Takehito Yokoyama
We investigate a superconducting state irradiated by a laser beam with spin and orbital angular momentum. It is shown that superconducting vortices are created by the laser beam due to heating effect and transfer of angular momentum of light. Possible experiments to verify our prediction are also discussed.
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We comment on the work by Moler, et al., [Science 279, 1193 (1998)] concerning determination of the c-axis penetration depth in the single-layer compound Tl-2201 through measuring the magnetic flux due to isolated interlayer Josephson vortices.
Nucleation of vortices in a superconductor below the first critical field can be assisted by transverse sound in the GHz frequency range. Vortices will enter and exist the superconductor at the frequency of the sound. We compute the threshold parameters of the sound and show that this effect is within experimental reach.
The screening currents induced in a superconducting film by a magnetic annulus whose magnetization is perpendicular to the superconductor are calculated. We show that close to the superconductor transition temperature $T_c$ particular values of the magnetization and radii of the annulus make the creation of superconducting vortices energy favorable. We also show that the magnetic annulus offers an alternative tool for vortex pinning in the superconductor. Statistical mechanical properties of this system are discussed.
Vortices in a narrow superconducting strip with a square array of pinning sites are studied. The interactions of vortices with other vortices and with external sources (applied magnetic field and transport current) are calculated via a screened Coulomb model. The edge barrier is taken into account and shown to have an important role on the system dynamics. Numerical simulations in this approach show that the field dependent magnetic moment presents peaks corresponding to history dependent configurations of the vortex lattice. Some effects of the edge barrier on the I-V characteristics are also…
The interaction between a superconducting vortex or antivortex in a superconducting film and a magnetic dipole with in- or out-of-plane magnetization is investigated within the London approximation. The dependence of the interaction energy on the dipole-vortex distance and the film thickness is studied and analytical results are obtained in limiting cases. We show how the short range interaction with the magnetic dipole makes the co-existence of vortices and antivortices possible. Different configurations with vortices and antivortices are investigated.
We demonstrate a unique prospect for inducing anisotropic vortex pinning and manipulating the directional motion of vortices using the stripe domain patterns of a uniaxial magnetic film in a the superconducting/ferromagnetic hybrid. Our observations can be described by a model, which considers interactions between magnetic charges of vortices and surface magnetic charges of domains resulting in the enhanced pinning of vortices on domain walls.
Pancake vortices in stacks of thin superconducting films or layers are considered. It is stressed that in the absence of Josephson coupling topological restrictions upon possible configurations of vortices are removed and various examples of structures forbidden in bulk superconductors are given. In particular, it is shown that vortices may skip surface layers in samples of less than a certain size R_c which might be macroscopic. The Josephson coupling suppresses R_c estimates.
In the present paper we develop an algorithm to solve the time dependent Ginzburg-Landau (TDGL) equations, by using the link variables technique, for circular geometries. In addition, we evaluate the Helmholtz and Gibbs free energy, the magnetization, and the number of vortices. This algorithm is applied to a circular sector. We evaluate the superconduting-normal magnetic field transition, the magnetization, and the superconducting density. Furthermore, we study the nucleation of giant and multi-vortex states for that geometry.
Here we describe a development of computer algorithm to simulate the Time Dependent Ginzburg-Landau equation (TDGL) and its application to understand superconducting vortex dynamics in confined geometries. Our initial motivation to get involved in this task was trying to understand better our experimental measurements on dynamics of superconductors with vortices at high frequencies leading to microwave stimulated superconductivity due to presence of vortex (Lara, et al., Scientific Reports, 59187 (2015)).
We show in this paper that it is possible to induce superconducting vortices in a gold nanowire connected to superconducting electrodes. The gold nanowire acquires superconductivity by the proximity effect. The differential magnetoresistance of the nanowire beyond a critical magnetic field shows uniform oscillations with increasing field with a period of \phi0/(2\pir^2) (\phi0 = h/2e is the superconducting flux quantum, r = 35 nm is the radius of the nanowire). We demonstrate that these periodic oscillations are the signatures of the sequential generation and moving of vortices across the gold…
On the basis of the time-dependent Ginzburg-Landau equation we studied the dynamics of the superconducting condensate in a wide two-dimensional sample in the presence of a perpendicular magnetic field and applied current. We could identify two critical currents: the current at which the pure superconducting state becomes unstable ($J_{c2}$ \cite{self1}) and the current at which the system transits from the resistive state to the superconducting state ($J_{c1}<J_{c2}$). The current $J_{c2}$ decreases monotonically with external magnetic field, while $J_{c1}$ exhibits a maximum at $H^*$. For suf…
In this article an analytical and numerical study of superconducting thin film with ferromagnetic textures of elliptical geometries in close proximity is presented. The screening currents induced in the superconductor due to the magnetic texture are calculated. Close to the superconducting transition temperature $T_c$ the spontaneous creation of superconducting vortices becomes energy favorable depending on the value of the magnetization and the geometrical quantities of the magnetic texture. The creation of vortices by elliptic dots is more energy favorable than those created by circular ones…
Manipulating individual vortices in a deterministic way is challenging, ideally, manipulation should be effective, local, and tunable in strength and location. Here, we show that vortices interact with strain fields generated by mechanical stress. We utilized this interaction to move individual vortices in thin superconducting films via local mechanical contact, without magnetic field or current. We used a scanning superconducting quantum interference device (SQUID) to image vortices and to apply local vertical stress with the tip of our sensor. Vortices were attracted to the contact point, re…
Interactions between vortices in planar superconducting films and a point magnetic dipole placed outside the film, and the creation of vortices by the dipole, are studied in the London limit. The exact solution of London equations for films of arbritrary thickness with a generic distribution of vortex lines, curved or straight, is obtained by generalizing the results reported by the author and E.H. Brandt (Phys. Rev. B {\bf 61}, 6370 (2000)) for films without the dipole. From this solution the total energy of the vortex-dipole system is obtained as a functional of the vortex distribution. The …
We address a recent experiment in which a strong decrease of the resistance of a superconducting film has been observed when a remote unbiased gate was placed above the film. Here we explain the experimental finding as a suppression of the vortex tunneling due to the Orthogonality Catastrophe of the electrons inside the gate. We interpret the change in the resistance of the film as a "metal-insulator" transition in the system of vortices induced by the gate.
We numerically solve the coupled Landau-Ginzburg-Maxwell equations for a model of a spin triplet px + ipy superconductor in which whole or half-quanta of flux thread through a hole. We recover the pattern of stable and unstable regions for the half-flux quanta observed in a recent experiment.
Microwave resonators with high quality factors have enabled many recent breakthroughs with superconducting qubits and photon detectors, typically operated in shielded environments to reduce the ambient magnetic field. Insufficient shielding or pulsed control fields can introduce vortices, leading to reduced quality factors, although increased pinning can mitigate this effect. A narrow slot etched into the resonator surface provides a straightforward method for pinning enhancement without otherwise affecting the resonator. Resonators patterned with such a slot exhibited over an order of magnitu…
Giant vortices with higher phase-winding than $2π$ are usually energetically unfavorable, but geometric symmetry constraints on a superconductor in a magnetic field are known to stabilize such objects. Here, we show via microscopic calculations that giant vortices can appear in intrinsically non-superconducting materials, even without any applied magnetic field. The enabling mechanism is the proximity effect to a host superconductor where a current flows, and we also demonstrate that antivortices can appear in this setup. Our results open the possibility to study electrically controllable topo…
Spontaneous nucleation and the consequent penetration of vortices into thin superconducting films and wires, subjected to a magnetic field, can be considered as a nonlinear stage of primary instability of the current-carrying superconducting state. The development of the instability leads to the formation of a chain of vortices in strips and helicoidal vortex lines in wires. The boundary of instability was obtained analytically. The nonlinear stage was investigated by simulations of the time-dependent generalized Ginzburg-Landau equation.
Anomalous Coherence Length of Majorana Zero Modes at Vortices in Superconducting Topological Insulators
研究发现超导拓扑绝缘体涡旋处马约拉纳零能模的相干长度由拓扑绝缘体本征性质决定、与超导序参量无关，揭示其与p波拓扑超导体的拓扑差异。
We propose a simple experiment to determine whether vortices persist above the superconducting transition temperature Tc in the pseudogap phase of high temperature cuprate superconductors. This involves using a magnetic dot to stabilize a vortex in a thin cuprate film beneath the dot. We calculate the magnetic field profile as a function of distance from the dot if a vortex is present, and discuss possible measurements that could be done to detect this. Finally, we comment on the temperature range where a stable vortex should be observable.
A scanning micro superconducting quantum interference device (microSQUID) microscope is used to directly image vortices in a superconducting Al thin film. We observe the temperature dependence of the vortex distribution in a regular defect (hole) array patterned into the Al film. The first direct observation of the localized superconducting state around the holes is shown as well as the effect of the hole size on nucleation of the superconducting state.
We have studied two nanomagnet systems with strong (Co/Pd multilayers) and weak (NdCo alloy films) stray magnetic fields by probing the out-of-plane magnetic states with superconducting vortices. The hybrid samples are made of array of nanomagnets embedded in superconducting Nb thin films. The vortex motion detects relevant magnetic state features, since superconducting vortices are able to discriminate between different magnetic stray field strengths and directions. The usual matching effect between the superconducting vortex lattice and the periodic pinning array can be quenched by means of …
The paper by G. R. Berdiyorov, M. V. Milosevic, and F. M. Peeters [Phys. Rev. B 81, 144511 (2010)] studies theoretically the dynamic properties of a superconducting loop. The authors claim that their consideration of asymmetric loop relates to our experimental results in this field. We point out that this claim is incorrect and explain shortly the true paradoxical essence of the results of our measurements of asymmetric superconducting loop.
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 revisit the critical penetration of Pearl vortices in narrow superconducting flat rings cooled in magnetic fields. Scanning superconducting quantum interference device microscopy measurements showed how magnetic field penetrates and vortices are trapped in flat rings made of amorphous MoGe thin films. Counting the number of trapped vortices for each image, we found that the vortices are completely excluded from the ring annulus when the applied field $H$ is below a threshold field $H_{\rm{p}}$: Above this field, the vortices increase linearly with field. The obtained values of $H_{\rm{p}}$ …
Quantized magnetic vortices driven by electric current determine key electromagnetic properties of superconductors. While the dynamic behavior of slow vortices has been thoroughly investigated, the physics of ultrafast vortices under strong currents remains largely unexplored. Here we use a nanoscale scanning superconducting quantum interference device to image vortices penetrating into a superconducting Pb film at rates of tens of GHz and moving with velocities up to tens of km/s, which are not only much larger than the speed of sound but also exceed the pair-breaking speed limit of supercond…
A plasma wave propagates inside an anisotropic superconducting film sandwiched between two semi-infinite non-conducting bounding dieletric media. Along the c-axis, perpendicular to the film surfaces, an external magnetic field is applied. We show how vortices, known to cause dissipation and change the penetration depth, affect the propagative mode. We obtain the complex wave number of this mode and, using YBCO at 4 K as an example, determine a region where vortex contribution is dominant and dissipation is small.
We have investigated the confinement of 3-D vortices in specific cases of Type-II ($κ= 2$) nano-superconducting devices. The emergent pattern of vortices greatly depends on the orientation of an applied magnetic field (transverse or longitudinal), and the size of the devices (a few coherence lengths $ξ$). Herein, cylindrical geometries are examined. The surface barriers become very significant in these nano-systems, and hence the characteristics of the vortices become highly sensitive to the shape of the system and direction of an applied field. It is observed that nano-cylindrical superconduc…
The superconducting state is formed by the condensation of a large number of Cooper pairs. The normal state electronic properties can give significant influence on the superconducting state. For usual type-II superconductors, the vortices are cylinder like with a round cross-section. For many two dimensional superconductors, such as Cuprates, 2H-NbSe$_2$ etc., albeit the in-plane anisotropy, the vortices generally have a round shape. In this paper we report results based on the scanning tunneling microscopy/spectroscopy measurements on a newly discovered superconductor Ta$_4$Pd$_3$Te$_{16}$. T…
Vortices in superconductors driven at microwave frequencies exhibit a response related to the interplay between the vortex viscosity, pinning strength, and flux creep effects. At the same time, the trapping of vortices in superconducting microwave resonant circuits contributes excess loss and can result in substantial reductions in the quality factor. Thus, understanding the microwave vortex response in superconducting thin films is important for the design of such circuits, including superconducting qubits and photon detectors, which are typically operated in small, but non-zero, magnetic fie…
Spin-orbit coupling (SOC) is known to play an important role in superconductor/ferromagnet heterostructures. Here we demonstrate that SOC results in the spontaneous generation of vortices in an \textit{s}-wave superconductor placed below a ferromagnetic metal with intrinsic or interfacial SOC. By using the generalized London and Ginzburg-Landau theories, we calculate the supercurrent amplitude in various types of S/F heterostructures at and below the superconducting critical temperature. In particular, we show that even in the limit of weak SOC, vortex-antivortex pairs are stabilized and that …
Inspired by the seminal, ground-breaking work of Abrikosov in 1957, we developed a new approximation to the interaction between two widely separated superconducting vortices. In contrast with Abrikosov's, we take into account the finite size of the vortices and their internal magnetic profile. We consider the vortices to be embedded within a superconducting, infinitely long hollow cylinder, in order to simplify the symmetry and boundary conditions for the mathematical analysis. We study this system in the context of a magnetic Ginzburg-Landau functional theory, by solving for the magnetic fiel…
Vortices play a crucial role in determining the properties of superconductors as well as their applications. Therefore, characterization and manipulation of vortices, especially at the single vortex level, is of great importance. Among many techniques to study single vortices, scanning tunneling microscopy (STM) stands out as a powerful tool, due to its ability to detect the local electronic states and high spatial resolution. However, local control of superconductivity as well as the manipulation of individual vortices with the STM tip is still lacking. Here we report a new function of the ST…
We consider a thin superconducting film with a magnetic dot with permanent magnetization (normal to the film) placed on it by a method based on London-Maxwell equations. For sufficiently high dot magnetization a single vortex appears in the ground state. Further increase of magnetization is accompanied with the appearance of antivortices and more vortices in the film. We study analytically conditions for the appearance of a vortex--antivortex pair for a range of parameters. The phase diagram with diversity of vortex--antivortex states is calculated numerically. When appear in the ground state,…
Interaction between a Bloch wall in a ferrite-garnet film and a vortex in a superconductor is analyzed in the London approximation. Equilibrium distribution of vortices formed around the Bloch wall is calculated. The results agree quantitatively with magneto-optical experiment where an in-plane magnetized ferrite-garnet film placed on top of NbSe2 superconductor allows observation of individual vortices. In particular, our model can reproduce a counter-intuitive attraction observed between vortices and a Bloch wall having the opposite polarity. It is explained by magnetic charges appearing due…
We consider a two-dimensional array of ultra-small superconducting grains, weakly coupled by Josephson junctions with large charging energy. We start from an effective action based on a microscopic tunneling Hamiltonian, which includes quasiparticle degrees of freedom, and study the resulting dissipative dynamics of quantum vortices. The equation of motion for a single vortex is deduced, and compared with a commonly adopted phenomenological model.
We show that the entropy per quantum vortex per layer in superconductors in external magnetic fields is bounded by the universal value k_B ln 2, which explains puzzling results of recent experiments on the Nernst effect.
Multi-quanta, or giant, vortices (GVs) are known to appear in very small superconductors near the superconducting transition due to strong confinement of magnetic flux. Here we present evidence for a new, pinning-related, mechanism for the formation of GVs. Using Bitter decoration to visualise vortices in small Nb disks, we show that confinement in combination with strong disorder causes individual vortices to merge into clusters or even GVs well below Tc and Hc2, in contrast to well-defined shells of individual vortices found in the absence of pinning.
In superconductors, the search for special vortex states such as giant vortices focuses on laterally confined or nanopatterned thin superconducting films, disks, rings, or polygons. We examine the possibility to realize giant vortex states and states with non-uniform vorticity on a superconducting spherical nanoshell, due to the interplay of the topology and the applied magnetic field. We derive the phase diagram and identify where, as a function of the applied magnetic field, the shell thickness and the shell radius, these different vortex phases occur. Moreover, the curved geometry allows th…
Ratchet motion of superconducting vortices, which is a directional flow of vortices in superconductors, is highly useful for exploring quantum phenomena and developing superconducting devices, such as superconducting diode and microwave antenna. However, because of the challenges in the quantitative characterization of the dynamic motion of vortices, a phase diagram of the vortex ratchet motion is still missing, especially in the superconductors with low dimensional structures. Here we establish a quantitative phase diagram of the vortex ratchet motion in a highly anisotropic superlattice supe…
We report magneto-transport measurements on a superconducting molybdenum-germanium (MoGe) film of thickness $d$=50 nm in parallel magnetic fields and show evidence of a transition from a Meissner state to a resistive state of spontaneous perpendicular vortices generated by thermal fluctuations above a certain temperature $T>T_v(B)$. Here $T_v$ appears to match the vortex core explosion condition $d\approx 4.4ξ(T_v)$, where $ξ$ is the coherence length. For $T>T_v$, we observed that a nonlinear current-voltage ($IV$) response (Ohmic at low currents and the power law $V\propto I^β$ at higher $I$)…
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…
When the impurity mean free path is short, only spin-polarized Cooper pairs which are non-locally and antisymmetrically correlated in time may exist in a half-metallic ferromagnet. As a consequence, the half-metal acts as an odd-frequency superconducting condensate. We demonstrate both analytically and numerically that quantum vortices can emerge in half-metals despite the complete absence of conventional superconducting correlations. Because these metals are conducting in only one spin band, we show that a circulating spin supercurrent accompanies these vortices. Moreover, we demonstrate that…
Direct visualization of quasiparticle concentration around superconducting vortices
本文利用扫描隧道噪声显微镜直接可视化超导涡旋周围准粒子浓度，提出原子尺度非侵入检测方法，并警示涡旋型 Majorana 量子比特中涡旋间距减小会导致准粒子浓度快速上升。
We analyze the vortex dynamics in superconducting thin films with a periodic array of pinning centers. In particular, we study the effect of anisotropy for a Kagomé pinning network when longitudinal and transverse transport currents are applied. By solving the equations of motion for the vortex array numerically at zero temperature we find different phases for the vortex dynamics depending on the pinning and driving force. An unusual sequence of peaks along and perpendicular to the main hexagonal lattice axes is observed for the differential resistance reflecting the anisotropy of the transpor…
Lorentz force on superconducting vortices near line defects
本研究针对第二类超导体中线缺陷附近的涡旋洛伦兹力问题，结合解析与快速傅里叶变换数值模拟方法，揭示了线缺陷对涡旋受力的增强规律，为缺陷工程超导体的涡旋动力学研究提供了方法支撑。
We study experimentally and theoretically, the reorganization of superconducting vortices driven by oscillatory forces near the plastic depinning transition. We show that the system can be taken to configurations that are tagged by the shaking parameters but keep no trace of the initial conditions. In experiments performed in $NbSe_2$ crystals, the periodic drive is induced by ac magnetic shaking fields and the overall order of the resulting configuration is determined by non invasive ac susceptibility measurements. With a model of interacting particles driven over random landscapes, we perfor…
In superconducting thin films, engineered lattice of antidots (holes) act as an array of columnar pinning sites for the vortices and thus lead to vortex matching phenomena at commensurate fields guided by the lattice spacing. The strength and nature of vortex pinning is determined by the geometrical characteristics of the antidot lattice (such as the lattice spacing $a_0$, antidot diameter $d$, lattice symmetry, orientation, etc) along with the characteristic length scales of the superconducting thin films, viz., the coherence length ($ξ$) and the penetration depth ($λ$). There are at least tw…
We investigate the electronic properties of type-II superconducting Nb(110) in an external magnetic field. Scanning tunneling spectroscopy reveals a complex vortex shape which develops from circular via coffee bean-shaped to elliptical when decreasing the energy from the edge of the superconducting gap to the Fermi level. This anisotropy is traced back to the local density of states of Caroli-de-Gennes-Matricon states which exhibits a direction-dependent splitting. Oxidizing the Nb(110) surface triggers the transition from the clean to the dirty limit, quenches the vortex bound states, and lea…
We use a scanning nanometer-scale superconducting quantum interference device (SQUID) to image individual vortices in amorphous superconducting MoSi thin films. Spatially resolved measurements of the magnetic field generated by both vortices and Meissner screening satisfy the Pearl model for vortices in thin films and yield values for the Pearl length and bulk penetration depth at 4.2 K. Flux pinning is observed and quantified through measurements of vortex motion driven by both applied currents and thermal activation. The effects of pinning are also observed in metastable vortex configuration…
Images of flux vortices in superconductors acquired by transmission electron microscopy should allow a quantitative determination of their magnetic structure but so far, only visual comparisons have been made between experimental images and simulations. Here, we make a quantitative comparison between Fresnel images and simulations based on the modified London equation to investigate the magnetic structure of flux vortices in MgB2. This technique gives an absolute, low-field (~30 Oe) measurement of the penetration depth from images of single vortices. We found that these simulations gave a good…
We have investigated the intermediate mixed state of a superconducting niobium sample by Very Small Angle Neutron Scattering.We show that this state is stabilized through a sequence where a regular vortex lattice appears, which then coexists with vortex clusters before vanishing at low temperature. Vortices in clusters have a constant periodicity regardless of the applied field, exhibit a temperature dependence close to the one of the penetration depth. The clusters disappear in the high temperature limit. All the results agree with an explanation in terms of vortex attraction due non local ef…
Short-range columnar defects essentially influence the magnetic properties of a mesoscopic superconducting disc.They help the penetration of vortices into the sample, thereby decrease the sample magnetization and reduce the upper critical field. Even the presence of weak defects split a giant vortex state (usually appearing in a clean disc in the vicinity of the transition to a normal state) into a number of vortices with smaller topological charges. In a disc with a sufficient number of strong enough defects vortices are always placed onto defects. The presence of defects lead to the appearan…
We discuss a recent experiment in which the resistance of a superconducting film has been measured in magnetic field. A strong decrease of the superconducting film resistance has been observed when a metallic gate is placed above the film. We study how the magnetic coupling between vortices in a thin superconducting film and electrons in a remote unbiased gate suppresses the tunneling rate of the vortices. We examine two general approaches to analyze tunneling in the presence of slow low-energy degrees of freedom: the functional-integral and scattering formalisms. In the first one, the respons…
Here we report a study of vortex states in a thin superconducting film with a magnetic dot grown upon it by means of a method based on London-Maxwell equations. Vortices with single quantum flux ($Φ_0 = h c / 2 e$), giant vortices (vortices with multiple flux) as well as antivortices (vortices with negative vorticity) are taken into consideration. It turns out that giant vortices occur, when the dot's size is sufficiently smaller ($R \leq 4.5 ξ$) than the effective penetration depth $Λ$. In the case of a dot with sufficiently large size ($R \geq 6 ξ$), the vortices with single quantum flux dom…
Majorana zero-modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($π$-phase domain walls) are creat…
Using Ginzburg-Landau theory, we have performed detailed studies of vortices in the presence of artificial defect arrays, for a thin film geometry. We show that when a vortex approaches the vicinity of a defect, an abrupt transition occurs in which the vortex core develops a ``string'' extending to the defect boundary, while simultaneously the supercurrents and associated magnetic flux spread out and engulf the defect. Current induced depinning of vortices is shown to be dominated by the core string distortion in typical experimental situations. Experimental consequences of this unusual depinn…
In this paper I show how to calculate the effect of a nearby Pearl vortex or antivortex upon the critical current $I_c(B)$ when a perpendicular magnetic induction $B$ is applied to a planar Josephson junction in a long, thin superconducting strip of width $W$ much less than the Pearl length $Λ= 2λ^2/d$, where $λ$ is the London penetration depth and $d$ is the thickness ($d < λ$). The theoretical results provide a qualitative explanation of unusual features recently observed experimentally by Golod {\it et al.}\cite{Golod10} in a device with a similar geometry.
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