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The heavy fermion superconductor $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$ is a candidate for chiral, time-reversal symmetry-breaking superconductivity with a nodal gap structure. Here, we microscopically visualized superconductivity and spatially inhomogeneous ferromagnetism in $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$. We observed linear-$T$ superfluid density, consistent with $d$-wave pairing symmetries including chiral $d$ wave, but did not observe the spontaneous magnetization expected for chiral $d$ wave. Local vortex pinning potentials had either four- or twofold rotational symmetries with various orientations at different locations. Taken together, these data support a nodal gap structure in $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$ and suggest that chirality either is not present or does not lead to detectable spontaneous magnetization.

Scanning Superconducting Quantum Interference Device (SQUID) Susceptometry simultaneously images the local magnetic fields and susceptibilities above a sample with sub-micron spatial resolution. Further development of this technique requires a thorough understanding of the current, voltage, and flux ( I V &Phi, ) characteristics of scanning SQUID susceptometers. These sensors often have striking anomalies in their current&ndash, voltage characteristics, which we believe to be due to electromagnetic resonances. The effect of these resonances on the performance of these SQUIDs is unknown. To explore the origin and impact of the resonances, we develop a model that qualitatively reproduces the experimentally-determined I V &Phi, characteristics of our scanning SQUID susceptometers. We use this model to calculate the noise characteristics of SQUIDs of different designs. We find that the calculated ultimate flux noise is better in susceptometers with damping resistors that diminish the resonances than in susceptometers without damping resistors. Such calculations will enable the optimization of the signal-to-noise characteristics of scanning SQUID susceptometers.

Gate-tunable semiconductor-superconductor nanowires with superconducting leads form exotic Josephson junctions that are a highly desirable platform for two types of qubits: those with topological superconductivity (Majorana qubits) and those based on tunable anharmonicity (gatemon qubits). Controlling their behavior, however, requires understanding their electrostatic environment and electronic structure. Here we study gated InAs nanowires with epitaxial aluminum shells. By measuring current-phase relations (CPR) and comparing them with analytical and numerical calculations, we show that we can tune the number of modes, determine the transparency of each mode, and tune into regimes in which electron-electron interactions are apparent, indicating the presence of a quantum dot. To take into account electrostatic and geometrical effects, we perform microscopic self-consistent Schrodinger-Poisson numerical simulations, revealing the energy spectrum of Andreev states in the junction as well as their spatial distribution. Our work systematically demonstrates the effect of device geometry, gate voltage and phase bias on mode behavior, providing new insights into ongoing experimental efforts and predictive device design., Comment: Main: 16 pages, 4 figures. Supplement: 8 pages, 4 figures

Strong vortex pinning in FeSe could be useful for technological applications and could provide clues about the coexistence of superconductivity and nematicity. To characterize the pinning of individual, isolated vortices, we simultaneously apply a local magnetic field and image the vortex motion with scanning SQUID susceptibility. We find that the pinning is highly anisotropic: the vortices move easily along directions that are parallel to the orientations of twin domain walls and pin strongly in a perpendicular direction. These results are consistent with a scenario in which the anisotropy arises from vortex pinning on domain walls and quantify the dynamics of individual vortex pinning in FeSe., 7 pages, 7 figures

We report on a study of epitaxially grown ultrathin Pb films that are only a few atoms thick and have parallel critical magnetic fields much higher than the expected limit set by the interaction of electron spins with a magnetic field, that is, the Clogston-Chandrasekhar limit. The epitaxial thin films are classified as dirty-limit superconductors because their mean-free paths, which are limited by surface scattering, are smaller than their superconducting coherence lengths. The uniformity of superconductivity in these thin films is established by comparing scanning tunneling spectroscopy, scanning superconducting quantum interference device (SQUID) magnetometry, double-coil mutual inductance, and magneto-transport, data that provide average superfluid rigidity on length scales covering the range from microscopic to macroscopic. We argue that the survival of superconductivity at Zeeman energies much larger than the superconducting gap can be understood only as the consequence of strong spin-orbit coupling that, together with substrate-induced inversion-symmetry breaking, produces spin splitting in the normal-state energy bands that is much larger than the superconductor's energy gap.

Topological superconductivity is of both fundamental and technological interest as a novel state of matter with potential applications in quantum information processing. Among the most promising materials for exhibiting intrinsic topological superconductivity is Sr${}_{2}$RuO${}_{4}$, which has shown evidence of triplet pairing and time-reversal symmetry breaking. Here, the authors use a scanning SQUID microscope to measure the local change in the critical temperature under application of anisotropic strain to test for a linear cusp at small strains, a key prediction of existing models for the superconducting state. The reported absence of such a cusp constrains future descriptions of this enigmatic material.

The diamagnetic susceptibility of a superconductor is directly related to its superfluid density. Mutual inductance is a highly sensitive method for characterizing thin films; however, in traditional mutual inductance measurements, the measured response is a non-trivial average over the area of the mutual inductance coils, which are typically of millimeter size. Here we image localized, isolated features in the diamagnetic susceptibility of {\delta}-doped SrTiO3, the 2-DES at the interface between LaAlO3 and SrTiO3, and Nb superconducting thin film systems using scanning superconducting quantum interference device susceptometry, with spatial resolution as fine as 0.7 {\mu}m. We show that these features can be modeled as locally suppressed superfluid density, with a single parameter that characterizes the strength of each feature. This method provides a systematic means of finding and quantifying submicron defects in two-dimensional superconductors., Comment: 11 pages, 10 figures

Magnetizing a topological insulator Inducing magnetism in a topological insulator can lead to exotic effects. The usual experimental route is to introduce magnetic dopants into the material, but that approach is intricate and creates unwanted disorder. Wang et al. used a simpler technique: They fabricated a bilayer consisting of Bi 2 Se 3 , a topological insulator, and EuS, a magnet. The physical proximity of EuS induced magnetism on the surface of Bi 2 Se 3 . This approach allowed for the creation of magnetic domains at will and the detection of characteristic current flowing along the domains' edges. Science , this issue p. 948

Semiconductor nanowires with superconducting leads are considered promising for quantum computation. The current–phase relation is systematically explored in gate-tunable InAs Josephson junctions, and is shown to provide a clean handle for characterizing the transport properties of these structures. Gate-tunable semiconductor nanowires with superconducting leads have great potential for quantum computation1,2,3 and as model systems for mesoscopic Josephson junctions4,5. The supercurrent, I, versus the phase, φ, across the junction is called the current–phase relation (CPR). It can reveal not only the amplitude of the critical current, but also the number of modes and their transmission. We measured the CPR of many individual InAs nanowire Josephson junctions, one junction at a time. Both the amplitude and shape of the CPR varied between junctions, with small critical currents and skewed CPRs indicating few-mode junctions with high transmissions. In a gate-tunable junction, we found that the CPR varied with gate voltage: near the onset of supercurrent, we observed behaviour consistent with resonant tunnelling through a single, highly transmitting mode. The gate dependence is consistent with modelled subband structure that includes an effective tunnelling barrier due to an abrupt change in the Fermi level at the boundary of the gate-tuned region. These measurements of skewed, tunable, few-mode CPRs are promising both for applications that require anharmonic junctions6,7 and for Majorana readout proposals8.

Proximity to a superconductor is predicted to induce exotic quantum phases in topological insulators. Here, scanning superconducting quantum interference device (SQUID) microscopy reveals that aluminum superconducting rings with topologically insulating Bi2Se3 junctions exhibit a conventional, nearly sinusoidal 2π-periodic current-phase relations. Pearl vortices occur in longer junctions, indicating suppressed superconductivity in aluminum, probably due to a proximity effect. Our observations establish scanning SQUID as a general tool for characterizing proximity effects and for measuring current-phase relations in new materials systems.

We use superconducting quantum interference device microscopy to characterize the current-phase relation (CPR) of Josephson junctions from the three-dimensional topological insulator HgTe (3D HgTe). We find clear skewness in the CPRs of HgTe junctions ranging in length from 200 to 600 nm. The skewness indicates that the Josephson current is predominantly carried by Andreev bound states with high transmittance, and the fact that the skewness persists in junctions that are longer than the mean free path suggests that the effect may be related to the helical nature of the Andreev bound states in the surface of HgTe. These experimental results suggest that the topological properties of the normal state can be inherited by the induced superconducting state, and that 3D HgTe is a promising material for realizing the many exciting proposals that require a topological superconductor.

We present transport and scanning SQUID measurements on InAs/GaSb double quantum wells, a system predicted to be a two-dimensional topological insulator. Top and back gates allow independent control of density and band offset, allowing tuning from the trivial to the topological regime. In the trivial regime, bulk conductivity is quenched but transport persists along the edges, superficially resembling the predicted helical edge-channels in the topological regime. We characterize edge conduction in the trivial regime in a wide variety of sample geometries and measurement configurations, as a function of temperature, magnetic field, and edge length. Despite similarities to studies claiming measurements of helical edge channels, our characterization points to a non-topological origin for these observations.

Recent measurements of the superconducting penetration depth in Ba(Fe 1− x Co x ) 2 As 2 appeared to disagree on the magnitude and curvature of Δ λ ab ( T ), even near optimal doping. These measurements were carried out on different samples grown by different groups. To understand the discrepancy, we use scanning SQUID susceptometry and a tunnel diode resonator to measure the penetration depth in a single sample. The penetration depth observed by the two techniques is identical with no adjustments. We conclude that any discrepancies arise from differences between samples, either in growth or crystal preparation.

Scanning Superconducting QUantum Interference Device (SQUID) microscopy provides valuable information about magnetic properties of materials and devices. The magnetic flux response of the SQUID is often linearized with a flux-locked feedback loop, which limits the response time to microseconds or longer. In this work, we present the design, fabrication, and characterization of a novel scanning SQUID sampler with a 40-ps time resolution and linearized response to periodically triggered signals. Other design features include a micron-scale pickup loop for the detection of local magnetic flux, a field coil to apply a local magnetic field to the sample, and a modulation coil to operate the SQUID sampler in a flux-locked loop to linearize the flux response. The entire sampler device is fabricated on a 2 mm × 2 mm chip and can be scanned over macroscopic planar samples. The flux noise at 4.2 K with 100 kHz repetition rate and 1 s of averaging is of order 1 mΦ0. This SQUID sampler will be useful for imaging dynamics in magnetic and superconducting materials and devices.

Quantum spin Hall devices with edges much longer than several microns do not display ballistic transport: that is, their measured conductances are much less than $e^2/h$ per edge. We imaged edge currents in InAs/GaSb quantum wells with long edges and determined an effective edge resistance. Surprisingly, although the effective edge resistance is much greater than $h/e^2$, it is independent of temperature up to 30 K within experimental resolution. Known candidate scattering mechanisms do not explain our observation of an effective edge resistance that is large yet temperature-independent., Comment: Version accepted to Physical Review Letters (http://prl.aps.org/). 12 pages, 3 figures. Supplementary Online Materials available at http://stanford.edu/group/moler/papers/Spanton_InAsGaSb_imaging_SI_v2.pdf

We report a scanning superconducting quantum interference device (SQUID) microsusceptometer with a spatial resolution of 8 μm, tested by measuring the susceptibility of individual 3 μm diam tin disks. Images of the disks agree well with numerical modeling based on the known geometry of the SQUID microsusceptometers. The low-field spin sensitivity between 1.5 and 6 K is 1×105 μB/Hz while scanning.

Vibrations can cause noise in scanning probe microscopies. Relative vibrations between the scanning sensor and the sample are important but can be more difficult to determine than absolute vibrations or vibrations relative to the laboratory. We measure the noise spectral density in a scanning SQUID microscope as a function of position near a localized source of magnetic field, and show that we can determine the spectra of all three components of the relative sensor-sample vibrations. This method is a powerful tool for diagnosing vibrational noise in scanning microscopies., 4 pages, 3 figures

We present measurements of the specific heat of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ single crystals from 2 to 10 K and from 0 to 10 T, including both twinned and untwinned crystals with various oxygen doping (7-\ensuremath{\delta}=6.95, 6.97, and 6.99). The zero-field specific heat includes a linear-T term which is small compared to other ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ samples, but large compared to the ${\mathrm{T}}^{2}$ electronic specific heat predicted for a superconductor with clean lines of nodes in the gap function. This zero-field linear-T term is found to decrease with decreasing concentration of twin boundaries and oxygen vacancies \ensuremath{\delta}. The specific heat increases in a magnetic field and has a strong dependence on field orientation. Several possible field-dependent contributions are compared to the data, including a Schottky anomaly, vortex-vortex interactions, a traditional mixed-state HT term associated with vortex cores, and a $\sqrt{H}$T term predicted by G. Volovik for the mixed state in superconductors with lines of nodes in the gap function. The field dependence of the specific heat, according to a comparison of fits based on these possibilities, supports the existence of lines of nodes and suggests the presence of quasiparticle excitations outside the vortex core.

As part of a joint Stanford/IBM effort to build a scanning SQUID microscopy user facility at Stanford, we have designed and fabricated three types of scanning SQUID microscope sensors. The first is a SQUID susceptometer, with a symmetric, gradiometric design, pickup loops with 0.1 micrometer minimum feature size integrated into the SQUID body through coaxially shielded leads, integrated flux modulation coils, and counterwound one-turn field coils. The second is a SQUID sampler, in which a picosecond current pulse generated on chip is inductively coupled into a hysteric scanning SQUID sensor. The feedback flux to keep the average SQUID voltage at a constant value is proportional to the flux through the sensor pickup loop at a fixed time delay. JSPICE simulations indicate that time resolutions below 10 picosec can be obtained. The third type is a dispersive SQUID, in which the capacitance and Josephson inductance of a one-junction SQUID are chosen so it has an LC resonance in the GHz range. The Josephson inductance depends on the magnetic flux through the SQUID. The magnetic flux is sensed through phase shifts in the reflected microwave signals at resonance. Calculations indicate spin sensitivities better than 1 Bohr magneton per root Hz for a 0.3 micrometer pickup loop diameter, with bandwidths of about 100 MHz possible.

The quantum spin Hall (QSH) state is a genuinely new state of matter characterized by a non-trivial topology of its band structure. Its key feature is conducting edge channels whose spin polarization has potential for spintronic and quantum information applications. The QSH state was predicted and experimentally demonstrated to exist in HgTe quantum wells. The existence of the edge channels has been inferred from the fact that local and non-local conductance values in sufficiently small devices are close to the quantized values expected for ideal edge channels and from signatures of the spin polarization. The robustness of the edge channels in larger devices and the interplay between the edge channels and a conducting bulk are relatively unexplored experimentally, and are difficult to assess via transport measurements. Here we image the current in large Hallbars made from HgTe quantum wells by probing the magnetic field generated by the current using a scanning superconducting quantum interference device (SQUID). We observe that the current flows along the edge of the device in the QSH regime, and furthermore that an identifiable edge channel exists even in the presence of disorder and considerable bulk conduction as the device is gated or its temperature is raised. Our results represent a versatile method for the characterization of new quantum spin Hall materials systems, and confirm both the existence and the robustness of the predicted edge channels., Comment: 11 pages, 4 figures; v2: link to supplementary information added, see http://www.stanford.edu/group/moler/publications.html

We have used a high-resolution scanning SQUID microscope to directly image conventional ( $h/2e$) Josephson vortices trapped in grain boundaries, and half-integer ( $h/4e$) Josephson vortices trapped at the tricrystal point, of the high- ${T}_{c}$ superconductor ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ grown on tricrystal substrates of SrTi${\mathrm{O}}_{3}$. Our observation of the half-integer vortex at the tricrystal point is the first direct demonstration of the positive paramagnetic Meissner effect in a polycrystalline high- ${T}_{c}$ film. In addition, our images provide the first direct measurement of the Josephson penetration depth.

The total specific heat of YBa2Cu3O6.95 single crystals includes contributions from phonons and spin-1/2 particles, as well as electronic contributions. The electronic specific heat is described by a quadratic term αT2 in zero field and a linear term [γ(0)+γ⊥(H)]T which is increased when a magnetic field H is applied perpendicular to the CuO2 planes. In agreement with d-wave superconductivity, we find that α≈γn/Tc and γ⊥(H)≈γn(H/Hc2)1/2, where γn is the coefficient of the normal-state linear term. The H1/2 dependence of the density of states at the Fermi level was predicted by G. Volovik for lines of nodes in the gap: the quasiparticles which contribute to this density of states are close to the nodes in momentum space and are located outside the vortex core.

In the past, magnetic images acquired using scanning Superconducting Quantum Interference Device (SQUID) microscopy have been interpreted using simple models for the sensor point spread function. However, more complicated modeling is needed when the characteristic dimensions of the field sensitive areas in these sensors become comparable to the London penetration depth. In this paper we calculate the response of SQUIDs with deep sub-micron pickup loops to different sources of magnetic fields by solving coupled London's and Maxwell's equations using the full sensor geometry. Tests of these calculations using various field sources are in reasonable agreement with experiments. These calculations allow us to more accurately interpret sub-micron spatial resolution data obtained using scanning SQUID microscopy., 15 pages, 9 figures

We describe a simple method to study vortex dynamics that can determine or set an upper limit on the vortex mass. The specific heat of the vortex lattice in layered superconductors has a classical limit of 1 kB per pancake vortex if the vortex mass is zero. If the vortex mass mv is finite, a new Einstein branch of normal modes will appear with a crossover temperature ΘE ∝ mv−1, and the specific heat will saturate at a new classical limit of 2 kB per pancake vortex.

The magnetic field dependence of the electronic density of states at the Fermi level, $N({E}_{F},H)$, is determined in single-crystal Y${\mathrm{Ba}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6.95}$ by specific heat measurements. The total specific heat is best described by including two predictions for the electronic specific heat of $d$-wave superconductivity: a ${T}^{2}$ term in zero field and an increased linear term in a magnetic field applied perpendicular to the Cu${\mathrm{O}}_{2}$ planes. The additional linear term, which implies a finite $N({E}_{F},H)$, obeys $N({E}_{F},H)\ensuremath{\propto}{(\frac{H}{{H}_{c2}})}^{\frac{1}{2}}$ as predicted by Volovik for superconductivity with lines of nodes in the gap.

We demonstrate the controlled, reversible manipulation of individual vortices in a superconductor with a locally applied magnetic field. The local field is supplied by a field coil on a superconducting quantum interference device (SQUID). The SQUID is used to image the vortices before and after moving. This device can be used both to push individual vortices and to create individual vortex–antivortex pairs. We calculate the force applied on a rigid vortex and find that ∼0.5 pN is necessary to move vortices in underdoped single crystals of YBa2Cu3O6.354 with Tc∼ 12 K.

Rings are a model system for studying phase coherence in one dimension. Superconducting rings have states with uniform phase windings that are integer multiples of 2$\pi$ called fluxoid states. When the energy difference between these fluxoid states is of order the temperature so that phase slips are energetically accessible, several states contribute to the ring's magnetic response to a flux threading the ring in thermal equilibrium and cause a suppression or downturn in the ring's magnetic susceptibility as a function of temperature. We review the theoretical framework for superconducting fluctuations in rings including a model developed by Koshnick$^1$ which includes only fluctuations in the ring's phase winding number called fluxoid fluctuations and a complete model by von Oppen and Riedel$^2$ that includes all thermal fluctuations in the Ginzburg-Landau framework. We show that for sufficiently narrow and dirty rings the two models predict a similar susceptibility response with a slightly shifted Tc indicating that fluxoid fluctuations are dominant. Finally we present magnetic susceptibility data for rings with different physical parameters which demonstrate the applicability of our models. The susceptibility data spans a region in temperature where the ring transitions from a hysteretic to a non hysteretic response to a periodic applied magnetic field. The magnetic susceptibility data, taken where transitions between fluxoid states are slow compared to the measurement time scale and the ring response was hysteretic, decreases linearly with increasing temperature resembling a mean field response with no fluctuations. At higher temperatures where fluctuations begin to play a larger role a crossover occurs and the non-hysteretic data shows a fluxoid fluctuation induced suppression of diamagnetism below the mean field response that agrees well with the models.

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.

We measure the penetration depth ${\ensuremath{\lambda}}_{ab}(T)$ in $\mathrm{Ba}({\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}{)}_{2}{\mathrm{As}}_{2}$ using local techniques that do not average over the sample. The superfluid density ${\ensuremath{\rho}}_{s}(T)\ensuremath{\equiv}1/{\ensuremath{\lambda}}_{ab}(T{)}^{2}$ has three main features. First, ${\ensuremath{\rho}}_{s}(T=0)$ falls sharply on the underdoped side of the dome. Second, ${\ensuremath{\lambda}}_{ab}(T)$ is flat at low $T$ at optimal doping, indicating fully gapped superconductivity, but varies more strongly in underdoped and overdoped samples, consistent with either a power law or a small second gap. Third, ${\ensuremath{\rho}}_{s}(T)$ varies steeply near ${T}_{c}$ for optimal and underdoping. These observations are consistent with an interplay between magnetic and superconducting phases.

Microscopic variations in composition or structure can lead to nanoscale inhomogeneity in superconducting properties such as the magnetic penetration depth, but measurements of these properties are usually made on longer length scales. We solve a generalized London equation with a non-uniform penetration depth $\ensuremath{\lambda}(\mathbit{r})$, obtaining an approximate solution for the disorder-averaged Meissner screening. We find that the effective penetration depth is different from the average penetration depth and is sensitive to the details of the disorder. These results indicate the need for caution when interpreting measurements of the penetration depth and its temperature dependence in systems which may be inhomogeneous.

LaAlO3 and SrTiO3 are insulating, nonmagnetic oxides, yet the interface between them exhibits a two-dimensional electron system with high electron mobility,1 superconductivity at low temperatures,2-6 and electric-field-tuned metal-insulator and superconductorinsulator phase transitions.3,6-8 Bulk magnetization and magnetoresistance measurements also suggest some form of magnetism depending on preparation conditions5,9-11 and suggest a tendency towards nanoscale electronic phase separation.10 Here we use local imaging of the magnetization and magnetic susceptibility to directly observe a landscape of ferromagnetism, paramagnetism, and superconductivity. We find submicron patches of ferromagnetism in a uniform background of paramagnetism, with a nonuniform, weak diamagnetic superconducting susceptibility at low temperature. These results demonstrate the existence of nanoscale phase separation as suggested by theoretical predictions based on nearly degenerate interface sub-bands associated with the Ti orbitals.12,13 The magnitude and temperature dependence of the paramagnetic response suggests that the vast majority of the electrons at the interface are localized, and do not contribute to transport measurements.3,6,7 In addition to the implications for magnetism, the existence of a 2D superconductor at an interface with highly broken inversion symmetry and a ferromagnetic landscape in the background suggests the potential for exotic superconducting phenomena., Comment: Edited version to appear in Nature Physics

We present scanning superconducting quantum interference device microscopy data on the superconductors Sr2RuO4 Tc=1.5 K and PrOs4Sb12 Tc=1.8 K. In both of these materials, superconductivity-related timereversal symmetry-breaking fields have been observed by muon spin rotation; our aim was to visualize the structure of these fields. However, in neither Sr2RuO4 nor PrOs4Sb12 do we observe spontaneous superconductivity-related magnetization. In Sr2RuO4, many experimental results have been interpreted on the basis of a pxipy superconducting order parameter. This order parameter is expected to give spontaneous magnetic induction at sample edges and order parameter domain walls. Supposing large domains, our data restrict domain wall and edge fields to no more than 0.1% and 0.2% of the expected magnitude, respectively. Alternatively, if the magnetization is of the expected order, the typical domain size is limited to 30 nm for random domains or 500 nm for periodic domains.

Superconducting quantum interference device microscopy shows stripes of increased diamagnetic susceptibility in the superconducting state of twinned, orthorhombic, underdoped crystals of $\text{Ba}{({\text{Fe}}_{1\ensuremath{-}x}{\text{Co}}_{x})}_{2}{\text{As}}_{2}$, but not in tetragonal overdoped crystals. These stripes are consistent with enhanced superfluid density on twin boundaries.

We use magnetic force microscopy (MFM) to measure the local penetration depth $\ensuremath{\lambda}$ in $\text{Ba}{({\text{Fe}}_{0.95}{\text{Co}}_{0.05})}_{2}{\text{As}}_{2}$ single crystals and use scanning superconducting quantum interference device susceptometry to measure its temperature variation down to 0.4 K. We observe that superfluid density ${\ensuremath{\rho}}_{s}$ over the full temperature range is well described by a clean two-band fully gapped model. We demonstrate that MFM can measure the important and hard-to-determine absolute value of $\ensuremath{\lambda}$, as well as obtain its temperature dependence and spatial homogeneity. We find ${\ensuremath{\rho}}_{s}$ to be uniform on the submicron scale despite the highly disordered vortex pinning.

We use low-temperature magnetic force microscopy (MFM) to study the hopping motion of vortices in an array of artificial pinning centers (APCs). The array consists of nanoscale holes etched in a niobium thin film by Ar-ion sputtering through an anodic aluminum-oxide template. Variable-temperature magnetometry shows a transition temperature of 7.1 K and an enhancement of the magnetization up to the third matching field at 5 K. Using MFM with attractive and repulsive tip-vortex interaction, we measure the vortex-pinning strength and investigate the motion of individual vortices in the APC array. The depinning force for individual vortices at low field ranged from 0.7 to 1.2 pN. The motion of individual vortices was found to be reproducible and consistent with movement between adjacent holes in the film. The movements are repeatable but the sequence of hops depends on the scan direction. This asymmetry in the motion indicates nonuniform local pinning, a consequence of array disorder and hole-size variation.

We use magnetic force microscopy to both image and manipulate individual vortex lines threading single crystalline ${\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{6.4}$, a layered superconductor. We find that when we pull the top of a pinned vortex, it may not tilt smoothly. Occasionally, we observe a vortex to break into discrete segments that can be described as short stacks of pancake vortices, similar to the ``kinked'' structure proposed by Benkraouda and Clem. Quantitative analysis gives an estimate of the pinning force and the coupling between the stacks. Our measurements highlight the discrete nature of stacks of pancake vortices in layered superconductors.

We measure changes in the penetration depth {lambda} of the T{sub c} {approx} 6 K superconductor LaFePO. In the process scanning SQUID susceptometry is demonstrated as a technique for accurately measuring local temperature-dependent changes in {lambda}, making it ideal for studying early or difficult-to-grow materials. {lambda} of LaFePO is found to vary linearly with temperature from 0.36 to {approx} 2 K, with a slope of 143 {+-} 15 {angstrom}/K, suggesting line nodes in the superconducting order parameter. The linear dependence up to {approx} T{sub c}/3 is similar to the cuprate superconductors, indicating well-developed nodes.

Susceptibility measurements of patterned thin films at sub-K temperatures were carried out using a scanning SQUID microscope that can resolve signals corresponding to a few hundred Bohr magnetons. Several metallic and insulating thin films, even oxide-free Au films, show a paramagnetic response with a temperature dependence that indicates unpaired spins as the origin. The observed response exhibits a measurable out-of-phase component, which implies that these spins will create 1/f-like magnetic noise. The measured spin density is consistent with recent explanations of low frequency flux noise in SQUIDs and superconducting qubits in terms of spin fluctuations, and suggests that such unexpected spins may be even more ubiquitous than already indicated by earlier measurements. Our measurements set several constraints on the nature of these spins., Journal version. 4 pages, 2 figures. See http://www.stanford.edu/group/moler/publications.html for the auxiliary document containing additional data and discussion (Ref. 15). Changes w.r.t. v1: Clarified some details, moved some technical points to auxiliary document

Fluctuations are important near phase transitions, where they can be difficult to describe quantitatively. Superconductivity in mesoscopic rings is particularly intriguing because the critical temperature is an oscillatory function of magnetic field. There is an exact theory for thermal fluctuations in one-dimensional superconducting rings, which are therefore expected to be an excellent model system. We measure the susceptibility of many rings, one ring at a time, using a scanning SQUID that can isolate magnetic signals from seven orders of magnitude larger background applied flux. We find that the fluctuation theory describes the results and that a single parameter characterizes the ways in which the fluctuations are especially important at magnetic fields where the critical temperature is suppressed., Comment: Reprinted with permission from AAAS

The ability to wiggle and stretch individual superconducting vortices with nanoscale precision enables unprecedented insight into their dynamics and the properties of the superconductor that supports them. Superconductors often contain quantized microscopic whirlpools of electrons, called vortices, that can be modelled as one-dimensional elastic objects1. Vortices are a diverse area of study for condensed matter because of the interplay between thermal fluctuations, vortex–vortex interactions and the interaction of the vortex core with the three-dimensional disorder landscape2,3,4,5. Although vortex matter has been studied extensively1,6,7, the static and dynamic properties of an individual vortex have not. Here, we use magnetic force microscopy (MFM) to image and manipulate individual vortices in a detwinned YBa2Cu3O6.991 single crystal, directly measuring the interaction of a moving vortex with the local disorder potential. We find an unexpected and marked enhancement of the response of a vortex to pulling when we wiggle it transversely. In addition, we find enhanced vortex pinning anisotropy that suggests clustering of oxygen vacancies in our sample and demonstrates the power of MFM to probe vortex structure and microscopic defects that cause pinning.

We have fabricated and characterized micro-SQUID susceptometers for use in low-temperature scanning probe microscopy systems. The design features the following: a 4.6 mum diameter pickup loop; an integrated field coil to apply a local field to the sample; an additional counterwound pickup-loop/field-coil pair to cancel the background signal from the applied field in the absence of the sample; modulation coils to allow setting the SQUID at its optimum bias point (independent of the applied field), and shielding and symmetry that minimizes coupling of magnetic fields into the leads and body of the SQUID. We use a SQUID series array preamplifier to obtain a system bandwidth of 1 MHz. The flux noise at 125 mK is approximately 0.25 mu Phi 0/ sqrt Hz above 10 kHz, with a value of 2.5 mu Phi 0/ sqrt Hz at 10 Hz. The nominal sensitivity to electron spins located at the center of the pickup loop is approximately 200 muB/ sqrt Hz above 10 kHz, in the white-noise frequency region.

We report controlled local manipulation of single vortices by low temperature magnetic force microscope (MFM) in a thin film of superconducting Nb. We are able to position the vortices in arbitrary configurations and to measure the distribution of local depinning forces. This technique opens up new possibilities for the characterization and use of vortices in superconductors.

It is widely believed that the perovskite Sr2RuO4 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 Sr2RuO4 single crystals using scanning Hall bar and superconducting quantum interference device 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 2 orders of magnitude. We speculate on the causes and implications of the lack of large spontaneous supercurrents in this very interesting superconducting system.

Various local probes have been applied to understanding current flow through superconducting films, which are often surprisingly inhomogeneous. Here we show that magnetic imaging allows quantitative reconstruction of both current density, J, and electric field, E, resolved in time and space, in a film carrying subcritical ac current. Current reconstruction entails inversion of the Biot-Savart law, while electric fields are reconstructed using Faraday's law. We describe the corresponding numerical procedures, largely adapting existing work to the case of a strip carrying ac current, but including new methods of obtaining the complete electric field from the inductive portion determined by Faraday's law. We also delineate the physical requirements behind the mathematical transformations. We then apply the procedures to images of a strip of YBa2Cu3O7 (YBCO) carrying an ac current at 400 Hz. Our scanning Hall probe microscope produces a time-series of magnetic images of the strip with 1 micron spatial resolution and 25 microsecond time resolution. Combining the reconstructed J and E, we obtain a complete characterization including local critical current density, E-J curves, and power losses. This analysis has a range of applications from fundamental studies of vortex dynamics to practical coated conductor development., 13 pages, 8 figures. Submitted to Physical Review B. Movies and higher resolution figures at http://www.stanford.edu/group/moler/rdinner

A recent advance in improving the spatial resolution of magnetic force microscopy (MFM) uses as sensor tips carbon nanotubes grown at the apex of conventional silicon cantilever pyramids and coated with a thin ferromagnetic layer. Magnetic images of high density vertically recorded media using these tips exhibit a doubling of the spatial frequency under some conditions. Here we demonstrate that this spatial frequency doubling is due to the switching of the moment direction of the nanotube tip. This results in a signal which is proportional to the absolute value of the signal normally observed in MFM. Our modeling indicates that a significant fraction of the tip volume is involved in the observed switching, and that it should be possible to image very high bit densities with nanotube magnetic force sensors., 10 pages

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.