resonator

Abstract

A method for forming a modified resonator (22), the method comprising: a. obtaining a resonator (2) over a substrate (1), the resonator (2) comprising niobium or tantalum, thereby forming an interface region (21) between a bottom face of the resonator (2) and a top face of the substrate (1), and b. contacting the resonator (2) with the substrate (1) with a liquid acidic etching solution, the liquid acidic etching solution being selected so as to have a higher etching rate on the substrate (1) than on the resonator (2) and a non-zero etching rate on the resonator (2).

Description

Technical Field

[0001] This invention relates to the field of resonators and their fabrication. More specifically, this invention relates to the field of resonators for realizing qubits. Background Technology

[0002] Resonators, especially superconducting resonators, are one of the fundamental components used to realize qubits for quantum computing. Generally, the resonator of a quantum circuit is a microwave transmission line specifically designed to support resonant oscillations (i.e., resonance) within the line under specific conditions (i.e., a resonant microwave transmission line). Resonators for superconducting qubit devices are typically implemented as coplanar waveguides (CPWs), a special type of microwave transmission line. Quantum computing requires low-loss operation of the resonator. Preferably, the resonator of a quantum computer has a quality factor of at least 1,000,000, meaning that the resonator preferably loses less than 1 / 1,000,000 of the peak energy stored in the resonator per radian of the oscillation period. Since extremely low microwave power is stored in the resonator, for example in atvars (10⁻¹⁸ watts), such low loss is necessary to achieve a long qubit lifetime (i.e., a long coherence time set at a temperature of ~20 mK). High quality factor, i.e. low-loss resonators, is crucial for achieving long coherence times in superconducting qubits, storing arbitrary quantum states in superconducting resonators at very low fields (i.e., single-photon states) where the electric field is much smaller than the critical field, and low-loss coupling between two qubits mediated by low-loss resonators.

[0003] In addition, low-loss resonators are also necessary for quantum-confined microwave amplifiers, which are extremely sensitive amplifiers for, for example, gravitational wave detection and multiplexed readouts for microwave dynamic inductive photonic detectors (MKIDs) (e.g., in astronomy).

[0004] Typically, resonators consist of superconducting materials and are mounted on an insulating substrate. Typical superconducting materials used in resonators are Al, TiN, Nb, NbN, and NbTiN, and crystalline sapphire or high-resistivity silicon substrates are commonly used for the insulating substrate.

[0005] At low microwave field levels and low temperatures (e.g., 20 mK, particularly in superconducting quantum information applications), the quality factor of resonators in the prior art is significantly reduced due to parasitic two-stage system (TLS) losses. TLS losses arise from defects in the material near the resonator (e.g., the substrate or native oxide layer). These defects can be specifically represented by atoms tunneling between two non-degenerate minimums of the potential provided by the atomic environment, thus forming TLSs. These TLSs can interact with and absorb photons in the resonator, subsequently leading to energy loss. Amorphous materials, oxides, and dielectrics at the resonator surface and / or interfaces primarily contribute to TLS losses, and oxides on the substrate surface may also contribute.

[0006] To reduce TLS loss, in the prior art, specifically, materials contributing to TLS loss near the resonator can be removed. For example, in Woods et al., Physical Review Applied 12 (2019) 014012, the interface region between the bottom and top surfaces of the resonator on the substrate is effectively reduced by forming photoresist on the resonator and selectively etching the substrate relative to the photoresist. The etching technique used is isotropic etching. The substrate below the resonator is etched, i.e., the resonator is under-etched, thereby reducing the interface region. Finally, a combination of ashing solvent resist strips is used to remove the photoresist. However, the use of photoresist used in the prior art for under-etching the resonator is undesirable because the process of removing the photoresist can introduce defects on the surface of the resonator: these defects may lead to further resonator loss.

[0007] Furthermore, the edges of resonators in the prior art are sharp, which the inventors of this invention consider to be another source of loss. Indeed, intuitively, it can be recognized that waves propagating on the surface of a resonator cannot propagate along the sharp angles of the cavity edges, thus leading to radiation losses of energy in the resonator.

[0008] Furthermore, control over the substrate and resonator profiles formed by etching (i.e., etch control) is particularly advantageous. Specifically, etch control is advantageous when multiple resonators on a single substrate are etched simultaneously, as it minimizes the interface region and prevents collapse of at least a portion of the resonator. Additionally, the resonators preferably possess the same predictable properties, such as well-resolved resonator frequencies and quality factors. Etching is therefore preferably highly reproducible. Finally, the resonators generally include a natural oxide layer on their surface, which can lead to further TLS losses.

[0009] Therefore, there is a need in the prior art for the present invention to solve one or more of the above-mentioned problems. Summary of the Invention

[0010] The objective of this invention is to provide a good method for reducing energy loss in resonators. Another objective of this invention is to provide resonators that can be obtained through these methods.

[0011] The above objectives are achieved by the method and apparatus according to the present invention.

[0012] An advantage of the various embodiments of the method in this invention is that the interface region between the resonator and the substrate is reduced. The substrate inherently includes defects, which leads to two-stage system (TLS) losses in the resonator. An advantage of the embodiments of the invention is that they provide resonators with low TLS losses. Resonators modified by embodiments of the first aspect of the invention tend to have smoother surfaces after modification. This is beneficial for low TLS losses. Another advantage of the embodiments of the invention is that rounded edges can be achieved, which leads to a further reduction in losses in the resonator. Therefore, resonators according to embodiments of the invention have particularly low losses and thus have long coherence times and high quality factors.

[0013] An advantage of embodiments of the present invention is that photoresist is not required in this method. Another advantage of embodiments of the present invention is that by selecting an etch solution with a higher etch rate on the substrate than that on the resonator, a significant reduction in the interface region can be achieved without photoresist. The acidic nature of the etch solution ensures that the etching of the substrate is isotropic and that its rate is non-zero for resonators comprising niobium or tantalum. An advantage of embodiments of the present invention is that this method is particularly well-suited for resonators comprising niobium or tantalum. In particular, niobium is widely used in resonators.

[0014] An advantage of embodiments of the present invention is that the native oxides ultimately present on the surface of the resonator can be removed. Surface oxides can cause TLS losses. An advantage of embodiments of the present invention is that TLS losses due to native oxides on the resonator are reduced. An advantage of embodiments of the present invention is the use of wet etching, which results in limited substrate surface roughness and limited damage to the substrate crystal structure compared to dry etching techniques.

[0015] The advantage of embodiments of the present invention is that the method is reproducible, reproducibly producing resonators with approximately the same resonator frequency and quality factor.

[0016] In a first aspect, the present invention relates to a method for forming a modified resonator, the method comprising:

[0017] a. A resonator, comprising niobium or tantalum, is obtained on a substrate, thereby forming an interface region between the bottom surface of the resonator and the top surface of the substrate, and

[0018] b. Use a liquid acidic etching solution to contact the resonator with the substrate, selecting the liquid acidic etching solution to have a higher etching rate on the substrate and a non-zero etching rate on the resonator than the etching rate on the resonator.

[0019] According to a second aspect, the present invention relates to a resonator on a substrate, the resonator comprising niobium or tantalum and at least one rounded edge having a radius of curvature from 15 nm to 40 nm, wherein the bottom surface of the resonator has a larger area than the interface region between the bottom of the resonator and the top surface of the substrate.

[0020] Specific and preferred aspects of the invention are set forth in the appended independent and dependent claims. Features from the dependent claims may be suitably combined with features of the independent claims and other dependent claims, and not merely as expressly set forth in the claims.

[0021] Although devices in the field are constantly being improved, changed, and developed, it is believed that the concepts of this invention represent a sufficiently novel and original advancement, including deviations from prior practice, thereby providing devices with this property that are more efficient, stable, and reliable.

[0022] The above and other features, characteristics, and advantages of the present invention will become apparent from the following detailed description taken in conjunction with the accompanying drawings, which illustrate the principles of the invention by way of example. This description is given for illustrative purposes only and does not limit the scope of the invention. The accompanying drawings are referenced below. Attached Figure Description

[0023] Figure 1A and Figure 1B It is a schematic vertical cross-section of a resonator with certain edges having different radii of curvature.

[0024] Figure 2A It is a schematic vertical cross-section and Figure 2B This is a SEM image of a resonator on a substrate according to an embodiment of the present invention.

[0025] Figure 3A It is a schematic vertical cross-section and Figure 3B This is a SEM image of a resonator on a substrate after applying isotropic etching according to existing technology.

[0026] Figure 4A It is a schematic vertical cross-section and Figure 4B This is a SEM image of a resonator on a substrate after anisotropic etching has been applied according to existing technology.

[0027] Figure 5A It is a schematic vertical cross-section and Figure 5BThis is a SEM image of a resonator on a substrate after anisotropic etching and then isotropic etching, according to an embodiment of the present invention.

[0028] In different accompanying drawings, the same reference numerals refer to the same or similar elements. Detailed Implementation

[0029] The invention will be described with reference to specific embodiments and particular drawings, but the invention is not limited thereto but is defined only by the claims. The described drawings are merely illustrative and not restrictive. In the drawings, for illustrative purposes, the dimensions of some elements may be enlarged and not drawn to scale. Scale and relative scale do not correspond to an actual reduction in the implementation of the invention.

[0030] Furthermore, the terms first, second, and third, etc., used in the specification and claims are used to distinguish similar elements and are not necessarily used to describe a temporal, spatial, arrangement, or any other chronological order. It should be understood that the terms thus used are interchangeable where appropriate, and the embodiments of the invention described herein can be operated in a different order than those described or illustrated herein.

[0031] Furthermore, the terms top, bottom, above, below, etc., used in the specification and claims are for descriptive purposes and not necessarily for describing relative positions. It should be understood that such terms are interchangeable where appropriate, and the embodiments of the invention described herein can operate in orientations different from those described or illustrated herein.

[0032] It should be noted that the term "comprising" as used in the claims should not be construed as limiting itself to the devices listed thereafter; it does not exclude other elements or steps. Therefore, the term should be interpreted as specifying the presence of the stated features, integers, steps, or components as mentioned, but does not exclude the presence or addition of one or more other features, integers, steps, or components, or groups thereof. Thus, the term "comprising" covers the case where only the stated feature is present, as well as the case where these features are present along with one or more other features. The term "comprising" according to the invention therefore also includes an embodiment where no other components are present. Therefore, the scope of the statement "an apparatus comprising devices A and B" should not be construed as limited to an apparatus consisting only of components A and B. This means that for the present invention, the only relevant components of the apparatus are A and B.

[0033] Throughout this specification, the reference to "an embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with that embodiment is included in at least one embodiment of the invention. Therefore, the appearance of the phrase "in an embodiment" or "in an embodiment" throughout this specification does not necessarily refer to the same embodiment, but may refer to the same embodiment. Furthermore, in one or more embodiments, as will be obvious to those skilled in the art from this disclosure, particular features, structures, or characteristics can be combined in any suitable manner.

[0034] Similarly, it should be understood that in the description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, drawing, or description for the purpose of simplifying the disclosure and aiding in the understanding of one or more inventive aspects. However, this method of disclosure should not be construed as reflecting an intention to claim more features than are expressly recited in each claim. Rather, as reflected in the appended claims, inventive aspects exist in fewer features than all the features of a single foregoing disclosed embodiment. Therefore, the claims appended following the detailed description are thus explicitly incorporated into this detailed description, wherein each claim itself represents a separate embodiment of the invention.

[0035] Furthermore, while some embodiments described herein include features that are included in other embodiments but not others, it will be understood by those skilled in the art that combinations of features from different embodiments are intended to fall within the scope of the invention and form different embodiments. For example, any embodiment of the claimed embodiments in the appended claims can be used in any combination.

[0036] Numerous specific details are set forth in the description provided herein. However, it should be understood that embodiments of the invention can be practiced without these specific details. In other instances, well-known methods, structures, and techniques have not been shown in detail so as not to obscure the understanding of this description.

[0037] In a first aspect, the present invention relates to a method for forming a modified resonator, the method comprising:

[0038] a. A resonator, comprising niobium or tantalum, is obtained on a substrate, thereby forming an interface region between the bottom surface of the resonator and the top surface of the substrate, and

[0039] b. Use a liquid acidic etching solution to contact the resonator with the substrate, selecting the liquid acidic etching solution to have a higher etching rate on the substrate and a non-zero etching rate on the resonator than the etching rate on the resonator.

[0040] A resonator can typically take the shape of a coplanar waveguide resonator. This resonator may include signal lines separated from a ground plane by two typically identical gaps. Both the signal lines and the ground plane lie in the same plane above the substrate and are both made of a conductive material (e.g., a superconductor). Therefore, when the term "resonator" is used, the term can refer to both the signal lines and the ground plane.

[0041] In step a, a resonator can be obtained by depositing a metal layer on the substrate and forming the resonator through that layer. For example, the resonator can be obtained by performing a photolithography step involving forming a mask on the layer and a dry plasma etching step through the mask.

[0042] Unless otherwise provided, when a resonator is mentioned in the remainder of the specification without specifying whether it is the resonator obtained in step a or after step b, it means that both the resonator obtained in step a and the resonator obtained after step b are mentioned. Indeed, many properties (e.g., chemical properties, dimensions, overall shape when the smoothness of the edges is ignored, etc.) remain substantially unchanged between steps a and b.

[0043] In various embodiments, the resonator comprises one of the following materials: niobium, tantalum, NbN, and NbTiN. In a preferred embodiment, the resonator comprises niobium or is composed of niobium. The dimensions of the resonator are preferably such that a standing wave can exist and be contained within the resonator, wherein the standing wave has a frequency from 1 GHz to 10 THz (e.g., from 4 GHz to 8 GHz). In various embodiments, the signal line of the resonator may have a width from 125 nm to 30 μm, preferably from 500 nm to 20 μm, more preferably from 5 to 15 μm. In various embodiments, the resonator (i.e., the signal line and the ground plane) may have a height from 35 to 140 nm. In various embodiments, the resonator (i.e., the signal line and the ground plane) may have a length from 0.1 to 1 cm, preferably from 2 to 5 mm. In various embodiments, the resonator (i.e., the signal line and the ground plane) may have a cuboid shape. In various embodiments, the shape of the signal line may be irregular. In a preferred embodiment, the resonator comprises a cuboid shape.

[0044] In various embodiments, the substrate may include silicon, silicon germanium, or gallium arsenide, preferably silicon. These materials are widely used in semiconductor manufacturing. Furthermore, these materials are etched faster with acidic solutions than niobium or tantalum. In various embodiments, the substrate comprises a bulk of silicon, silicon germanium, or gallium arsenide and an optional oxide top surface. Preferably, the substrate is crystalline. The crystallinity of the substrate reduces TLS loss due to a relatively lower defect concentration compared to amorphous materials.

[0045] The resonator obtained in step a typically includes at least one edge. Although the method of the first aspect is useful even for resonators without edges due to its surface smoothing effect, it is most useful for modifying resonators with at least one edge due to its edge rounding effect.

[0046] When at least one edge exists in a resonator, the edge is typically formed by the intersection of two adjacent surfaces of the resonator's elements (e.g., two adjacent surfaces of its signal lines). Examples of surfaces that can intersect to form at least one edge are the bottom and side surfaces of the signal line, or the top and side surfaces of the signal line. Typically, the surfaces forming at least one edge are flat.

[0047] Although the surfaces forming at least one edge theoretically meet on a single line defining said edge, in practice, at least one edge of the resonator obtained in step a may have already exhibited some degree of rounding. In other words, at least one edge obtained in step a may be a curved surface connecting two flat surfaces forming that at least one edge.

[0048] In embodiments where at least one edge is a curved surface, the surface may have a curvature equal to the angle between adjacent surface planes connected by the curved surface.

[0049] In the embodiment where the resonator obtained in step a consists of cuboid resonator elements (i.e., the signal line and the ground plane each have essentially cuboid resonators) and where the edges are curved surfaces with a curvature of 90°.

[0050] For example, a cuboid resonator element including a top surface, a bottom surface, and four sides may have eight edges, each edge being defined by the intersection between two surfaces or by a curved surface connecting the two surfaces. For example, the resonator element may have a first edge defined by a line at the intersection between the top surface and one of the side surfaces of the resonator or by a curved surface connecting the top surface and one of the side surfaces of the resonator.

[0051] The edges of the resonator prior to step b of the method (e.g., obtained in steps a, a', or a”, see below) may be sharp in various embodiments: sharp edges can lead to radiation loss of the wave resonating in the resonator.

[0052] Therefore, preferably, the edges of the resonator are rounded during step b.

[0053] As used here, a sharp edge is a line at the intersection of two adjacent (e.g., flat) surfaces or a curved surface connecting two adjacent flat surfaces, the curved surface having a radius of curvature of less than 10 nm.

[0054] The radius of curvature is the radius of a circle that best fits the curved surface of the edge. In various embodiments, the method of the first aspect may introduce curvature in at least one edge (when the at least one edge is defined by a line at the intersection of two surfaces) or increase the radius of curvature of the at least one edge (when the at least one edge is defined by a curved surface at the intersection of two surfaces). In various embodiments, the method of the first aspect may reduce the sharpness of the at least one edge and may produce a rounded edge, i.e., an edge in which curvature has been introduced or increased. In various embodiments, at least one edge of a rounded resonator includes at least two edges of the rounded resonator, such as all edges of a signal line or even all edges included in the resonator. In various embodiments, at least one edge of a rounded resonator includes increasing the average radius of curvature of the edges of the resonator. The average radius of curvature of the edges of the resonator is the average radius of curvature of the edges of the resonator. In other words, the average radius of curvature of the edges of the resonator is defined by the sum of the radii of curvature of each edge of the resonator divided by the number of edges. In various embodiments, the average radius of curvature of the edges after step b is larger than the average radius of curvature of the edges before step b. In various embodiments, at least one edge of the smoothed resonator includes smoothing at least one edge of the signal line by increasing the average radius of curvature of each edge of the signal line. The average radius of curvature of each edge of the signal line is the average radius of curvature of each edge of the signal line. In other words, the average radius of curvature of each edge of the signal line is defined by the sum of the radii of curvature of each edge of the signal line divided by the number of edges. In various embodiments, the average radius of curvature of each edge of the signal line after step b is larger than the average radius of curvature of each edge of the signal line before step b. In various embodiments, the radius of curvature of at least one edge after step b is 1.2 to 10 times, such as 1.5 to 5 times, such as 1.5 to 3 times, the radius of curvature of the at least one edge before step b. In various embodiments, the average radius of curvature of each edge of the resonator after step b is 1.2 to 10 times, such as 1.5 to 5 times, such as 1.5 to 3 times, the average radius of curvature of each edge of the resonator before step b.

[0055] In various embodiments, the average radius of curvature of each edge of the signal line after step b is 1.2 to 10 times, such as 1.5 to 5 times, or such as 1.5 to 3 times, the average radius of curvature of each edge of the signal line before step b. In various embodiments, the radius of curvature of at least one edge before step b may be less than 10 nm and may be from 15 nm to 40 nm after step b. For example, in various embodiments, the average radius of curvature of each edge of the resonator before step b may be less than 10 nm and may be from 15 nm to 40 nm after step b.

[0056] For example, in various embodiments, the average radius of curvature of each edge of the signal line before step b can be less than 10 nm and can be from 15 nm to 40 nm after step b.

[0057] In various embodiments, at least one edge of the smooth resonator (e.g., all edges of the signal line or all edges of the resonator) produces a low-loss resonator, i.e., a resonator with a quality factor of at least 1,000,000.

[0058] The resonator and substrate are in direct contact with each other, i.e., the resonator is on top of the substrate, thereby forming an interface between the bottom surface of the resonator and the top surface of the substrate. In various embodiments, the complete bottom surface of the resonator obtained in step a is in direct contact with the substrate. The interface region is the interface area between the bottom surface of the resonator and the top surface of the substrate. In various embodiments, the size of the interface region differs from the product of the lateral dimensions of the resonator (i.e., the surface area of ​​the bottom surface of the resonator) by no more than 5%, preferably no more than 1%, and most preferably equal to that product. The interface may include a native oxide layer belonging to the substrate. The native oxide at the interface can induce TLS loss. Although the top surface of the substrate can be cleaned, for example by using an HF etching solution, before the resonator is deposited on the substrate to substantially remove the native oxide from the substrate, a small amount of native oxide may still remain after cleaning. In various embodiments, the native oxide may still be present at the interface after the resonator is formed on the substrate. Since extremely low loss is necessary for the resonator (e.g., for quantum computing), even TLS loss due to a small amount of native oxide at the interface can be problematic. Therefore, the interface region is preferably reduced mechanically as much as possible during step b. However, if the interface region becomes too small, the resonator may collapse on the substrate. Therefore, a balance is preferably sought between limiting the TLS loss of the resonator and preventing the resonator from collapsing on the substrate. In various embodiments, step b reduces the interface region by 5% to 95%, preferably 50% to 95%, relative to the interface region formed in step a. Reducing the interface region includes removing a portion of the substrate beneath the resonator.

[0059] The liquid acidic etching solution is chosen to have a higher etching rate on the substrate than on the resonator. Therefore, it is advantageous that no photoresist is needed in this method to prevent the liquid acidic etching solution from substantially etching the resonator. Although the etching rate on the substrate is higher than that on the resonator, the etching rate on the resonator is non-zero. This non-zero etching rate on the resonator is typically enabled for edge smoothing. Another advantage of the non-zero etching rate on the resonator is that it improves the surface smoothness of the resonator. In various embodiments, the liquid acidic etching solution comprises HF and HNO3. In various embodiments, the liquid acidic etching solution may further comprise a diluent, such as H2O or CH3COOH, preferably H2O. In these embodiments, preferably, the liquid acidic etching solution comprises 30 to 70 parts by volume of HNO3, such as 60 to 70 parts, such as 65 to 69 parts by volume, and 0.1 to 5 parts by volume of HF, such as 0.1 to 2 parts, preferably 0.4 to 0.6 parts by volume, and additional volume parts of diluent such that the volume parts of HNO3, HF, and diluent add up to 100. In a preferred embodiment, the temperature of the etching solution is constant, which results in a constant and therefore controllable etching rate. In a preferred embodiment, the temperature ranges from 20°C to 70°C. These embodiments have the effect that, due to the presence of HNO3, the resonator is constantly oxidized so that the surfaces of the resonator remain oxidized after step b. Therefore, there will be an oxide layer present on various aspects of the resonator after step b. The solution comprising HF and HNO3 has a non-zero etching rate for the resonator, but typically has a significantly higher etching rate for the substrate than for the resonator. This is true for most substrates, especially for silicon, silicon germanium, or gallium arsenide. In various embodiments, the ratio of the etching rate of the substrate to the etching rate of the resonator is at least 5, such as from 5 to 500, preferably from 10 to 100.

[0060] This ratio can be optimized by improving the mixing ratio of acids included in the liquid acid etching solution and / or by adding additives. For example, in various embodiments, increasing the concentration of HF can increase the etching rate for both the resonator and the substrate, but the etching rate for the resonator can be increased more than the etching rate for the substrate, thus lowering the ratio. In various embodiments, conversely, decreasing the concentration of HF can increase the ratio. The liquid acid etching solution typically etches the substrate isotropically, meaning that the liquid acid etching solution typically etches at the same etching rate in all directions, i.e., the etching rate is independent of the etching direction. In various embodiments, the etching caused by the liquid acid etching solution creates a cavity with an isotropic profile in the substrate after the solution is applied. For example, the cavity may have a truncated spherical shape.

[0061] In various embodiments, step b may oxidize the surface of the resonator so that oxides are present on the surface of the resonator. Oxides on the surface of the resonator can cause TLS loss. Therefore, in various embodiments, the method includes step c, which removes any oxides from the surface of the resonator after step b. In various embodiments, step c may also remove any oxides from the exposed top surface of the substrate (i.e., at the interface between the substrate and air). Oxides on the exposed top surface of the substrate can also cause TLS loss. In various embodiments, step c may include applying a liquid etching solution comprising HF but excluding HNO3. HF is typically applied in the art to remove native oxides from the surface.

[0062] In various embodiments, the method includes step a', which involves selectively etching the substrate relative to the resonator by applying a liquid alkaline etchant after step a and before step b, thereby anisotropically etching the substrate and reducing the interface region. This embodiment is particularly useful when the substrate comprises a crystal structure to be etched. Using a liquid alkaline etchant solution has the advantage that it anisotropically etches the substrate along a well-defined crystal substrate plane. This allows for a good level of control over the interface portions removed by the etching process. In other words, the inherent directionality of liquid alkaline etching allows the etching of the substrate to be self-aligned with the crystal substrate plane.

[0063] Although the liquid acidic etching solution from step b etches within the material at a similar etching rate in all directions (i.e., the etching rate is orientation-independent), the liquid alkaline etching solution exhibits anisotropic etching rates, meaning it depends on the etching direction within the substrate, for example, due to different etching rates for different crystal planes. For instance, in silicon, the liquid alkaline etching solution exhibits a particularly low etching rate for Si(111) planes. In various embodiments, the temperature of the liquid alkaline etching solution can be set above room temperature. Advantageously, in this way, the etching rate for Si(111) planes can also be increased, thereby reducing the required etching time. This increase in the etching rate for Si(111) planes can also be observed when the liquid alkaline etching solution is stirred using a mechanical stirrer during step a'.

[0064] In various embodiments, the resonator may include at least one curved surface.

[0065] Since the etching rate of the liquid alkaline etching solution depends on the etching direction within the material, in embodiments where the resonator includes a curved surface, the etching rate of the substrate at the interface with the resonator may differ locally. That is, the first portion of reducing the interface region in step a' may be carried out at a different rate than the second portion reducing the interface region. This difference in etching rate can be advantageously used in some embodiments to optimize the parameters of the interface etching. Alternatively, the etching rate can be homogenized over the interface region by adjusting the resonator design or substrate orientation (e.g., by using a Si(110) substrate).

[0066] In various embodiments, the profile of the cavity in the substrate after applying the liquid alkaline etching solution in step a' can be anisotropic. For example, the surface of the etched cavity can be flat along a crystal plane. In various embodiments, the liquid alkaline etching of the substrate is self-aligned. This is advantageous because it means that the etching is anisotropic and highly controllable. This is advantageous because it reduces the need for photomasks, for example, in the case of anisotropic (dry) plasma etching or reactive ion etching. In various embodiments, the liquid alkaline etching solution comprises hydroxides, such as one of the following: tetramethylammonium hydroxide, ammonium hydroxide, and sodium hydroxide. In various embodiments, the liquid alkaline etching solution selectively etches the substrate relative to the resonator. In various embodiments, the liquid alkaline etching solution substantially does not etch the resonator. Combining the liquid acidic etching solution with the liquid alkaline etching solution has the advantage of providing additional parameters for optimizing the profile of the resonator and / or the substrate. Indeed, in various embodiments, the profile of the substrate can be etched first in step a' to take an anisotropic profile, which is then further isotropically etched in step b. The resulting substrate profile can be a combination of anisotropic and isotropic profiles in various embodiments. Combining liquid acidic etch solutions with liquid alkaline etch solutions in various embodiments can provide additional parameters for optimizing the interface region (such as the interface region size). Indeed, although liquid acidic etch solutions are considered advantageous for non-zero etch rates of resonators, in some embodiments it may be difficult to sufficiently reduce the interface region without simultaneously etching too many resonators (i.e., when using only liquid acidic etch solutions). Etching too many resonators can cause, for example, a shift in the resonator's resonant frequency. In various embodiments, step a' reduces the interface region by 5% to 95%, preferably 50% to 95%, relative to the interface region formed in step a.

[0067] In embodiments according to the first aspect, the method may include, after step a and before step b, a step a” instead of step a’, selectively etching the substrate relative to the resonator by applying dry isotropic etching to thereby reduce the interface region. In various embodiments, the dry isotropic etching may be an inductively coupled plasma etching (ICPE) technique, wherein the dry isotropic etching includes NF3 and H2 gases. ICPE with NF3 and H2 gases does not etch various metals, including niobium and tantalum; that is, it selectively etches the substrate relative to the resonator. In various embodiments, the dry isotropic etching is essentially… The resonator is not etched. Combining the liquid acid etching solution with a dry isotropic etching phase in various embodiments can provide additional parameters for optimizing the interface region (such as the interface region size). Indeed, although the non-zero etch rate of the resonator by the liquid acid etching solution is considered advantageous, in some embodiments it may be difficult to sufficiently reduce the interface region without simultaneously etching too many resonators (i.e., when using only the liquid acid etching solution). In various embodiments, step a” reduces the interface region by 5% to 95%, preferably 50% to 95%, relative to the interface region formed in step a.

[0068] The modified resonator formed by the various embodiments of the method in this invention typically has the same composition and dimensions as the resonator obtained in step a, but with rounded edges. In various embodiments, the average radius of curvature of the modified resonator is from 15 nm to 40 nm. In various embodiments, the interface between the resonator and the substrate has the same material as the interface between the modified resonator and the substrate, but the size of the interface region is different. In various embodiments, the interface region can be 5% to 95%, preferably 50% to 95%, of the bottom surface region of the modified resonator. In various embodiments, the resonator can include or be composed of one of the following materials: niobium, tantalum, NbN, and NbTiN, preferably niobium or tantalum. In various embodiments, the resonator is composed of niobium. In various embodiments, the dimensions of the modified resonator, i.e., the thickness and each of the two lateral dimensions, can be the same as or differ by less than 1% from the dimensions of the resonator obtained in step a. In various embodiments, the surface of the modified resonator may be substantially free of oxides, such as native oxides. In various embodiments, the profile of the substrate beneath the modified resonator may include one of the following profiles: an isotropic profile, an anisotropic profile, and a combination of an isotropic and anisotropic profiles.

[0069] According to a second aspect, the present invention relates to a resonator on a substrate, the resonator comprising niobium or tantalum and at least one rounded edge having a radius of curvature from 15 nm to 40 nm, wherein the bottom surface of the resonator has a larger area than the interface region between the bottom of the resonator and the top surface of the substrate.

[0070] The present invention also relates to a (modified) resonator that can be obtained through any embodiment of the first aspect of the invention.

[0071] Any features of the resonator and substrate in the second aspect may be defined as those of the modified resonator and substrate in any embodiment of the first aspect of the invention.

[0072] In other words, the resonator and substrate according to the second aspect can have the same characteristics as the modified resonator and substrate of the device formed according to any embodiment of the method of the first aspect. The characteristics of the resonator and substrate according to the second aspect can have the same advantages as the characteristics of the resonator and substrate formed by the method of the first aspect.

[0073] The following are some examples of the characteristics that a resonator in the second aspect may have. Other examples can be found in the description of the modified resonator in the first aspect.

[0074] In various embodiments, the average radius of curvature of the smoothed edges of the resonator can range from 15 nm to 40 nm. This large average radius of curvature can significantly reduce radiation losses of waves, particularly those propagating on the surface of the resonator.

[0075] In various embodiments, the interface region between the bottom of the resonator and the top surface of the substrate can be 5% to 95% of the bottom surface region of the resonator, preferably 50% to 95%. In various embodiments, the resonator may be composed of niobium, and the substrate may include or be composed of silicon, silicon germanium, or gallium arsenide, preferably silicon.

[0076] The invention will now be described in detail through several embodiments thereof. It will be apparent that other embodiments of the invention can be configured based on the knowledge of those skilled in the art without departing from the technical teachings of the invention, which is limited only by the terms of the appended claims.

[0077] Example

[0078] The reference image shows the vertical cross-sections of the first cuboid resonator 3 and the second cuboid resonator 4. Figure 1A and Figure 1B Each cuboid resonator has bottom surfaces 31 and 41, top surfaces 32 and 42, and side surfaces 33 and 43. First edges 313 and 413, including edge lines, connect a first surface plane including bottom surfaces 31 and 41 to a second surface plane including side surfaces 33 and 43. The first and second surface planes are adjacent planes.

[0079] Furthermore, in both resonators, the second edges 323 and 423 include edge surfaces connecting the third surface plane, including the top surfaces 32 and 42, to the second surface plane, where the second and third surface planes are adjacent surface planes. The angle between the second and third surface planes is equal to 90° in both cases. In this embodiment, the curvature of the second edge is equal to 90°. A circle that best fits the surface of the second edges 323 and 423 is shown in each case. The radius of the circle (e.g., ...) Figure 1A In this context, R) is equal to the radius of curvature of the second edges 323 and 423. The radius of curvature of the second edge 323 of the first resonator 3 is larger than that of the second edge 423 of the second resonator 4. That is, the second edge 343 of the first resonator 3 is not as sharp as the second edge 423 of the second resonator 4, but is smoother. For example, applying the liquid acid etching solution in step b of the method of the present invention to the second resonator 4 can produce the first resonator 3 in various embodiments.

[0080] Reference Figure 2A , Figure 2A This is a schematic vertical cross-section of a resonator 2 on a substrate 1 according to an embodiment of the present invention: three resonators 2 are shown on the substrate 1. Figure 2A Following step a in various embodiments of the method of the first aspect of the invention, a resonator 1 is shown on a substrate 2 before any etching is applied. In this example, the resonator 1 has a cuboid shape. An interface region 21 exists between the top surface of the substrate 1 and the bottom surface of the resonator. The interface region 21 may include oxide (e.g., from the substrate), which may be a first source of TLS loss for any oscillation field in the resonator 2. Furthermore, the top surface of the substrate may include oxide 11, which may be a second source of TLS loss for any oscillation field in the resonator 2. Additionally, the surface of the resonator 2 may include oxide 22, which may be a third source of TLS loss for any oscillation field in the resonator 2. The objective of various embodiments of the method of the invention is to reduce these three sources of TLS loss. Referring now to... Figure 2B , Figure 2B An SEM image of a resonator 2 on a substrate 1 according to an embodiment of the present invention is shown. As can be observed, the edges 20 of the resonator are still quite sharp, i.e., they have a small radius of curvature.

[0081] Now refer to Figure 3ADry isotropic etching is applied to resonator 2 and substrate 1, wherein the dry isotropic etching selectively etches substrate 1 relative to resonator 2, thereby reducing interface region 21, i.e., an embodiment of step a” of the method of the present invention. Due to the high selectivity of the etching, the edges of resonator 2 are not rounded. In this example, in the middle case of the three resonators 2, the interface region 21 after step a” has been reduced by about one-third, i.e., 33%, compared to the interface region 21 after step a. In addition, the surface of substrate 12 has taken an isotropic profile, i.e., forming part of a sphere. This is a result of the dry isotropic etching having an equal etching rate in all directions of substrate 1. Referring now to Figure 3B The image shows a SEM image of the resonator 2 on the substrate after isotropic etching. In this specific example, the interface region 21 after isotropic etching is reduced by approximately 90% compared to the interface region 21 after step a.

[0082] Now refer to Figure 4A Instead, a liquid alkaline etching solution is applied to the resonator 2 and substrate 1 obtained after step a (i.e., according to the embodiment of step a'), instead of... Figure 3A and Figure 3B Isotropic dry etching. The liquid alkaline etching solution produces a highly anisotropic profile 12 of substrate 1, i.e., substantially flat along the crystal surface of substrate 1. In this example, the liquid alkaline etching solution has reduced interface region 21 by approximately two-thirds, or 66%, compared to the interface region after step a. Now refer to Figure 4B , Figure 4B The SEM image of the resonator 2 on the substrate 1 after step a' is shown.

[0083] Now refer to Figure 5A After step a', that is, in Figure 4A A liquid acid etching solution is applied to the resonator on the substrate shown, i.e., an embodiment of step b of the method according to the invention. This further reduces the interface region 21, i.e., reduces it to about 30% of the surface region after step a, i.e., corresponds to a 70% reduction in the interface region 21 compared to the interface region 21 after step a. The profile of the substrate 1 is a combination of an isotropic profile 121 and an anisotropic profile 122. The liquid acid etching solution has a non-zero etching rate on the resonator 2, so that the resonator now has a rounded edge 20, i.e., the resonator, i.e., the edge 20, has a radius of curvature much larger than that of the resonator after step a. Furthermore, in an embodiment of step c of the method according to the invention, any oxides that may have formed on the surface of the resonator and on the surface of the substrate 1 during the implementation of step b have been removed, for example, by HF etching. Figure 5A Therefore, an embodiment of the resonator 22 according to an embodiment of the present invention is shown. Figure 5BThis is a SEM image of a resonator 22 formed on a substrate 1 according to an embodiment of the present invention and in an embodiment of the method of the present invention.

[0084] It is understood that although preferred embodiments, specific structures and configurations, and materials have been discussed herein with respect to the device according to the invention, various changes or modifications in form and detail may be made without departing from the scope of the invention. For example, any molecular formulas given above represent only steps that can be used. Functions can be added or removed from the block diagrams, and operations can be interchanged between function blocks. Steps can be added or removed from the method within the scope of the invention.

Claims

1. A method for forming a modified resonator (22), the method comprising: a. A resonator (2) is obtained on a substrate (1), the resonator (2) having at least one edge (20) and comprising niobium or tantalum, thereby forming an interface region (21) between the bottom surface of the resonator (2) and the top surface of the substrate (1), and b. The resonator (2) is brought into contact with the substrate (1) using a liquid acidic etching solution, wherein the liquid acidic etching solution is selected such that the etching rate of the substrate (1) is higher than that of the resonator (2) and the etching rate of the resonator (2) is non-zero, thereby smoothing the edge (20).

2. The method of claim 1, wherein the liquid acid etching solution comprises HF and HNO3.

3. The method as described in any of the preceding claims, including step c, which removes any oxides from the surface of the resonator (2) after step b.

4. The method of claim 1, wherein the resonator (2) is composed of niobium.

5. The method of claim 1, wherein the substrate (1) comprises silicon, silicon germanium, or gallium arsenide.

6. The method of claim 1, wherein step b reduces the interface region (21) by 5% to 95% relative to the interface region (21) formed in the preceding step.

7. The method of claim 6, wherein step b reduces the interface region (21) by 80% to 95% relative to the interface region (21) formed in the preceding step.

8. The method of claim 1, wherein the rounded edge (20) of the modified resonator (22) after step b has an average radius of curvature that is 1.5 to 3 times the average radius of curvature of the edge (20) obtained in step a.

9. The method of claim 1, comprising step a' after step a and before step b, selectively etching the substrate relative to the resonator (2) by contacting the resonator (2) with the substrate (1) using a liquid alkaline etching solution to thereby reduce the interface region (21), and wherein step b further reduces the interface region (21).

10. The method of claim 1, comprising step a'' after step a and before step b, selectively etching the substrate (1) relative to the resonator (2) by performing dry isotropic etching to thereby reduce the interface region (21), and wherein step b further reduces the interface region (21).

11. The method of claim 10, wherein step a'' reduces the interface region (21) by 5% to 95% relative to the interface region (21) formed in step a.

12. A resonator (22) on a substrate (1), the resonator (22) comprising niobium or tantalum, wherein the bottom surface of the resonator (22) has a larger area than the interface region (21) between the bottom of the resonator (22) and the top surface of the substrate (1), characterized in that The resonator (22) has at least one rounded edge (20) with a radius of curvature ranging from 15 nm to 40 nm.

13. The resonator (22) of claim 12, wherein the average radius of curvature of all edges (20) of the resonator (22) is from 15 nm to 40 nm.

14. The resonator of claim 12 or 13, wherein the interface region (21) is 5% to 95% of the bottom surface region of the resonator (22).

15. The resonator of claim 12 or 13, wherein the resonator (22) is composed of niobium, and wherein the substrate (1) comprises silicon, silicon germanium, or gallium arsenide.

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