Method for producing superconducting flux diodes in artificial spin ice

Abstract

The application discloses a method for generating superconducting magnetic flux diodes in artificial spin ice medium, belongs to the technical field of superconducting diodes, and can generate obvious odd-symmetry magnetic transport effects by breaking space inversion antisymmetry. The effects have never appeared in superconducting materials before. The abnormal non-reciprocal effect is driven by a magnetic field and can be non-volatile regulated, thereby generating reversible magnetic field driven magnetic flux diodes. The findings show that the antisymmetry breaking plays a key role in exploring unconventional superconducting characteristics, and provide a new principle for designing new superconducting electronic devices.

Description

Technical Field

[0001] This invention belongs to the field of superconducting diode technology, and specifically relates to a method for a superconducting flux diode driven by a reversible magnetic field generated by antisymmetry breaking in an ASI dielectric. Background Technology

[0002] Symmetry breaking plays a crucial role in revealing novel properties and functions of various systems. Notably, the breaking of spatial and temporal symmetries leads to special phenomena such as the superconducting diode effect and odd-symmetric magnetoresistance. However, generating and controlling these effects presents significant challenges, limiting their practical applications.

[0003] Generally, charge transport effects in materials exhibit even symmetry with respect to an external magnetic field. For example, resistance typically remains constant between positive and negative magnetic fields. However, in special cases where time-reversal symmetry is broken, odd components of magnetoresistance, termed odd-symmetric magnetoresistance, have been observed in some systems. This unusual field-driven non-reciprocal effect is usually quite subtle. A recent study reported a pronounced odd-symmetric magnetoresistance in InAs quantum wells, highlighting this interesting effect in non-superconducting materials. However, until now, this effect has not been explored in superconductors. In most superconductors, charge transport properties are symmetric with respect to a magnetic field B, resulting in resistance and various superconducting parameters (such as critical temperature and critical current) being even functions of the magnetic field B.

[0004] The recent discovery of the superconducting diode effect has attracted considerable attention due to its potential in dissipative electronics. It exhibits superconductivity when current flows in one direction and normal resistance when current flows in the opposite direction. This effect typically depends on the simultaneous breaking of spatial and temporal inversion symmetries. Notably, the polarity of the diode can be reversed by reversing the applied magnetic field. This phenomenon can be described by magnetic chirality anisotropy, which includes a nonlinear resistance term (γBI) dependent on the current I and the magnetic field B (γ represents the magnetic chirality anisotropy coefficient). Therefore, superconducting diodes often exhibit non-reciprocal superconductivity in their response to current and magnetic fields. The complex relationship between these superconducting non-reciprocal charge transport effects and symmetry breaking remains unclear and requires further investigation.

[0005] In type II superconductors, magnetic flux penetrates the material in quantized units (called flux quanta or magnetic flux). The movement of magnetic flux directly affects the electromagnetic properties of the superconductor. Recently, a type of magnetic nanostructure called artificial spin ice (ASI) has emerged as a general tool for manipulating magnetic flux movement. ASIs consist of arrays of mutually coupled magnetic nanomagnets fabricated using electron beam lithography. They have been used as reconfigurable pinning potential landscapes in superconductors, producing some interesting phenomena such as the reversible flux-quantum Hall effect, the switchable flux-quantum diode effect, and reconfigurable geometric frustration. Now, this invention introduces a novel ASI-superconducting heterostructure to generate even more exotic superconducting effects. Summary of the Invention

[0006] The technical problem solved by this invention: This invention proposes a significant superconducting odd-symmetric magnetic transport effect, which is generated by breaking the spatial inversion antisymmetry, and realizes a programmable magnetic field driven superconducting flux diode.

[0007] Technical Solution: To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0008] A method for generating superconducting flux diodes in artificial spin ice media is proposed. This method generates a superconducting odd-symmetric magnetic transport effect by breaking spatial inversion antisymmetry. This odd-symmetric magnetic transport effect is driven by a magnetic field and can be non-volatilely modulated, thereby generating a programmable magnetic field-driven superconducting flux diode.

[0009] Furthermore, the superconducting flux diode is located in the superconducting Mo... 0.79 Ge 0.21 The square ASI composition on the thin film, the inversion antisymmetry leads to the symmetrical transport effect of the flux diode on the magnetic field B, thus producing a superconducting flux diode without magnetic chirality anisotropy.

[0010] Furthermore, the symmetric transport effect originates from the spatial inversion antisymmetry of the magnetic charge potential, expressed as:

[0011]

[0012] In the formula, P and -P represent opposite positive / negative pinning potentials, respectively, and -r represents spatial inversion operation. Since the interaction between magnetic charge and magnetic flux / anti-magnetic flux is opposite, the following relationship is obtained:

[0013]

[0014] In the formula, P(B) and P(-B) represent the pinning potentials of magnetic flux and anti-magnetic flux with the same magnetic charge mode, respectively. The sign reversal in P indicates that the negative potential or attractive potential of the magnetic flux is transformed into the positive potential or repulsive potential of the anti-magnetic flux. For a given pinning potential P, the following relationship holds:

[0015]

[0016] In the formula, V is the dissipated voltage, representing the movement of the magnetic flux, and I represents the current. From equation (3), we know that the energy dissipation of the magnetic flux moving within a certain pinning potential is equal to that of the anti-magnetic flux moving within the same pinning potential. From equation (4), we know that under the spatial inversion operation, the movement of the magnetic flux within the pinning potential is equivalent to its movement under the reverse driving current within the original potential. Combining equations (1) to (4), we get:

[0017] V(-B) = V(B)

[0018] It was demonstrated that the existence of spatial inversion antisymmetry leads to a symmetric transport effect on the magnetic field B, thereby generating a superconducting flux diode without magnetic chirality anisotropy.

[0019] Furthermore, ASI composed of asymmetric nanomagnets is used to create superconducting devices to break the antisymmetry of the potential barrier and realize odd-symmetric magnetic transport effects.

[0020] Furthermore, the narrow end of the asymmetric nanomagnet has a larger magnetic charge than the wide end. Since they come from the same nanomagnet, the positive and negative magnetic charges produce equal total flux. The smaller magnetic charge produces a more localized and stronger pinning force than the larger magnetic charge. The mismatch of the magnetic potential between the positive and negative magnetic charges breaks the antisymmetry of the pinning potential through P(-r) ≠ -P(r). Therefore, asymmetric superconductivity with respect to the magnetic field appears, leading to odd-symmetric magnetic transport effects.

[0021] Furthermore, the odd-symmetric magnetic transport effect is observed through the critical current I under both positive and negative fields. c Regarding the characterization using magnetic field curves, I c The odd components are represented as:

[0022] .

[0023] Furthermore, when the magnetization state reverses, the polarity of the magnetic charge switches, but its magnitude and intensity remain unchanged. Under demagnetized ASI, The sign of the magnet is reversed, thus realizing an in-situ reversible magnetic field driven superconducting flux diode.

[0024] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0025] (1) This invention realizes a flux diode by introducing and breaking a controllable method of spatial inversion antisymmetry. Experiments of this invention reveal a flux diode effect with respect to magnetic field symmetry caused by inversion antisymmetry. More notably, by breaking the antisymmetry, a significant odd-symmetric magnetic transport effect emerges. Furthermore, this anomalous non-reciprocal effect is driven by a magnetic field and can be non-volatilely modulated, thereby generating a reversible magnetic field-driven flux diode. This highlights the crucial role of antisymmetry breaking in conceptual studies of unlocking unconventional superconducting properties and provides promising prospects for innovative functions in superconducting electronics.

[0026] (2) The method of generating superconducting flux diodes in ASI dielectrics of the present invention, breaking this asymmetry has proven to be an effective strategy for generating odd-symmetry magnetic transport effects, thereby producing reversible field-driven diodes. These findings provide insights into the inversion of antisymmetry and antisymmetry breaking related properties. The observed field-driven non-reciprocal magnetic transport effects have potential applications in various (quasi-)particle systems beyond superconducting flux. These examples include magnetic particles, skyrmions, and fluxes, whose polarities can be controlled by magnetic fields. Furthermore, by extending the operation to electric (quasi-)particles, introducing a unique potential field with antisymmetry breaking can open up possibilities for studying innovative transport phenomena. This can be achieved by fabricating nanostructured gate electrodes in two-dimensional materials such as graphene, which will pave the way for the discovery of new principles and functions, thereby driving future advancements in electronics. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of a current-driven flux diode with spatial inversion antisymmetry and magnetic field symmetry; in which,

[0028] a is a schematic diagram of an ASI-mediated superconductor;

[0029] b is a scanning electron microscope (SEM) image showing a square ASI structure with symmetrical nanomagnets, scale bar: 700 nm, with the dashed box highlighting the magnetic vertices containing four nanorods;

[0030] c is a magnetic force microscopy (MFM) image of the ASI structure in Figure b. White / black dots represent positive / negative magnetic charges, and arrows indicate the magnetization direction of single-domain nanomagnets. The inset shows an MFM image of one vertex in a micromagnetic simulation.

[0031] d is the dissipated voltage caused by the movement of magnetic flux, which depends on the current and the magnetic field. The magnetic field unit is obtained through the matching field B. m =78.9 Gs normalized, the dashed line represents the critical current curve extracted using the ±1 μV voltage criterion;

[0032] e is a schematic diagram illustrating the interaction between the magnetic charge and magnetic flux (top) and the antimagnetic flux (bottom) of the nanomagnet;

[0033] f and g are comparison graphs of the critical current curves for positive and negative current (f) and positive and negative magnetic field (g);

[0034] h is the spatial inversion antisymmetric plot proving the magnetic potential, and the blue / red dots represent the attractive / repulsive pinning potentials to the magnetic flux.

[0035] Figure 2 This is a correlation diagram of a flux diode driven by a magnetic field exhibiting antisymmetry breaking; among which,

[0036] a is a SEM image of the ASI structure with asymmetric nanomagnets, scale bar: 700 nm;

[0037] b is the MFM image of the ASI structure in (a), and the arrows show the magnetization direction of the nanomagnets at the vertices highlighted in the dashed boxes in (a). The inset is a micromagnetic simulation of the MFM image of one vertex.

[0038] c is a schematic diagram of the magnetic charge of an asymmetric nanomagnet and its interaction with magnetic flux and antimagnetic flux;

[0039] d represents the breaking of antisymmetry in magnetic potential spatial inversion;

[0040] e is a comparison of the critical current curves for positive and negative magnetic fields, and the inset shows the odd-symmetric component of the critical current.

[0041] f and g are and The diagram shows the motion of magnetic flux / anti-magnetic flux. The blue / red dots represent the attraction / repulsion pinning potentials of magnetic flux (white dots) and anti-magnetic flux (black dots). The dashed line represents the trajectory (f) of the magnetic flux motion, while the anti-magnetic flux is captured (g).

[0042] h is the field-dependent magnetoresistance measured at I = 1.4 mA.

[0043] Figure 3 It is a reconfigurable odd-symmetric magnetic transport effect; among which,

[0044] Image 'a' is a SEM image showing the ASI structure composed of asymmetric nanomagnets with inverted geometry. Figure 2 a forms a contrast, scale bar: 700nm;

[0045] b and c are MFM images of ASI (a) with positive (b) and negative (c) magnetization, respectively, with arrows indicating the magnetization state of the nanomagnets. The insets are MFM images obtained from micromagnetic simulations;

[0046] d and e are schematic diagrams of the magnetic charge of nanomagnets with two opposite magnetization directions;

[0047] f and g are a comparison of the critical current curves for positive and negative magnetic fields, and the inset shows the odd symmetric component of the critical current.

[0048] Figure 4 This is a schematic diagram related to magnetic transport in relation to potential symmetry; among which,

[0049] a is the magnetic nail potential under positive (a1) and negative (a2) magnetization at the vertex;

[0050] b is the magnetic pinning potential of the reverse nanomagnet;

[0051] c represents the critical current curves under different pinning potentials in (a) and (b);

[0052] d is the critical current curve normalized using equation (9).

[0053] Figure 5 It is an optical image of the sample.

[0054] Figure 6 This is an explanation of formula (3): the equivalence of magnetic flux and antimagnetic flux motion.

[0055] Figure 7 This is an explanation of formula (4): In a reversal potential, the magnetic flux movement in the opposite direction is equivalent.

[0056] Figure 8 It is the dissipated voltage caused by the movement of magnetic flux and related to current and magnetic field.

[0057] Figure 9 This is an explanation of formula (7): magnetic flux and anti-magnetic flux motion are equivalent.

[0058] Figure 10 This is an explanation of formula (8): the equivalence of magnetic flux and antimagnetic flux motion.

[0059] Figure 11 It is a temperature-dependent resistance.

[0060] Figure 12 It corrects the residual magnetic field through a reference channel.

[0061] Figure 13 It is the correction of the odd-symmetric component by the reference channel. Detailed Implementation

[0062] The present invention will be further illustrated below with reference to specific embodiments. These embodiments are implemented based on the technical solutions of the present invention, and it should be understood that these embodiments are only used to illustrate the present invention and are not intended to limit the scope of the present invention.

[0063] This invention relates to a method for generating superconducting flux diodes in artificial spin ice media (ASI). By breaking the spatial inversion antisymmetry, a superconducting odd-parity magnetotransport effect is generated. The odd-parity magnetotransport effect is driven by a magnetic field and non-volatilely modulated, thereby generating a programmable magnetic field-driven superconducting flux diode.

[0064] In this embodiment, we first introduce the symmetrical flux diode caused by antisymmetry. For example... Figure 1 As shown, this illustrates how a non-magnetic chiral anisotropic flux diode is generated. Figure 1 The schematic diagram of the ASI dielectric superconducting device is shown in Figure a. Figure 5 A photograph of the device. It is constructed from superconducting Mo. 0.79 Ge 0.21 Square ASI on the thin film ( Figure 1 b) Composition. The magnetic charge pattern of ASI was directly observed using magnetic force microscopy (MFM) images. Figure 1 c). Figure 1 Figure d shows a color diagram of the dissipated voltage caused by magnetic flux movement. An asymmetric effect with respect to the applied current is observed. Figure 1 f), leading to a flux diode effect, also known as the flux ratcheting effect. This current-driven non-reciprocal effect arises from the breaking of the spatial inversion symmetry of the flux pinning potential. In this case, the symmetry breaking is caused by the positive and negative magnetic charges of nanomagnets, which exert attractive and repulsive pinning forces on the flux / anti-flux, respectively. Figure 1 e). Furthermore... Figure 1 f shows the diode in the first matched magnetic field B. m The polarity reverses at 78.9 Gs, where one nanomagnet corresponds to one magnetic flux. This reversal occurs because the magnetic flux attracting the pinning center exhibits a repulsive potential at B > B. m Therefore, when the magnetic field passes through B m At that time, the pinning landscape changes, causing the sign in the flux diode to reverse. Similar effects are widely observed in flux diodes.

[0065] Similar to the non-reciprocal superconducting normal transition effect in superconducting diodes, most existing flux diodes exhibit asymmetry in both I and B, including descriptions of magnetic chirality anisotropy. In contrast, the flux diode of this invention exhibits a symmetrical transport effect with respect to the magnetic field B. Figure 1 f and 1g). This unique symmetric transport effect originates from the spatial inversion antisymmetry of the magnetic charge potential (f and 1g). Figure 1 h), can be represented as:

[0066]

[0067] In the formula, P and -P represent opposite positive / negative pinning potentials, respectively, and -r represents the spatial inversion operation. Furthermore, due to the opposite interaction between magnetic charge and magnetic flux / anti-flux ( Figure 1 e), has the following relationship:

[0068]

[0069] In the formula, P(B) and P(-B) represent the pinning potentials of magnetic flux and anti-magnetic flux with the same magnetic charge mode, respectively. The sign reversal in P indicates that the negative (or attractive) potential of the magnetic flux is transformed into the positive (or repulsive) potential of the anti-magnetic flux. Furthermore, for a given pinning potential P, the following relationship holds:

[0070]

[0071] In the formula, V is the dissipation voltage, which directly represents the movement of the magnetic flux. Since negative resistance is not allowed, the sign of V needs to be reversed whenever the sign of the applied current I changes. From equation (3), it can be seen that the energy dissipation of the magnetic flux moving within a certain pinning potential is equal to that of the reverse magnetic flux moving within the same pinning potential (driven by a reverse current). See [link to equation (3)] for details. Figure 6 As can be seen from equation (4), under spatial inversion operation, the magnetic flux motion within the pinned potential is equivalent to its motion under the reverse driving current within the original potential. (See details...) Figure 7 Combining equations (1) to (4), we can obtain:

[0072] V(-B) = V(B)

[0073] Therefore, this demonstrates that the existence of spatial inversion antisymmetry leads to a symmetric transport effect on the magnetic field B, resulting in a unique flux diode without magnetic chirality and anisotropy. This phenomenon can be visually demonstrated through molecular dynamics (MD) simulations.

[0074] Explanation of Formula (3): The equivalence of magnetic flux and anti-magnetic flux motion. Figure 6 The motion of a magnetic flux (generated by +B) and an anti-magnetic flux (generated by -B) is described. When these (anti-)magnetic fluxes move in the same direction within the same pinning potential P, their motion is equivalent. However, to keep the driving forces of the magnetic and anti-magnetic fluxes consistent, currents need to be applied in opposite directions. Therefore, this results in a sign reversal of the dissipated voltage.

[0075] Explanation of Formula (4): In a reverse potential, the reverse magnetic flux motion is equivalent. Figure 7 The diagram illustrates the magnetic flux motion within the pinned potential P(r) (right figure) and its reversed counterpart P(-r) (left figure). When the magnetic flux moves in opposite directions within these potentials, their motions are indistinguishable. To maintain the opposing driving forces, a reverse current is applied, resulting in a reversal of the sign of the dissipated voltage.

[0076] Next, this embodiment introduces the generation of odd-symmetric magnetic transport by antisymmetry breaking. In ASI dielectric superconductors, the antisymmetry of the fixed potential barrier is crucial for realizing the magnetically symmetric diode effect. To achieve the odd-symmetric magnetic transport effect, this invention breaks the antisymmetry of the potential barrier by creating a superconducting device using a unique ASI structure composed of asymmetric nanomagnets. Figure 2 a). MFM image ( Figure 2 b) and micromagnetic simulation ( Figure 2 The insertion of b) reveals the difference in the magnitude of the magnetic charge and the field strength at both ends of the asymmetric nanomagnet. For example... Figure 2 As shown in c, the narrow end has a larger magnetic charge than the wide end. Since they originate from the same nanomagnet, the positive and negative magnetic charges produce equal total flux, and the smaller magnetic charge produces a more localized but stronger pinning force than the larger magnetic charge. Figure 2 c). The antisymmetric reversal of the pinning potential is broken between positive and negative magnetic charges through P(-r) ≠ -P(r). Figure 2 d). Therefore, the asymmetric superconductivity with respect to the magnetic field leads to an odd-symmetric magnetic transport effect. This effect can be observed through the critical current I under positive and negative fields. c Characterized by magnetic field curves ( Figure 2 e). I c odd symmetric components (in) Figure 2 (as shown in the illustration of e) is represented as:

[0077]

[0078] First Matching Field B m A significant odd-symmetric component is observed around and / or below. The negative odd-symmetric component in the figure indicates that the negative field I... c The amplitude is larger than that of the positive field. Furthermore, due to current-driven non-reciprocal transport effects, the odd-symmetric component is asymmetric with respect to the current (see...). Figure 2 e). Odd-symmetric magnetic transport effects are readily observed in MD simulations. For example... Figure 2 As shown in f and 2g, the (anti)magnetic flux moves only in a single direction of the applied magnetic field. This behavior is similar to that of a current-driven flux diode, where the flux moves only in the direction of the applied current, indicating a magnetic field-driven flux diode effect.

[0079] In existing technologies, the odd-symmetric magnetoresistance in InAs quantum wells reaches as high as 27% under a high magnetic field of 10 Tesla. In the superconducting device of this invention, the zero-field resistance R(B=0) ≈ 0 is negligible (less than the experimental sensitivity of ~10⁻⁵ Ω), such as... Figure 2 As shown in h. This results in B mAt an extremely low magnetic field of 78.9, the odd-symmetric magnetoresistance is exceptionally large, exceeding 4000% (within experimental resolution). Furthermore, in previous studies of superconducting diodes and flux diodes, based on the requirement of magnetic chirality anisotropy, the asymmetric response to the magnetic field can be reversed by changing the current direction. However, in the experiments of this invention, The polarity cannot be reversed by switching the polarity of the current. Figure 2 (e. Insert figure). This is because although the reverse current changes the direction of (anti)magnetic flux flow (compare videos 2 and 3), it does not change the pinning strength determined by the magnetic charge of ASI. This distinguishes the odd symmetry magnetic transport effect caused by antisymmetry breaking from the chiral magnetic anisotropy effect caused by typical symmetry breaking in previously reported superconducting diodes and flux diodes.

[0080] Next, we introduce reversible magnetic field-driven flux diodes. This is because antisymmetry breaking is caused by the asymmetric geometry of the nanomagnets ( Figure 2 c) To reverse the odd-symmetric transport effect, a direct method is to invert the asymmetric geometry of the nanomagnet, such as... Figure 3 a and Figure 2 As shown in a. This change alters the magnitudes of the positive and negative magnetic charges (and...). Figure 2 Compared to 2c, see b and 2c respectively. Figure 3 (b and 3d), thus altering the attractive / repulsive pinning strengths of the magnetic flux and anti-magnetic flux. Figure 2 The negative observed in e on the contrary, Figure 3 f describes having a positive The value exhibits an inverse odd-symmetric magnetic transport effect. However, adjusting the geometry of nanomagnets is not an in-situ method because their geometry cannot be altered once the sample is fabricated. Nevertheless, this invention, by applying in-plane polarization fields, provides the advantage of in-situ reconfigurability of magnetic charge modes in ASIs, thereby enabling programmable control of magnetic flux. Figure 3 c shows the relationship with Figure 3 Compared to b, the magnetic charge mode in the demagnetized state. When the magnetization state is reversed, the polarity of the magnetic charge switches, but its magnitude and intensity remain unchanged. Figure 3 d and 3e). Transport experiments show that under reverse magnetized ASI, The sign is reversed (with) Figure 3 Compared to f, Figure 3 g), thus realizing an in-situ reversible magnetic field driven flux diode.

[0081] Reversing the local geometry and reversing the magnetization of ASI can both reverse the polarity of the odd-symmetry magnetic transport effect, but their effects differ. Reversing the geometry does not switch the amplitude of the odd-symmetry component between positive and negative currents. In this case, The amplitude under positive current is always greater than the amplitude under negative current. Figure 2 (Illustrations for e and 3f). However, the reverse magnetization of ASI switches between reverse currents. Amplitude ( Figure 3 (Illustrations for f and 3g). This is because the current-related non-reciprocal transport effects are primarily determined by the symmetry breaking introduced by the distribution of positive and negative magnetic charges, while the polarity of the positive and negative magnetic charges does not change when the geometry of the nanomagnet is reversed. On the other hand, current-driven non-reciprocal effects can be reversed by reversing the magnetization of the ASI. This is because when the magnetization of the ASI is reversed, the asymmetry of the pinning potential is reversed.

[0082] Finally, we introduce magnetic transport related to potential symmetry. To gain a deeper understanding of the influence of magnetic barrier symmetry on current and field-driven transport effects, we will discuss... Figure 4 Figures a and 4b illustrate various pinning potentials under different ASI magnetization intensities. , corresponding to Figure 2 Nanomagnet samples with flippable asymmetric geometries in a and 3a. Figure 4 c displays the corresponding I c The curves show a clear symmetry between the current and the magnetic field. A close correlation can be derived between the symmetry / antisymmetry of the pinning potential and the resulting transport effect. From... Figure 4 From a and 4b, we obtain:

[0083]

[0084] like Figure 8 The diagram shows the dissipation voltage related to current and magnetic field caused by magnetic flux movement. Figures a1, a2, b1, and b2 correspond to... Figure 4 a and Figure 4 The results are obtained from the four pinning potentials shown in b. The magnetic field unit is obtained through the first matching field B. m =78.9Gs normalized. The dashed line represents the critical current curve extracted using the ±1 μV voltage criterion. Therefore, from equations (3) and (5), we can obtain:

[0085]

[0086] This indicates that the motion of magnetic flux in the magnetic charge potential is equivalent to the motion of antimagnetic flux under reverse driving current and opposite magnetization magnetic charge potential.

[0087] like Figure 9 As shown, the magnetic potential P a1 (+B) and P a2(-B) represent the potentials experienced by the magnetic flux in the positive magnetization state (left figure) and the antimagnetic flux in the negative magnetization state (right figure), respectively. These potentials are identical, as shown in equation (5), and are generated by the interaction between magnetic charges: negative magnetic charges attract magnetic flux, and positive magnetic charges attract antimagnetic flux, as shown in equation (5). Figure 1 As shown in e. Figure 6 As shown, referring to formula (3), it can be clearly seen that the motion of one magnetic flux and one anti-magnetic flux is equivalent in the same direction and within the same pinning potential. To ensure that the driving forces of the two magnetic fluxes are consistent, the currents need to be applied in opposite directions. Therefore, this leads to a reversal of the sign of the dissipation voltage. Therefore, from equations (4), (6), and (7), we can obtain that...

[0088]

[0089] This means that the motion of magnetic flux in the magnetic charge potential is equivalent to the motion of anti-magnetic flux in the reversed asymmetric geometric magnetic potential. Equations (7) and (8) establish the magnetic flux motion related to the symmetry of the magnetic potential. Since the critical current is extracted from the dissipation voltage curve related to the current, the following relationship can be derived from equations (7) and (8):

[0090]

[0091] According to equation (9), all critical current curves can be scaled together, such as Figure 4 As shown in d. This fully confirms the relationship described in equations (7) and (8), enabling a comprehensive understanding of magnetic flux motion.

[0092] The complete experimental procedure of this embodiment is as follows:

[0093] (1) Sample fabrication. The experimental sample consisted of a superconducting Mo0.79Ge0.21 (MoGe) microbridge with various square ASIs patterns on it. Figure 5 The microbridge consists of four parts (). Figure 5 ): Regular symmetrical nanomagnetic part ( Figure 1 b), Asymmetric nanomagnetic portion ( Figure 2 a), the asymmetrically reversed nanomagnetic portion ( Figure 3a), and a reference portion without any nanomagnets. To fabricate the device, a 50 nm thick MoGe film was sputtered onto a silicon substrate with an oxide layer. Microbridges with a width of 50 μm were fabricated using standard photolithography. ASIs composed of permalloy (Ni0.80Fe0.20) nanomagnets were then fabricated using electron beam lithography and electron beam evaporation. The symmetrical rectangular nanomagnets have dimensions of 300 nm (length) × 80 nm (width) × 25 nm (thickness), while the asymmetrical nanomagnets are isosceles trapezoids with dimensions of 300 nm (height) × 40 nm (bottom) × 120 nm (bottom) × 25 nm (thickness). The lattice constant of the square ASI is 724 nm. The critical temperature of the superconducting MoGe film is 6.7 K ( Figure 11 ).

[0094] (2) Experiments. Before each measurement, the sample was magnetized with an in-plane magnetic field exceeding 1000 Gs. After magnetization, the magnetic field was zeroed. MFM (magnetic force microscopy) images were obtained using the ATTOCUBE MFM system and the NANOSENSORS MFM probe (PPP-MFMR). Transport measurements were performed at 5.7 K in a triaxial superconducting vector magnet capable of providing a magnetic field in any desired 3D direction. Only out-of-plane magnetic fields were applied during transport experiments. To ensure the accuracy of the measurements, the transport results were carefully corrected based on data obtained from the reference section. Figure 12 and Figure 13 ).

[0095] (3) Molecular dynamics simulation. The simulation was conducted in a two-dimensional system with periodic boundary conditions. The sample size was L×L, where L = 25.6λ (λ is the London penetration depth). Magnetic charges at both ends of the rod magnet acted as attractive or repulsive pinning centers for the magnetic flux and anti-magnetic flux. The interaction energy between the magnetic flux and individual pinning points was modeled using a Gaussian function. The attractive (repulsive) pinning sites introduced negative (positive) interaction energy into the system. Attractive pinning force Derived from the pinning potential energy gradient, its form is as follows:

[0096]

[0097] In the formula, Let be the standard deviation of the Gaussian function. The pinning factor is... and This indicates the center location of the attraction pinning point k. Repulsive force. The forms are the same, but the signs are reversed. For symmetrical nanomagnets, use In asymmetric nanomagnets, small magnetic charge Large magnetic charge For more details on the simulation, see Trahms, M., Melischek, L., Steiner, JF, Mahendru, B., Tamir, I., Bogdanoff, N., Peters, O., Reecht, G., Winkelmann, CB, von Oppen, F., & Franke, KJDiode effect in Josephson junctions with a single magnetic atom, Nature 615, 628 (2023).

[0098] (4) Micromagnetic simulation. Micromagnetic simulation was performed using MuMax3 code. The material parameters of the permalloy were: saturation magnetization Ms = 8.6 × 10⁵ A / m, exchange stiffness Aex = 13 pJ, and damping factor α = 0.01. After the magnetization degraded to the fixed state (ground state), the z-component of the demagnetization field Bdemag and the MFM image at 50 nm above the nanomagnet were extracted to plot the magnetic charge diagram.

[0099] This invention has demonstrated that introducing spatial inversion antisymmetry and antisymmetry breaking can produce unconventional superconducting properties. Specifically, the introduction of spatial inversion antisymmetry leads to the emergence of flux diodes with non-magnetic chirality and anisotropy. More notably, intentionally breaking this asymmetry has proven to be an effective strategy for generating odd-symmetry magnetic transport effects, resulting in reversible field-driven diodes. These findings provide insights into properties related to inversion antisymmetry and antisymmetry breaking. The observed field-driven non-reciprocal magnetic transport effects have potential applications in various (quasi-)particle systems beyond superconducting flux. These examples include magnetic colloids, skyrmions, and fluxes, whose polarities can be controlled by magnetic fields. Furthermore, by extending the operation to electric (quasi-)particles, introducing unique potentials with antisymmetry breaking can open the possibility of studying innovative transport phenomena. This can be achieved by fabricating nanopatterned gated electrodes in two-dimensional materials such as graphene, whose charge polarities (electrons and holes) can be tuned using electric fields. This will pave the way for the discovery of new principles and functions, thereby driving future advancements in electronics.

[0100] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for generating a superconducting flux diode in an artificial spin ice medium, characterized in that: By breaking the spatial inversion antisymmetry, a superconducting odd-symmetric magnetic transport effect is generated. This odd-symmetric magnetic transport effect is driven by a magnetic field and can be non-volatilely controlled, thereby generating a programmable magnetic field-driven superconducting flux diode. The superconducting flux diode consists of a superconducting Mo0.79Ge0.21 microbridge with a square artificial spin ice medium pattern. The microbridge includes four parts: a regular symmetrical nanomagnet part, an asymmetrical nanomagnet part, an asymmetrical inverted nanomagnet part, and a reference part without any nanomagnets. The asymmetrical nanomagnets are in the shape of isosceles trapezoids. The fabrication process of the superconducting flux diode is as follows: MoGe thin film is sputtered onto a silicon substrate with an oxide layer, microbridges are fabricated using standard photolithography, and then artificial spin ice medium composed of permalloy nanomagnets is fabricated using electron beam exposure and electron beam evaporation. Superconducting devices are created by using an artificial spin ice medium composed of asymmetric nanomagnets to break the antisymmetry of the potential barrier and achieve odd-symmetric magnetic transport effects. The narrow end of the asymmetric nanomagnet has a larger magnetic charge than the wide end. Since they come from the same nanomagnet, the positive and negative magnetic charges produce equal total flux. The smaller magnetic charge produces a more localized and stronger pinning force than the larger magnetic charge. The magnetic potential mismatch between the positive and negative magnetic charges breaks the antisymmetry of the pinning potential through P(-r) ≠ -P(r), and the asymmetric superconductivity with respect to the magnetic field appears, leading to the odd-symmetric magnetic transport effect.

2. The method for generating a superconducting flux diode in an artificial spin ice medium according to claim 1, characterized in that: Odd-symmetric magnetic transport effect is observed through the critical current I under positive and negative fields. c Regarding the characterization using magnetic field curves, I c The odd components are represented as: 。 3. The method for generating a superconducting flux diode in an artificial spin ice medium according to claim 2, characterized in that: When the magnetization state reverses, the polarity of the magnetic charge switches, but its magnitude and intensity remain unchanged, in an artificially spin-ice medium that is demagnetized. The sign of the magnet is reversed, thus realizing an in-situ reversible magnetic field driven superconducting flux diode.

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