a full width at half maximum (FWHM) of 16 ps are shown in orange and blue. For (a) E = 0 MV / cm, (b) E = 0.3 MV / cm, (c) E = 0.6 MV / cm. Large Py fluctuations are observed, corresponding to the domain wall motion. For E = 0.3 MV / cm and E = 0.6 MV / cm, the domain wall hopping is faster, leading to sharper and larger oscillations of Py. [Diagram 32]MD simulations of polarization autocorrelation functions. Total time-polarized autocorrelation functions (ACFs) of the Py component for individual simulations at E = 0 MV / cm (black), E = 0.3 MV / cm (green), and E = 0.6 MV / cm (red). Large oscillations are visible in all cases, and it is clear that the ACF oscillations become more frequent with higher fields. The sampling quality of the ACF decreases with increasing time, with the ACF for the E = 0 MV / cm trajectory (≈ 8 ns) showing earlier divergence than the ACFs for E = 0.3 MV / cm and E = 0.6 MV / cm trajectories (≈ 12 ns). [Diagram 33] Loss tangents from MD simulations. Low-frequency tan δ(f) obtained from the Fourier transform of the first 9 ns of the ACFs of the individual MD simulations (black and red) and the average tan δ(f) (blue) for (a) E = 0 MV / cm, (b) E = 0.3 MV / cm, (c) E = 0.6 MV / cm. The tan δ(f) of the two simulations are similar to each other, indicating that the 14 ns trajectory is long enough to produce reproducible results for the location of the peaks and valleys in tan δ(f). However, the peak and valley values ​​show some variation, indicating the uncertainty in the obtained tan δ(f) and Q values. This is especially significant for Q, since a small difference in tan δ(f) can lead to a very large difference in Q. The values ​​of tan δ(f) converge better for E = 0.3 MV / cm and E = 0.6 MV / cm. Nevertheless, even for the E=0.3 MV / cm and E=0.6 MV / cm data, negative tan δ(f) values ​​and sharp tan δ(f) oscillations can be seen, indicating the presence of noise and the need for further smoothing of the data. [Diagram 34]Smoothed loss tangents from MD simulations. Low-frequency tan δ(f) (black and red) obtained from the average of tan δ(f) of the individual MD simulations, and smoothed tan δ(f) obtained using a Gaussian window function with FWHM of 0.2 (green) and 0.3 GHz (magenta) for (a) E = 0 MV / cm, (b) E = 0.3 MV / cm, (c) E = 0.6 MV / cm. The use of a smoothing function eliminates anti-physical negative tan δ(f) values ​​and abrupt large oscillations in tan δ(f). In some cases, smoothing with a FWHM of 0.05 is sufficient, while in other cases smoothing with a larger FWHM is necessary. [Figure 35A] Q(f) from MD simulations. Q(f) is obtained from the inverse of the average tan δ(f) of the individual MD simulations using the Fourier transform of different lengths of ACF. Tan δ(f) from ACFs of 3, 4, 5, 6, and 7 ns are shown in black, red, green, blue, and yellow for (a) E = 0 MV / cm, (b) E = 0.3 MV / cm, and (c) E = 0.6 MV / cm, respectively. The Q(f) curves are obtained from tan δ(f) smoothed using a Gaussian window function with FWHM of 0.05 and 0.1 GHz. Q(f) from ACFs of 3, 7, 8, 9, 10, 11, and 12 ns are shown in black, yellow, brown, orange, purple, cyan, and magenta for (d) E = 0 MV / cm, (e) E = 0.3 MV / cm, and (f) E = 0.6 MV / cm, respectively. For E = 0 MV / cm, the Q(f) curves obtained using ACFs over 7 ns show a wide change in the position of the loss peak (dropping of Q), while for E of 0.3 MV / cm and 0.6 MV / cm the position of the loss peak and the position of the peak Q are found to be largely preserved. The absolute value of Q varies with the length of the ACF, reaching ≈104 in some cases. Nevertheless, despite the uncertainty in the absolute value of Q, it is clear that resonance peaks above the baseline exist in the MD system and can even become larger than the extrapolation of the 1 / f high frequency rise of Q with decreasing f. [Figure 35B] Same as above. [Figure 35C]Same as above. [Figure 35D] Same as above. [Figure 35E] Same as above. [Fig. 35F] Same as above. [Diagram 36] Comparison of Q(f) from MD simulations of different electric fields. Q(f) was obtained from the inverse of the average tan δ(f) of the individual MD simulations using Fourier transform for E=0 MV / cm (black), E=0.3 MV / cm (green), and E=0.6 MV / cm (red). Q(f) of the Fourier transform performed for (a) 7 ns ACF, (b) 9 ns ACF, and (c) 11 ns ACF is shown. A shift from the baseline to higher frequencies and a larger rise of the resonance peak as well as a higher maximum Q are observed upon application of the field. [Figure 37] Autocorrelation functions of different layers in MD simulations. Autocorrelation functions of the poling times of different layers in the domain wall region (layers 74-86) and bulk region (layers 100-113) obtained from MD simulation trajectories for E=0.6MV / cm. The ACF DW layers show large oscillations with long periods, while the bulk layer ACF shows small oscillations with short periods, appearing as cyan lines around zero in the plot. [Figure 38] Shift in DW position as a function of time, showing MD supercell and domain fluctuations at Ex = 0 MV / cm (a) and Ex = 0.3 MV / cm (b), with domains with Py>0 shown in black and domains with Py<0 shown in red. [Figure 39] Figure 1 shows a comparison of Q(f) obtained from stochastic model simulations using two different parameter sets, with Q(f) for parameter set A shown in black and Q(f) for parameter set B shown in red. Similar to the Q(f) obtained with MD shown in Figure 3, the slight shift to higher frequency and higher Q peaks can be obtained by changes in the stochastic model potential. [Diagram 40](a) Schematic of the out-of-plane piezoelectric resonant oscillation of wave vector k3 in the film resulting from in-plane bias and polarization achieved using an LFE type device, i.e., IDC electrodes, and (b) the calculated resonant and anti-resonant frequencies for a Ba0.8Sr0.2TiO3 thin film with t=100 nm for 0% and 0.05% in-plane strain us from the substrate. [Diagram 41] (b) Schematic of (a) in-plane piezoelectric resonant oscillation of wave vector k1 in the film due to in-plane bias and polarization achieved using an IDC electrode containing n fingers with finger gap d, width w, and length l, and periodicity 2W=2(w+d), and (c) calculated resonant and antiresonant frequencies for 2W=20.6 μm and 34.4 μm. [Diagram 42] Optical micrograph images collected at different magnifications of representative devices with different electrode finger / gap periodicities (F10 and F06) as described above. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0028] This disclosure relates to a framework for microwave dielectric oscillation media based on atomic-scale domain wall fluctuations that enable new devices. These new metadielectric thin film materials allow low losses that overcome the inherent limitations specific to the materials. These polar media are distinguished by their high density of specially engineered planar defects that oscillate at some selected frequency under a selected DC bias. The vibration axes of the internal oscillators are not random, but instead oriented along one or more preferred directions, indicating that DC field-driven collective oscillations can support EM wave propagation. In addition, experimental observations and model calculations show that the oscillation frequency can be controlled and tuned. The availability of solid-state microwave media, whose resonant frequencies depend a priori on tunable nanoscale oscillators arising within the medium, rather than on geometric dimensions and acoustic modes, opens up a wide range of possibilities for frequency selectivity, spectral management, and reduced power requirements through the nonlinear response properties of the material.

[0029] A material having ferroelectric instability, being in or near a ferroelectric phase, and having a high density of ferroelectric domain walls, a. The domain walls oscillate or fluctuate in their position over time under the application of a DC or AC field of sufficient magnitude and selected frequency, or under zero field (e.g., obtaining a spectrum having resonant frequencies associated with the motion of the ferroelectric domain walls). b. The timescale or rate of these fluctuations varies depending on the electrostatic potential landscape, and on the domain width or domain wall density, applied field, temperature, strain (coherent or relaxed), and / or stress). c. The corresponding frequency spectrum associated with these fluctuations exhibits one or more minima in the dielectric loss (or peaks in the reciprocal loss Q) of the material along an axis parallel to the axis of the domain wall and perpendicular to the fluctuations. d. The width of the domains separating the domain walls fluctuates. E. The material can be ferroelectric within its ferroelectric phase (normal ferroelectric, inexact ferroelectric, hybrid inexact ferroelectric, relaxor ferroelectric), any of which can be ferroelectric with a temperature of 50,000 nm or less. 2 The density of domain walls ranges from 1 to 100 per unit area. f. Examples of ferroelectric materials are BaTiO3, (Ba,Sr)TiO3 (including combinations of Ba and / or Sr), PbTiO3, PZT, (Pb,Sr)TiO3, BiFeO3, Bi(Fe,Mn)O3, and numerous other compounds, with combinations of compositions and strain states that allow for the high domain wall densities described above, whereby domain structures having polarization components exist fully or partially within the plane of the film, even in the presence of weak or strong (e.g., 1 MV / cm) ordering electric fields. As non-limiting examples, ferroelectric materials can include perovskites BaxSr1-xTiO3 (BSTx), PbTiO3, Pb(Zr,Ti)O3, (Pb,Sr)TiO3, BiFeO3, Bi(Fe,Mn)O3, and related solid solutions, the Ruddelson-Popper phases An+1BnX3n+1, or more generally An+1A'2BnX3n+1, where A and A' represent alkali and / or alkaline earth metals and B represents a rare earth metal, such as A=Sr or Ba, B=Ti, and X=O, or other ferroelectrics, such as SrBi2Ta2O9, and related solid solutions, BaTiO3, (Ba,Sr)TiO3, PbTiO3, PZT, (Pb,Sr)TiO3, BiFeO3, Bi(Fe,Mn)O3. g. The temperature is within the ferroelectric phase transition temperature, Tc, of 100°C. h. The domain walls can be of any type, including c+ / c- / c+ / c-, a1 / a2 / a1 / a2, aa1 / aa2, r1 / r2 / r1 / r2, a / c / a / c, ca1 / ca2 / ca1 / ca2, ca* / aa* / ca* / aa*, c / a / c / a, or other structures identified in the Appendix, or any mixture thereof. i.50,000 nm 2 Domain wall density of 1 to 100 per 40,000 nm 2 1~100 or 60,000 nm per 2 Other domain wall densities can be used, such as 1-100 per 100, or per 100.

[0030] The domain wall oscillations produce one or more frequencies or frequency bands that can make the dielectric material losses very low, and the material Q exceeds the intrinsic limit, and can include the following characteristics: a. These frequencies, which correspond to high Q, can be held fixed by an applied field or can be shifted to higher or lower frequencies by an applied field. b. The frequency can range from 0.01 GHz to 300 GHz depending on the material, domain structure, domain wall density, strain, temperature, and applied field.

[0031] Materials that contain oscillations of domain walls (DWs) and thus have low dielectric loss / high Q are anisotropic (do not necessarily occur in all directions under an applied field), thereby allowing microwave and RF band electromagnetic energy to propagate with significantly less loss in one or more preferred directions.

[0032] The microwave cavity supports the propagation of transverse electromagnetic (TEM) waves, carried and / or modulated by domain wall oscillations, with little or no dissipation.

[0033] This disclosure relates to (a) the origin of the anomalously large experimentally observed Q spike using a comparison of experimental data with data obtained from bond-valence potential molecular dynamics (MD) simulations of a model BaTiO3 (BTO) system (Methods), (b) the voltage (or electric field) tunability of the observed Q spike that is also exceptionally large, and (c) the occurrence of both the (frequency tunability) Q spike and the exceptionally high voltage tunability of the dielectric constant in the RF and microwave bands in the same material.

[0034] Thermodynamic Ginzburg-Landau-Devonshire (GLD) model calculations support the hypothesis that large in-plane dielectric constant values ​​can be obtained through in-plane domains. Application of the phenomenological GLD model shows that the in-plane distortion u can be reduced by several additional domain variants, which are expected for BST films (Figure 1). sThis allows the calculation of a temperature (T)-polarization (P) phase diagram (Figure 1) ("superdomain" [S. Matzen, O. Nesterov, G. Rispens, JA Heuver, M. Biegalski, HM Christen, and B. Noheda, Super switching and control of in-plane ferroelectric nanodomains in strained thin films, Nature Commun. 5, 4415 (2014)]). Focusing on x=0.8, for example, produces a vertex in the phase diagram, where several domain-wall variant phases are expected to intersect near room temperature (Figure 1). Showing close proximity and high accessibility between the different variants, this region of the phase diagram can be referred to as a manifold domain-wall-variant material (MDVM).

[0035] Zero- and finite-field phase-field model calculations for three selected strain states (represented by yellow dashed lines I, II, and III in Fig. 1) reveal the predicted c + / c - The results confirm the structure of the I and III films under moderate tensile strain, and the in-plane domain structure of the films under moderate tensile strain (III) (Figure 2). Application of moderately large fields (0.1 MV / cm along <0100>) leaves the domain structures of I and III essentially unchanged (Figure 2). For case II, corresponding to MDVM materials, aa1 / aa2 domain wall variant structure is expected at zero field, suggesting the coexistence of multiple domain wall variants consistent with its position in the phase diagram. Despite the softer three-dimensional potential energy landscape of II when compared to I and III, the domain structure is not eliminated at moderate fields (Figure 1), consistent with reports on epitaxial films that cannot eliminate the domain structure under applied electric fields [Griggio, F. et al. Composition dependence of local piezoelectric nonlinearity in (0.3)Pb(Ni 0.33 Nb 0.67 )O3-(0.7)Pb(Zr x Tu1-x )O3films.J.Appl. Phys.110,044109(2011)].

[0036] The dielectric constant values ​​of MDVM engineered films exceed the composition-specific state-of-the-art for dielectric thin films: theoretically, the zero-field relative dielectric constant ε 11 The expected value of / ε0 easily exceeds 10,000, and the expected value of 10 for the selected combinations is 5 The higher dielectric constant leads to an extended dielectric and capacitance tunability n(E)=ε r、max / ε r、min (=C max / C min ), where ε r is the real part of the dielectric constant, C max and C min is the capacitance at zero and applied electric field E, driven by the proximity to the phase boundary.

[0037] The theoretically calculated quasistatic in-plane tunability of MDVM films can be significantly larger. For example, the coherently strained x = 0.8 film (u s ≈0.05%, case II) has tunability of n(E1)>20 at E1=0.3MV / cm, while the n of films on SrTiO3(I) and BaTiO3(III) is expected to be significantly weaker (Fig. 3).

[0038] Experimental results support the GLD theoretical predictions. Epitaxial films with x = 0.8 and thicknesses of 100 and 400 nm were deposited by pulsed laser deposition on SmScO3(110) and characterized using various techniques.

[0039] The smaller out-of-plane lattice constant in our films compared to the bulk supports in-plane domain formation, and plane-normal and lateral dual amplitude resonant tracking (DART™) piezoelectric force microscopy (PFM) confirms the presence of in-plane oriented domains, with domain walls aligned along either

[0100] or

[0010] , consistent with an aa1 / aa2 / aa1 / aa2 domain structure (SI).

[0040] The potential-dependent capacitance data for the coplanar geometry (Figure 4) at selected frequencies across the measurement range demonstrate high capacitance tunability at moderate fields (Figure 5), consistent with our calculations, and persists to above 20 GHz. This capacitance tunability is consistent with the equivalent field Ruddelson-Popper (RP) Sr7Ti6O 19 This is significantly higher than the current state of the art in molecular beam epitaxy grown films, including [C.-H. Lee, ND Orloff, T. Birol, Y. Zhu, V. Goian, E. Rocas, R. Haislmaier, E. Vlahos, J A Mundy, LF Kourkoutis, et al., Exploiting dimensionality and defect mitigation to create tunable microwave dielectrics, Nature 502, 532-536 (2013)] and (Ba,Sr)TiO3 [C J G Meyers, CR Freeze, S. Stemmer, and RAY Ork, (Ba,Sr)TiO3 tunable capacitors with RF commutation quality factors exceeding 6000, Appl. Phys. Lett. 109, 112902 (2016)] (Figure 5).

[0041] Remarkably, n(f) remains above 13 (at 0.67 MV / cm) over almost the entire frequency range studied, peaking at n≈18.5 at 15.2 GHz (FIG. 21). The deposited films also exhibit low loss (high Q). This is in contrast to the high loss typically observed with high tunability. The MDVM films exhibit low Q at zero field, but high Q at maximum field (<Q(f)> ≈1200, averaged over frequencies from 0.1 to 20 GHz). The Q at the highest applied field is typically 10 over the range 2 to 10 GHz. 2 ~10 3 (Figure 22).

[0042] A closer look reveals a special feature of the thinner membrane: the field and frequency combinations at which Q oscillates easily with frequency are T C The frequency-dependent bulk limit of BaTiO3 is (10 3 Less than or approximately equal to, Figure 28) and easily exceeding 10 5and even exceeding it (Figure 6). Considering such Q values, including ferroelectrics considering high Q dielectrics [Budimir, M. Damjanovic, D. and Setter, N. Extension of the dielectric tunability range in ferroelectric materials by electric bias field antiparallel to polarization. Appl. Phys. Lett. 88, 082903 (2006); Rojac, T. et al. Piezoelectric nonlinearity and frequency dispersion of the direct piezoresponse of BiFeO3 ceramics. J. Appl. Phys. 112, 064114 (2012); Vorobiev, A. et al., J. Appl. Phys. 110, 054102 (2011)], are larger than the best reported to date in intrinsically tunable film materials [Meyers, CJ Get al., (Ba,Sr)TiO3 tunable capacitors with RF commutation quality factors exceeding 6000]. Appl. Phys. Lett. 109, 112902 (2016); Vorobiev, A. et al. Correlations between microstructure and Q-factor of tunable thin film bulk acoustic wave resonators. J. Appl. Phys. 110, 054102 (2011)], and much larger than in AlN films, which are excellent non-ferroelectric (i.e., not inherently tunable) piezoelectric materials [Rinaldi, M. et al. Super-high two-port AlN contour-mode resonators for RF applications. IEEE Trans. Ultrason. Ferroelectr. Freq.Control 57,38-45(2010); Rinaldi, M. et al. 5-10 GHz AlN contour-mode nanoelectromechanical resonators. in 2009 IEEE 22nd International Conference on Micro Electro Mechanical Systems 916-919 (IEEE, 2009).] The experimentally determined Q-factors are indeed comparable to measurements for bulk single crystal quartz [Krupka, J, et al. Extremely high-Q factor dielectric resonators for millimeter-wave applications. IEEE Trans. Microw. Theory Tech. 53, 702-712 (2005); Harnett, JGet al., Room temperature measurement of the anisotropic loss tangent of sapphire using the whispering gallery mode technique. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53 34-38 (2006)], and ZnO [Magnusson, EBet al. Surface acoustic wave devices on bulk ZnO crystals at low temperature. Appl. Phys. Lett. 106, 063509 (2015)]. The resonant frequency f. r The field dependence of (E) shows exceptional variation of more than a factor of 10 across the L (1-2 GHz), S (2-4 GHz), and C (4-8 GHz) bands, up to the X-band (8-12 GHz), all in a single device. The rectification quality factor CQF(f)=(n(f)-1) 2Q(0,f) Q(E,f) / n(f) [Vendik, IB et al., Commutation quality factor of two-state switchable devices. IEEE Trans. Microw. Theory Tech. 48, 802-808(2000)] is the key metric incorporating n(E) and Q(E) and is larger than the best reported BST film [Meyers, CJ Get al., Appl. Phys. Lett. 109, 112902(2016)].

[0043] Bulk dielectric and membrane resonators rely on electromechanical coupling of microwave power through piezoelectric oscillations, which manifest as resonance and anti-resonance features that can be voltage tuned by <4.5% in the best tunable materials [Berge, J. and Gevorgian, S. Tunable bulk acoustic wave resonators based on Ba 0.25 Sr 0.75 [TiO3 thin films and a HfO2 / SiO2 Bragg reflector. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 2768-2771 (2011)]. Considering the change in the piezoelectric coupling coefficient, in our experimental geometry, 0.8 Sr 0.2 The calculated bias field dependence of the resonance and antiresonant frequencies of the in-plane piezoelectric oscillations of TiO3 reach below ≈3% for 0–0.6 MV / cm (Figure 42), which is several hundred times lower than those observed in our device.

[0044] Furthermore, the design of piezoelectric resonators utilizing in-plane piezoelectric oscillations operating at fundamental (or higher mode) frequencies relies on the electrode periodicity of interdigital capacitors (IDCs) [Gevorgian, SS, Tagantsev, AK and Vorobiev, AK Tunable Film Bulk Acoustic Wave Resonators (Springer, New York, 2013)]. Comparing the spectra obtained in devices with different electrode finger widths instead reveals that the spectra are essentially the same (Figure 6), further demonstrating that the observed spectra are highly unlikely to arise from piezoelectric oscillations.

[0045] The origin of the experimentally observed peculiar Q spike can be observed using data obtained from molecular dynamics (MD) simulations of a model BaTiO3 (BTO) system (Methods). The analytical theory of the intrinsic dielectric response [Tagantsev, 2003] predicts a 1 / f dependence of Q(f), as also found in our single-domain MD simulations (Figure 28), indicating that the peculiar f dependence of Q is due to external influences. Investigation of the static dielectric response shows that a peak dielectric constant value is observed in the FE phase, and such a DW-driven shift of the dielectric response peak into the FE phase is due to T CAs previously experimentally observed in BaTiO3 in the FE phase close to [Hoshia,T.et al.Domain size effect on dielectric properties of barium titanate ceramics.,Jpn.J.Appl.Phys.47,7607-7611(2008);Wang,YLet al.Giant domain wall contribution to the dielectric susceptibility in BaTiO3.Appl.Phys.Lett.91,062905(2007)], we find that reversible domain wall oscillations also lead to peak dielectric constants in the FE phase of our model aa1 / aa2DW supercell in our MD simulations (Fig. 2 and Fig. 8). The main reason for this domain wall contribution to the dielectric constant is the presence of ultra-low energy modes localized to the 2D domain walls. The strong effect of domain wall oscillations on the dielectric response and the presence of a high density of domain walls in the superdomain state of our samples suggest that these oscillations could also be the cause of Q(f) oscillations.

[0046] To understand the relationship between the reversible domain wall dynamics and Q(f), long (14 ns) simulations are performed using a model system containing two domain walls aa1 / aa2 in a 120 × 10 × 10 supercell (Figure 8) at 50 K, which is below the FE-PE transition temperature, and then Q(f) can be obtained from the fluctuations of the total polarization of the supercell. As can be seen in Figure 8, at this domain length (24 nm), a clear difference is observed between the domain walls and the bulk-like regions of the sample, so this size can be chosen. In addition, the GLD theory predicts that the domain size should be around 30 nm.

[0047] A comparison of the experimental and MD obtained Q(f) shows some similar features (Fig. 9). First, at zero DC bias, a linear or nearly linear rise in the Q value with decreasing f is followed by a flattening of the Q with mild oscillations due to the onset of relaxation at about 18 GHz (indicated by the black arrow). This observation is consistent with the expectation that the presence of domain walls leads to higher losses and lower Q, as can be seen from the fact that the Q vs. f in the low f region (<2 GHz experimental and <18 GHz MD) is much lower than expected from the intrinsic 1 / f Q dependence. Second, at higher bias, higher Q peaks than the baseline appear at certain frequencies (indicated by the blue arrow) where the Q curve shifts to higher frequencies with higher DC bias. Finally, at higher bias, a larger number of narrow Q peaks are observed. The Q from E=0MV / cm and E=0.6MV / cm y (f) Data, Q of MD simulation y (f) is qualitatively similar to the experimentally obtained Q(f) data for E = 0.09 and E = 0.25 MV / cm (Fig. 9), albeit at higher frequencies due to differences between the experimental BST and calculated BTO systems. The uniform shift to higher Q with higher DC bias is not observed for the MD simulations, and this difference is likely due to the difference between the simple model used in the MD simulations and the much more complicated E-field profile in the experimental sample.

[0048] Analysis of the Q(f) of the individual layers shows that the bulk-like layers (i.e., the layers in the middle of the domain that do not show switching) exhibit a bulk-like 1 / f of Q versus f, while the DW layers exhibit a Q(f) spike and a flattening of Q(f) at low f, similar to the experimentally observed data and the computationally obtained Q(f) for the whole system (Fig. 10). Comparison of the autocorrelation functions (ACFs) for the bulk-like and DW layers (SI) shows that the bulk-like layer ACF exhibits the usual behavior of a rapid decay followed by small fluctuations around zero, while the DW layers exhibit a slow ACF decay as well as large amplitudes and long oscillation periods due to the very large magnitude of the fluctuations P of the DW layers between the two sides of the double well potential compared to the oscillations of P inside the well. Thus, the DW fluctuations dominate the dielectric response at low f.

[0049] (-P y From +P y Analysis of the polarization switching speed (to and vice versa) shows that the hopping speed increases with increasing DC bias (Fig. 11), which is also evident from the oscillations of the overall polarization time autocorrelation function (Fig. 12). Application of a DC bias therefore accelerates the speed of DW oscillations, leading to a shift of the Q(f) curve to higher f. In the absence of DW oscillations, a bulk-like 1 / fQ spectrum is obtained, whereas in the case of slow DW hopping, a relaxation-driven flattening is observed along with a mild oscillation of Q, and in the case of faster hopping, a sharp Q peak is obtained. This strongly suggests that the experimental Q spectrum with mild oscillations at zero bias is due to the slow oscillations of a dense DW, and that the appearance of the experimentally observed sharp Q peak is due to the acceleration of DW hopping with the application of a DC bias.

[0050] To show that the DW fluctuation mechanism alone can give rise to the observed sharp Q(f) peaks, we performed stochastic simulations using a simple model of coupled bistable oscillators with domain walls (SIs). We found that by tuning the double-well parameters of the oscillators (SIs), we could obtain DW position oscillations and Q(f) profiles that are qualitatively similar to those obtained in MD, demonstrating that DW oscillations can give rise to the observed sharp changes in Q(f).

[0051] The hypothesis that fluctuations in the domain wall positions give rise to the anomalous Q observed at high static bias in the experiment explains why such Q characteristics have not been observed before. To obtain Q oscillations, a high domain wall density, corresponding to a domain size of <100 nm, is needed, since the high Q arising from the domain walls would otherwise be averaged out by the normal behavior of the bulk of the domains. Secondly, this effect is only seen in the T region when the DW thickness is larger and the barrier to switching is very low. C GHz frequencies, and can only occur near y At lower T, the P of the layer y The energy barrier to switching between the two alternating P y The time required to traverse the barriers between states becomes too long, and high Q is only observed below f in the MHz regime, and such effects cannot be manifested by the high Q of the bulk dielectric response at such low f. Finally, very high quality films are required to observe these effects, since changes in frequency of the ultra-low dielectric loss resonances due to defects, grain boundaries, and compositional changes lead to averaging out of the low losses and disappearance of the high Q peaks.

[0052] The product of Q and frequency, f, is one of the most frequently quoted metrics for all dielectric microwave resonators, and in the Akhiezer limit for phonon-phonon scattering, α ∝ f 2The acoustic attenuation parameterized by leads to Qf being equal to a material-specific constant. We find that the product of Qf in the material is 1 for our experimental membrane. <f r We note that for <10 GHz there is a departure from the usual monotonic Q(f) dependence, showing a large increase in Qf in this range, since the effective scattering rate by thermal phonons increases with f r This suggests that the frequency band of voltage-tuned domain-wall resonances is much lower than that observed experimentally for BST, providing additional experimental evidence that our domain-wall resonance films overcome the intrinsic losses in this range. On the other hand, the simulations for BTO indicate that the expected frequency band of voltage-tuned domain-wall resonances is material-specific and can be higher than that observed experimentally for BST.

[0053] Thus, these experimental and computational simulation results show that engineered domain structures can indeed be exploited for ultralow loss and exceptional frequency selectivity without piezoelectric resonance and with very large voltage tunability of capacitance, as well as without hysteresis. Materials are defined not simply by chemical composition, but rather by the proximity and accessibility between thermodynamically predicted strain-induced ferroelectric domain wall variants to achieve gigahertz microwave tunability and dielectric loss that are one to two orders of magnitude greater than the current best film devices, which achieve values ​​comparable to bulk single crystals but in intrinsically tunable materials [Pertsev,NA et al.Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films.Phys.Rev.Lett.80 1988-1991(1998)]. The nearly isotropic free energy polarization landscape of these materials (and the corresponding lower barrier to polarization rotation) is expected to lead to rich phase diagrams and large responses to applied electric fields. The measured magnitude of the quality factor Q exceeds the theoretically predicted zero-field intrinsic limit due to domain wall fluctuations rather than normal piezoelectric oscillations. Tuning of the resonant frequency across the L, S, and C microwave bands was achieved in individual devices, approximately 100 times higher than the best currently intrinsically tuned materials. The tunable dielectric material of the comfortably driven MDVM achieves a T that exceeds the intrinsic limit without piezoelectric oscillations. C "These results demonstrate a Q value near 100 nm, promising for achieving similar Q values ​​over a wider range of frequencies. These results point to a rich phase space of possible nanodomain structures that can be used to overcome current limitations and demonstrate fundamentally new and promising strategies for ultrahigh frequency agility and low-loss microwave devices."

[0054] The present disclosure includes at least the following aspects: 1. An article (e.g., may be included in a resonator, oscillator, device, etc.) that includes a ferroelectric material in its ferroelectric phase, the article having ultra-low dielectric loss (10 3 <Q<10 7 or 10 -3 >tanδ>10 -7 .) 2. An article having a ferroelectric material containing one or more (thermally) dense ferroelectric domain wall perturbations, the article having ultra-low dielectric loss (10 3 <Q<10 7 or 10 -3 >tanδ>10 -7 ), and the domain wall density is 50,000 nm 2 Items range from 1 to 100 per item. 3. An article comprising a ferroelectric material in its ferroelectric phase, the article exhibiting ultra-low dielectric loss (10 3 <Q<10 7 or 10 -3 >tanδ>10 -7 ). 4. An article comprising a ferroelectric material in its ferroelectric phase in thin film form, the composition and strain of the material being selected to stabilize the material in or with respect to two or more energetically equivalent, or nearly energetically equivalent, thermodynamically predicted domain wall variant types for a given temperature, as specified by domain wall variant boundaries or vertices that intersect the boundaries, thereby providing ultra-low dielectric loss (10 3 <Q<10 7 or 10 -3 >tanδ>10 -7 ) to enable efficient signal propagation. 5. A field-tunable article comprising a ferroelectric material, wherein the range and / or value of one and / or more article frequencies are controlled based on a change in domain wall oscillation frequency in response to an electric field applied to the ferroelectric material. 6. The article of any one of aspects 1 to 5, wherein the selected frequency is from 0.01 GHz to 300 GHz. 7. The article of any one of the preceding embodiments, wherein the range and / or value of one or more article frequencies of the article is controlled based on the density of the domain walls. 8. The article of any one of the preceding embodiments, wherein the magnitude of the quality factor, Q, is controlled by and increases with domain wall density. 9. The article of any one of embodiments 1-5, wherein the range and / or value of the one or more article frequencies is controlled based on one and / or more types of ferroelectric domain wall variants. 10. The article of any one of the preceding claims, wherein the range and / or value of one or more article frequencies is controlled based on the degree of distortion. 11. The article of any one of the preceding aspects, wherein the ferroelectric material is in a phase comprising one of: a normal ferroelectric, a mismatched ferroelectric, a hybrid mismatched ferroelectric, a relaxor ferroelectric, an early ferroelectric phase, or a multiferroic ferromagnetic or antiferromagnetic ferroelectric. 12. The article of any one of embodiments 1-11, wherein the range and / or value of the one and / or more article frequencies is controlled based on a change in domain wall oscillation in response to a magnetic field applied across a multiferroic ferromagnetic (or antiferromagnetic) ferroelectric material due to ferroelectric polarization via multiferroic coupling of the magnetic field. 13. The article of any one of the preceding claims, wherein the chemical composition of the ferroelectric material includes BaTiO3, (Ba,Sr)TiO3, PbTiO3, PZT, (Pb,Sr)TiO3, BiFeO3, and related solid solutions. 14. Under selected or zero DC bias, at selected frequencies, and within 100°C of T of ferroelectric materials C Oscillates at 50,000 nm2 1. An article comprising a ferroelectric material having 1 to 100 engineered planar two-dimensional topological defects per one piece. 15. The article of embodiment 14, wherein the selected frequency is between 0.1 GHz and 300 GHz. 16. The article of any one of embodiments 14-15, wherein the vibration axes of the ferroelectric domain walls are oriented along one or more directions and exhibit collective oscillation capable of supporting EM wave propagation in the presence and absence of a DC bias field. 17. The article of any one of embodiments 14-16, wherein the planar two-dimensional topological defects comprise domain walls, and the domain walls oscillate or fluctuate in their position with respect to time under an applied DC or AC field or under zero DC or AC field. 18. The article of embodiment 17, wherein the timescale or rate of fluctuations varies as a function of the electrostatic potential landscape, and the domain width or domain wall density, applied field, temperature, strain (coherent or relaxed), and / or stress. 19. The article of embodiment 18, wherein a corresponding frequency spectrum associated with the fluctuations exhibits one or more minima in the dielectric loss (or peaks in the reciprocal loss Q) of the material along an axis parallel to the axis of the domain wall and perpendicular to the fluctuations. 20. The article of embodiment 19, wherein the width of the domains separating the domain walls varies. 21. The article of any one of aspects 14-20, wherein the ferroelectric material is in its ferroelectric or paraelectric phase (regular ferroelectric, inferroelectric, hybrid inferroelectric, relaxor ferroelectric, early ferroelectric, multiferroic or ferroelectric antiferromagnetic material). 22. The article of any one of aspects 14-22, wherein the ferroelectric material comprises BaTiO3, (Ba,Sr)TiO3, PbTiO3, PZT, (Pb,Sr)TiO3, BiFeO3, Bi(Fe,Mn)O3. 23. A microwave or millimeter wave cavity that supports the propagation of transverse electromagnetic (TEM) waves with dissipation less than that of the inherent limit of the material forming the cavity, the TEM waves being carried and / or modulated by oscillation of one or more domain walls and at or near the Tc of the material forming the microwave cavity, and the density of the domain walls is less than 50,000 nm 2 The range is 1 to 100 per device. 24. The apparatus of embodiment 23, wherein the microwave cavity comprises a ferroelectric material comprising one or more domain walls that oscillate at a selected frequency under zero bias or a selected finite DC bias. 25. An apparatus as described in embodiment 24, wherein the vibration axes of one or more domain walls are oriented along one or more directions, and wherein the zero-field or finite DC field driven collective oscillation thereof is capable of supporting EM wave propagation. 26. The device of any one of aspects 23-25, wherein the ferroelectric material is in its ferroelectric or paraelectric phase (regular ferroelectric, inferroelectric, hybrid inferroelectric, relaxor ferroelectric, early ferroelectric, multiferroic or ferroelectric antiferromagnetic material). 27. The device of any one of aspects 23-26, wherein the ferroelectric material comprises BaTiO3, (Ba,Sr)TiO3, PbTiO3, PZT, (Pb,Sr)TiO3, BiFeO3, Bi(Fe,Mn)O3. 28. The device of any one of aspects 23-27, wherein Q increases with increasing temperature and may depend on proximity to Tc. 29. An apparatus as described in any one of aspects 23 to 28, wherein the magnitude of Q may depend on the amplitude of ambient stochastic noise (given by temperature) related to the amplitude of a drive signal that probes the transmission and / or reflection of RF, microwave, or millimeter wave energy through the article. 30. A method of making the article of any one of embodiments 1-22. 31. A method of making a device according to any one of embodiments 23 to 29. EXAMPLES

[0055] Transducers. Domain wall oscillating materials (DWOs) can be the basis for highly efficient transduction of electromechanical energy at one or more resonant frequencies for sensing and / or actuation via coupling to mechanical and / or electromagnetic waves. The change in resonant frequency associated with attaching an analyte to its surface, and its effect on the thermodynamic landscape and DW oscillation conditions, changing the surface boundary conditions, is different from traditional bulk and / or surface acoustic waves, or other similar devices whose natural frequencies are influenced by geometry.

[0056] Communications. The extremely high Q at room temperature, as well as other temperature availability, enabled by DWO-based devices, allows for the encoding, detection, and sensing of information with significantly higher fidelity than current solid-state oscillator materials. This includes use as highly frequency-selective voltage-tuned filters, antennas, or oscillators.

[0057] Position, Navigation, and Timing. The availability of extremely high Q at room temperature as well as other temperatures possible in DWO-based devices allows for a more precise relationship (higher fidelity) between Q and the variables that define position, navigation, and timing, and frequency selectivity is the means for establishing the values ​​of these values, requiring less power to transmit and receive signals related to position, navigation, and timing.

[0058] Programmability of the domain structure to reconfigure the DW orientation, oscillation vector, and wave propagation by application of local or non-local DC or AC fields, strain, and / or temperature.

[0059] Although metamaterials and articles have been described herein with reference to preferred embodiments and / or preferred methods, it should be understood that the terms used herein are terms of description and illustration, rather than of limitation, and that the scope of the present disclosure is not intended to be limited to those matters, but is meant to extend to all structures, methods, and / or uses of metamaterials described herein. Those skilled in the art, having the benefit of the teachings herein, can effect numerous modifications to the metamaterials described herein, and changes can be made without departing from the scope and spirit of the present disclosure, e.g., as set forth in the appended claims. As one example, the conventional concept of ferroelectrics with polarization-field hysteresis is not supported, as it is suppressed on a macroscopic scale by the high domain density of the present disclosure.

[0060] An oscillator, or a system / group of oscillators coupled to a ferroelectric material, can be constructed in accordance with the present invention to exhibit resonance at an odd integer multiple of the fundamental domain wall frequency, with noise-induced fluctuations of the system between the two sides of the double well switching the resonant frequency.

[0061] The resonator may comprise an article for which one or more of the dimension(s) of a volume or cavity containing a medium bounding the apparatus and the mechanical and electrical boundary conditions are selected according to one or more of the domain wall resonant frequencies to promote efficient flow of mechanical and / or electromagnetic energy, thereby enabling constructive interference with wave energy according to changes in the modulus and / or magnetic susceptibility of the material in which the domain wall is renormalized (or which is dominated by).

Claims

1. 1. An article comprising a ferroelectric crystalline material within or near its ferroelectric phase and having ferroelectric domains bounded by engineered domain walls, the domain walls oscillate or fluctuate in their position over time under the application of a DC or AC field of a selected frequency, or under zero field; The ferroelectric crystalline material exhibits a quality factor Q of 10 or greater than 10 at one or more selected frequencies due to oscillation or fluctuation of domain walls. ３ <Q<10 ７ or 10 －３ >tan δ>10 －７ The article is configured to enable low loss signal propagation, exhibiting a dielectric loss represented by:

2. The density of the domain walls is 50,000 nm ２ The article of claim 1, wherein the number of particles per unit area is in the range of 1 to 100.

3. The ferroelectric crystal material has a quality factor Q of tan δ or its reciprocal that is greater than or equal to 10 at or within 20% of the Curie temperature (Tc) of the ferroelectric crystal material, when the Curie temperature (Tc) is measured in Kelvin. ３ <Q<10 ７ or 10 －３ >tan δ>10 －７ 13. The article of claim 1 configured to enable low loss signal propagation, exhibiting a dielectric loss represented by:

4. 2. The article of claim 1, wherein the composition and strain of the ferroelectric crystalline material are selected to stabilize the ferroelectric crystalline material in or for two or more energetically equivalent thermodynamically predicted domain wall variant types for a given temperature, as specified by domain wall variant boundaries or vertices that intersect the boundaries.

5. The article of claim 1 , wherein the range and / or value of one or more of the fluctuation frequencies is controlled based on a change in domain wall oscillation frequency in response to an electric field applied to the ferroelectric crystal material.

6. 10. The article of claim 1, wherein the one or more selected frequencies are from 0.01 GHz to 300 GHz.

7. The article of claim 1 , wherein the range and / or value of one or more wobble frequencies of the article are controlled based on the density of the domain walls.

8. 2. The article of claim 1, wherein the magnitude of the quality factor Q is controlled by and increases with domain wall density.

9. The article of claim 1 , wherein the range and / or value of the one or more wobble frequencies is controlled based on one and / or more types of ferroelectric domain wall variants.

10. The article of claim 1 , wherein the range and / or value of the one or more wobble frequencies is controlled based on a degree of strain in the ferroelectric crystal material.

11. The article of claim 1 , wherein the ferroelectric crystalline material is in a ferroelectric-containing phase.

12. 12. The article of claim 11, wherein the range and / or value of the one and / or more fluctuation frequencies are controlled based on a change in domain wall oscillations in response to a magnetic field applied across the multiferroic ferromagnetic (or antiferromagnetic) ferroelectric material due to ferroelectric polarization via multiferroic coupling of a magnetic field.

13. The ferroelectric crystalline material is at the one or more selected frequencies under a selected electrical DC bias or zero electrical DC bias, and at the T Ｃ Oscillates at 40,000 nm ２ 13. The article of claim 12 having 1 to 100 engineered planar two-dimensional topological defects per surface.

14. 14. The article of claim 13, wherein the selected frequency is from 0.1 GHz to 300 GHz.

15. 14. The article of claim 13, wherein the planar two-dimensional topological defects comprise domain walls, the domain walls oscillating or fluctuating in their position with respect to time under an applied DC or AC electric field or under zero DC or AC electric field.

16. 16. The article of claim 15, wherein the timescale or rate of the fluctuations varies as a function of the electrostatic potential landscape, and domain width or domain wall density, applied field, temperature, strain (coherent or relaxed), and / or stress.

17. The ferroelectric crystal material is a perovskite, Ba ｘ Sr １－ｘ TiO ３ (B.S.T. ｘ ), PbTiO ３ , Pb(Zr,Ti)O ３ , (Pb, Sr)TiO ３ , BiFeO ３ , Bi(Fe,Mn)O ３ , or Ruddelson-Popper A ｎ＋１ B ｎ X ３ｎ＋１ , or Ruddelson-Popper phase A ｎ＋１ A' ２ B ｎ X ３ｎ＋１ wherein A and A' represent alkali and / or alkaline earth metals, B represents a rare earth metal, and X represents O or other ferroelectrics, or a combination thereof.

18. A method of making the article of claim 1.

19. A resonator comprising the article of claim 1.

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