Patent Description:
Resonators, in particular superconducting resonators, are one of the fundamental components for the realization of quantum bits, i.e. "qubits", for quantum computing. In general, a resonator of a quantum circuit is a microwave transmission line that is deliberately designed to support resonant oscillations (i.e. resonance) within the line, under certain conditions (i.e. a resonant microwave transmission line). Resonators of superconducting qubit devices are typically implemented as coplanar waveguides (CPWs), which is a particular type of a microwave transmission line. Quantum computing requires low-loss operations of the resonators. Preferably, the resonator of a quantum computer has a quality factor of at least <NUM>,<NUM>,<NUM>, i.e. in each radian of a cycle of oscillation, the resonator preferably loses less than <NUM>/<NUM>,<NUM>,<NUM> of peak energy stored in the resonator. As extremely low microwave powers are stored in the resonator e.g. in the order of attowatts, that is <NUM>-<NUM> watts, such a low loss is necessary to reach a long qubit lifetime i.e. long coherence time even at temperatures of ~<NUM> mK. High-quality factor i.e. low-loss resonators are of fundamental importance for reaching long coherence times in superconducting qubits, for storage of arbitrary quantum states in superconducting resonators at very low fields i.e. in the single-photon regime, where the electric field is much less than the critical field, and for low-loss coupling between two qubits mediated by a low-loss resonator.

Furthermore, low-loss resonators are also required for quantum-limited microwave amplifiers, which are extremely sensitive amplifiers used e.g. for gravitational waves detection, and for multiplexed readout of microwave kinetic inductance photon detectors (MKID) e.g. in astronomy.

Typically, the resonator comprises a superconductor material and is on an insulating substrate. Typical superconductor materials used for resonators are Al, TiN, Nb, NbN, and NbTiN, and for the insulating substrate usually crystalline sapphire or high-resistivity silicon substrates are used.

At low microwave field levels and low temperatures e.g. <NUM> mK used in particular in superconducting quantum information applications, the quality factors of the resonator in the state-of-the-art are reduced in particular by parasitic two-level systems (TLS) losses. TLS losses are due to defects in a material, e.g. the substrate or a native oxide layer, in the proximity of the resonator. The defect may in particular be represented by an atom that tunnels between two nondegenerate minima of a potential provided by the environment of the atom, thereby forming the TLS. The TLS may interact with and absorb a photon in the resonator, consequently resulting in loss of energy. Predominantly amorphous materials, oxides, and dielectrics at surfaces and/or interfaces of the resonator contribute to TLS losses, and also oxides on the surface of the substrate may contribute to the TLS losses.

To reduce TLS losses, in the state-of-the-art, in particular, the material in the proximity of the resonator contributing to the TLS losses may be removed. For instance, in <NPL>, an interface area between a bottom surface of the resonator on the substrate and a top surface of the substrate is effectively reduced by forming a photoresist on the resonator and etching the substrate selectively with respect to the photoresist. The etching technique used is an isotropic etching technique. The substrate underlying the resonator is etched i.e. underetching of the resonator occurs, thereby reducing the interface area. Finally, the photoresist is removed using a combination of ashing solvent resist strips. The use of photoresists, which are generally used in the state-of-the-art to underetch resonators, is however unwanted, because the process of removing the photoresist may induce defects on surfaces of the resonator: these defects may give rise to further resonator losses. Resonators and fabrication thereof involving undercut etching using liquid base etchants according to the preamble of independent claims <NUM> and <NUM>, respectively, are known from <CIT> and <NPL>, for example.

Besides, the edges of the resonators of the state-of-the-art are sharp, which the present inventors believe to be a source of further losses. Indeed, intuitively, it may be realized that a wave propagating on the surface of the resonator cannot follow the sharp angles at the cavity edges, hence resulting in radiative loss of the energy in the resonator.

Furthermore, control of the substrate and resonator profiles formed by the etching, i.e. control of the etching, is particularly advantageous. That is, in particular when many resonators on a single substrate are etched at the same time, the control of etching is advantageous to at the same time minimize the interface area and prevent the collapse of at least part of the resonators. Moreover, resonators preferably have the same, predictable, properties such as well-resolved resonator frequency and high-quality factor. Etching is therefore preferably highly reproducible. Finally, resonators generally comprise a native oxide layer on their surfaces which may give rise to further TLS losses.

Therefore, there is a need in the state-of-the-art for an invention that addresses one or more of the above problems.

It is an object of the present invention to provide good methods for reducing energy losses in a resonator. It is a further object of the present invention to provide a resonator obtainable by the methods.

The above objective is accomplished by a method and device according to present independent claims <NUM> and <NUM>, respectively.

It is an advantage of embodiments of the method of the present invention that the interface area between the resonator and the substrate is reduced. The substrate inherently comprises defects, which lead to two-level system (TLS) losses in the resonator. It is an advantage of embodiments of the present invention that resonators are provided with low TLS losses. The resonators modified by embodiments of the first aspect of the present invention tend to have smoother surfaces after modification. This favors low TLS losses. It is a further advantage of embodiments of the present invention that rounded edges can be achieved, which results in a further reduction of losses in the resonator. Therefore, resonators according to embodiments of the present invention have particularly low losses and therefore a long coherence time and a high-quality factor.

It is an advantage of embodiments of the present invention that no photoresist needs to be used in the method. It is an advantage of embodiments of the present invention that by selecting the etching solution so that it has a higher etching rate towards the substrate than towards the resonator, a considerable reduction of the interface area can be achieved without the need for a photoresist. The acidic nature of the etching solution assures that the etching of the substrate is isotropic and that its rate toward the niobium or tantalum-comprising resonator is nonzero. It is an advantage of embodiments of the present invention that the method is particularly well suited for resonators comprising niobium or tantalum. In particular, niobium is widely used in resonators.

It is an advantage of embodiments of the present invention that native oxides eventually present on the surface of the resonator may be removed. The oxides on the surface may induce TLS losses. It is an advantage of embodiments of the present invention that TLS losses due to natives oxides on the resonator surface are reduced. It is an advantage of embodiments of the present invention that a wet etch is used resulting in limited substrate surface roughness and limited damage to the substrate crystal structure when compared to dry etch techniques.

It is an advantage of embodiments of the present invention that the method is reproducible, reproducibly yielding resonators with approximately the same resonator frequency and quality factor.

According to a first aspect, the present invention provides a method for forming a modified resonator, the method comprising:.

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 of from <NUM> to <NUM>, wherein a bottom surface of the resonator has an area larger than an interface area between the bottom of the resonator and a top surface of the substrate.

This description is given for the sake of example only, without limiting the scope of the invention which is defined by the claims.

Furthermore, the terms first, second, third, and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner.

Moreover, the terms top, bottom, over, under, and the like in the description and the claims are used for descriptive purposes and not necessarily for describing relative positions.

The word "comprising" according to the invention therefore also includes as one embodiment that no further components are present.

In other instances, well-known methods, structures, and techniques have not been shown in detail in order not to obscure an understanding of this description.

According to a first aspect, the present invention relates to a method for forming a modified resonator, the method comprising:.

A resonator may typically take the shape of a coplanar waveguide resonator. It may comprise a signal line separated from a ground plane by two gaps which are typically identical. The signal line and the ground plane all lie in the same plane over a substrate and are all made from electrically conductive materials (e.g. from superconductors). When the term "resonator" is used, it may therefore relate to the signal line and a ground plane.

In step a, the resonator may be obtained on the substrate by depositing a metal layer and form the resonator from that layer. For instance, the resonator may be obtained by performing a lithography step involving the formation of a mask over the layer and a dry plasma etching step through the mask.

Unless provided otherwise, when a resonator is referred to in the rest of the description and it is not mentioned whether it is the resonator obtained in step a or after step b, it means that both, the resonator obtained in step a and after step b are referred to. Indeed, many characteristics (e.g. chemical nature, dimensions, overall shape when the rounding of the edges are neglected. ) remain substantially unchanged between steps a and b.

In embodiments, the resonator comprises one of the following materials: niobium, tantalum, NbN, and NbTiN. In preferred embodiments, the resonator comprises or consists 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 of from <NUM> to <NUM> THz, such as from <NUM> to <NUM>. In embodiments, the signal line of the resonator may have a width of from <NUM> to <NUM>, preferably from <NUM> to <NUM>, more preferably from <NUM> to <NUM>. In embodiments, the resonator (i.e. the signal line and the ground plane) may have a height of from <NUM> to <NUM>. In embodiments, the resonator (i.e. the signal line and the ground plane) may have a length of from <NUM> to <NUM>, preferably from <NUM> to <NUM>. In embodiments, the resonator (i.e. the signal line and the ground planes) may have a rectangular cuboid shape. In embodiments, the shape of the signal line may be irregular. In preferred embodiments, the resonator comprises a rectangular cuboid form.

In embodiments, the substrate may comprise silicon, silicon germanium, or gallium arsenide, preferably silicon. These materials are widely used in semiconductor manufacturing. Also, these materials are etched faster by acid solutions than niobium or tantalum. In embodiments, the substrate consists of a silicon, silicon germanium, or gallium arsenide bulk and optionally an oxide top surface. Preferably, the substrate is crystalline. Crystallinity of the substrate reduces TLS losses because of the relatively low concentration of defects compared to amorphous materials.

The resonator obtained in step a comprises at least one edge. Although the method of the first aspect would be useful even for resonators that do not have edges due to its surface smoothing effect, the method of the first aspect is most useful for modifying resonators having at least one edge due to its edge rounding effect.

When at least one edge is present in the resonator, it is typically formed by the intersection of two adjacent surfaces of an element of the resonator (e.g. two adjacent surfaces of its signal line). Examples of surfaces that can intersect to form the at least one edge are the bottom surface and a lateral surface of the signal line or the top surface and a lateral surface of the signal line. Typically, the surfaces forming the at least one edge are planar.

Although the surfaces forming the at least one edge theoretically meet in a single line defining said edge, in practice, the at least one edge of the resonator obtained in step a may already present some rounding. In other words, the at least one edge obtained in step a may be a curved surface joining both planar surfaces forming the at least one edge.

In embodiments where the at least one edge is a curved surface, this surface may have a degree of curvature that equals the angle between the adjacent surface planes connected by the curved surface.

In embodiments wherein the resonator obtained in step a is composed of rectangular cuboid resonator elements, i.e. a resonator which signal line and ground planes each substantially have the form of a rectangular cuboid, and wherein the edges are curved surfaces, these surfaces have a degree of curvature of <NUM>°.

For instance, a rectangular cuboid resonator element comprising a top surface, a bottom surface, and four lateral surfaces, may have eight edges, each being either defined by the intersection between two surfaces or by a curved surface connecting two surfaces. For instance, it may have a first edge defined either by a line at the intersection between the top surface of the resonator and one of the lateral surfaces, or by a curved surface that connects the top surface of the resonator and one of the lateral surfaces.

The edges of the resonator before step b (e.g. as obtained in step a, a' or a", see below) of the method may in embodiments be sharp: sharp edges may result in radiative loss of a wave resonating in the resonator.

Therefore, according to the present invention, the edges of the resonator are rounded during step b.

As used herein, a sharp edge is either a line at the intersection between two adjacent (e.g. planar) surfaces or a curved surface joining two adjacent planar surfaces, said curved surface having a radius of curvature below <NUM>.

The radius of curvature is a radius of a circle that best fits the curved surface of the edge. In embodiments, the method of the first aspect may introduce a curvature in the at least one edge (when the at least one edge was defined by a line at the intersection of two surfaces) or may increase the radius of curvature of the at least one edge (when the at least one edge was defined by a curved surface at the intersection of two surfaces). In embodiments, the method of the first aspect may reduce the sharpness of the at least one edge and may result in a rounded edge, i.e. an edge where a curvature has been introduced or increased. In embodiments, rounding at least one edge of the resonator comprises rounding at least two edges of the resonator, such as all edges of the signal line or even all edges comprised in the resonator. In embodiments, rounding at least one edge of the resonator comprises increasing the average radius of curvature of the edges of the resonator. The average radius of the curvature of the edges of the resonator is the radius of curvature averaged over 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 radius of curvature of each edge of the resonator, divided by the number of edges. In 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 embodiments, rounding at least one edge of the resonator comprises rounding at least one edge of the signal line by increasing the average radius of curvature of the edges of the signal lines. The average radius of the curvature of the edges of the signal line is the radius of curvature averaged over the edges of the signal line. In other words, the average radius of curvature of the edges of the signal line is defined by the sum of the radius of curvature of each edge of the signal line, divided by the number of edges. In embodiments, the average radius of curvature of the edges of the signal line after step b is larger than the average radius of curvature of the edges of the signal line before step b. In embodiments, the radius of curvature of the at least one edge after step b is from <NUM> to <NUM>, such as from <NUM> to <NUM>, such as from <NUM> times to <NUM> times the radius of curvature of the at least one edge before step b. In embodiments, the average radius of curvature of the edges of the resonator after step b is from <NUM> to <NUM>, such as from <NUM> to <NUM>, such as from <NUM> times to <NUM> times the average radius of curvature of the edges before step b.

In embodiments, the average radius of curvature of the edges of the signal line after step b is from <NUM> to <NUM>, such as from <NUM> to <NUM>, such as from <NUM> times to <NUM> times the average radius of curvature of the edges of the signal line before step b. In embodiments, the radius of curvature of the at least one edge before step b may be below <NUM>, and after step b be from <NUM> to <NUM>. For instance, in embodiments, the average radius of curvature of the edges of the resonator before step b may be below <NUM>, and after step b be from <NUM> to <NUM>.

For instance, in embodiments, the average radius of curvature of the edges of the signal line before step b may be below <NUM>, and after step b be from <NUM> to <NUM>.

In embodiments, rounding at least one edge (e.g. all edges of the signal line or all edges of the resonator) of the resonator yields a low-loss resonator, i.e. a resonator having a quality factor of at least <NUM>,<NUM>,<NUM>.

The resonator and the 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 embodiments, the complete bottom surface of the resonator obtained in step a is in direct contact with the substrate. The interface area is the area of the interface between the bottom surface of the resonator and the top surface of the substrate. In embodiments, the magnitude of the interface area is within <NUM>% of, preferably within <NUM>% of, most preferably equals, the product of the lateral dimensions of the resonator, i.e. the surface area of the bottom surface of the resonator. The interface may comprise a native oxide layer belonging to the substrate. The native oxide of the interface may induce TLS losses. Although the top surface of the substrate may be cleaned, e.g. by using a HF etching solution, before deposition of the resonator on the substrate, thereby substantially removing the native oxide from the top of the substrate, a small amount of the native oxide may still be left after the cleaning. In embodiments, after the formation of the resonator on top of the substrate, the native oxide may still be present at the interface. Because of the extremely low loss required for resonators, e.g. for quantum computing, even TLS losses due to the small amount of the native oxide at the interface may be problematic. Therefore, the interface area is preferably reduced as much as it is mechanically possible during step b. If the interface area becomes too small, however, the resonator may collapse on the substrate. Therefore, a balance is preferably sought between limiting TLS losses of the resonator and preventing the collapse of the resonator on top of the substrate. In embodiments, step b reduces the interface area by <NUM> to <NUM>%, preferably by <NUM> to <NUM>% with respect to the interface area formed in step a. Reducing the interface area comprises removing part of the substrate underneath the resonator.

The liquid acidic etching solution is selected so that it has a higher etch rate towards the substrate than towards the resonator. Therefore, advantageously, no photoresist to prevent the liquid acidic etching solution from substantially etching the resonator is required in this method. Although the etch rate towards the substrate is higher than towards the resonator, the etch rate towards the resonator is nonzero. The nonzero etch rate towards the resonator typically enables in the rounding of the edges. An additional advantage of the nonzero etch rate towards the resonator is that it may improve the surface smoothness of the resonator. In embodiments, the liquid acidic etching solution comprises HF and HNO<NUM>. In embodiments, the liquid acidic etching solution may further comprise a diluent such as H<NUM>O or CH<NUM>COOH, preferably H<NUM>O. In these embodiments, preferably, the liquid acidic etching solution comprises from <NUM> to <NUM>, such as from <NUM> to <NUM>, such as from <NUM> to <NUM> parts by volume of HNO<NUM>, and from <NUM> to <NUM>, such as from <NUM> to <NUM>, preferably from <NUM> to <NUM> parts by volume of HF, and additional parts by volume of the diluent so that the parts by volume of HNO<NUM>, HF, and the diluent add up to <NUM>. In preferred embodiments, the temperature of the etching solution is constant, which may result in a constant, hence controllable, etch rate. In preferred embodiments, the temperature is from <NUM> to <NUM>. These embodiments have the effect that, due to the presence of HNO<NUM>, the resonator is constantly oxidized so that the surfaces of the resonator remain oxidized after step b. There will therefore be an oxide layer present at the surfaces of the resonator after step b. The solution comprising HF and HNO<NUM> has a nonzero etch rate towards the resonator but has typically considerably higher etch rate towards the substrate than towards the resonator. This is true for most substrates and in particular for silicon, silicon germanium, or gallium arsenide. In embodiments, a ratio of the etch rate towards the substrate to the etch rate towards the resonator is at least <NUM>, such as from <NUM> to <NUM>, preferably from <NUM> to <NUM>.

The ratio may be optimized by optimizing the mix ratio of the acids comprised in the liquid acidic etching solution and/or by adding additives. For instance, in embodiments, by increasing the concentration of HF, the etch rate towards both the resonator and the substrate may increase, but the etch rate towards the resonator may increase more than the etch rate towards the substrate, so that the ratio becomes lower. In embodiments, conversely, by reducing the concentration of HF, the ratio may increase. The liquid acidic etching solution typically isotropically etches the substrate, i.e. the liquid acidic etching solution typically etches at the same etch rate in all directions, that is, the etch rate is independent of the etching direction. In embodiments, the etching caused by the liquid acidic etching solution results in a cavity in the substrate having an isotropic profile after application of the solution. For instance, the cavity may have the shape of a truncated sphere.

In 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 may induce TLS losses. Therefore, in embodiments, the method comprises a step c after step b of removing any oxides on the surface of the resonator. In embodiments, in step c, also any oxides on an exposed top surface of the substrate, i.e. at an interface of the substrate and air, may be removed. Oxides on the exposed top surface of the substrate may also induce TLS losses. In embodiments, step c may comprise applying a liquid etching solution comprising HF and no HNO<NUM>. HF is generally applied in the field to remove native oxides on a surface.

In embodiments, the method comprises a step a' after step a and before step b of etching the substrate, selectively with respect to the resonator, by applying a liquid alkaline etching solution, thereby anisotropically etching the substrate and reducing the interface area. This embodiment is particularly useful when the substrate comprises a crystalline structure to be etched. The use of a liquid alkaline etching solution has the advantage that it etches the substrate anisotropically along well-defined crystallographic substrate planes. This allows a good level of control of the portion of the interface removed by the etching process. In other words, the intrinsic directionality of the liquid alkaline etch, allows for the etching of the substrate to be self-aligned to crystallographic substrate planes of the substrate.

Although the liquid acidic etching solution from step b etches, within a material, at a similar etch rate in all directions i.e. the etch rate is independent of the direction, the liquid alkaline etching solution has an etch rate that is anisotropic, i.e. which depends on the etch direction within the substrate, e.g. due to different etch rates towards different crystal planes. For instance, in silicon, the etch rate of the liquid alkaline etching solution towards Si(<NUM>) planes is particularly low. In embodiments, the temperature of the liquid alkaline etching solution may be set above room temperature. Advantageously, in this way, the etch rate also towards the Si(<NUM>) planes may be increased, thereby reducing the required etching time. An increase in the etch rate toward the Si(<NUM>) planes can also be observed when the liquid alkaline etching solution is stirred, e.g. using a mechanical stirrer, during step a'.

In embodiments, the resonator may comprise at least one bent surface.

Since the etch rate of the liquid alkaline etching solution depends on the etch direction within the material, in embodiments wherein the resonator comprises a bent surface, the speed at which the liquid alkaline etching solution etches the substrate at the interface with the resonator may differ locally. That is, reducing a first part of the interface area may in embodiments of step a' occur at a different rate than reducing a second part of the interface area. This difference in etch rate may in some embodiments be advantageously used as a parameter to optimize the etching of the interface. Alternatively, the etching rate may be homogenized for the interface area by adapting the resonator design or the substrate orientation (e.g. by using a Si(<NUM>) substrate).

In embodiments, the profile of the cavity in the substrate after application of the liquid alkaline etching solution of step a' may be anisotropic. For instance, a surface of the etched cavity may be flat along a crystal plane. In embodiments, the liquid alkaline etching of the substrate is self-aligning. This is advantageous because it means that the etching is anisotropic and highly controllable. This is advantageous because it reduces the need for a lithographic mask, as would be the case for instance in the case of anisotropic (dry) plasma etching or reactive-ion etching. In embodiments, the liquid alkaline etching solution comprises a hydroxide, such as one of the following hydroxides: tetramethylammonium hydroxide, ammonium hydroxide, and sodium hydroxide. In embodiments, the liquid alkaline etching solution etches the substrate selectively with respect to the resonator. In embodiments, the liquid alkaline etching solution substantially does not etch the resonator. Combining the liquid acidic etching solution and the liquid alkaline etching solution has the advantage to provide a further parameter to optimize the profile of the resonator and/or the substrate. Indeed, in embodiments, the profile of the substrate may be first etched in step a' to assume the anisotropic profile, which is then further etched isotropically in step b. The resulting profile of the substrate may in embodiments be a combination of the anisotropic profile and the isotropic profile. Combining the liquid acidic etching solution and the liquid alkaline etching solution may in embodiments provide a further parameter to optimize the interface area, such as the magnitude of the interface area. Indeed, although the nonzero etch rate of the liquid acidic etching solution towards the resonator is considered an advantage, in some embodiments, it may be difficult to sufficiently reduce the interface area without at the same time etching too much of the resonator i.e. when only the liquid acidic etching solution is used. Etching too much of the resonator may result e.g. in a shift of the oscillation frequency of the resonator. In embodiments, step a' reduces the interface area by <NUM> to <NUM>%, preferably <NUM> to <NUM>% with respect to the interface area formed in step a.

In embodiments according to the first aspect, instead of the step a', the method may comprise a step a" after step a and before step b of etching the substrate, selectively with respect to the resonator, by applying a dry isotropic etch, thereby reducing the interface area. In embodiments, the dry isotropic etch may be an Inductively Coupled Plasma Etching (ICPE) technique, wherein the dry isotropic etch comprises NF<NUM> and H<NUM> gas. ICPE with NF<NUM> and H<NUM> gas does not etch various metals, including niobium and tantalum, i.e. it etches the substrate selectively with respect to the resonator. In embodiments, the dry isotropic etch substantially does not etch the resonator. Combining the liquid acidic etching solution and the dry isotropic etch may, in embodiments, provide a further parameter to optimize the interface area, such as the magnitude of the interface area. Indeed, although the nonzero etch rate of the liquid acidic etching solution towards the resonator is considered an advantage, in some embodiments, it may be difficult to sufficiently reduce the interface area without at the same time etching too much of the resonator i.e. when only the liquid acidic etching solution is used. In embodiments, step a" reduces the interface area by <NUM> to <NUM>%, preferably <NUM> to <NUM>% with respect to the interface area formed in step a.

The modified resonator formed by embodiments of the method of the present invention has typically the same composition and dimensions as the resonator obtained in step a, but has rounded edges. In embodiments, the average radius of curvature of the modified resonator is from <NUM> to <NUM>. In embodiments, the interface between the resonator and the substrate is of the same material as the interface between the modified resonator and the substrate but the magnitude of the interface area is different. In embodiments, the interface area may be from <NUM>% to <NUM>%, preferably from <NUM>% to <NUM>% of the area of the bottom surface of the modified resonator. In embodiments, the resonator may comprise or consists of one of the following materials: niobium, tantalum, NbN, and NbTiN, preferably niobium or tantalum. In embodiments, the resonator consists of niobium. In embodiments, the dimensions, i.e. each of the thickness and the two lateral dimensions, of the modified resonator may be the same or may be within <NUM>% of the dimensions of the resonator obtained in step a. In embodiments, the surface of the modified resonator may comprise substantially no oxides such as native oxides. In embodiments, the profile of the substrate underneath the modified resonator may comprise one of the following profiles: the isotropic profile, the anisotropic profile, and a combination of the isotropic profile and the anisotropic profile.

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 edges having a radius of curvature of from <NUM> to <NUM>, wherein a bottom surface of the resonator has an area larger than an interface area between the bottom of the resonator and a top surface of the substrate.

The present invention also relates to a (modified) resonator obtainable by any embodiment of the first aspect of the present invention.

Any feature of the resonator and the substrate of the second aspect may be as defined for the modified resonator and the substrate in any embodiment of the first aspect of the present invention.

In other words, the resonator and the substrate according to the second aspect may have the same features as the modified resonator and the substrate according to the device formed by any embodiment of the method of the first aspect. The features of the resonator and the substrate according to the second aspect may have the same advantages as features of the resonator and the substrate formed by the method of the first aspect.

Herebelow are some examples of features that the resonator of the second aspect may have. Other examples are available in the description of the modified resonator of the first aspect.

In embodiments, the average radius of curvature of the rounded edges of the resonator may be from <NUM> to <NUM>. Such a large average radius of curvature may significantly reduce the radiative losses of, in particular, a wave propagating on the resonator surface.

In embodiments, the interface area between the bottom of the resonator and a top surface of the substrate may be from <NUM>% to <NUM>%, preferably <NUM>% to <NUM>% of the area of the bottom surface of the resonator. In embodiments, the resonator may consist of niobium, and the substrate may comprise or consist of silicon, silicon germanium, or gallium arsenide, preferably silicon.

Reference is made to <FIG> that show a vertical cross-section of a first <NUM> and a second rectangular cuboid resonator <NUM>. Each of the rectangular cuboid resonators has a bottom surface <NUM> and <NUM>, a top surface <NUM> and <NUM> and a lateral surface <NUM> and <NUM>. A first edge <NUM> and <NUM> comprising an edge line connects a first surface plane that comprises the bottom surface <NUM> and <NUM> with a second surface plane that comprises the lateral surface <NUM> and <NUM>. The first and second surface plane are adjacent planes.

Furthermore, in both resonators, a second edge <NUM> and <NUM> comprises an edge surface that connects a third surface plane comprising the top surface <NUM> and <NUM> with the second surface plane, wherein the second and third surface planes are adjacent surface planes. The angle between the second and third surface planes in both cases equals <NUM>°. In this resonator, the degree of curvature of the second edge equals <NUM>°. A circle that best fits the surface of the second edge <NUM> and <NUM> is shown in each case. The radius of the circle, e.g. R in <FIG>, equals a radius of curvature of the second edge <NUM> and <NUM>. The radius of curvature of the second edge <NUM> of the first resonator <NUM> is larger than the radius of curvature of the second edge <NUM> of the second resonator <NUM>. That is, the second edge <NUM> of the first resonator <NUM> is less sharp and more rounded than the second edge <NUM> of the second resonator <NUM>. For instance, applying a liquid acidic etching solution of step b of the method of the present invention to second resonator <NUM> may in embodiments yield the first resonator <NUM>.

Reference is made to <FIG>, which is a schematic vertical cross-section of a resonator <NUM> on top of a substrate <NUM>: three resonators <NUM> are shown on top of the substrate <NUM>. <FIG> shows the resonator <NUM> on top of the substrate <NUM> after step a of embodiments of the method of the first aspect of the present invention, i.e. before the application of any etch. The resonator <NUM> in this example has the form of a rectangular cuboid. An interface area <NUM> exists between the top surface of the substrate <NUM> and the bottom surface of the resonator <NUM>. The interface area <NUM> may comprise oxides e.g. from the substrate, wherein the oxides may be a first source of TLS losses for any oscillating field in the resonator <NUM>. Furthermore, the top surface of the substrate may comprise oxides <NUM>, which may be a second source of TLS losses for any oscillating field in the resonator <NUM>. Also, the surface of the resonator <NUM> may comprise oxides <NUM>, which may be a third source of TLS losses for any oscillating field in the resonator <NUM>. An aim of embodiments of the method of the present invention is to reduce the three sources of TLS losses. Reference is now made to <FIG>, which shows an SEM image of the resonator <NUM> on top of the substrate <NUM>.

As can be observed, the edges <NUM> of the resonator are still substantially sharp i.e. they have a small radius of curvature.

Reference is now made to <FIG>. A dry isotropic etch is applied to the resonator <NUM> and the substrate <NUM>, wherein the dry isotropic etch selectively etches the substrate <NUM>, selectively with respect to the resonator <NUM>, thereby reducing the interface area <NUM>, i.e. an embodiment of step a" of the method of the present invention. Due to the high selectivity of the etch, the edges of the resonator <NUM> are not rounded. In this example, in the case of the middle of the three resonators <NUM>, the interface area <NUM> after step a" has been reduced by approximately one-third, i.e. <NUM>%, compared to the interface area <NUM> after step a. In addition, the surface of the substrate <NUM> has assumed an isotropic profile, i.e. forming part of a sphere. This is a result of the dry isotropic etch having equal etch rates in all directions of the substrate <NUM>. Reference is now made to <FIG>, showing a SEM image of the resonator <NUM> on the substrate after the application of an isotropic etch. In this particular example, the interface area <NUM> after the isotropic etch has been reduced by approximately <NUM>% compared to the interface area <NUM> after step a.

Reference is now made to <FIG>. Instead of the isotropic dry etch of <FIG>, instead, a liquid alkaline etching solution i.e. according to embodiments of step a' is applied to the resonator <NUM> and substrate <NUM> obtained after step a. The liquid alkaline etching solution results in a highly anisotropic profile <NUM> of the substrate <NUM> i.e. substantially flat along a crystalline plane of the substrate <NUM>. In this example, the liquid alkaline etching solution has reduced the interface area <NUM> with approximately two-third i.e. <NUM>% compared to the interface area after step a. Reference is now made to <FIG>, which shows a SEM image of the resonator <NUM> on top of the substrate <NUM> after step a'.

Reference is now made to <FIG>. After step a', i.e. on the resonator on top of the substrate shown in <FIG>, a liquid acidic etching solution i.e. according to embodiments of step b of the method of the present invention, is applied. This further reduces the interface area <NUM>, i.e. to approximately <NUM>% of the surface area after step a, i.e. corresponding to a reduction of the interface area <NUM> with <NUM>% compared to the interface area <NUM> after step a. The profile of the substrate <NUM> is a combination of an isotropic profile <NUM> and an anisotropic profile <NUM>. The liquid acidic etching solution has a nonzero etch rate towards the resonator <NUM>, so that the resonator now has rounded edges <NUM>, i.e. the edges <NUM> have a radius of curvature that is considerably larger than the radius of curvature of the resonator after step a. Also, any oxides on the surface of the resonator and on the surface of the substrate <NUM>, possibly formed on the application of step b, have been removed, e.g. with a HF etch, according to embodiments of step c of the method of the present invention. <FIG> hence shows an embodiment of a resonator <NUM> according to embodiments of the present invention. <FIG> is an SEM image of a resonator <NUM> on top of the substrate <NUM> according to embodiments of the present invention, and formed in embodiments of the method of the present invention.

Claim 1:
A method for forming a modified resonator (<NUM>), the method comprising:
a. obtaining a resonator (<NUM>) on top of a substrate (<NUM>), the resonator (<NUM>) having at least one edge (<NUM>) and comprising niobium or tantalum, thereby forming an interface area (<NUM>) between a bottom surface of the resonator (<NUM>) and a top surface of the substrate (<NUM>), characterised in that the method further comprises a step b of
b. contacting the resonator (<NUM>) and the substrate (<NUM>) with a liquid acidic etching solution selected so as to have a higher etch rate towards the substrate (<NUM>) than towards the resonator (<NUM>) and a nonzero etch rate towards the resonator (<NUM>), thereby rounding the edge (<NUM>).