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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 $\gamma=\lambda_2/\lambda_1>\sqrt{2}$ in some parts of the $(x,y)$ plane., Comment: 8 pages, 12 figures

It is of interest to determine the exit angle of a vortex from a superconducting surface, since this affects the intervortex interactions and their consequences. Two ways to determine this angle are to image the vortex magnetic fields above the surface, or the vortex core shape at the surface. In this work we evaluate the field h(x, y, z) above a flat superconducting surface x, y and the currents J(x,y) at that surface for a straight vortex tilted relative to the normal to the surface, for both the isotropic and anisotropic cases. In principle, these results can be used to determine the vortex exit tilt angle from analyses of magnetic field imaging or density of states data., Comment: 5 pages, 4 figures

A magnetic domain boundary on the surface of a three-dimensional topological insulator is predicted to host a chiral edge state, but direct demonstration is challenging. Here, we used a scanning superconducting quantum interference device to show that current in a magnetized EuS/Bi2Se3 heterostructure flows at the edge when the Fermi level is gate-tuned to the surface band gap. We further induced micron-scale magnetic structures on the heterostructure, and detected a chiral edge current at the magnetic domain boundary. The chirality of the current was determined by magnetization of the surrounding domain and its magnitude by the local chemical potential rather than the applied current. Such magnetic structures, provide a platform for detecting topological magnetoelectric effects and may enable progress in quantum information processing and spintronics.

Scanning SQUID susceptometry images the local magnetization and susceptibility of a sample. By accurately modeling the SQUID signal we can determine the physical properties such as the penetration depth and permeability of superconducting samples. We calculate the scanning SQUID susceptometry signal for a superconducting slab of arbitrary thickness with isotropic London penetration depth, on a non-superconducting substrate, where both slab and substrate can have a paramagnetic response that is linear in the applied field. We derive analytical approximations to our general expression in a number of limits. Using our results, we fit experimental susceptibility data as a function of the sample-sensor spacing for three samples: 1) delta-doped SrTiO3, which has a predominantly diamagnetic response, 2) a thin film of LaNiO3, which has a predominantly paramagnetic response, and 3) a two-dimensional electron layer (2-DEL) at a SrTiO3/AlAlO3 interface, which exhibits both types of response. These formulas will allow the determination of the concentrations of paramagnetic spins and superconducting carriers from fits to scanning SQUID susceptibility measurements., Comment: 11 pages, 13 figures

We discuss the Meissner response to a known field source of superconductors having inhomogeneities in their penetration depth. We simplify the general problem by assuming that the perturbations of the fields by the penetration depth inhomogeneities are small. We present expressions for inhomogeneities in several geometries, but concentrate for comparison with experiment on planar defects, perpendicular to the sample surfaces, with superfluid densities different from the rest of the samples. These calculations are relevant for magnetic microscopies, such as Scanning Superconducting Quantum Interference Device (SQUID) and Magnetic Force Microscope, which image the local diamagnetic susceptibility of a sample., Comment: 9 pages, 4 figures

In this article, we wish to give a historical and personal account of the advent of our tricrystal phase-sensitive symmetry experiments using scanning SQUID microscope for establishing d-wave pairing symmetry in the cuprate superconductors. On the occasion of commemorating the 100th anniversary of the discovery of superconductivity by Kamerlingh Onnes, we are honored to present this article as part of the Jubilee Celebration., Comment: 19 pages, 10 figures; Invited paper submitted for a book entitled "100 Years of Superconductivity", edited by H. Rogalla and P. Kes; Corresponding author: Chang C. Tsuei (tsuei at us.ibm.com)

We use scanning SQUID microscopy to investigate the behavior of vortices in the presence of twin boundaries in the pnictide superconductor Ba(Fe1-xCox)2As2. We show that the vortices avoid pinning on twin boundaries. Individual vortices move in a preferential way when manipulated with the SQUID: they tend to not cross a twin boundary, but rather to move parallel to it. This behavior can be explained by the observation of enhanced superfluid density on twin boundaries in Ba(Fe1-xCox)2As2. The observed repulsion from twin boundaries may be a mechanism for enhanced critical currents observed in twinned samples in pnictides and other superconductors.

In this review I discuss the application of scanning magnetic imaging to fundamental studies of superconductors, concentrating on three scanning magnetic microscopies - scanning SQUID microscopy (SSM), scanning Hall bar microscopy (SHM), and magnetic force microscopy (MFM). I briefly discuss the history, sensitivity, spatial resolution, invasiveness, and potential future developments of each technique. I then discuss a selection of applications of these microscopies., Comment: 59 pages, 31 figures, to be published in Reports on Progress in Physics

We calculate the change in susceptibility resulting from a thin sheet with reduced penetration depth embedded perpendicular to the surface of an isotropic superconductor, in a geometry applicable to scanning Superconducting QUantum Interference Device (SQUID) microscopy, by numerically solving Maxwell's and London's equations using the finite element method. The predicted stripes in susceptibility agree well in shape with the observations of Kalisky et al. of enhanced susceptibility above twin planes in the underdoped pnictide superconductor Ba(Fe1-xCox)2As2 (Ba-122). By comparing the predicted stripe amplitudes with experiment and using the London relation between penetration depth and superfluid density, we estimate the enhanced Cooper pair density on the twin planes, and the barrier force for a vortex to cross a twin plane. Fits to the observed temperature dependence of the stripe amplitude suggest that the twin planes have a higher critical temperature than the bulk, although stripes are not observed above the bulk critical temperature., Comment: 16 pages, 9 figures

Using tunneling spectroscopy we have measured the spectral density of states of the mobile, two-dimensional electron system generated at the LaAlO3-SrTiO3 interface. As shown by the density of states the interface electron system differs qualitatively, first, from the electron systems of the materials defining the interface and, second, from the two-dimensional electron gases formed at interfaces between conventional semiconductors.

We employed a scanning Hall probe microscope to detect the hopping of individual vortices between pinning sites along grain boundaries in YBCO thin films in the presence of an applied current. Detecting the motion of individual vortices allowed us to probe the current-voltage (I-V) characteristics of the grain boundary with voltage sensitivity below a femto-volt. We find a very sharp onset of dissipation with V~I^n with an unprecedented high exponent of n~290 that shows essentially no dependence on temperature or grain boundary angle. Our data have no straightforward explanation within the existing grain boundary transport models.

Hybrid magnetometers based on a normal conducting sensor and a superconducting flux concentrator have been investigated. When this sensor is operated in an unshielded environment flux vortices can be trapped in the superconducting body when the sensor is cooled. Thermal hopping of the trapped vortices gives rise to 1/f noise. The mechanism for vortex trapping has been investigated. A new model for the critical field for vortex trapping and vortex densities has been derived. Experimental verification on YBa2Cu3O7-delta (YBCO) strips with a Scanning SQUID Microscope shows very good agreement between measurements and the new theory. The results have been applied in a hybrid magnetometer design based on a bismuth Hall sensor and an YBCO flux concentrator. Measurements show a field gain up to G=7 with a large effective range of 175uT.

Vortex trapping is investigated in thin-film strips of superconducting material. We present a model for the critical field above which vortex trapping occurs in these strips. This model includes the pairing energy of vortex-antivortex pairs in addition to the energy of single vortices. Experimental verification of the model with a scanning SQUID microscope shows very good agreement between the model and experiments on YBa2Cu3O7-delta and Nb strips. Statistical analysis of the vortex distribution in the strips above the critical field has been performed and a comparison has been made between Nb and YBa2Cu3O7-delta for the distributions in the lateral and longitudinal directions.

A scanning SQUID microscope was used to image vortex trapping as a function of the magnetic induction during cooling in thin-film YBCO strips for strip widths W from 2 to 50 um. We found that vortices were excluded from the strips when the induction Ba was below a critical induction Bc. We present a simple model for the vortex exclusion process which takes into account the vortex - antivortex pair production energy as well as the vortex Meissner and self-energies. This model predicts that the real density n of trapped vortices is given by n=(Ba-BK)/Phi0 with BK = 1.65Phi0/W^2 and Phi0 = h/2e the superconducting flux quantum. This prediction is in good agreement with our experiments on YBCO, as well as with previous experiments on thin-film strips of niobium. We also report on the positions of the trapped vortices. We found that at low densities the vortices were trapped in a single row near the centers of the strips, with the relative intervortex spacing distribution width decreasing as the vortex density increased, a sign of longitudinal ordering. The critical induction for two rows forming in the 35 um wide strip was (2.89 + 1.91-0.93)Bc, consistent with a numerical prediction.

It is widely believed that the perovskite Sr$_2$RuO$_4$ is an unconventional superconductor with broken time reversal symmetry. It has been predicted that superconductors with broken time reversal symmetry should have spontaneously generated supercurrents at edges and domain walls. We have done careful imaging of the magnetic fields above Sr$_2$RuO$_4$ single crystals using scanning Hall bar and SQUID microscopies, and see no evidence for such spontaneously generated supercurrents. We use the results from our magnetic imaging to place upper limits on the spontaneously generated supercurrents at edges and domain walls as a function of domain size. For a single domain, this upper limit is below the predicted signal by two orders of magnitude. We speculate on the causes and implications of the lack of large spontaneous supercurrents in this very interesting superconducting system., Comment: 9 pages

We have theoretically investigated macroscopic quantum tunneling (MQT) and the influence of nodal quasiparticles and zero energy bound states (ZES) on MQT in s-wave/ d-wave hybrid Josephson junctions. In contrast to d-wave/d-wave junctions, the low-energy quasiparticle dissipation resulting from nodal quasiparticles and ZES is suppressed due to a quasiparticle-tunneling blockade effect in an isotropic s-wave superconductor. Therefore, the inherent dissipation in these junctions is found to be very weak. We have also investigated MQT in a realistic s-wave/d-wave (Nb/Au/YBCO) junction in which Ohmic dissipation in a shunt resistance is stronger than the inherent dissipation and find that MQT is observable within the current experimental technology. This result suggests high potential of s-wave/d-wave hybrid junctions for applications in quantum information devices., Comment: 4 pages, 3 figures

Remarkably rich physics is involved in the behavior of hybrid Josephson junctions, connecting high-Tc and low-Tc superconductors. This relates in particular to the different order parameter symmetries underlying the formation of the superconducting states in these materials. Experiments on high-Tc/low-Tc contacts have also played a crucial role in settling the decade-long d-wave versus s-wave debate in cuprate superconductors. Recently, such hybrid junctions have enabled more detailed pairing symmetry tests. Furthermore, with these junctions, complex arrays of pi-rings have been realized, enabling studies on spontaneously generated fractional flux quanta and their mutual interactions. Steps toward novel superconducting electronic devices are taken, utilizing the phase-shifts inherent to the d-wave superconducting order parameter. This paper is intended to reflect the current status of experiments using high-Tc and low-Tc Josephson contacts., Comment: 25 pages, 10 figures, to appear in "Electron Correlation in New Materials and Nanosystems", NATO Science Series II, Springer, 2006

We have made current-voltage (IV) measurements of artificially layered high-$T_c$ thin-film bridges. Scanning SQUID microscopy of these films provides values for the Pearl lengths $\Lambda$ that exceed the bridge width, and shows that the current distributions are uniform across the bridges. At high temperatures and high currents the voltages follow the power law $V \propto I^n$, with $n=\Phi_0^2/8\pi^2\Lambda k_B T+1$, and at high temperatures and low-currents the resistance is exponential in temperature, in good agreement with the predictions for thermally activated vortex motion. At low temperatures, the IV's are better fit by $\ln V$ linear in $I^{-2}$. This is expected if the low temperature dissipation is dominated by quantum tunneling of Pearl vortices., Comment: 5 pages, 7 figs

We report experiments in which one dimensional (1D) and two dimensional (2D) arrays of YBa2Cu3O7-x-Nb pi-rings are cooled through the superconducting transition temperature of the Nb in various magnetic fields. These pi-rings have degenerate ground states with either clockwise or counter-clockwise spontaneous circulating supercurrents. The final flux state of each ring in the arrays was determined using scanning SQUID microscopy. In the 1D arrays, fabricated as a single junction with facets alternating between alignment parallel to a [100] axis of the YBCO and rotated 90 degrees to that axis, half-fluxon Josephson vortices order strongly into an arrangement with alternating signs of their magnetic flux. We demonstrate that this ordering is driven by phase coupling and model the cooling process with a numerical solution of the Sine-Gordon equation. The 2D ring arrays couple to each other through the magnetic flux generated by the spontaneous supercurrents. Using pi-rings for the 2D flux coupling experiments eliminates one source of disorder seen in similar experiments using conventional superconducting rings, since pi-rings have doubly degenerate ground states in the absence of an applied field. Although anti-ferromagnetic ordering occurs, with larger negative bond orders than previously reported for arrays of conventional rings, long-range order is never observed, even in geometries without geometric frustration. This may be due to dynamical effects. Monte-Carlo simulations of the 2D array cooling process are presented and compared with experiment., Comment: 10 pages, 15 figures

We introduce a new Josephson junction circuit for which quantum operations are realized by low-bandwidth, nearly adiabatic magnetic-flux pulses. Coupling to the fundamental mode of a superconducting transmission line permits a stabilization of the rotation angle of the quantum operation against flux noise. A complete scheme for one-qubit rotations, and high-visibility Ramsey-fringe oscillations, is given. We show that high visibility depends on passing through a portal in the space of applied fluxes, where the width of the portal is proportional to the ramp-up rate of the flux pulse., Comment: 5 pages, pdflatex

The current-voltage characteristics of Superconducting Quantum Interference Devices (SQUIDs) are known to modulate as a function of applied magnetic field with a period of one flux quantum $\Phi_0=h$/2e. Here we report on the fabrication and properties of SQUIDs modulating with a period of $1/2\times\Phi_0$. The characteristics of these bicrystal SQUIDs are consistent with either a strong sin(2$\phi$) component of the current-phase relation of the Josephson current, or with an interaction between the Cooper-pairs, causing an admixture of quartets to the condensate., Comment: 15 pages, 4 Figures

We have used scanning SQUID magnetometry to image vortices in ultra-thin [Ba$_{0.9}$Nd$_{0.1}$CuO$_{2+x}$]$_m$/[CaCuO$_2$]$_n$ (CBCO) high temperature superconductor samples, with as few as three superconducting CuO$_2$ planes. The Pearl lengths ($\Lambda = 2\lambda_{L}^2/d$, $\lambda_{L}$ the London penetration depth, $d$ the superconducting film thickness) in these samples, as determined by fits to the vortex images, agree with those by local susceptibility measurements, and can be as long as 1mm. The in-plane penetration depths $\lambda_{ab}$ inferred from the Pearl lengths are longer than many bulk cuprates with comparable critical temperatures. We speculate on the causes of the long penetration depths, and on the possibility of exploiting the unique properties of these superconductors for basic experiments., Comment: 4 pages, 3 figures. to be published in Physical Review Letters

Although initially quite controversial, it has been widely accepted that the Cooper pairs in optimally doped cuprate superconductors have predominantly dx2-y2 wavefunction symmetry. The controversy has now shifted to whether the high-Tc pairing symmetry changes away from optimal doping. Here we present phase-sensitive tricrystal experiments on three cuprate systems: Y0.7Ca0.3Ba2Cu3O7-x (Ca-doped Y-123), La2-xSrxCuO4 (La-214) and Bi2Sr2CaCu2O8+x (Bi-2212),with doping levels covering the underdoped, optimal and overdoped regions. Our work implies that time-reversal invariant, predominantly dx2-y2 pairing symmetry is robust over a large variation in doping, and underscores the important role of on-site Coulomb repulsion in the making of high-temperature superconductivity., Comment: 4 pages, 4 figures

A detailed investigation of the magnetic response of YBaCuO grain boundary Josephson junctions has been carried out using both radio-frequency measurements and Scanning SQUID Microscopy. In a nominally zero-field-cooled regime we observed a paramagnetic response at low external fields for 45 degree asymmetric grain boundaries. We argue that the observed phenomenology results from the d-wave order parameter symmetry and depends on Andreev bound states., Comment: To be published in Phys. Rev. B

We have measured the dynamics of individual magnetic fluxoids entering and leaving photolithographically patterned thin film rings of the underdoped high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, using a variable sample temperature scanning SQUID microscope. These results can be qualitatively described using a model in which the fluxoid number changes by thermally activated nucleation of a Pearl vortex in, and transport of the Pearl vortex across, the ring wall., Comment: 9 pages, 10 figures, fixed typos

We have observed spontaneous fluxoid generation in thin-film rings of the amorphous superconductor Mo$_3$Si, cooled through the normal-superconducting transition, as a function of quench rate and externally applied magnetic field, using a variable sample temperature scanning SQUID microscope. Our results can be explained using a model of freezout of thermally activated fluxoids, mediated by the transport of bulk vortices across the ring walls. This mechanism is complementary to a mechanism proposed by Kibble and Zurek, which only relies on causality to produce a freezout of order parameter fluctuations., Comment: 4 pages, 3 figures

From an analysis of the specific heat of one of the cuprate superconductors it is shown, that even if a large part of the experimental specific heat associated with the superconducting phase transition is due to fluctuations, this part must be counted when one tries to extract the condensation energy from the data. Previous work by Chakravarty, Kee and Abrahams, where the fluctuation part was subtracted, has resulted in an incorrect estimation of the condensation energy., Comment: 4 pages, 5 encapsulated Postscript figures, uses ReVTeX.sty

We report on the dependence on field and temperature of the critical current of micro-SQUIDs: SQUIDs with diameters as small as 1 micron, using Dayem bridges as weak links. We model these SQUIDs by solving the Ginzburg-Landau equations with appropriate boundary conditions to obtain the supercurrent-phase relationships. These solutions show that the phase drops and depression of the order parameter produced by supercurrent flow are often distributed throughout the micro-SQUID structure, rather than being localized in the bridge area, for typical micro-SQUID geometries and coherence lengths. The resultant highly non-sinusoidal current-phase relationships $I_c(\phi)$ lead to reduced modulation depths and triangular dependences of the micro-SQUID critical currents on applied magnetic flux $I_c(\Phi)$. Our modelling agrees well with our measurements on both Al and Nb micro-SQUIDs., Comment: 7 pages 8 figures

A direct probe of superparamagnetism was used to determine the complete anisotropy energy distribution of Co nanoparticle films. The films were composed of self-assembled lattices of uniform Co nanoparticles 3 nm or 5 nm in diameter, and a variable temperature scanning-SQUID microscope was used to measure temperature-induced spontaneous magnetic noise in the samples. Accurate measurements of anisotropy energy distributions of small volume samples will be critical to magnetic optimization of nanoparticle devices and media., Comment: 4 pages, 4 figures. Submitted to Physical Review Letters

We have measured the dynamics of individual magnetic fluxoids entering and leaving thin film, photolithographically patterned rings of underdoped high-temperature superconductor BSCCO using a variable sample temperature scanning SQUID microscope. These measurements can be understood within a phenomenological model in which the fluxoid number changes by thermal activation of a Pearl vortex in the ring wall. We place upper limits on the "vison" binding energy in these samples from these measurements., Comment: 4 pages, 4 figures

We present various concepts and experimental procedures to produce biepitaxial YBaCuO grain boundary Josephson junctions. The device properties have an interesting phenomenology related in part to the possible influence of "pai loops". The performance of our junctions and Superconducting Quantum Interference Devices indicates significant improvement in the biepitaxial technique. Further, we propose methods for fabricating circuits in which "0-" and "pai-loops" are controllably located on the same chip., Comment: 10 figures to be published in Phys. Rev. B

Phase-sensitive tests of pairing symmetry have provided strong evidence for predominantly d-wave pairing symmetry in both hole- and electron-doped high-Tc cuprate superconductors. Temperature dependent measurements in YBCO indicate that the d-wave pairing dominates, with little if any imaginary component, at all temperatures from 0.5K through Tc. In this article we review some of this evidence and discuss the implications of the universal d-wave pairing symmetry in the cuprates., Comment: 4 pages, M2S 2000 conference proceedings

We have observed spontaneous magnetic moments with random signs in c-axis oriented thin films of the high-Tc cuprate superconductor YBCO, imaged with a scanning SQUID microscope. These moments arise when the samples become superconducting, and appear to be associated with non-ferromagnetic defects in the films. In contrast with granular high-Tc samples, which also show spontaneous magnetization with random signs, these samples shield diamagnetically., Comment: 4 pages, 4 figures

We present phase-sensitive evidence that the electron-doped cuprates Nd1.85Ce0.15CuO4-y (NCCO) and Pr1.85Ce0.15CuO4-y (PCCO) have d-wave pairing symmetry. This evidence was obtained by observing the half-flux quantum effect, using a scanning SQUID microscope, in c-axis oriented films of NCCO or PCCO epitaxially grown on tricrystal [100] SrTiO3 substrates designed to be frustrated for a d(x2-y2) order parameter. Samples with two other configurations, designed to b unfrustrated for a d-wave superconductor, do not show the half-flux quantum effect., Comment: 4 pages, Latex, 4 figures

The Intlerlayer Josephson coupling between the planes of Tl2Ba2CuO6 was determined using infrared spectroscopy and magnetic flux vortex imaging. These methods give a consistent value of $\omega_J$= 28 cm$^{-1}$ which, when combined with the condensation energy produces a discrepancy of at least an order of magnitude with deductions based on the interlayer tunneling model., Comment: 5 pages, article, epsf, 3 encapsulated postscript figures

We present the solution to London's equations for the magnetic fields of a vortex oriented parallel to the plane, and normal to a crystal face, of a layered superconductor. These expressions account for flux spreading at the superconducting surface, which can change the apparent size of the vortex along the planes by as much as 30%. We compare these expressions with experimental results., Comment: 13 pages, 5 figures

We have magnetically imaged interlayer Josephson vortices emerging from an ac face of single crystals of the single layer cuprate high-Tc superconductor (Hg,Cu)Ba2CuO4+d. These images provide a direct measurement of the c-axis penetration depth of about 10 microns. This length is a factor of 10 longer than predicted by the interlayer tunneling model for the mechanism of superconductivity in layered compounds, indicating that the condensation energy available through this mechanism is 100 times smaller than is required for superconductivity., Comment: 10 pages, 4 figures

We have imaged the spatial distribution of magnetic flux on a granular sample of the high-temperature superconductor Bi2Sr2CaCu2Ox using a scanning SQUID microscope. Our results establish the presence of spontaneous orbital magnetic moments which were suggested to be the origin of the paramagnetic response of these materials. The signature of the orbital magnetic moments is a rather broad distribution of local magnetic fields at the surface of the sample. A simple model for the distribution is presented., Comment: 10 pages, 4 figures

The spontaneous flux generation and magnetic field modulation of the critical current in a 0-pi Josephson junction are calculated for different ratios of the junction length to the Josephson penetration depth, and different ratios of the 0-junction length to the pi-junction length. These calculations apply to a Pb-YBCO c-axis oriented junction with one YBCO twin boundary, as well as other experimental systems. Measurements of such a junction can provide information on the nature of the c-axis Josephson coupling and the symmetry of the order parameter in YBCO. We find spontaneous flux even for very short symmetric 0-pi junctions, but asymmetric junctions have qualitatively different behavior., Comment: 13 pages, TEX,+ 7 figures, postscript