The pairing temperature of superconducting thin films is expected to display, within the Bardeen-Cooper-Schrieffer theory, oscillations as a function of the film thickness. We show that the pattern of these oscillations switches between two different periodicities at a density-dependent value of the superconducting coupling. The transition is most abrupt in the anti-adiabatic regime, where the Fermi energy is less than the Debye energy. To support our numerical data, we provide new analytical expressions for the chemical potential and the pairing temperature as a function of thickness, which only differ from the exact solution at weak coupling by exponentially-small corrections.
We present experiments on single Cooper-pair transistors made of two different superconducting materials. We chose Ti and Al to create an energy gap profile such that the island has a higher gap than the leads, thereby acting as a barrier to quasiparticle tunneling. Our transport measurements demonstrate that quasiparticle poisoning is suppressed in all our TiAlTi structures (higher gap for the island) with clear 2e periodicity observed, whereas full quasiparticle poisoning is observed in all AlTiAl devices (higher gap for the leads) with e periodicity.
We report on the effect of the back-action of a Single Cooper Pair Transistor electrometer (E) on the measurement of charge on the island of a superconducting box (B). The charge is e-periodic in the gate bias of B when E is operated near voltages 2Delta/e or 4Delta/e. We show that this is due to quasiparticle poisoning of B at a rate proportional to the number of quasiparticle tunneling events in E per second. We are able to eliminate this back action and recover 2e charge periodicity using a new measurement method based on switching current modulation of E.
The periodic response of a metallic or a superconducting ring to an external magnetic flux is one of the most evident manifestations of quantum mechanics. It is generally understood that the oscillation period hc/2e in the superconducting state is half the period hc/e in the metallic state, because the supercurrent is carried by Cooper pairs with a charge 2e. On the basis of the Bardeen-Cooper-Schrieffer theory we discuss, in which cases this simple interpretation is valid and when a more careful analysis is needed. In fact, the knowledge of the oscillation period of the current in the ring pr…
The giant vortex states of a multiply connected superconductor, with radius comparable to the penetration depth and the coherence length, are theoretically investigated based on the nonlinear Ginzburg-Landau theory, in which the induced magnetic field by the super-currents is accurately taken into account. The solutions of Ginzburg-Landau equations are found to be actually independent of the angular momentum L in a gauge invariant point of view, provided that the hole is in the center. Different cases with the paramagnetic current, the diamagnetic current, and the coexistence of the above two,…
The magnetic flux dependence of order parameter and supercurrent is studied in a hollow d-wave superconducting cylinder. It is shown that the existence of line nodal quasiparticles in a pure $d_{x^2-y^2}$ pairing state gives rise to an $hc/e$ periodicity in the order parameter and a first-order quantum phase transition for a large system size. We demonstrate that the flux periodicity in the supercurrent is sensitive to the detailed electronic band structure and electron filling factor. In particular, we find that, in cooperation with the increase of the cylinder circumference, the $hc/2e$ peri…
This review article describes the developments over the last 30 months in the thin film growth and junction fabrication of superconducting MgB2, including a brief summary the chemistry and physics of MgB2.
Explicit analytical expressions for conductivity of a superconducting film above and below critical temperature in an arbitrary electric field are derived in the frameworks of the time dependent Ginzburg-Landau theory. It is confirmed that slightly below critical temperature the differential conductivity of superconducting film can become negative for small enough values of electric field. This fact may cause generation of electromagnetic oscillations if the superconducting film is appropriately coupled of with a resonator. Their maximal frequency is proportional to the value of critical tempe…
The presumably unconventional superconductor Beta-FeSe was deposited by radio frequency sputtering and molecular beam epitaxy (MBE) from two elementary sources. Superconducting thin films were grown in (001)-orientation on MgO(100) and YAlO3(010) substrates. The morphology of the samples was studied and directly related to the superconducting properties of the Beta-FeSe thin films. The MBE grown thin films show microcracks down to the substrate and a roughness of about 100~nm. In contrast, sputter deposited superconducting thin films show a smooth surface with almost no precipitates. In both c…
We have studied the electronic transport properties of homogeneously disordered superconducting tantalum thin films in magnetic fields. The films exhibit three distinct transport regimes in the zero temperature limit which we identify as superconducting, metallic, and insulating phases. The metallic phase is unexpected. The transport characteristics of this metallic phase are found to be similar to those of MoGe films and high mobility dilute two-dimensional electrons or holes confined in semiconductor interface or transistor geometry.
Scanning tunneling spectroscopy measurements of superconducting thin lanthanum films grown on a normal metal tungsten substrate reveal an extraordinarily large broadening of the coherence peaks. The observed broadening corresponds to very short electron-like quasiparticle lifetimes in the tunneling process. A thorough analysis considering the different relaxation processes reveals that the dominant mechanism is an efficient quasiparticle relaxation at the interface between the superconducting film and the underlying substrate. This process is of general relevance to scanning tunneling spectros…
The small size vortex-antivortex pairs proliferation in type II superconducting film is considered for the wide interval of temperatures below Tc. The corresponding contribution to free energy is calculated. It is shown that these fluctuations give the main contribution to the heat capacity of the film both at low temperatures and in the vicinity of transition.
Wave functions of low-energy quasiparticle subgap states in d-wave superconducting rings, threaded by Aharonov-Bohm magnetic flux, are found analytically. The respective energies are closest to the midgap position at small magnetic fluxes and deviate from the Fermi surface due to the Doppler shift, produced by the supercurrent. The Doppler-shifted zero-energy states result in a paramagnetic response of the ring at small fluxes. The states exist only for even angular momenta of the center of mass of Cooper pairs, in agreement with recent numerical studies of the problem. This macroscopic quantu…
While the electrodynamic treatment of a superconducting thin film by Yeh et al. [Phys. Rev. B 48, 9861 (1993)] ostensibly has application for the "thin-film limit," their work is shown to employ bulk-like instead of thin-film electrodynamics, and as such obtains an ostensible violation of the first law of thermodynamics and a surface resistance independent of the resistivity. Other theoretical and experimental difficulties are briefly noted.
Whether or not epitaxially grown superconducting films have the same bulk-like superconducting properties is an important concern. We report the structure and the electronic properties of epitaxially grown Ba(Fe1-xCox)2As2 films using scanning tunneling microscopy and scanning tunneling spectroscopy (STS). This film showed a different surface structure, (2sqrtx2sqrt2)R45 reconstruction, from those of as-cleaved surfaces from bulk crystals. The electronic structure of the grown film is different from that in bulk, and it is notable that the film exhibits the same superconducting transport prope…
The in-situ growth of superconducting MgB2 thin films was examined from various perspectives. The paper discusses (1) growth temperature, (2) the effect of excess Mg, (3) the effect of residual gas during growth, (4) the effect of in-situ annealing, (5) thickness dependence and (6) the effect of substrates. Our results provide a guide to the preparation of high-quality superconducting MgB2 films for potential electronics applications.
Precursor MgB2 thin films were prepared on sapphire substrates by magnetron sputtering. Influence of ex-situ annealing process on superconducting MgB2 thin films roughness is discussed. Optimized annealing process of MgB precursor thin films in vacuum results in smooth superconducting MgB2 thin films with roughness below 10 nm, critical temperature Tcon = 31 K and transition width DTc less than 1 K. Nano-bridges based on the superconducting MgB2 thin films using optical and Focused Ion Beam lithography were prepared. Critical current density jc (4.2 K) measured on 50 nm wide strip was 7.3x106 …
This paper presents a general model for calculating the density of states and the Cooper pair potential in proximity coupled superconducting bi- and tri-layer films. The typical example of superconducting tunnel junctions is used to show that the model is working.
Superconducting MgB2 thin films were prepared on Al2O3(0001) and MgO(100) substrates. Boron thin films were deposited by the electron-beam evaporation followed by post-annealing process with magnesium. Proper post annealing conditions were investigated to grow good superconducting MgB2 thin films. The X-ray diffraction patterns showed randomly orientated growth of MgB2 phase in our thin films. The surface morphology was examined by scanning electron microscope (SEM) and atomic force microscope (AFM). Critical current density (Jc) measured by transport method was about 10^7 A/cm^2 at 15 K, and …
In the present work we study an anisotropic layered superconducting film of finite thickness. The film surfaces are considered parallel to the $bc$ face of the crystal. The vortex lines are oriented perpendicular to the film surfaces and parallel to the superconducting planes. We calculate the local field and the London free energy for this geometry. Our calculation is a generalization of previous works where the sample is taken as a semi-infinite superconductor. As an application of this theory we investigate the flux spreading at the superconducting surface.
In this paper we investigate the elastic properties of the vortex lattice for a superconducting film of finite thickness. We derive an analytic expression for the compression modulus. The shear modulus is evaluated numerically by using both the Pearl interaction potential, valid in the limit of very thin film, and a potential for films of arbitrary thickness. A comparative study of the shear moduli is carried out.
It is shown that there are a significant number of quasiparticles present in the superconductor even at the temperature far below the transition temperature and that those quasiparticles seriously contribute to the characteristics of superconducting thin-film resonators.
The phase transition between the intermediate and normal states in type-I superconducting films is investigated using magneto-optical imaging. Magnetic hysteresis with different transition fields for collapse and nucleation of superconducting domains is found. This is accompanied by topological hysteresis characterized by the collapse of circular domains and the appearance of lamellar domains. Magnetic hysteresis is shown to arise from supercooled and superheated states. Domain-shape instability resulting from long-range magnetic interaction accounts well for topological hysteresis. Connection…
We have successfully synthesized the c-axis oriented YBa2Cu4O8 (Y-124) film on a SrTiO3 (1 0 0) substrate via a coprecipitation and a dip-coating method. The precipitations including Y, Ba and Cu ions were obtained using their metal nitrates solution and the aqueous solution of sodium hydroxide and oxalic acid. Superconducting transition of the synthesized film was observed at 64-70 K corresponding to bulk Y-124 in the magnetization and resistance measurement. The developed method is suitable to synthesize superconducting Y-124 films for technological applications.
We fabricated superconducting MgB2 thin films on (001) MgO substrates. The samples were prepared by magnetron rf and dc co-sputtering on heated substrates. They were annealed ex-situ for one hour at temperatures between 450°C and 750°C. We will show that the substrate temperature during the sputtering process and the post annealing temperatures play a crucial role in forming MgB2 superconducting thin films. We achieved a critical onset temperature of 27.1K for a film thickness of 30nm. The crystal structures were measured by x-ray diffraction.
A superconducting substrate is not able to shrink drastically domains in a ferromagnetic film, contrary to the prediction of Bulaevskii and Chudnovsky [Phys. Rev. B, 63, issue1 (2001)]. This is shown on the basis of the exact solution for the stripe domain structure.
Vortex dynamics is strongly connected with the mechanisms responsible for the photon detection of superconducting devices. Indeed, the local suppression of superconductivity by photon absorption may trigger vortex nucleation and motion effects, which can make the superconducting state unstable. In addition, scaling down the thickness of the superconducting films and/or the width of the bridge geometry can strongly influence the transport properties of superconducting films, e.g. affecting its critical current as well as its switching current into the normal state. Understanding such instabilit…
We propose a mechanism that helps stabilize a superconducting state with broken time-reversal symmetry, which was predicted to realize in a d-wave superconducting film [A. B. Vorontsov, Phys. Rev. Lett. 102, 177001 (2009)]. In this superconducting phase, the time-reversal symmetry breaking is accompanied by spontaneous breaking of the translation symmetry along the film surface. We examine how the normal-superconducting phase boundary in the thickness-temperature phase diagram of the film is modified depending on the Fermi surface shape. In particular, the nonuniform superconducting phase is f…
ctical superconducting material for power-electric products such as superconducing tapes because of its high superconducting transition temperature (Tc) of 39 K and its high critical current density (Jc) under magnetic fields. Superconducting films of MgB2 have been, however, fabricated only on planar substrates with areas of several tens of a square centimeter at the largest so far due to the technical limitation of the conventional fabrication methods. In this report, we describe a technique for the fabrication of MgB2 films by means of electrolysis on molten compounds of Mg and B, that is, …
The propagation of a normal zone in a long superconducting film on a dielectric or conducting substrate is analyzed analytically. The obtained expressions for the propagation velocity allow one to analyze the effect of the thermal resistance between the film and substrate and the width of the substrate. The results are expended into the cases of the normal zone propagation in a meander and in parallel films and can be approximately applied to multifilamentary coated superconductors.
We report on the observation of the rearrangement of vortices in the ns time scale following a fast magnetic field variation due to selective local heating of a high Tc superconducting film. Detailed simulations describing the measured photoinduced voltage signals along the film lead to a better understanding of vortex dynamics in fast regimes, where mean vortex velocities reach 10000 m/s
The precondition for the BKT transition in thin superconducting films, the logarithmic intervortex interaction, is satisfied at distances short relative to $Λ=2λ^2/d$, $λ$ is the London penetration depth of the bulk material and $d$ is the film thickness. For this reason, the search for the transition has been conducted in samples of the size $L<Λ$. It is argued below that film edges turn the interaction into near exponential (short-range) thus making the BKT transition impossible. If however the substrate is superconducting and separated from the film by an insulated layer, the logarithmic in…
Superconducting thin films of MgB2 were deposited by Pulsed Laser Deposition on magnesium oxide and sapphire substrates. Samples grown at 450C in an argon buffer pressure of about 10-2 mbar by using a magnesium enriched target resulted to be superconducting with a transition temperature of about 25 K. Film deposited from a MgB2 sintered pellet target in ultra high vacuum conditions showed poor metallic or weak semiconducting behavior and they became superconducting only after an ex-situ annealing in Mg vapor atmosphere. Up to now, no difference in the superconducting properties of the films ob…
Superconducting thin-film rings smaller than the film penetration depth (the Pearl length) are considered. The current distribution, magnetic moment, and thermodynamic potential ${\cal F}(H,N,v)$ for a flat, washer-shaped annular ring in a uniform applied field $H$ perpendicular to the film are solved analytically within the London approach for a state with winding number $N$ and a vortex at radius $v$ between the inner and outer radii.
The authors of the paper indicated in the title derived an erroneous effective free energy of superconducting slabs (films) and this error led them to entirely wrong conclusions. Here we shall make clear the genesis of this mistake.
We investigated the microwave conductivity spectrum of FeSe$_{0.5}$Te$_{0.5}$ epitaxial films on CaF$_2$ in the vicinity of the superconducting transition. We observed the critical behavior of the superconducting fluctuations in these films with a dimensional crossover from two-dimensional to three-dimensional as the film thickness increased. From the temperature dependence of the scaling parameters we conclude that the universality class of the superconducting transition in FeSe$_{0.5}$Te$_{0.5}$ is that of the 3D-XY model. The lower limit of the onset temperature of the superconducting fluct…
We obtain the vortex configurations, the matching fields and the magnetization of a superconducting film with a finite cross section. The applied magnetic field is normal to this cross section, and we use London theory to calculate many of its properties, such as the local magnetic field, the free energy and the induction for the mixed state. Thus previous similar theoretical works, done for an infinitely long superconducting film, are recovered here, in the special limit of a very long cross section.
Superconducting MgB2 thin films were fabricated on Al2O3(0001) substrates under an ex-situ processing conditions. Boron thin films were deposited by pulsed laser deposition followed by a post-annealing process. Resistance measurements of the deposited MgB2 films show Tc of about 39K, while scanning electron microscopy and x-ray diffraction analysis indicate that the films consist of well-crystallized grains with a highly c-axis-oriented structure.
We demonstrate that good superconductor/insulator/normal-metal tunnel junctions can be fabricated on as-grown superconducting MgB2 thin films. The as-grown films were prepared by coevaporation at low growth temperatures of around 280deg. The insulating barrier was formed by an Mg overlayer, which is subsequently oxidized in air. The tunneling spectra for Au/MgOx/MgB2 junctions showed a reproducible and well-defined superconducting gap (delta= ~ 2.5 meV). The resultant 2delta/ kBTc was significantly smaller than the predicted BCS value of 3.52.
We study coherent quantum transport through a superconducting film connected to normal-metal electrodes. Simple expressions for the differential conductance and the local density of states are obtained in the clean limit and for transparent interfaces. Quasiparticle interference causes periodic vanishing of the Andreev reflection at the energies of geometrical resonances, subgap transport, and gapless superconductivity near the interfaces. Application of the results to spectroscopic measurements of the superconducting gap and the Fermi velocity is analyzed.
We consider the tunneling Density of States (DoS) of superconducting films driven to the paramagnetic phase by the Zeeman splitting. We show that there is minimum in the DoS whose position depends on the orientation of the applied field. This dependence, not predicted by previous theoretical calculations, is in agreement with a recent experiment.
We investigate transition to the superconducting state in the ultrathin titanium nitride films on approach to superconductor-insulator transition. Building on the complete account of quantum contributions to conductivity, we demonstrate that the resistance of thin superconducting films exhibits a non-monotonic temperature behaviour due to the competition between weak localization, electron-electron interaction, and superconducting fluctuations. We show that superconducting fluctuations give rise to an appreciable decrease in the resistance even at temperatures well exceeding the superconductin…
Thin superconducting films form a unique platform for geometrically-confined, strongly-interacting electrons. They allow an inherent competition between disorder and superconductivity, which in turn enables the intriguing superconducting-to-insulator transition and believed to facilitate the comprehension of high-Tc superconductivity. Furthermore, understanding thin film superconductivity is technologically essential e.g. for photo-detectors, and quantum-computers. Consequently, the absence of an established universal relationships between critical temperature ($T_c$), film thickness ($d$) and…
Transition-metal nitrides (e.g., TiN, ZrN, TaN) are incredible materials with excellent complementary-metal-oxide-semiconductor compatibility and remarkable performance in refractory plasmonics and superconducting quantum electronics. Epitaxial growth of flexible transition-metal nitride films, especially at wafer-scale, is fundamentally important for developing high-performance flexible photonics and superconducting electronics, but the study is rare thus far. This work reports the high-quality epitaxy of 2-inch titanium nitride (TiN) films on flexible fluorophlogopite-mica (F-mica) substrate…
We have studied the Hall effect in superconducting tantalum nitride films. We find a large contribution to the Hall conductivity near the superconducting transition, which we can track to temperatures well above $T_c$ and magnetic fields well above the upper critical field, $\text{\text{H}}_{c2}(0)$. This contribution arises from Aslamazov-Larkin superconducting fluctuations, and we find quantitative agreement between our data and recent theoretical analysis based on time dependent Ginzburg-Landau theory.
The temperature dependences of the enhanced critical current in wide and thin Sn films exposed to the microwave field have been investigated experimentally and analyzed. It was found that the microwave field stabilizes the current state of a wide film with respect to the entry of Abrikosov vortices. The stabilizing effect of irradiation increases with frequency. Using similarity between the effects of microwave enhancement of superconductivity observed for homogeneous (narrow films) and inhomogeneous (wide films) distributions of the superconducting current over the film width, we have succeed…
A one-dimensional equation describing the distribution of the effective vector potential $\bar A(y)$ across a film width, which holds for thin ($d<λ$) and thick ($d>λ$) films alike, is derived based on the analysis of a 2D Maxwell-Londons equation for superconducting films in a perpendicular magnetic field. The validity of this equation for a finite-thickness film is verified by a numerical analysis. An approximation dependence $\bar A(y)$, finite (with all of its derivatives) across the entire film width, is found for films, being in the Meissner state. The flux-entry field is evaluated for a…
By using multi-bands BCS theory, we have calculated the superconductivity energy gap and the critical temperature of a thin-film metallic superconductor. The thermodynamic superconducting characteristics such as critical magnetic field, specific heat, as well as the tunneling conductance are investigated for varying film thickness and temperature. We find the oscillation of thermodynamic superconducting properties with the film thickness, including the thermodynamic critical field $H_c$, the specific heat of normal, superconducting state, and position of the differential tunneling conductance …
Thin films of Fe(1+delta)Se(1-x)Te(x) (delta ~ 0.18 & x ~ 0.5) have been successfully grown on (100) oriented single crystalline SrTiO3 and LaAlO3 substrates by pulsed laser deposition. The crystal structure was characterized by x-ray diffraction, and the superconducting properties by electrical resistivity measurements. X-ray diffraction analysis establishes the epitaxial growth of the films with c-axis orientation. Atomic force microscopy showed a smooth surface morphology for the films grown on SrTiO3 substrates. All the films are observed to be superconducting with a Tc of ~ 8-14 K dependi…
We develop an explicitly gauge-invariant semiclassical approach to investigate nonlinear response of superconductors to monochromatic THz radiation. We demonstrate that in clean superconductors charge conservation forbids non-linear response to a uniform field. We apply our approach to quasi-two-dimensional films and obtain an explicit expression for the photoinduced current. We find that the photoinduced current exhibits a strong dependence on polarization and incidence angle of the radiation. Our predictions may be directly verified in experiments with quasi-two-dimensional superconducting f…
The vortex-state electrodynamics of s-wave superconductors has been studied by infrared spectroscopy. Far-infrared transmission and reflection spectra of superconducting NbTiN and NbN thin films were measured in a magnetic field perpendicular to the film surface, and the optical conductivity was extracted. The data show clear reduction of superconducting signature. We consider the vortex state as a two-component effective medium of normal cores embedded in a BCS superconductor. The spectral features are well explained by the Maxwell-Garnett theory. Our analysis supports the presence of magneti…
A simple method has been developed for manufacturing a thin film superconducting quantum interferometer (SQI) with ultralow inductance (~10^-13 H). Current-voltage and voltage-field characteristics of the SQI are presented. The basic design equations are obtained and confirmed experimentally. The SQI has been used for the first time to determine the penetration depth of a magnetic field into a film of 50% In-50% Sn alloy.
We demonstrate that films of unconventional (d-wave) superconductor at temperatures $T\lesssim 0.43 T_c$ can exhibit unusual superconducting phases. The new ground states beside the broken gauge and the point group symmetries can spontaneously break (i) continuous translational symmetry and form periodic order parameter structures in the plane of the film, or (ii) time-reversal symmetry and develop supercurrent flowing along the film. These states are result of the strong transverse inhomogeniety present in films with thickness of several coherence lengths. We show a natural similarity between…
The magnetic field of vortices in anisotropic superconducting films is considered in the framework of anisotropic London approach. It is found that at distances large relative to the core size, the magnetic field normal to the film surface may change sign. We find that the magnetic field attenuates at large distances as $1/r^3$ as it does in isotropic films, but the anisotropy induces an angular dependence to the supercurrents which causes the sign of the field to change for anisotropy parameters $γ=λ_2/λ_1>\sqrt{2}$ in some parts of the $(x,y)$ plane.
We have fabricated high-quality FeSe1-x superconducting films with a bulk Tc of 11-12 K on different substrates, Al2O3(0001), SrTiO3(100), MgO(100), and LaAlO3(100), by using a pulsed laser deposition technique. All the films were grown at a high substrate temperature of 610 oC, and were preferentially oriented along the (101) direction, the latter being to be a key to fabricating of FeSe1-x superconducting thin films with high Tc. According to the energy dispersive spectroscopy data, the Fe:Se composition ratio was 1:0.90+-0.02. The FeSe1-x film grown on a SrTiO3 substrate showed the best qua…
We have investigated the crystal structures and superconducting properties of thin films of FeSe$_{0.5}$Te$_{0.5}$ grown on eight different substrates. Superconductivity is not correlated with the lattice mismatch; rather it is correlated with the degree of in-plane orientation and with the lattice parameter ratio $c/a$. The best superconducting properties were observed in films on MgO and LaAlO$_3$ ($T_\mathrm{c}^\mathrm{zero}$ of 9.5 K). TEM observation showed that the presence or absence of an amorphous-like layer at the substrate surface plays a key role in determining the structural and s…
The results of experimental investigations of the critical currents and certain nonequilibrium phenomena in thin Sn films of different width w are analyzed. Usually, thin superconducting films are divided into two groups: narrow channels w<lambda and wide films w>lambda. A wide transitional region where the condition w>lambda holds with a large margin and at the same time cannot be explained from the standpoint of the theory of the appearance of a vortex state has been found. This shows that the generally accepted criterion w \sim lambda for dividing films into wide and narrow does not work. T…
The Halperin-Lubensky-Ma (HLM) effect of a fluctuation-induced change of the order of phase transition in thin films of type I superconductors with relatively small Ginzburg-Landau number $κ$ is considered. Numerical data for the free energy, the order parameter jump, the latent heat, and the specific heat of W, Al and In are presented to reveal the influence of film thickness and material parameters on the properties of the phase transition. We demonstrate for the first time that in contrast to the usual notion the HLM effect occurs in the most distinct way in superconducting films with high …
We theoretically investigate the response of a superconducting film to line currents flowing in linear wires placed above the film, and we present analytic expressions for the magnetic-field and current distributions based on the critical state model. The behavior of the superconducting film is characterized by the sheet-current density $K_z$, whose magnitude cannot exceed the critical value $j_cd$, where $j_c$ is the critical current density and $d$ is the thickness of the film. When the transport current $I_0$ flowing in the wire is small enough, $|K_z|$ is smaller than $j_cd$ and the magnet…
