Compact superconducting resonator based on space filling curve, preparation method, design method and application

Abstract

The invention discloses a compact superconducting resonator based on a space filling curve, a preparation method, a design method and application, and the resonator comprises a space filling curve resonator main body which is composed of a superconducting thin film, and the space filling curve continuously traverses a limited plane area in a non-crossed manner; the coplanar waveguide transmission line comprises a central conductor belt and ground planes positioned on two sides of the central conductor belt; and the space filling curve resonator main body and the coplanar waveguide transmission line are coupled through a capacitor or an inductor so as to realize signal reading. The compact superconducting resonator can be widely applied to quantum bit reading, microwave filters and low-temperature quantum microwave devices, and has the advantages of high integration level, low loss and good expandability.

Description

Technical Field

[0001] This invention relates to the field of superconducting quantum chip integration technology, and in particular to a compact superconducting resonator based on a space-filling curve, its fabrication method, design method, and applications. Background Technology

[0002] Superconducting quantum chips are fundamental components for realizing quantum computing and quantum information processing. Their core circuits typically include superconducting qubits, readout resonators, coupling lines, and control wiring. In superconducting circuits, microwave resonators serve as the elements for qubit readout and quantum state measurement, and their performance affects readout fidelity and quantum coherence time.

[0003] Existing superconducting resonators mostly employ lumped LC parameter structures or distributed parameter structures, such as quarter-wavelength resonators (λ / 4 resonators), half-wavelength resonators (λ / 2 resonators), and their variations. These structures are typically implemented using coplanar waveguides (CPWs), with the resonant frequency determined by the equivalent electrical length of the transmission line. At typical operating frequencies (approximately 4-8 GHz), the physical size of such resonators typically reaches hundreds of micrometers to several millimeters, making them among the most space-consuming components on a chip. This large space occupation severely limits the integration density and multi-qubit expansion capabilities of quantum chips.

[0004] Therefore, without altering the resonator's operating frequency band and electrical performance, achieving a longer equivalent electrical length within a limited chip area, thereby reducing the resonator's footprint, has become a critical issue that urgently needs to be addressed in superconducting quantum circuit design. Improving the space utilization of the resonator circuitry within a limited area from a geometric perspective has become one of the important ways to achieve resonator miniaturization.

[0005] Space-filling curves are geometric paths that can continuously and non-intersectingly traverse the entire plane within a finite two-dimensional region, exhibiting high space utilization and scalable iterative characteristics. Typical examples include Hilbert curves and Peano curves. Superconducting resonators built based on space-filling curves can significantly reduce the footprint while maintaining the equivalent electrical length or without changing the resonant frequency, thereby achieving compact layout and high-density integration.

[0006] On the other hand, with the increasing complexity of quantum circuits, relying solely on single-chip planar wiring is no longer sufficient to meet the requirements of multi-node interconnection and electromagnetic isolation. Flip-chip structures, as a three-dimensional integration method, can achieve vertical interconnection through metal pillars (such as indium pillars) between the upper and lower chips, thereby shortening signal paths, suppressing parasitic modes, and improving circuit stability. However, existing flip-chip structures typically suffer from complex manufacturing processes, high alignment precision, and the formation of non-superconducting phases at the interface, leading to uneven grounding and additional microwave losses at low temperatures.

[0007] Therefore, there is an urgent need to propose a compact superconducting resonator structure and its fabrication method based on space-filling curves. This structure can achieve long electrical length resonance characteristics within a limited area, and can be configured using two-dimensional planar grounding or three-dimensional flip-chip grounding. High space utilization can be achieved through controllable geometric iteration of the space-filling curve, and the device area can be further reduced while maintaining the resonant frequency. This invention proposes an improved solution to address the above problems by applying space-filling curves (such as Hilbert curves or Peano curves) to the design of two-dimensional and three-dimensional integrated superconducting resonators, forming a multilayer superconducting microwave device architecture that combines miniaturization, high-reliability grounding, and flexible coupling characteristics. Summary of the Invention

[0008] The purpose of this invention is to propose a compact superconducting resonator based on a space-filling curve, its fabrication method, design method, and applications, in order to solve the problems of large size, limited integration, complex three-dimensional interconnection, and uneven grounding of existing superconducting resonators.

[0009] The technical solution for achieving the objective of this invention is: a compact superconducting resonator based on a space-filling curve, comprising:

[0010] The main body of the space-filled curve resonator is composed of a superconducting thin film, wherein the space-filled curve traverses the defined planar region continuously and without intersection.

[0011] And a coplanar waveguide transmission line, the coplanar waveguide transmission line including a central conductor strip and ground planes located on both sides of the central conductor strip;

[0012] The space-filling curve resonator body and the coplanar waveguide transmission line are coupled capacitively or inductively to achieve signal readout.

[0013] Furthermore, the space-filling curve is a Hilbert curve or a Peano curve; the material of the superconducting thin film is selected from at least one of tantalum, niobium, niobium nitride, titanium niobium nitride, titanium nitride, or aluminum.

[0014] Furthermore, the resonator is a two-dimensional planar structure, comprising: a substrate (101), a space-filled curve resonator body (102) disposed on the substrate (101), and a coplanar waveguide transmission line (103) coupled thereto; the space-filled curve resonator body (102) is an open-circuit structure at both ends, and the ground plane of the coplanar waveguide transmission line (103) is disposed around the space-filled curve resonator body (102) and does not enter the traversal region where the body is located.

[0015] Furthermore, the resonator is a three-dimensional flip-chip structure, comprising an upper chip and a lower chip;

[0016] The lower chip includes a lower substrate (210), a coplanar waveguide transmission line (211) disposed on the lower substrate (210), and a plurality of metal pillars (212) formed on the ground planes on both sides of the coplanar waveguide transmission line (211).

[0017] The upper chip includes an upper substrate (220), a space-filled curve resonator body (221) disposed on the upper substrate (220), and a bridging superconducting thin film (222).

[0018] The metal pillar (212) is bonded to the bridging superconducting film (222), so that the bridging superconducting film (222) and the metal pillar (212) together form a superconducting air bridge that spans the central conductor strip of the coplanar waveguide transmission line (211), thereby realizing a three-dimensional superconducting ground interconnection between the upper chip and the lower chip.

[0019] The space-filled curve resonator body (221) is coupled to the coplanar waveguide transmission line (211) through inter-chip capacitance or inter-chip inductance.

[0020] Furthermore, the metal pillar (212) is an indium pillar with a diameter of 5 micrometers to 600 micrometers and a height of 4 micrometers to 15 micrometers; the bridging superconducting thin film (222) is rectangular or square in shape.

[0021] Furthermore, in the vertical projection overlap region of the space-filled curve resonator body (221) and the coplanar waveguide transmission line (211), the width of the central conductor strip of the coplanar waveguide transmission line (211) and its distance from the ground plane are not less than twice the corresponding size of the non-overlapping region.

[0022] A method for fabricating a compact superconducting resonator includes the following steps:

[0023] The space-filled curve resonator body (102) and coplanar waveguide transmission line (103) are patterned on the superconducting thin film by etching or stripping processes.

[0024] The etching process includes: first depositing a superconducting thin film, and then forming a pattern through photolithography, development, and dry or wet etching; the lift-off process includes: first forming a pattern through photolithography, then depositing a superconducting thin film and lifting it off to obtain the pattern.

[0025] A method for fabricating a compact superconducting resonator includes the following steps:

[0026] A space-filled curve resonator body (221) and a bridging superconducting thin film (222) are fabricated on the upper substrate (220).

[0027] Coplanar waveguide transmission lines (211) and metal pillars (212) are fabricated on the lower substrate (210).

[0028] The upper and lower chips are flip-chip aligned and pressed together to connect the metal pillar (212) with the bridging superconducting thin film (222) to form a three-dimensional superconducting grounding structure.

[0029] A method for designing the resonant frequency of a compact superconducting resonator includes the following steps:

[0030] The design parameters of the space-filling curve resonator body are determined, and the design parameters include at least: the iteration order of the space-filling curve, the conductor width, the ground spacing, and the side length of the area occupied by the resonator body;

[0031] The design requirements for the target resonant frequency can be met by adjusting at least one of the following parameters: iteration order, conductor width, ground spacing, and side length.

[0032] Applications of a compact superconducting resonator based on a space-filling curve for use in superconducting quantum bit readout resonators, superconducting parametric amplifiers, or low-temperature quantum microwave devices.

[0033] Compared with the prior art, the significant advantages of this invention are:

[0034] (1) By introducing space-filling fractal curves (including Hilbert curves, Peano curves, and combinations thereof) into the resonator body design, the high space-filling rate of the fractal iteration curves within a limited region can be utilized to significantly reduce the size of the resonator body while maintaining the same resonant frequency. By adjusting the iteration order, linewidth, and spacing of the curves, a longer equivalent electrical length can be obtained in a smaller area, thereby achieving miniaturization and high integration of the resonator body.

[0035] (2) The proposed compact superconducting resonator can flexibly adopt a two-dimensional planar structure or a three-dimensional flip-chip structure according to the application scenario: In the two-dimensional structure, the space-filled curve resonator body and the coplanar waveguide transmission line are arranged in the same plane. The space-filled curve resonator body is open at both ends, which is similar to a half-wavelength resonator in structure. One end of the space-filled curve resonator body can be connected to a quantum bit node, which is suitable for planar circuit integration; In the three-dimensional flip-chip structure, the upper chip is set with the space-filled curve resonator body and the bridging superconducting thin film, and the lower chip is set with the coplanar waveguide transmission line and the metal pillar. The metal pillar is connected to the bridging superconducting thin film to form a three-dimensional grounding and electrical interconnection structure between the upper chip and the lower chip. This structure can improve the ground potential balance, thereby effectively suppressing parasitic wave modes. The space-filled curve resonator body on the upper substrate and the coplanar waveguide transmission line on the lower substrate are connected by inter-chip capacitive coupling or inter-chip inductive coupling to achieve signal readout, which can further improve the spatial integration.

[0036] (3) A stable superconducting air bridge structure is formed by designing a bridging superconducting thin film on the upper chip and using small-sized indium pillars on the lower chip to connect the bridging superconducting thin film and cross the center conductor strip of the coplanar waveguide transmission line to connect the ground planes on both sides. This structure realizes superconducting connectivity between the ground planes of the upper and lower chips, effectively suppresses parasitic wave modes that are easily excited in flip-chip interconnects, improves ground potential balance, and enhances the stability and consistency of the entire microwave circuit.

[0037] (4) In the process of flip-chip bonding, in addition to achieving precise alignment through the flip alignment marks, the small indium pillars of the lower chip and the bridging superconducting film of the upper chip are also designed to maintain a strict correspondence, which can provide an auxiliary alignment effect during the bonding process, thereby further improving the alignment accuracy and bonding stability of the upper and lower chips.

[0038] (5) Metal pillars are fabricated only on the lower chip, and no metal pillars are set on the upper chip. This greatly reduces the impact of the metal pillar process on the performance of the upper space-filling curve resonator. In addition, most of the upper chip is a substrate area except for the space-filling curve resonator and the bridging superconducting film. This structure avoids the problem of forming a non-superconducting alloy layer at the interface between the aluminum and other metal oxide layers and the indium pillars, and ensures the interface reliability and superconducting continuity of the flip-chip connection.

[0039] (6) The proposed compact superconducting resonator is compatible with a variety of superconducting material systems, including Ta, Nb, NbTiN, NbN, TiN, Al, etc. Different material systems can be selected according to application requirements to achieve higher integration and better performance of quantum circuit design. Attached Figure Description

[0040] Figure 1The diagram shows a schematic diagram and a magnified view of a partial area of ​​the two-dimensional planar grounding structure (a) and the three-dimensional inverted grounding structure (b) provided in the embodiments of the present invention.

[0041] Figure 2 Examples of space-filling curves involved in this invention are Hilbert curves of orders 1 to 6.

[0042] Figure 3 Examples of space-filling curves involved in this invention are Peano curves of orders 1 to 4 (generated using the L-system);

[0043] Figure 4 This is a process flow diagram of the fabrication process of the two-dimensional planar structure and the three-dimensional flip-chip structure superconducting resonator in the embodiments of the present invention;

[0044] Figure 5 S is the two-dimensional planar resonator with resonant frequency fr = 5.7158 GHz in Example 1. 21 Parameter test results graph;

[0045] Figure 6 The graph shows the test results of the intrinsic quality factor of the resonator with resonant frequency fr = 5.7158 GHz in Example 1 as a function of the average number of photons in the cavity <n>.

[0046] Figure 7 The S of the three-dimensional flip-chip resonator with resonant frequency fr = 4.8583 GHz in Example 2 is... 21 Parameter test results graph;

[0047] Figure 8 The graph shows the test results of the intrinsic quality factor of the resonator with resonant frequency fr = 4.8583 GHz in Example 2 as a function of the average number of photons in the cavity <n>.

[0048] Figure 9 The resonant frequency f in Example 3 r The graph shows a monotonically decreasing inverse proportional relationship between the length of the graph and the side length l1. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0050] This invention provides a compact superconducting resonator based on a space-filling curve. Its core lies in introducing a space-filling geometric path to significantly reduce the size of the resonator body. At the same time, it flexibly adopts a two-dimensional planar structure or a three-dimensional flip-chip grounding structure according to the circuit integration requirements, thereby balancing compact design with the overall performance of superconducting quantum chips.

[0051] I. Resonator Structure

[0052] like Figure 1 As shown, a compact superconducting resonator based on a space-filling curve is disclosed. The compact superconducting resonator includes a space-filling curve resonator body made of a superconducting thin film and a coplanar waveguide transmission line. The space-filling curve resonator body is composed of a space-filling curve path, which continuously and without intersection traverses a defined planar region. The coplanar waveguide transmission line includes a central conductor strip and ground planes located on both sides of the central conductor strip. The space-filling curve resonator body and the coplanar waveguide transmission line are capacitively or inductively coupled to achieve signal readout.

[0053] Preferably, the space-filling curve is a Hilbert curve, a Peano curve, or other space-filling curve to achieve adjustable iteration order and area compression. Preferably, the superconducting thin film material is Ta, Nb, NbN, NbTiN, TiN, or Al to balance low surface loss and high superconducting transition temperature characteristics.

[0054] Furthermore, the compact superconducting resonator can take either a planar structure or a three-dimensional flip-chip structure, as detailed below:

[0055] (1) Two-dimensional planar structure

[0056] like Figure 1 As shown in (a), a compact superconducting resonator with a two-dimensional planar structure includes: a substrate (101), a space-filled curve resonator (102) fabricated on the substrate, and a coplanar waveguide transmission line (103) coupled thereto. The space-filled curve resonator body (102) is open at both ends, structurally similar to a half-wavelength resonator, and one end of the space-filled curve resonator body can be used to connect superconducting qubits. The ground plane is distributed only around the space-filled curve resonator body and does not enter its interior.

[0057] (2) Three-dimensional inverted structure

[0058] like Figure 1As shown in (b), the compact superconducting resonator with a three-dimensional flip-chip structure includes an upper chip and a lower chip. The lower chip includes a lower substrate (210), a coplanar waveguide transmission line (211) disposed on the lower substrate, and a metal pillar (212). The upper chip includes an upper substrate (220), a space-filled curve resonator body (221) disposed on the upper substrate, and a bridging superconducting thin film (222). The upper chip and the lower chip are three-dimensionally grounded and electrically connected through the metal pillar and the bridging superconducting thin film to suppress parasitic modes.

[0059] Furthermore, the metal pillar is preferably an indium pillar structure with a diameter ranging from 5 to 600 μm and a height ranging from 4 to 15 μm, while also serving as a support and spacing control function.

[0060] Furthermore, when a three-dimensional flip-chip structure is adopted, in the vertical projection overlap area of ​​the space-filled curve resonator body (221) and the coplanar waveguide transmission line (211), the width of the central conductor strip of the coplanar waveguide transmission line and its distance from the ground plane are increased relative to the non-overlapping area, and are respectively not less than twice the corresponding size of the non-overlapping area, so as to enhance the inter-chip capacitive coupling.

[0061] II. Preparation Method

[0062] (1) Two-dimensional planar structure

[0063] A method for fabricating a compact superconducting resonator based on a two-dimensional planar structure and a space-filling curve includes the following steps:

[0064] S1. Using an etching process or a lift-off process, a patterned structure of a space-filled curve resonator body (102) and a coplanar waveguide transmission line (103) is formed on a superconducting thin film. The superconducting thin film can be deposited by magnetron sputtering or electron beam evaporation, and the preferred thickness range is 5-300 nm. The etching process involves first depositing a superconducting thin film on a substrate (101), and then obtaining the structure of the space-filled curve resonator body (102) and the coplanar waveguide transmission line (103) through homogenization, photolithography, development, and etching. The lift-off process involves first homogenizing and photolithography to form a pattern on a substrate (101), then depositing a superconducting thin film and lifting it to obtain the structure of the space-filled curve resonator body (102) and the coplanar waveguide transmission line (103).

[0065] If an etching process is used, the preferred process flow is as follows: first, a superconducting thin film is deposited on the substrate (101), and then the space-filling curve resonator body (102) and coplanar waveguide transmission line (103) are formed by homogenization, photolithography, development and etching. The etching method can be reactive ion etching (RIE), inductively coupled plasma etching (ICP) or wet etching. The photoresist can be a single-layer photoresist structure, preferably using ultraviolet positive photoresist such as S1813 series, such as S1813 and AZ1500 photoresist.

[0066] If a stripping process is used, the preferred process flow is as follows: first, perform homogenization, photolithography, and development to form a pattern, then deposit a superconducting thin film and strip it to obtain the space-filled curve resonator body (102) and the coplanar waveguide transmission line (103); the photoresist adopts a double-layer photoresist structure, the bottom layer (first layer) photoresist close to the substrate (101) is LOR series photoresist or other photoresist special for stripping process, and the top layer (second layer) photoresist above the bottom layer (first layer) photoresist is S series or AZ series positive photoresist, and the double-layer photoresist structure is, for example, LOR5A + S1813 or LOR10B + S1805.

[0067] Since the thickness of the bottom sacrificial layer photoresist is greater than the thickness of the superconducting film to be deposited, it is used to form a sufficient undercut structure to ensure the reliability of the stripping. The thickness of the top positive photoresist is used to meet the requirements of pattern resolution and exposure contrast. When the line width and spacing are small, a thinner top photoresist is preferred, and when the line width and spacing are large, a thicker top photoresist can be used.

[0068] Preferably, in the above stripping process, MF319 developer is selected, and low-temperature static development is adopted. The preferred development temperature is 10-15℃, so as to obtain a space-filling curve resonator structure with clear edges and uniform line width.

[0069] S2. A compact superconducting resonator with a two-dimensional planar structure is obtained by etching or stripping, wherein the space-filled curve resonator body (102) and the coplanar waveguide transmission line (103) are capacitively or inductively coupled to achieve signal readout.

[0070] After the compact superconducting resonator is fabricated, further surface chemical cleaning can be performed to remove photolithography residues and improve interface cleanliness. Cleaning methods may include organic solvent cleaning, piranha solution cleaning, or buffered oxide etchant (BOE) cleaning, to ensure superconducting surface quality and reduce dielectric loss.

[0071] (2) Three-dimensional inverted structure

[0072] This invention also provides a method for fabricating a compact superconducting resonator with a three-dimensional flip-chip structure, comprising the following steps:

[0073] S1'. A space-filled curve resonator body (221) and a bridging superconducting thin film (222) are fabricated on an upper substrate (220). The fabrication process can be carried out by etching or lift-off. The etching process involves first depositing a superconducting thin film, and then obtaining the space-filled curve resonator body (221) and the bridging superconducting thin film (222) structure through photolithography, development and etching. The lift-off process involves first forming a pattern through photolithography, then depositing a superconducting thin film and then lifting it to obtain the space-filled curve resonator body (221) and the bridging superconducting thin film (222) structure.

[0074] Preferably, in step S1', the etching or stripping process used in the fabrication of the upper layer chip of the three-dimensional flip structure is the same as that used in the two-dimensional structure.

[0075] If an etching process is used, the preferred process flow is as follows: first, a superconducting thin film is deposited on the upper substrate (220), and then the space-filling curve resonator body (221) and the bridging superconducting thin film (222) are formed by photolithography, development and etching. The etching method can be reactive ion etching (RIE), inductively coupled plasma etching (ICP) or wet etching. The photoresist can adopt a single-layer adhesive structure, and it is preferred to use ultraviolet positive photoresist such as S1813 series, such as S1813 and AZ1500 photoresist.

[0076] If a lift-off process is used, the preferred process flow is as follows: first, photolithography and development are performed to form a pattern, then a superconducting thin film is deposited and lifted off to obtain the space-filling curve resonator body (221) and the bridging superconducting thin film (222); the photoresist adopts a double-layer photoresist structure, the bottom layer (first layer) photoresist near the upper substrate (220) is LOR series photoresist or other photoresist specifically for lift-off processes, and the top layer (second layer) photoresist above the bottom layer (first layer) photoresist is S series or AZ series positive photoresist, and the double-layer photoresist structure is, for example, LOR5A + S1813 or LOR10B + S1805. Since the thickness of the bottom sacrificial layer photoresist is greater than the thickness of the superconducting thin film to be deposited, it is used to form a sufficient undercut structure to ensure the reliability of lift-off; the thickness of the top layer positive photoresist is used to meet the requirements of pattern resolution and exposure contrast. When the line width and spacing are small, a thinner top layer photoresist is preferred, and when the line width and spacing are large, a thicker top layer photoresist can be used.

[0077] Preferably, in the above stripping process, MF319 developer is selected, and low-temperature static development is adopted. The preferred development temperature is 10-15℃, so as to obtain a space-filling curve resonator structure with clear edges and uniform line width.

[0078] Furthermore, after the upper-layer chip is fabricated, protective adhesive can be spin-coated and the wafer dicing can be performed to obtain multiple upper-layer chips, which are then chemically cleaned to remove surface contaminants. Cleaning processes can include organic solvent cleaning, piranha solution cleaning, or BOE cleaning, to ensure interface cleanliness.

[0079] S2'. A coplanar waveguide transmission line (211) and a metal pillar (212) are fabricated on a lower substrate (210). The metal pillar is made of indium and is formed by an evaporation process.

[0080] Preferably, in step S2', the fabrication process of the lower-layer chip includes:

[0081] S2'-1. Deposit a superconducting thin film on the lower substrate (210);

[0082] S2'-2. Coplanar waveguide transmission lines are formed by photolithography and etching (211).

[0083] S2'-3. The metal pillar (212) pattern is formed again by photolithography;

[0084] S2'-4. Indium is deposited by evaporation or electroplating, with a thickness ranging from 4 to 15 μm, and a metal pillar structure is obtained by a stripping process.

[0085] Preferably, the metal pillar (212) is made of indium and has a diameter ranging from 5 to 600 μm. It is used to control the spacing between the upper and lower chips and provide mechanical support. At the same time, it connects the bridging superconducting thin film (222) and crosses the central conductor strip of the coplanar waveguide transmission line (211) to connect the ground planes on both sides, thereby achieving ground potential balance and parasitic mode suppression.

[0086] S3'. The upper and lower chips are flip-chip aligned so that the metal pillar (212) and the bridging superconducting thin film (222) correspond precisely, and a superconducting interconnect and three-dimensional grounding structure between the upper and lower chips is formed by cold welding.

[0087] Preferably, in step S3', the upper and lower chips are precisely aligned using flip-chip alignment marks, which can be designed with three collinear alignment marks. After alignment, the chips are pressed together by cold soldering in a nitrogen or vacuum environment to achieve electrical and mechanical connection between them. Preferably, the cold soldering process is performed at room temperature or slightly above room temperature, with a soldering pressure range of 10-200 N to ensure the formation of a continuous and stable indium-superconducting thin film interface and maintain good superconducting contact characteristics.

[0088] III. Resonant Frequency Design Method

[0089] This invention also provides a method for designing the resonant frequency of a compact superconducting resonator based on a space-filling curve; by adjusting four design parameters—the iteration order, conductor width, ground spacing, and side length—of the space-filling curve resonator body, the design requirements for the target resonant frequency can be met.

[0090] (1) Two-dimensional planar structure

[0091] like Figure 1 As shown in (a), the design parameters of the compact superconducting resonator with a two-dimensional planar structure include: iteration order n1, conductor width w1, ground spacing d1, and side length l1; where:

[0092] The iteration order n1 refers to the iteration order of the selected space-filling curve.

[0093] The conductor width w1 refers to the width of the conductor in the body (102) of the space-filled curve resonator;

[0094] The ground spacing d1 refers to the distance between the edge of the space-filled curve resonator body (102) and the ground plane.

[0095] The side length l1 refers to the side length of the body (102) of the space-filled curve resonator.

[0096] (2) Three-dimensional inverted structure

[0097] like Figure 1 As shown in (b), the design parameters of the compact superconducting resonator with a three-dimensional flip-chip structure include: iteration order n2, conductor width w2, ground spacing d2, and side length l2;

[0098] The iteration order n2 refers to the iteration order of the space-filling curve used.

[0099] The conductor width w2 refers to the width of the conductor in the body (221) of the space-filled curve resonator;

[0100] The ground spacing d2 refers to the vertical distance between the space-filled curve resonator body (221) of the upper chip and the ground plane of the lower chip, and also refers to the vertical distance between the space-filled curve resonator body (221) of the upper chip and the coplanar waveguide transmission line (211) of the lower chip.

[0101] The side length l2 refers to the side length of the body (221) of the space-filled curve resonator.

[0102] (3) General design method

[0103] The design method includes the following steps:

[0104] Step 1: Draw the space-filling curve of the resonator body

[0105] By setting different design parameters, fractal algorithm scripts can be written using automated layout generation tools (such as the gdspy package in the Python environment) to draw mask files (such as GDSII format, DXF format, etc.) of the space-filling curve resonator body. To improve design efficiency, any three design parameters can be set as constants, and the remaining design parameter can be set as a variable. For example, the values ​​of the iteration order, conductor width, and ground spacing can be set as constants, and the side length can be set as a variable to draw layouts of space-filling curve resonators of different sizes.

[0106] Step 2: Verify the resonant frequency f using electromagnetic simulation r

[0107] Import the generated layout file (e.g., DXF format) into finite element analysis software (e.g., COMSOL Multiphysics) to build an electromagnetic simulation model and obtain the resonant frequency f corresponding to different side lengths. r ;

[0108] Step 3: Calculate the fitting formula between the variables in the design parameters and the resonant frequency.

[0109] Based on multiple sets of data obtained from simulations, and under the premise of fixed iteration order, conductor width, and ground spacing, an empirical formula for the resonant frequency as a function of side length is derived through numerical analysis. Designers can directly calculate the required physical side length based on the target frequency without having to repeat tedious finite element simulations, significantly improving the design efficiency of compact superconducting resonators and saving computational resources.

[0110] To verify the effectiveness of the present invention, the following experimental design was conducted.

[0111] Example 1

[0112] 1. Structure and Design:

[0113] This embodiment prepares as follows: Figure 1 (a) shows a compact two-dimensional planar superconducting resonator. Sapphire is used as the substrate (101). The space-filling curve resonator body (102) adopts a sixth-order Hilbert curve design, with specific parameters as follows: conductor width w1 = 3 μm, ground spacing d1 = 3 μm, and side length l1 = 508 μm. A coplanar waveguide transmission line (103) is coupled to it through edge capacitance.

[0114] 2. Preparation:

[0115] The preparation is carried out using the aforementioned etching process, and the preparation method includes the following steps:

[0116] S1. Deposit a superconducting thin film on a substrate (101).

[0117] In step S1, before coating, the substrate is first baked at 120 °C and cleaned with ultraviolet ozone to remove moisture, organic residues and surface contaminants; then a 200 nm low-loss α-phase tantalum (Ta) thin film is sputtered.

[0118] S2. Using a process of homogenization, photolithography, development, and etching, a pattern structure of a space-filled curve resonator body (102) and a coplanar waveguide transmission line (103) is formed on the superconducting thin film;

[0119] In step S2, positive photoresist S1813 is used for pattern transfer. The spin coating speed is 3000 rpm, the spin coating time is 60 s, the baking temperature is 115 ℃, and the baking time is 120 s. Maskless exposure is performed using a laser direct-write exposure system (MicroWriter ML3, Durham Magneto Optics) with an exposure resolution of 0.6 μm and an exposure dose of 160 mJ / cm². Development is performed using positive photoresist developer MF319 for approximately 50 s, followed by fixing with deionized water for 45 s and drying with nitrogen. After pattern development, RIE etching is performed using CF4 gas for approximately 4 min.

[0120] S3. A compact superconducting resonator with a two-dimensional planar structure is obtained after etching, wherein the space-filled curve resonator body (102) and the coplanar waveguide transmission line (103) are capacitively coupled.

[0121] In step S3, after etching, a protective coating is applied, followed by scrubbing and subsequent cleaning. After scrubbing with a DS616 scrubber, the sample is cleaned sequentially with N-methylpyrrolidone (NMP) and isopropanol (IPA): first, the sample is immersed in NMP at 80 °C for 4 min, then sonicated in NMP for 5 min, followed by sonication in IPA for 5 min, and finally dried with nitrogen. To ensure a clean surface free of organic residue, the sample is further cleaned with piranha solution (98% H2SO4 : 30% H2O2 = 7 : 3) for 20 min, and then cleaned with BOE solution (HF:NH4F = 6 : 1) for 12 min to remove the oxide layer.

[0122] Finally, the compact superconducting resonator sample with a two-dimensional planar structure from Example 1 was placed in a sample box, encapsulated, and wire-connected. It was then placed in a dilution refrigerator for testing. When the ambient temperature of the sample box inside the refrigerator reached or fell below 30 mK, the S-value of the compact superconducting resonator based on the space-filling curve from Example 1 was read using a vector network analyzer. 21 The parameters, and their amplitude and phase spectrum results are as follows: Figure 5As shown, the resonator frequency of Example 1 is fr = 5.7158 GHz.

[0123] To further verify the performance of the compact superconducting resonator based on the space-filling curve, the frequency response of the resonator under different input powers was measured and fitted using the DCM algorithm, which yielded the intrinsic quality factor variation curves of the resonator under different average photon numbers in the cavity. Figure 8 The test results show the intrinsic quality factor of the resonator as a function of the average number of photons <n> in the cavity at a resonant frequency of fr = 5.7158 GHz. By comparing the fitted curves with the experimental data, it is demonstrated that the compact superconducting resonator based on the planar structure of this invention has excellent performance, maintaining a high quality factor at both high power and single-photon levels, and can be effectively applied to fields such as quantum information processing.

[0124] Example 2

[0125] 1. Structure and Design:

[0126] This embodiment prepares as follows: Figure 1 (b) shows a compact superconducting superconducting resonator with a three-dimensional flip-chip structure. Sapphire is used as the upper substrate (220) and the lower substrate (210). The space-filling curve resonator body (221) of the upper chip adopts a sixth-order Hilbert curve design, with specific parameters as follows: conductor width w2 = 3 μm, ground spacing d2 = 8 μm, and side length l2 = 580 μm. The coplanar waveguide transmission line (211) of the lower chip is coupled to the space-filling curve resonator body (221) of the upper chip through inter-chip capacitance.

[0127] The coplanar waveguide transmission line (211) has a center conductor width of 10 μm and a ground spacing of 5 μm. In the vertical projection overlap region between the space-filling curve resonator body (221) and the coplanar waveguide transmission line (211), the width of the center conductor strip of the coplanar waveguide transmission line and its spacing with the ground plane are increased relative to the non-overlapping region, and are 10 times the corresponding size of the non-overlapping region, to enhance inter-chip capacitive coupling.

[0128] Both the upper and lower chips use low-loss α-phase tantalum (Ta) films as their superconducting thin film material. The thickness of the superconducting thin film on the lower chip is 200 nm, and the thickness of the superconducting thin film on the upper chip is also 200 nm.

[0129] The metal pillars (212) are made of indium (In) and include two types: small-sized indium pillars and large-sized indium pillars. Small-sized indium pillars are used for inter-chip interconnect support and ground potential balance, while large-sized indium pillars are used for support and spacing control. Small-sized indium pillars have a diameter of about 60 μm and a height of about 8 μm; large-sized indium pillars have a diameter of about 500 μm and a height of about 8 μm.

[0130] 2. Preparation:

[0131] The fabrication method of a compact superconducting resonator with a three-dimensional flip-chip structure includes the following steps:

[0132] The upper-layer chip is fabricated using an etching process, the main steps of which are as follows:

[0133] S1'-1. A tantalum thin film with a thickness of 200 nm is deposited on the lower substrate (210) by magnetron sputtering.

[0134] S1'-2. The substrate with the superconducting thin film deposited was spin-coated with S1813 positive photoresist at a speed of 3000 rpm and baked at 115 ℃ for 2 min. The laser direct writing maskless exposure technology was used with an exposure dose of 160 mJ / cm². The developer was MF319 and the development time was 50 s. After development, the Ta film was etched by reactive ion etching (RIE) with CF4 as the etching gas and an etching time of about 4 min to form the space-filled curve resonator body (221) and the bridging superconducting thin film (222) pattern.

[0135] S1'-3. After etching, a protective adhesive is spin-coated, and the wafers are diced into individual chips using a DS616 dicing machine. The chips are then immersed in N-methylpyrrolidone (NMP) solution at 80 °C for 4 h, followed by ultrasonic cleaning in NMP for 5 min, and then ultrasonic cleaning in isopropanol (IPA) for 5 min. Finally, they are dried with nitrogen. To further improve interface cleanliness, the wafers are cleaned with piranha solution (H2SO4:H2O2 = 7:3) for 20 min and immersed in BOE (HF:NH4F = 6:1) for 12 min to remove organic residues and oxide layers.

[0136] The steps for fabricating the lower-layer chip are as follows:

[0137] S2'-1. Substrate pretreatment and superconducting thin film deposition:

[0138] After the upper substrate (220) was cleaned by CIF ultraviolet ozone (120 °C, 30 min), a 200 nm tantalum film was deposited by magnetron sputtering.

[0139] S2'-2. Photolithography and etching to form a coplanar waveguide transmission line (211):

[0140] Spin-coat S1813 photoresist (3000 rpm, 115 ℃, 2 min), exposure dose 135 mJ / cm², development 50 s. Pattern etched using CF4 gas reactive ion etching (RIE) process, followed by photoresist removal.

[0141] S2'-3. Photolithography to form metal pillar patterns

[0142] A dual-layer AZ4620 photoresist structure was used, with each layer approximately 7 μm thick, for a total thickness of approximately 14 μm. Baking was performed at 95 °C and 105 °C. The exposure dose was 1500 mJ / cm², and overlay was completed using inverted alignment marks. ZJX-100 positive photoresist developer was used, and development was completed for 2 min 30 s followed by fixing with deionized water.

[0143] S2'-4. Indium deposition and stripping

[0144] After cleaning the exposed area with argon ion milling, an 8 μm thick indium layer was deposited by evaporation. It was then stripped by immersion in 80 °C NMP solution for 4 h and followed by low-power ultrasonic cleaning.

[0145] S2'-5. Scraping and Cleaning

[0146] Spin-coating protective adhesive, followed by dicing and separation to remove the adhesive, yields multiple lower-layer chips.

[0147] S3'. Use a flip-chip alignment soldering machine to flip the lower chip so that the side of the lower chip with the coplanar waveguide transmission line (211) faces down. Achieve precise alignment by using the flip-chip alignment marks on the upper chip and the lower chip.

[0148] After alignment, cold welding and pressing are performed in a nitrogen atmosphere at a pressure of 50 N for 10 minutes.

[0149] After lamination, small indium pillars are connected to bridge the superconducting thin film and cross the central conductor strip of the coplanar waveguide transmission line to connect the ground planes on both sides, forming a stable superconducting air bridge structure that effectively balances the ground potential and suppresses parasitic modes.

[0150] Finally, the compact superconducting resonator sample with the three-dimensional flip-chip structure from Example 2 was placed in a sample box, encapsulated, and wire-connected. It was then placed in a dilution refrigerator for testing. When the ambient temperature of the sample box inside the refrigerator reached 30 mK, the S-value of the compact superconducting resonator based on the space-filling curve from Example 2 was read using a vector network analyzer. 21 The parameters, and their amplitude and phase spectrum results are as follows: Figure 7 As shown, the resonator frequency of Example 2 is fr = 4.8583 GHz.

[0151] To further verify the performance of the compact superconducting resonator based on the space-filling curve, the frequency response of the resonator under different input powers was measured and fitted using the DCM algorithm, which yielded the intrinsic quality factor variation curves of the resonator under different average photon numbers in the cavity. Figure 8The test results show the intrinsic quality factor of the resonator as a function of the average number of photons <n> in the cavity at a resonant frequency of fr = 4.8583 GHz. By comparing the fitted curves with the experimental data, it is demonstrated that the compact superconducting resonator based on the three-dimensional flip-chip structure of this invention has excellent performance and maintains a stable quality factor at the single-photon level, which can be effectively applied to fields such as quantum information processing.

[0152] Example 3

[0153] like Figure 1 As shown in (a), this embodiment provides a method for designing the resonant frequency of a compact superconducting resonator based on a two-dimensional planar structure and a space-filling curve. The design requirements for the target resonant frequency are met by adjusting four design parameters: the iteration order of the space-filling curve resonator body, the conductor width, the ground spacing, and the side length.

[0154] like Figure 1 As shown in (a), the design parameters of the two-dimensional planar structure design method include: iteration order n1, conductor width w1, ground spacing d1, and side length l1.

[0155] The iteration order n1 refers to the iteration order of the selected Hilbert space-filling curve.

[0156] The conductor width w1 refers to the width of the conductor in the body (102) of the space-filled curve resonator;

[0157] The ground spacing d1 refers to the distance between the edge of the space-filled curve resonator body (102) and the ground plane.

[0158] The side length l1 refers to the side length of the body (102) of the space-filled curve resonator.

[0159] The design method includes the following steps:

[0160] Step 1: Draw the space-filling curve of the resonator body

[0161] A fractal algorithm script is written using an automated layout generation tool (such as the gdspy package in Python) to draw the GDSII format mask file of the space-filling curve resonator body, and then the GDSII format file is converted into a DXF file. Based on design experience, the iteration order of the Hilbert space-filling curve is usually set to 4-6. This is because it has been found in practice that as the order of the Hilbert space-filling curve increases, the number of bends increases, and the configurable conductor width decreases. When the order is set to 7, the allowable conductor width is in the sub-micron range, which is not suitable for implementation using ultraviolet lithography. Higher precision lithography techniques, such as EBL (electron beam lithography), are required. To balance processing efficiency and the performance of a compact superconducting resonator, this embodiment preferably uses an iteration order n1 of 5, a conductor width w1 of 3 μm, and a ground spacing d1 of 3 μm. The layout of the space-filling curve resonator body with different resonant frequencies is drawn by changing the side length l1.

[0162] Step 2: Verify the resonant frequency f using electromagnetic simulation r

[0163] Import the generated layout file (e.g., DXF format) into finite element analysis software (e.g., COMSOL Multiphysics) to build an electromagnetic simulation model and obtain the resonant frequency f corresponding to different side lengths l1. r ;

[0164] Step 3: Calculate the fitting formula between the variables in the design parameters and the resonant frequency.

[0165] Based on multiple sets of data obtained from simulations, and under the premise of fixed iteration order n1, conductor width w1, and ground spacing d1, the resonant frequency f is obtained through numerical analysis and fitting. r An empirical formula that varies with side length l1. For example... Figure 9 As shown, the resonant frequency f can be seen. r The relationship between the resonant frequency and the side length l1 is monotonically decreasing and inversely proportional. This is because increasing the side length l1 directly increases the effective electrical length of the space-filling curve, thereby increasing the equivalent inductance and causing the resonant frequency to shift downward. Through the fitted empirical formula, designers can directly calculate the required physical side length l1 based on the target frequency, eliminating the need for tedious finite element simulations and significantly improving the design efficiency of compact superconducting resonators.

[0166] Within the process window determined in this embodiment (n1 = 5, w1 = 3 μm, d1 = 3 μm), the fitting empirical formula covers the mainstream operating frequency band of superconducting quantum chips from 4 GHz to 8 GHz. The side length l1 of the corresponding space-filling curve resonator body can be directly calculated based on the target frequency.

[0167] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0168] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A compact superconducting resonator based on a space-filling curve, characterized in that, include: The main body of the space-filled curve resonator is composed of a superconducting thin film, wherein the space-filled curve traverses the defined planar region continuously and without intersection. And a coplanar waveguide transmission line, the coplanar waveguide transmission line including a central conductor strip and ground planes located on both sides of the central conductor strip; The space-filling curve resonator body and the coplanar waveguide transmission line are coupled capacitively or inductively to achieve signal readout.

2. The compact superconducting resonator according to claim 1, characterized in that, The space-filling curve is a Hilbert curve or a Peano curve; the material of the superconducting thin film is selected from at least one of tantalum, niobium, niobium nitride, titanium niobium nitride, titanium nitride, or aluminum.

3. The compact superconducting resonator according to claim 1 or 2, characterized in that, The resonator is a two-dimensional planar structure, comprising: a substrate (101), a space-filled curve resonator body (102) disposed on the substrate (101), and a coplanar waveguide transmission line (103) coupled thereto; the space-filled curve resonator body (102) is an open-circuit structure at both ends, and the ground plane of the coplanar waveguide transmission line (103) is disposed around the space-filled curve resonator body (102) and does not enter the traversal region where the body is located.

4. The compact superconducting resonator according to claim 1 or 2, characterized in that, The resonator is a three-dimensional flip-chip structure, consisting of an upper chip and a lower chip; The lower chip includes a lower substrate (210), a coplanar waveguide transmission line (211) disposed on the lower substrate (210), and a plurality of metal pillars (212) formed on the ground planes on both sides of the coplanar waveguide transmission line (211). The upper chip includes an upper substrate (220), a space-filled curve resonator body (221) disposed on the upper substrate (220), and a bridging superconducting thin film (222). The metal pillar (212) is bonded to the bridging superconducting film (222), so that the bridging superconducting film (222) and the metal pillar (212) together form a superconducting air bridge that spans the central conductor strip of the coplanar waveguide transmission line (211), thereby realizing a three-dimensional superconducting ground interconnection between the upper chip and the lower chip. The space-filled curve resonator body (221) is coupled to the coplanar waveguide transmission line (211) through inter-chip capacitance or inter-chip inductance.

5. The compact superconducting resonator according to claim 4, characterized in that, The metal pillar (212) is an indium pillar with a diameter of 5 micrometers to 600 micrometers and a height of 4 micrometers to 15 micrometers; the bridging superconducting film (222) is rectangular or square in shape.

6. The compact superconducting resonator according to claim 4 or 5, characterized in that, In the vertical projection overlap area of ​​the space-filling curve resonator body (221) and the coplanar waveguide transmission line (211), the width of the central conductor strip of the coplanar waveguide transmission line (211) and the distance between it and the ground plane are not less than twice the corresponding size of the non-overlapping area.

7. A method for fabricating a compact superconducting resonator according to claim 3, characterized in that, Including the following steps: The space-filled curve resonator body (102) and coplanar waveguide transmission line (103) are patterned on the superconducting thin film by etching or stripping processes. The etching process includes: first depositing a superconducting thin film, and then forming a pattern through photolithography, development, and dry or wet etching; the lift-off process includes: first forming a pattern through photolithography, then depositing a superconducting thin film and lifting it off to obtain the pattern.

8. A method for fabricating a compact superconducting resonator as described in claim 4 or 5, characterized in that, Including the following steps: A space-filled curve resonator body (221) and a bridging superconducting thin film (222) are fabricated on the upper substrate (220). Coplanar waveguide transmission lines (211) and metal pillars (212) are fabricated on the lower substrate (210). The upper and lower chips are flip-chip aligned and pressed together to connect the metal pillar (212) with the bridging superconducting thin film (222) to form a three-dimensional superconducting grounding structure.

9. A method for designing the resonant frequency of a compact superconducting resonator according to any one of claims 1-6, characterized in that, Including the following steps: The design parameters of the space-filling curve resonator body are determined, and the design parameters include at least: the iteration order of the space-filling curve, the conductor width, the ground spacing, and the side length of the area occupied by the resonator body; The design requirements for the target resonant frequency can be met by adjusting at least one of the following parameters: iteration order, conductor width, ground spacing, and side length.

10. Use of a compact superconducting resonator based on a space-filling curve according to any one of claims 1-6, characterized in that, Used in superconducting quantum bit readout resonators, superconducting parametric amplifiers, or low-temperature quantum microwave devices.

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