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Fundamental studies of superconductors using scanning magnetic imaging

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2010 Rep. Prog. Phys. 73 126501

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IOP PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 73 (2010) 126501 (36pp) doi:10.1088/0034-4885/73/12/126501

Fundamental studies of superconductorsusing scanning magnetic imagingJ R Kirtley

Center for Probing the Nanoscale, Stanford University, Stanford, CA, USA

Received 26 June 2010, in final form 4 August 2010Published 18 November 2010Online at stacks.iop.org/RoPP/73/126501

AbstractIn this review I discuss the application of scanning magnetic imaging to fundamental studies ofsuperconductors, concentrating on three scanning magnetic microscopies—scanning SQUIDmicroscopy (SSM), scanning Hall bar microscopy (SHM) and magnetic force microscopy(MFM). I briefly discuss the history, sensitivity, spatial resolution, invasiveness and potentialfuture developments of each technique. I then discuss a selection of applications of thesemicroscopies. I start with static imaging of magnetic flux: an SSM study provides deeperunderstanding of vortex trapping in narrow strips, which are used to reduce noise insuperconducting circuitry. Studies of vortex trapping in wire lattices, clusters and arrays ofrings and nanoholes show fascinating ordering effects. The cuprate high-Tc superconductorsare shown to have predominantly d-wave pairing symmetry by magnetic imaging of thehalf-integer flux quantum effect. Arrays of superconducting rings act as a physical analog forthe Ising spin model, with the half-integer flux quantum effect helping to eliminate one sourceof disorder in antiferromagnetic arrangements of the ring moments. Tests of the interlayertunneling model show that the condensation energy available from this mechanism cannotaccount for the high critical temperatures observed in the cuprates. The strong divergence inthe magnetic fields of Pearl vortices allows them to be imaged using SSM, even for penetrationdepths of a millimeter. Unusual vortex arrangements occur in samples comparable in size tothe coherence length. Spontaneous magnetization is not observed in Sr2RuO4, which isbelieved to have px ± ipy pairing symmetry, although effects hundreds of times bigger thanthe sensitivity limits had been predicted. However, unusual flux trapping is observed in thissuperconductor. Finally, unusual flux arrangements are also observed in magneticsuperconductors. I then turn to vortex dynamics: imaging of vortices in rings of highlyunderdoped cuprates places limits on spin-charge separation in these materials. Studies ofspontaneous generation of fluxoids upon cooling rings through the superconducting transitionprovide clues to dynamical processes relevant to the early development of the universe, whilestudies of vortex motion in cuprate grain boundaries allow the measurement of current–voltagecharacteristics at the femtovolt scale for these technologically important defects. ScanningSQUID susceptometry allows the measurement of superconducting fluctuations on samplescomparable in size to the coherence length, revealing stripes in susceptibility believed to beassociated with enhanced superfluid density on the twin boundaries in the pnictidesuperconductor Co doped Ba-122, and indicating the presence of spin-like excitations, whichmay be a source of noise in superconducting devices, in a wide variety of materials. Scanningmagnetic microscopies allow the absolute value of penetration depths to be measured locallyover a wide temperature range, providing clues to the symmetry of the order parameter inunconventional superconductors. Finally, MFM tips can be used to manipulate vortices,providing information on flux trapping in superconductors.

(Some figures in this article are in colour only in the electronic version)

This article was invited by Sean Washburn.

0034-4885/10/126501+36$90.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

http://dx.doi.org/10.1088/0034-4885/73/12/126501

http://stacks.iop.org/RoPP/73/126501

Rep. Prog. Phys. 73 (2010) 126501 J R Kirtley

Contents

1. Introduction 22. Fundamentals 3

2.1. Scanning SQUID microscopy 32.2. Hall bar microscopy 72.3. Magnetic force microscopy 72.4. Summary 10

3. Applications 103.1. Static imaging of magnetic flux 10

3.2. Fluxoid dynamics 19

3.3. Local susceptibility measurements 22

3.4. Penetration depths 25

3.5. Manipulation of individual vortices 274. Conclusions 28Acknowledgments 28References 28

1. Introduction

Superconductivity, the complete absence of electricalresistance in some metals below a critical temperature Tc, isone of the best understood phenomena in solid-state physics.As described in the Bardeen–Cooper–Schrieffer (BCS) theory[1], superconductivity occurs in conventional superconductorsthrough phonon-mediated pair-wise interaction of chargecarriers to form Cooper pairs. The characteristic energy neededto form a single particle excitation from the superconductingcondensate is the energy gap �. The Cooper pairs form amacroscopic quantum state characterized by a wave functionψ(�r) = |ψ(�r)|eiφ(�r), where φ is the Cooper pair phase,with the density of Cooper pairs ns(�r) = |ψ(�r)|2. Themacrosopic quantum nature of the pairing state leads to a widevariety of behaviors: persistent currents, Meissner screening,flux quantization, the Josephson effects and superconductingquantum interference. Superconductivity has become a multi-billion dollar industry, with applications in medical imaging,cell phone filters, electrical motors, power transmission,transportation, magnetic manipulation, magnetic sensing, etc.

However, there is still much to learn about super-conductivity: although superconductivity in the copper-oxidebased perovskites [2, 3] was discovered in 1986, despitealmost 25 years of intense effort a consensus on the pairingmechanism in the cuprates has yet to emerge. The heavyFermion superconductors [4, 5] have large carrier masses,strong interaction between spin and charge degrees offreedom, and potentially a wide variety of Cooper pairingsymmetries. The non-cuprate perovskite superconductorSr2RuO4 [6] is believed to have a p-wave pairing state thatbreaks time-reversal symmetry [7]. Interest in unconventionalsuperconductors has been reawakened with the discovery ofiron-based compounds with high critical temperatures [8].

Scanning magnetic microscopies have played, and willcontinue to play, an important part in our efforts to understandsuperconductivity. There are now a large number of scanningmagnetic microscopies. Although it is beyond the scope of thisreview to cover all of these techniques in detail, it is appropriateto briefly describe some, and provide a few citations as anintroduction to the literature.

In Lorentz microscopy [9–12], a coherent beam of highenergy (300 keV) electrons is passed through a thin sampleand then imaged slightly off the focal plane. Magnetic fieldsintroduce small phase shifts which cause interference in thedefocussed image. Lorentz microscopy can image vorticeswith deep sub-micrometer spatial resolution at video rates and

is sensitive not only to surface fields but also to fields withinthe sample. However, the sample must be thinned.

In magneto-optic microscopy [13–15], the sample isilluminated with linearly polarized light. It passes through amagneto-optically active layer, reflects off the sample surface,passes again through the magneto-optically active layer andthen through a crossed polarizer. Regions of the samplewith magnetic fields produce rotations of the polarization axisof the light, which is detected as a bright spot. Magneto-optic microscopy has the advantage of relative simplicity, buthas limited spatial resolution and sensitivity. Nevertheless,observation of individual superconducting vortices with0.8 µm spatial resolution has recently been reported [15].

In Sagnac interferometry [16–18], a single beam of light issplit into two components which travel along identical paths inopposite directions around a loop. Any effects which break thetime-reversal symmetry, including the magneto-optical effect,will cause interference, which can be sensitively detected.High-spatial resolution can be achieved using scanning near-field optical microscopy [17].

In spin polarized scanning tunneling microscopy [19–21]a scanning tunneling microscope has a tunneling tip witha ferromagnetic or antiferromagnetic atom from which thetunneling electrons emerge. The tunneling probability dependson the spin polarization of the sample, and therefore contrastis achieved if there are spatial variations in the atomic spinpolarizations. Spin polarized tunneling is useful for studyingsuperconductivity in, for example, the high-temperaturecuprate superconductors, since it is believed that magnetismand superconductivity are intimately related in these materials.However, the spin polarized tunneling current is not directlyrelated to the local magnetic fields.

In diamond nitrogen-vacancy microscopy [22–26], amagnetic field dependent shift in the energy levels of a singlenitrogen-vacancy center in diamond is detected optically.There are two basic schemes for detecting this energy shift.In the ‘dc’ technique, microwaves at a fixed frequency as wellas light are incident on the center. The intensity of the resultantfluorescence depends on the local magnetic field. This methodcan detect millitesla fields with, in principle, nanometer scalespatial resolution. In the ‘ac’ technique, spin-echo microwavetechniques are used to detect the modulation of the center’senergy levels by the local magnetic field. The ‘ac’ techniquehas a sensitivity of 60 nT, with in principle a spatial resolutionof a few nanometers, but has reduced sensitivity at zerofrequency. These techniques have the advantages of goodsensitivity, small invasiveness, room temperature operation,

2


inherently high speed and potentially high-spatial resolution.It remains to be seen whether single nitrogen-vacancies innanometer sized diamond crystallites can be placed at theend of scanning tips while still retaining long lifetimes. Thisis necessary to achieve high-spatial resolution and sensitivitysimultaneously.

A magnetic tunnel junction [27–29] is a planar structurewith two ferromagnetic electrodes. The tunneling currentthrough the junction depends on the relative alignment of themagnetic moments of the electrodes, and therefore dependson the magnetic field environment. Magnetic tunnel junctionssuitable for scanning microscopy are commercially available.They currently have lateral dimensions (and therefore spatialresolutions) of about 4 µm, and field noises at 100 Hz of about5 × 10−8 T Hz−1/2 [29]. They have the advantage of roomtemperature operation, but are not currently as sensitive asHall bars or SQUIDs.

There are already excellent reviews covering the topicof magnetic imaging of superconductors [30–32], and therehas been an enormous amount of work in this area. I willtherefore not attempt to survey the field exhaustively, but willonly cover recent work in three scanning magnetic microscopytechniques: scanning SQUID, scanning Hall bar and magneticforce microscopies. I will present a short introduction to thebasics, advantages and shortcomings of each technique, andthen present a selection of fundamental applications of thesemagnetic microscopies in the area of superconductivity.

2. Fundamentals

2.1. Scanning SQUID microscopy

Josephson [33] predicted in 1962 the possibility of coherenttunneling of Cooper pairs between two weakly coupledsuperconductors, and wrote down the fundamental relations:

Is = I0 sin ϕ,

V = h

2e

dϕ

dt, (1)

where Is is the supercurrent through the weak link, ϕ is thedifference between the Cooper pair phases on the two sides ofthe weak link, V is the voltage across the weak link and I0 is thecritical current, the maximum current that can pass through theweak link before a voltage develops. A physical Josephsonjunction can be modeled as an ideal Josephson element inparallel with a capacitor and a resistor in the resistively shuntedjunction (RSJ) model [34, 35]. This model obeys the equation

IB = h

2eR

dϕ

dt+ I0 sin ϕ +

hC

2e

d2ϕ

dt2+ In, (2)

where IB is the bias current through the weak link, R is theresistance, C is the capacitance and In is a noise current, givenfor the case of an ideal Johnson noise source resistor by

〈In〉2 = 4kBT �f/R, (3)

where 〈In〉2 is the time-averaged current noise squared and�f is the frequency band width over which the current noise

is measured. This equation has the form of the equation ofmotion of a driven, damped harmonic oscillator. The dynamicsof a Josephson weak link can be visualized as that of a particlemoving in a sinusoidal ‘washboard’ potential with oscillationamplitude I0�0/2π , where �0 = h/2e is the superconductingflux quantum, and the average slope of the potential is given by〈dE/dϕ〉 = �0(IB−In)/2π . In the absence of a noise current,the particle will escape from a local potential minimum and rundown the washboard potential if the bias current IB is greaterthan I0, in the process developing a voltage given by the secondof equations (1). The solutions to equation (2) can be dividedinto two classes, according to whether the Stewart–McCumberparameter βc = 2πI0R

2C/�0 is greater or less than 1. Ifβc < 1, a running particle retraps into a local potentialminimum (V = 0), as soon as IB = I0, and the current–voltage characteristic is non-hysteretic. On the other hand, ifβc > 1, a running particle is not retrapped until IB < I0, andthe current–voltage characteristic is hysteretic. In the presenceof noise, the SQUID can be thermally excited out of the zero-voltage state at bias current IB < I0 [36]. In addition, the weaklink can macroscopically quantum tunnel through the barrierfrom the zero-voltage to the voltage state [37].

A superconducting quantum interference device (SQUID)is a superconducting ring interrupted by at least one Josephsonweak link. For SQUID microscope applications a SQUID withtwo weak links is most often used. Such a SQUID is labelleda dc SQUID, since the modulation of the SQUID criticalcurrent with applied magnetic field can be observed usingtime-independent bias currents. Consider a SQUID with twoweak links 1,2 with critical currents Ic1, Ic2, phase drops acrossthe weak links ϕ1, ϕ2 and inductances in the two arms of theSQUID L1, L2. Neglecting for the moment the current throughthe resistive and capacitive elements of the junctions, the biascurrent across the SQUID IB and the magnetic flux throughthe SQUID � are given by

IB = I1 sin ϕ1 + I2 sin ϕ2,

� = �a − L1I1 + L2I2, (4)

where �a is the externally applied flux. Integrating allthe changes in phase around the SQUID loop along a pathsufficiently deep inside the superconductor that there are nosupercurrents along the integration path (i.e. they are shieldedout by the bulk of the superconductor), and using the factthat the canonical momentum of the Cooper pairs is given by�pcanonical = �p + q �A, where q = 2e is the charge on the Cooperpairs and �A is the vector potential, and finally by insisting thatthe Cooper pair wavefunction be single valued, leads to

2πn = ϕ2 − ϕ1 +2π

�0(�a + L2I2 − L1I1), (5)

where n an integer. Solutions to equations (4) and (5) areshown in figure 1 for the case of symmetric weak-link criticalcurrents (Ic1 = Ic2) and arm inductances (L1 = L2). Theparameter γ = 2πLI0/�0 (L = L1 + L2) determines thedepth of the modulation of the SQUID critical current with flux:larger SQUID inductances lead to smaller modulations. The

3


Figure 1. Plot of the critical current Ic of a symmetric(L1 = L2 = L/2, I1 = I2 = I0) 2-weak link SQUID as afunction of applied flux, for different values of the parameterγ = 2πLI0/�0. I0 is the critical current of one of the weak linksand L is the total inductance of the SQUID.

solutions for non-symmetric SQUIDs are more complicated,but qualitatively similar [38] to those shown here.

There are four relevant sources of noise in dc SQUIDs:Johnson noise arises from thermal fluctuations, assumed to bedominated by the shunt resistor in the RSJ model [39]; shotnoise arises from the discrete charge of the quasiparticles [40]and Cooper pairs [41] traversing the weak links; quantumnoise arises from zero point motion in the shunt resistors [42]and 1/f noise can arise from a number of sources. Tescheand Clarke calculate [39] that the signal-to-noise ratio in dcSQUIDs is optimized when βL = 2LI0/�0 ≈ 1. In addition,the Stewart–McCumber parameter βc should be about 1, sothat the SQUID is just non-hysteretic. If these two conditionsare met, the optimal flux noise power S� from the first threesources is given by

S� =

4kBT L(πLC)1/2 Johnson noise,hL shot noise,hL quantum noise.

(6)

Typical values for scanning SQUID sensors using aNb–Al2O3–Nb trilayer process are L = 100 pH, I0 =6 µA, C = 8.36 × 10−13 F, R = 5 and T = 4.2 K.These parameters lead to �Johnson = √

S� = 3 ×10−7�0 Hz−1/2 for the flux noise due to Johnson noise; �shot =1.2 × 10−7�0 Hz−1/2 for shot noise and �quantum = 5 ×10−8�0 Hz−1/2 for noise due to zero point motion.

In the frequency range of interest Johnson noise, shot noiseand quantum noise are ‘white’: the frequency distribution ofthe noise is independent of frequency f . 1/f noise, as its nameimplies, has a noise amplitude that follows a 1/f dependence.It almost always dominates the noise in SQUIDs at sufficientlylow frequencies. In the Dutta, Dimon and Horn model [43] 1/f

noise results from the superposition of ‘shot-like’ contributionsfrom two-level traps with a large distribution of trap energies.These traps could be, for example, defects in the oxide in atunnel junction, or trap sites for superconducting vortices in thesuperconducting bulk making up the SQUID. Koch et al [44]have suggested that 1/f noise in SQUIDs could result from

coupling of flux into the SQUID from electronic spins with awide distribution of characteristic times.

As can be seen in figure 1, the critical current of theSQUID is periodic in flux, with a period given by �0. Themost commonly used scheme for determining this magneticflux uses ac flux modulation in a flux locked loop [45, 46].If the SQUID is current biased just above the critical currentthe voltage varies sinusoidally with flux. If the dc componentof the total (external+feedback) flux is such that the SQUIDvoltage is at an extremum, the first harmonic voltage responsewill be zero and any flux change will give a linear response.Negative feedback on the dc component of the flux is then usedto keep the response zeroed: this feedback signal is linearlyproportional to the flux threading the SQUID, with a constantof proportionality that can easily be determined with greatprecision. Traditionally the voltage signal from the SQUID hasbeen amplified at low temperatures using a tuned LC circuit ora transformer, and at room temperature using phase-sensitivelock-in detection. However, SQUID array amplifiers can alsobe used [47, 48]. These have the advantage of bandwidths ofover 100 MHz, and no requirement for ac flux modulation.

For this review I will concentrate on high-spatialresolution magnetic microscopies. High-spatial resolution isachieved by placing a small sensor close to the sample. Inthe case of SQUID microscopy this means making the SQUIDsmall, or making a pickup loop integrated into the SQUIDsmall. The scaling of the flux signal and spatial resolutionwith sensor size depends on the type of field source. For thisreview I will concentrate on three basic field sources: a pointdipole �m generates the magnetic induction

�B = µ0

4π

3r(r · �m) − �mr3

(dipole field), (7)

where r is the direction and r is the magnitude of the vectorbetween the dipole source and the point of interest. The mag-netic flux �s through a square area of side d oriented parallel tothe xy plane and centered at �r = zz above a dipole at the originwith moment m oriented parallel to the z-axis is given by

�s = 2µ0md2

π√

d2/2 + z2(d2 + 4z2)(dipole flux). (8)

In the limit z → 0 �s → 2√

2µ0m/πd.A superconducting vortex generates the induction

�B = �0

2πr2r (monopole field), (9)

where �0 = h/2e is the total magnetic flux generated by thevortex. The magnetic flux �s through a square area of side d

oriented parallel to the xy plane and centered at �r = zz abovea vortex at the origin is given by

�s = 2�0

πtan−1

(d2

2z√

2d2 + 4z2

)(monopole flux).

(10)

In the limit z → 0 �s → �0. Finally, an infinitely long andnarrow line of current I in the y direction at x = 0, z = 0

4


Figure 2. Plot of the flux through a square area of side d a height z above a (a) dipole source with moment m oriented normal to the scanplane (b) a monopole source with flux �0 and (c) a line source with current I . For a given value of z/d the flux signal scales like d−1 for adipole source, is independent of d for a monopole source and scales like d for a current line source.

0.0 0.5 1.0 1.5 2.010–2

10–1

100

z/d

p/p

)1.0=d/z(

Current lineMonopoleDipole

0.0 0.5 1.0 1.5 2.01

2

3

z/d

d/x(a) (b)

Figure 3. (a) Plot of the ratio of the flux through a square area of side d at a height z to that at a height z/d = 0.1 for various sources offield. (b) Full width at half-maximum for the dipole and monopole field sources, and the peak-to-peak distance for the current line source, ofthe flux through a square area of side d as a function of height z.

generates the field

�B = µ0I

2πrθ (current field), (11)

where θ = (xz−zx)/r . The magnetic flux �s through a squarearea of side d oriented parallel to the xy plane at a height z

above this line has peaks at x = ±√d2 + z2/2 with magnitude

�s = µ0Id

4πln

[(d − √

d2 + z2)2 + 4z2

(d +√

d2 + z2)2 + 4z2

](current flux).

(12)

In the limit z → 0 the magnetic fluxes at the peaks divergelogarithmically.

Figure 2 plots the calculated flux through a square area ofside d, as a function of x, the lateral position of the SQUIDrelative to the field source, for various spacings z betweenpickup loop and sample, for these three different sources offield. For a given value of z/d for a dipole source the flux signalthrough the SQUID gets larger as d gets smaller, scaling like1/d. For a monopole source, such as a superconducting vortex,the peak flux is independent of d at constant z/d. For a line

of current, the peak flux signal increases like d as the pickuploop diameter increases. In all cases the spatial resolution isabout d for z � d.

Figure 3(a) plots the ratio of the peak flux at a sensorheight z to the peak flux at z/d = 0.1 for the three different fieldsources, while figure 3(b) plots the full width at half-maximum�x for the dipole and monopole field sources, and the peak-to-peak distance for the current line source, as a function of sensorheight z. The peak amplitude falls off nearly exponentiallywith sensor height for a dipole source, but falls off less quicklyfor a monopole or current line source. Similarly, the spatialresolution falls off quickly with sensor height for the dipolesource, but less quickly for the monopole and current linesources.

The same qualitative conclusions can be made for morecomplex sources of field [49]. Consider a geometry in whichthe sample takes up a half-space z < 0. For the static case themagnetic fields in the half-space z > 0 can be written as thegradient of a scalar potential ϕm: �B = ��ϕm, where �2ϕm = 0.If we resolve the fields at the surface of the sample into Fourier

5


components

�bk(z = 0) = 1

(2π)2

∫ ∞

−∞

∫ ∞

−∞�B(x, y, z = 0)ei(kxx+kyy) dx dy,

(13)

each of the components of the field for z > 0 will be of the form�bk(z) = �bk(z = 0)e−kz, where k =

√k2x + k2

y : The fields will

decay exponentially as z gets larger, with the higher Fouriercomponents decaying more rapidly. Therefore to get optimalsensitivity and spatial resolution, the sensor must be placedclose to the sample.

It was recognized almost immediately after the demon-stration of the Josephson effect [50] and superconductingquantum interference effects [51] that local magnetic fieldscould be imaged by scanning samples and SQUIDs relativeto each other [52]. However, it was not until 1983 that thefirst two-dimensional scanning SQUID microscope was builtby Rogers and Bermon at IBM Research [53]. This instrumentwas used to image superconducting vortices in association withthe IBM Josephson computer program. Other notable earlyefforts in SQUID microscopy were by the Wellstood group atthe University of Maryland [54–56], the van Harlingen groupat the University of Illinois [57, 58], the Wikswo group atVanderbilt University [59], the Clarke group at U.C. Berkeley[60] and the Kirtley group at IBM Research [61, 62]. Kirtleyand Wikswo [46] reviewed some of fundamentals and earlywork concerning scanning SQUID microscopy.

There are several competing strategies for achieving goodspatial resolution in a SQUID microscope sensor. The firststrategy is to make the SQUID very small (µ-SQUID), withnarrow and thin constrictions for the Josephson weak links[63, 64]. This strategy has the advantage of simplicity,since only one level of lithography is required. The originalµ-SQUIDs had hysteretic current–voltage characteristics.This meant that flux-locked-loop feedback schemes could notbe used, and the flux sensitivity of these SQUIDs was relativelypoor. However, recently non-hysteretic, sub-micrometer sizedSQUIDs have been made [65–67]. Hao et al [67] madetheir SQUIDs non-hysteretic using a second layer of tungstento shunt them, and reported white noise floor levels of0.2µ�0 Hz−1/2 in a SQUID with a diameter of about 370 nm.

A second strategy is a self-aligned SQUID recentlyreported by Finkler et al [68]. In this work three aluminumevaporations are made onto a quartz tube that has been pulledinto a sharp tip with apex diameter between 100 and 400 nm.The first two evaporations, performed at an angle of 100◦

degrees relative to the axis of the tube, form the leads. Athird evaporation along the tip axis forms a ring with two weaklinks to the leads, forming a dc SQUID. Finkler et al report non-hysteretic current–voltage characteristics and a flux sensitivityof 1.8×10−6�0 Hz−1/2 for SQUIDs with an effective area of0.34 µm2, operating at fields up to 0.6 T.

A third strategy is to make a more conventional SQUID,but to have well shielded superconducting leads to a smallpickup loop integrated into it [62, 69, 70] (see figure 4).This has the disadvantage of complexity, since multiplelevels of metal are required to shield the pickup loop leads,but has the advantages of reduced interaction between the

Figure 4. Advanced integrated scanning SQUID susceptometerdesign. (a) Diagram of a counterwound SQUID susceptometer.(b) Optical micrograph of pickup loop region of an optically definedSQUID susceptometer. The outer ring is the field coil and the innerring is the flux pickup loop of the SQUID. The edge of the siliconsubstrate is just visible on the left side. The inset shows an atomicforce microscopy cross-section of the structure along a horizontalline through the center of the pickup loop. The pickup loop isclosest to the sample when the tip is aligned at a 2.5◦ angle.(c) Design for a SQUID sensor in which the pickup loop is definedusing focussed ion beam (FIB) etching. In this case the pickup loopcan touch down first if the alignment angle is between 2◦ and 5◦.(d) Scanning electron microscopy images of the sensor after FIBdefinition of the pickup loop. Figure reprinted with permissionfrom [71]. Copyright 2008 by the American Institute of Physics.

SQUID and the sample and ease of using flux feedbackschemes. Figure 4 illustrates the properties of current advancedscanning SQUID sensor design using the integrated pickuploop strategy [71]. This SQUID sensor uses a Nb–AlOx–Nb trilayer for the junctions, two levels of Nb for wiring andshielding and SiO2 for insulation between the various levels,as well as local field coils integrated into the pickup loopregion, to allow local magnetic susceptibility measurements[69, 72]. High symmetry in the SQUID and field coil layout(see figure 4(a)) and tapered terminations are desirable todiscriminate against background fields and reduce unwantedelectromagnetic resonances [73]. Figure 4(b) is an opticalmicrograph of a scanning SQUID susceptometer in whichboth the field coil and pickup loop were defined using opticallithography. Figure 4(c) shows the layout for a device inwhich the pickup loop area is left as a ‘tab’ in the opticallithography step, and then patterned into a loop using focussedion beam lithography [74]. The completed device is shownin a scanning electron micrograph in figure 4(d). This deviceuses a non-planarized process, meaning that each successivelayer conforms to the topography of the previous one. Thisleads to constraints on the alignment angles that can be used toget the minimum spacing between pickup loop and sample, asillustrated in the insets in figures 4(b) and (c). These constraintscan be reduced using a planarized process [75], in which

6


Figure 5. Scanning electron microscope image of a 100 nm Hallprobe. The four leads are separated by narrow etch lines. The gateshields the Hall cross from stray electrical charges and allows themodulation of the 2DEG beneath. The Hall probe will touch thesample surface at the contact tip. Figure reprinted with permissionfrom [87]. Copyright 2007 by the American Institute of Physics.

the insulating layers are chemically–mechanically polished toplanes before successive Nb wiring levels are added.

If we assume a SQUID or SQUID pickup loop diameterof 1 µm, a SQUID-sample spacing of 0.1 µm, and a SQUIDnoise level of 10−6�0 Hz−1/2, and using the peak signalsfrom figure 2, we find a minimum detectable dipole signal of139 electron spins Hz−1/2, a minimum flux signal of 1.25 ×10−6�0 Hz−1/2 and a minimum detectable line of currentsignal of 1.65 nA Hz−1/2.

2.2. Hall bar microscopy

A Hall bar develops a transverse voltage Vx = −IyBz/n2de

when a current Iy passes through it in the presence of amagnetic induction Bz perpendicular to the plane of the Hallbar [30–32], where n2d is the carrier density per unit area of theHall bar. Therefore, materials with small carrier densities suchas semi-metals, semi-conductors or two-dimensional electrongases at the interface between semi-conductors with differentbandgaps develop larger Hall voltages. Early scanning Hallbar systems used evaporated films of bismuth [76], InSb[77] or GaAs [78]. More recently high sensitivity andspatial resolution have been achieved with GaAs/AlxGa1−xAsheterojunction structures [79–84]. In addition, 250 nm sizedscanning probes using GaSb/InAs/GaSb [83] and micrometer-scale Si/SiGe and InGaAs/InP Hall crosses have beencharacterized at low temperatures [85, 86]. Hicks et al [87]have fabricated Hall bars as small as 85 nm on a side (seefigure 5) from GaAs/AlxGa1−xAs heterojunction electron gasmaterial, and estimate a field noise of 500 µT Hz−1/2 and aspin sensitivity of 1.2 × 104µB Hz−1/2 at 3 Hz and 9 K forsensors 100 nm on a side. Sandhu et al have fabricated 50 nmbismuth Hall bar sensors with a noise of 0.8G Hz−1/2 [88].At present Hall bars from GaAs/AlxGa1−xAs heterojunctionsare less sensitive above about 100 K because of thermalexcitations. Scanning Hall bar sensors made of InAs have

been used to image single magnetic biomolecular labels atroom temperature [89].

Magnetic tunnel junctions and Hall bars in the ballisticregime [90–92] produce a signal which is proportional to 〈Bz〉,the magnetic field perpendicular to the sensor, averaged overthe sensor. The situation is more complicated for a Hall barin the diffusive regime [93], but may be qualitatively similar.The average fields 〈Bz〉 for the three sources of field canbe inferred from equations (8), (10) and (12) (figure 2), bysubstituting �s → 〈Bz〉d2 and in figure 3 by substituting�p → 〈Bz〉p, where 〈Bz〉p is the peak value of the fieldaveraged over the sensor area. For a given value of z/d, 〈Bz〉is proportional to 1/d3 for a dipole source, proportional to1/d2 for a monopole source and proportional to 1/d for a lineof current source. Boero et al [94] estimate that the optimalJohnson noise limited minimum detectable magnetic field in amicro-Hall cross geometry is given by

Bmin ≈√

4kBT R0

vsatw(thermal), (14)

where kB is Boltzmann’s constant, T is the operatingtemperature, R0 is the output resistance at zero magneticfield, vsat is the saturation carrier drift velocity and w is thewidth of the cross. This results in Bmin ∼ 2nT Hz−1/2

for w = 1 µm at 300 K for doped InSb. Thermal noiseis expected to be independent of sensor size, and thereforethe scaling for sensitivity is the same as for 〈Bz〉. At thelow temperatures used for imaging superconductors, transportin the Hall sensor is often quasi-ballistic and the signal tonoise is better characterized by a mobility and an optimumprobe current. The latter is typically defined as the maximumcurrent before the onset of pronounced 1/f noise due to one ofseveral possible sources, e.g. generation–recombination noiseat deep traps or across heterointerfaces, heating or impactionization. However, noise in micrometer-scale Hall barsis often dominated by 1/f noise [87, 92]. The minimumdetectable field can then be written as [89]

Bmin ≈ 1

µd

√αHGN�f

nf1/f, (15)

where µ is the mobility, d is the sensor size, αH is Hooge’s1/f noise parameter, GN ∼ 0.325 is a constant, n is thecarrier density, �f is the measurement bandwidth, and f is thefrequency. Equation (15) combined with equations (8), (10)and (12) imply that for a Hall bar dominated by 1/f noise,for a given z/d the minimum detectable dipole moment scaleslike d2, the minimum detectable flux scales like d, and theminimum detectable current is independent of d. It has beenreported that low frequency noise in Hall bar devices can besignificantly reduced by optimizing the voltage on a gate overthe Hall cross [87, 95] and that sufficiently small devices havenoise which is composed of a single Lorentzian spectrum [96].

2.3. Magnetic force microscopy

A magnetic force microscope [97–99] images small changes inthe resonance frequency of a cantilever due to the interactionbetween magnetic material on a sharp tip at the end of the

7


zt

zt

zt

Figure 6. MFM modeling using the point dipole tip approximation: force derivative curves for (a) a dipole source located at the origin withmagnetic moment ms oriented parallel to z (solid line) or parallel to x (dashed line), equation (7), (b) a monopole source located at the originwith magnetic flux �0, equation (9) and (c) a line of current I along the y-axis, equation (11). The tip is modeled as a point dipole withmoment mt oriented parallel to the z-axis scanning a height z above the xy plane along the x-axis (equations (17)).

cantilever and local sample magnetic fields. The signal from amagnetic force microscope is proportional to dFtip,z/dz, thederivative of the z-component of the force on the tip withrespect to its z-position. Since this force sums contributionsfrom all the magnetic material in the tip, the sensitivity andspatial resolution of MFM depends on the shape of the tip.For some applications and tip materials the tip can be modeledas having a magnetic monopole at its apex, with the forceapproximated by Ftip,z = qBz, where q is the effectivemagnetic monopole moment of the tip [32, 100]. However,Lohau et al have shown that their experimental MFM data canbe modeled either using a monopole or a dipole approximationfor the end of the tip, but that in either case the effective positionof the monopole or dipole is a function of the field gradient fromthe sample [101].

It is instructive to model the ultimate spatial resolutionand sensitivity of an MFM in the point monopole and dipoleapproximations for the tip. The z-component of the forcederivative on the tip from a field source can be written as [101]

∂Fz

∂z= − q

µ0

∂Bz

dz+ mt,x

∂2Bx

∂z2+ mt,y

∂2By

∂z2+ mt,z

∂2Bz

∂z2, (16)

where q is the magnetic monopole flux and mt,i are the dipolemoments of the tip in the x, y, z directions. Assume forsimplicity that the tip is magnetized only in the z-direction.Then mt,x = mt,y = 0 and the force derivatives due to thedipole, monopole and current line sources of equations (7),(9), and (11) are given by

∂Fz

∂z= 3mt,zµ0

4πr9[−5(ms,xx + ms,yy)z(3r2 − 7z2)

+ ms,z(3r4 − 30r2z2 + 35z4)] (dipole–dipole)

= 3�0mt,zz

2π

5z2 − 3r2

r7(dipole–monopole)

= −mt,zµ0Ix

π

x2 − 3z2

(x2 + z2)3(dipole–current line),

(17)where mt,z is the dipole moment of the tip.

Similarly, the expressions for the force derivative in themonopole approximation are given by

∂Fz

∂z= − 3q

4πr7[(xms,x + yms,y)(r

2 − 5z2)

+ ms,zz(3r2 − 5z2)] (monopole–dipole),

= − q�0

2πµ0

r2 − 3z2

r5(monopole–monopole),

= qI

π

xz

(x2 + z2)2(monopole–current line). (18)

The force derivative relations given by equations (17) and(18) are plotted in figures 6 and 7, respectively.

Within these approximations the force derivative signalstrength increases strongly with decreasing z—like 1/z5 fora dipole source, 1/z4 for a monopole source and 1/z3 for acurrent line source, in the point dipole tip approximation. In themonopole tip approximation the powers are 1/z4, 1/z3 and 1/z2

for the three sources, respectively. The widths of the predictedforce derivative features are proportional to the height z of thetip above the source. The constant of proportionality does notdepend strongly on source: the full width at half-maximum (ordistance between the maximum and minimum force derivativefor a line of current source) is �x/z=0.6, 0.66 and 0.74 for thedipole tip, and 0.74, 1 and 1.16 for the monopole tip, for thethree sources respectively. Of course, these conclusions wouldbe modified for a real tip with a finite size. It has been estimatedthat the effective tip–sample spacing and radius of curvaturefor conventional MFM tips are both about 10 nm [100].

In order to estimate the sensitivity of MFM, one needs toknow the minimum detectable force derivative, which dependson a number of characteristics of the MFM, including those ofthe cantilever. Using the lumped mass model for an MFMcantilever, the equation of motion of the tip end position z isthat of a driven, damped harmonic oscillator

md2z

dt2+

dz

dt+ kz = Fsignal(t) + Fthermal(t), (19)

8


Figure 7. MFM modeling using the monopole tip approximation: MFM force derivative curves for (a) a dipole source located at the originwith magnetic moment ms oriented parallel to z (solid line) or parallel to x (dashed line), equation (7), (b) a monopole source located at theorigin with magnetic flux �0, equation (9) and (c) a line of current I along the y-axis, equation (11). The tip is modeled as a point monopolewith magnetic flux q scanning along the x-axis a height z above the xy plane (equations (18)).

where m is the effective mass of the cantilever, = k/ω0Q isthe damping constant, k is the cantilever spring constant, ω0 isthe angular resonant frequency, Q is the quality factor, Fsignal

is the signal force (in this case the magnetic interaction betweenthe tip and sample) and Fthermal is the force due to thermalfluctuations. The equipartition theorem implies that the powerspectral density of thermal fluctuations SF = 4 kBT . Thenthe thermally limited minimum detectable force derivative isgiven by

∂F

∂z min= 1

A

√4kkBT BW

ω0Q, (20)

where BW is the measurement bandwidth and A is theamplitude of oscillation of the cantilever.

A major thrust in research in MFM is to reduce the spatialextent of the magnetic material in the cantilever tip to improvespatial resolution and spin sensitivity. This is often done byshaping previously deposited material using focussed ion beametching [102, 103]. Sharply defined magnetic tips can alsobe produced by evaporating magnetic material on an electronbeam deposited carbon and ion etched needle [104] or onto acarbon nanotube [105] at the end of a conventional Si cantilevertip (see figure 8). Small amounts of magnetic material can alsobe deposited directly onto cantilever tips through nanoscaleholes fabricated in a stencil mask [106]. One approach wouldbe to place nano-magnets directly on the apex of cantilevertips. This would serve three goals: improved spatial resolution,improved spin sensitivity and improved interpretability, sincethe magnetic moment of the deposited nano-magnets wouldhave a narrow distribution [107]. However, such tips couldbe especially susceptible to switching of the tip momentin sufficiently strong magnetic fields, reversing the contrastof the MFM images [108]. Although this complicates theinterpretation of the images, the tip magnetization state is alsoknown better, since the tip is generally close to saturationeither up or down except for a narrow window around theswitching points. The tip in MFM exerts a relatively strong

Figure 8. Scanning electron microscope (SEM) image of a MFMcantilever. Inset upper left: high-resolution SEM image of the apexof the pyramid, where a coated carbon nanotube tip (CCNT) isvisible. The arrow shows the direction of the metal evaporation.Figure reprinted with permission from [105]. Copyright 2004 by theAmerican Institute of Physics.

magnetic field, comparable to the saturation magnetization ofthe magnetic material used in the tip, on the sample. Thiscan cause switching of the local magnetic moment of thesample [109], but can also be taken advantage of for vortexmanipulation studies (see section 3.5).

Assume for the moment that it will be possible to locatesmall nanoparticles at the end of an MFM cantilever tip. A 7 nmdiameter cobalt nanoparticle will have a total dipole momentof mt = 2.5 × 10−19A − m2, assuming a magnetization ofM = 1.4 × 106 A m−1 [110]. Taking values of Q =50 000,ω0 = 2π × 50 kHz, A = 1 nm, k = 2 N m−1, and T =4 K, the thermally limited force gradient is ∂F/∂z|min =1.7 × 10−7 N m−1. Using the dipole approximation for thetip, the peak force due to a sample dipole ms with both tipand sample moments oriented parallel to the z-axis is givenby (equation (17), figure 6) ∂F/∂zpeak = 24µ0msmt/4πz5.Using z = 15 nm results in a minimum detectible samplemoment of 23 Bohr magnetons. From figure 6 the full widthat half-maximum of the force derivative response would be�x = 0.6z = 9 nm.

9


Table 1. Sensitivities and spatial resolutions.

Dipole Monopole Line of current

N (µB Hz−1/2) �x (nm) � (�0 Hz−1/2) �x (nm) I (A Hz−1/2) �x (nm)

MFM 23 9 3.4 × 10−5 10 2.6 × 10−5 11SHM 4.5 × 104 100 3 × 10−3 108 1.14 × 10−4 118SSM 78 600 8.8 × 10−7 650 5.5 × 10−9 710

2.4. Summary

Each of the three techniques described here has advantagesand disadvantages: MFM has the best spatial resolution(∼10–100 nm), SSM the worst (∼0.3–10 µm), with Hall barsintermediate (∼0.1–5 µm). The relative sensitivities of thethree techniques depend on the type of field source. As in allmicroscopies there are tradeoffs between sensitivity and spatialresolution. The spatial resolution for SQUIDs and Hall bars isroughly given by �x ≈ d for heights z � d , where d is thesensor size (figures 2 and 3). For MFM the spatial resolutionis roughly set by the tip–sample spacing z, in the limit wherethe radius of curvature of the tip end is smaller than z (figures 7and 6). The flux sensitivity of SQUIDs is nearly independentof SQUID or pickup loop size, so the field sensitivity is roughlyproportional to the inverse of the pickup loop or SQUID area.With Hall bars limited by thermal noise the field sensitivityis independent of sensor size, but Hall bars dominated by 1/f

noise are expected to have a field sensitivity that scales like 1/d.Current MFM’s measured field gradients, rather than fields,and the tradeoff between sensitivity and spatial resolution areless clear than for SSM and SHM, although larger amountsof magnetic material will produce larger sensitivity but willrequire a larger tip volume, reducing resolution. Table 1compares sensitivity and spatial resolution for a state of theart SQUID, with a pickup loop 0.6 µm in diameter and aflux noise of 0.7 × 10−6�0 Hz−1/2 (figure 4 [71]), a stateof the art Hall bar with sensor size 100 nm and a field noiseof 500 µT Hz−1/2 at 3 Hz (figure 5 [87], and a hypotheticalMFM with a 7 nm Co nanoparticle at the tip as discussed insection 2.3. These numbers should be treated with a great dealof caution. For example, I have assumed that z/d = 0.1 for theSQUID and SHM, and z = 15 nm for the MFM. These maybe unreasonably optimistic estimates. However, table 1 doesindicate how relative sensitivities depend on the field source.As an example, SSM and MFM have roughly comparablespin sensitivities, but SSM has nearly two orders of magnitudebetter sensitivity for a monopole source, and nearly 4 ordersof magnitude better sensitivity for a line of current source.

Table 2 provides a summary of the scaling exponents n

for the minimum detectable field sources for SSM, SHM, andMFM. In this table the minimum detectable dipole moment,flux and current are proportional todn for SSM and SHM, whilefor MFM these quantities are proportional to zn. For SSM andSHM it is assumed that z/d is held constant. For SSM it isassumed that thermal noise dominates. There are two columnsfor SHM, for noise dominated by thermal fluctuations or by1/f noise. For MFM there are two columns, for the monopoleand the dipole tip approximations.

Table 2. Scaling exponents n.

Hall Hall MFM MFMSSM 1/f thermal monopole dipole

Line of current −1 0 1 2 3Monopole 0 1 2 3 4Dipole 1 2 3 4 5

There are further considerations in deciding whichmagnetic microscopy is best for a given application: SSMrequires a cooled sensor, about 9 K for the most technicallyadvanced Nb SQUIDs, although Hall bars and MFM also havethe highest sensitivities at low temperatures. SSM is the moststraightforward to calibrate absolutely. Hall bars tend to bemechanically delicate and the most sensitive to destructionby electrostatic discharge. MFM applies the largest fieldsto the sample. In general Hall bars and MFM can toleratelarger applied magnetic fields than SQUIDs. As we shallsee in section 3, each has been very useful in the study ofsuperconductors.

3. Applications

3.1. Static imaging of magnetic flux

Understanding the trapping of vortices in superconductors is ofcrucial technological importance, both because vortex pinningis the primary mechanism for enhanced critical currents intype II superconductors [111], and also because trappedvortices are an important source of noise in superconductingelectronic devices. Superconducting vortices are especiallyeasy to image using magnetic imaging because they arehighly localized, quantized and have relatively large magneticfields. A number of different techniques have been used forimaging superconducting vortices, including SSM, SHM andMFM. MFM has the advantage for this application that it canimage both the vortex magnetic fields and sample topographysimultaneously, so that macroscopic pinning sites can beidentified. It has been used for a number of years to imagesuperconducting vortices [112–119]. Care must be taken,however, because the large magnetic fields exerted by the tipcan distort and dislodge vortices. As we will see in section 3.5,this can also be an advantage. An particularly striking exampleof simultaneous magnetic and topographic imaging of vorticesin Nb and YBCO is shown in figure 9 [113].

3.1.1. Narrow strips. The sensitivity of high-Tc

superconducting sensors such as SQUIDs and hybridmagnetometers based on high-Tc flux concentrators is limited

10


Figure 9. (a) Magnetic force microscope image of 0.3 µm × 0.3 µm areas for a Nb film and the corresponding topographic image (b).(d) MFM image of a single vortex (2 × 2 µm−2) in a YBCO film and (e) the corresponding topographic image. The white circles in (b) and(e) correspond to the distance at which the stray field emanating from the vortex has decreased to 1/e of its maximum value. The smalldotted circle in (b) defines an area with diameter 2ξ ≈ 22 nm. The lower panels (c) and ( f ) are cross-sections of the MFM data and thetopography profiles along the white lines indicated in (a), (b), (d) and (e). The dotted curves correspond to a Gaussian profile fitting of theMFM signal after filtering with a low-pass filter. Figure reprinted with permission from [113]. Copyright 2002 by Elsevier.

by 1/f noise. One source of this noise is the movementof vortices trapped in the sensor. One technique forreducing this source of noise is to trap vortices at a distancefrom the sensitive regions of the superconducting circuitryusing holes or motes [121, 122]. Jeffery et al [123] useda scanning SQUID microscope to image flux trapping insuperconducting electronic circuitry with motes. Veauvy et al[124] studied vortex trapping in regular arrays of nanoholesin superconducting aluminum. Noise from trapped vorticesin high-Tc SQUID washers and flux concentrators can beeliminated by dividing the high-Tc body into thin strips[125, 126]. Below a critical magnetic induction no vortex

trapping occurs in these strips for a given strip width. Anumber of models for the critical inductions of thin-film stripshave been proposed [120, 127–129]. Indirect experimentaltesting of these models was done by observing noise inhigh-Tc SQUIDs as a function of strip width and induction[125, 126, 130]. More direct experimental verification of these

models was presented by Stan et al [131] using scanningHall probe microscopy on Nb strips and Suzuki et al [132]using SQUID microscopy on NdBa2Cu3Oy thin-film patternswith slots. Both experiment and theory found that thecritical magnetic induction varied roughly as 1/W 2, whereW is the strip width. However, the experimental [131]and theoretical [120, 127] pre-factors multiplying this 1/W 2

dependence differ significantly.Figures 10(a)–(d) show SQUID microscope images

of 35 µm wide YBCO strips cooled in various magneticinductions. Figure 10(e) plots Bc+, the lowest induction atwhich vortices are trapped in the strips (squares), and Bc−, thehighest induction at which vortices are not trapped (dots) as afunction of strip width. The lines represent various theoreticalpredictions.

The Gibbs free energy for a vortex in a superconductingstrip of width W in an applied induction Ba perpendicular to

11


(a)

(c) (d)

(b) (e)

Figure 10. SQUID microscope images of 35 µm wide YBCO thin-film strips cooled in magnetic inductions of (a) 5, (b) 10, (c) 20 and(d) 50 µT. (e) Critical inductions for vortex trapping in YBCO thin films as a function of strip width. The squares (Bc+) represent the lowestinduction in which vortices were observed, and the dots (Bc−) are the highest inductions for which vortex trapping was not observed. Thedashed–dotted line is the metastable critical induction B0 (equation (23)) [120], the short-dashed and long-dashed lines are BL

(equation (22)), the critical induction calculated using an absolute stability criterion, and the solid line is BK (equation (24)), calculatedusing a dynamic equilibrium criterion between thermally activation and escape for vortices. Figure reprinted with permission from [133].Copyright 2008 by the American Physical Society.

Figure 11. Top row: scanning Hall probe microscopy images of vortex configurations in square areas 19.6 µm on a side in asuperconducting Nb wire grid of 0.95 µm × 0.95 µm square holes, at the filling fractions f indicated. The second row shows maps of thepositions in the grid occupied by vortices. Note the ordering observed near f = 1/3 (c), f = 1/2 (e) and f = 4/3 (h). Figure reprintedwith permission from [143]. Copyright 1993 by the American Physical Society.

the strip plane can be written as [133]

G(x) = �20

2πµ0�ln

[αW

ξsin

(πx

W

)]

∓ �0(Ba − n�0)

µ0�x(W − x), (21)

where � = 2λ2/d is the Pearl length of the strip, λ is theLondon penetration depth, α is a constant of order 1, n isthe areal density of vortices, ξ is the coherence length and x

is the lateral position of the vortex in the strip. The criticalinduction model of Likharev [127] states that in order to trapa vortex in a strip the vortex should be absolutely stable: thecritical induction then happens when the Gibbs free energy in

the middle of the strip is equal to zero, and leads to

BL = 2�0

πW 2ln

(αW

ξ

). (22)

This is plotted as the dashed and dotted lines in figure 10for two assumed values of α. A second model, proposedby Clem [120], considers a metastable condition, in whichthe induction is just large enough to cause a minimum in theGibbs energy at the center of the strip, d2G(W/2)/dx2 = 0,leading to

B0 = π�0

4W 2. (23)

This is plotted as the dotted–dashed line in figure 10. Finally,Kuit et al proposed that the critical induction should result from

12


Figure 12. (a) Schematic diagram for the tricrystal (1 0 0) SrTiO3 substrate used in the phase-sensitive pairing symmetry tests of Tsueiet al [61]. Four epitaxial YBa2Cu3O7−δ thin-film rings are interrupted by 0, 2 or 3 grain boundary Josephson weak links. (b) ScanningSQUID microscopy image of the four superconducting rings in (a), cooled in a field <5 mG. The central ring has �0/2 magnetic flux, where�0 is the superconducting flux quantum, spontaneously generated in it. The other 3 rings, which have no spontaneously generated flux, arevisible through a small change in the self-inductance of the SQUID when passing over the superconducting walls of the rings.

a dynamic equilibrium between vortex thermal generation andescape. This leads to the critical induction

BK = 1.65�0

W 2. (24)

It can be seen in figure 10 that the dynamic equilibriummodel (solid line) fits the data the best of the three models,with no fitting parameters. Kuit et al also found that n,the areal density of trapped vortices, increased linearly withapplied induction above the critical value, in agreement withthe dynamic equilibrium model, and that the trapping positionsshowed lateral ordering, with a critical field for formation of asecond row that was consistent with a numerical predictionof Bronson et al [134]. Similar good agreement betweenexperiment and the dynamic equilibrium model was found bythe same authors for Nb thin film strips.

3.1.2. Superconducting wire lattices, clusters, and nano-hole arrays. Two-dimensional arrays of Josephson junctions[135–140] have been studied extensively, not only because theyoffer the possibility of studying Kosterlitz–Thouless effects[141] in an ordered two-dimensional system, but also becauseof interesting effects that were predicted [142] and observed[135, 136] to occur when the applied field is a rational fractionof the field required to populate each cell with a quantum ofmagnetic flux. Scanning magnetic microscopy has been usedextensively for studying vortex trapping in two-dimensionalsuperconducting wire networks [58, 143], clusters [58] andring arrays [144, 145]. Superconducting wire networks are ofinterest because they are a physical realization of a frustratedxy model [142, 146, 147], where the variable is the phaseof the superconducting order parameter. The filling fractionf of vortices/site can be varied by cooling the network invarious fields, and detailed predictions have been made forthe ground state vortex configurations at rational f fractions[148]. Studies of these systems were at first performed using

transport [149–152], but direct imaging of the vortex positionsprovides more detailed information. Runge et al did magneticdecoration experiments on superconducting wire lattices [153].Although these experiments provided direct information on thevortex positions, only one set of conditions, such as magneticfield and temperature, can be explored for each sample usingthis technique. An example of scanning Hall bar microscopyon a superconducting wire network is shown in figure 11 [143].In this figure ordered vortex arrangements can be seen nearthe fractional filling factors of f = 1/3, 1/2 and 4/3, aspredicted by Teitel and Jayaprakash [146, 142], along withgrain boundaries between ordered domains and vacancies.

Field et al [154] studied vortex trapping in Nb films withlarge arrays of 0.3 µm holes on a square lattice. They foundstrong matching effects when the cooling fields correspondedto one or two vortices per site, as well as at several fractionalmultiples of the matching field. They also observed strikingdomain structure and grain boundaries between the domains.

3.1.3. Half-integer flux quantum effect. The discovery in1986 of superconductivity at high temperatures in the cuprateperovskites [2, 3] generated enormous excitement. Althoughthere was a great deal of evidence that the superconductinggap in the cuprates was highly anisotropic, with a significantdensity of states at low energies [155, 156], phase-sensitivemeasurements [157, 158] were required to distinguish, forexample, d-wave from highly anisotropic s-wave pairingsymmetry. The first such experiments [159, 160] used themagnetic field dependence of the critical current in singlejunctions and 2-junction SQUIDs between single crystals ofYBa2Cu3O7−δ (YBCO) and Pb to demonstrate a π -phase shiftbetween the component of the superconducting pairing orderparameter perpendicular to adjacent in-plane crystal faces (e.g.ignoring the effects of twinning, between the a- versus b-axisnormal faces) of the YBCO. Similar phase-sensitive Josephsoninterference experiments were performed by Brawner and Ott

13


[161], Iguchi and Wen [162], Miller et al [163] and Kouznetsovet al [164].

A phase-sensitive technique for determining pairingsymmetry that is complementary to Josephson interferenceis to image magnetic fields generated by spontaneoussupercurrents in superconducting rings that have an oddnumber of intrinsic π -phase shifts in circling them or inJosephson junctions that have intrinsic π -shifts along them[165, 166]. The first such experiments were done by Tseuiet al [61], using YBCO films grown epitaxially on tricrystalSrTiO3 substrates. A scanning SQUID microscope wasused to image the magnetic fields generated by supercurrentscirculating rings photolithographically patterned in the YBCOfilms. In the original experiments the central ring (figure 12),which has 3 grain boundary Josephson junctions, has either1 or 3 intrinsic π -phase shifts for a superconductor withpredominantly dx2−y2 pairing symmetry, and should thereforehave states of local energy minima, when no external magneticfield is applied, with the total flux through the ring of� = (n + 1/2)�0, n an integer, for sufficiently large LIc

products, where L is the inductance of the ring and Ic isthe smallest critical current of the junctions in the ring.This is the half-integer flux quantum effect. The two outerrings that cross the tricrystal grain boundaries have 0 or2 intrinsic π -phase shifts for a dx2−y2 superconductor, andshould therefore have local energy minima at � = n�0—conventional flux quantization. Similarly, the ring that doesnot cross any grain boundaries has no intrinsic phase drop,and also has local minimum energies for � = n�0. Theseexpectations were confirmed using a SQUID microscope usinga number of techniques [61], consistent with YBCO havingdx2−y2 pairing symmetry. Subsequent tricrystal experimentseliminated the possibility of a non-symmetry related origin forthe half-integer flux quantization in the 3-junction ring [167],ruled out extended s-wave symmetry [168], demonstratedthe existence of half-flux quantum Josephson vortices [169],showed results consistent with dx2−y2 pairing symmetry for thehole-doped high-Tc cuprate superconductors Tl2Ba2CuO6+δ

[170], Bi2Sr2CaCu2O8+δ [171] and La2−xSrxCuO4−y [172],for a range of doping concentrations [172], and theelectron-doped cuprates Nd1.85Ce0.15CuO4−y (NCCO) andPr1.85Ce0.15CuO4−y (PCCO) [173], and for a large range intemperatures [174]. SQUID magnetometry experiments ontwo-junction YBCO–Pb thin-film SQUIDs were performed byMathai et al [175], and on tricrystals by Sugimoto et al [176].Josephson interference, SQUID magnetometry and microwavemeasurements were made on biepitaxial all high-Tc YBCOjunctions by Cedergren et al [177].

Grain boundary junctions in the high-Tc cupratesuperconductors are Josephson weak links. As such they areused for fundamental and device applications. In addition,grain boundaries play an important role in limiting the criticalcurrent density in superconducting cables made from thecuprates [178]. Anomalies in the dependence on magnetic fieldof the critical current of cuprate grain boundary junctions withmisorientation angles near 45◦ can be understood in terms oftheir dx2−y2 pairing symmetry combined with faceting, whichnaturally occurs on a 10–100 nm scale in these grain boundaries

Figure 13. Scanning SQUID microscope image of an area of1024 × 256 µm2 along an asymmetric 45◦ [0 0 1] tilt YBa2Cu3O7−δ

bicrystal grain boundary (arrows). There are some ten bulk vorticesin the grains (film thickness ∼180 nm), but there is also flux of bothsigns spontaneously generated in the grain boundary. The bottomsection shows a cross-section through the data along the grainboundary measured in units of �0 penetrating the SQUID pickuploop. Figure reprinted with permission from [180] Copyright 1996by the American Physical Society.

Figure 14. Generation of half-flux quanta in connected andunconnected YBa2Cu3O7−δ–Au–Nb zigzag structures. Shown arescanning SQUID micrographs of (a) 16 antiferromagneticallyordered half-integer flux quanta at the corners of a connected zigzagstructure and (b) 16 ferromagnetically ordered half-flux quanta at thecorners of an unconnected zigzag structure (T = 4.2 K). The cornerto corner distance is 40 µm in both (a) and (b). Figure reprinted withpermission from [184]. Copyright 2003 Macmillan Publishers Ltd.

[179]. This combination produces a series of intrinsic π -phaseshifts along such grain boundaries. These π -shifts havespontaneous supercurrents and magnetic fields associated withthem [180]. An example of SQUID microscope imaging ofthese spontaneous fields is shown in figure 13. If the distancebetween intrinsic π -shifts is shorter than the Josephsonpenetration depth, the total magnetic flux associated with eachwill be less than �0/2 [181]; a high density of facets canresult in ‘splintered’ Josephson vortices with fractions of �0

of flux [182].Facetted (zigzag) junctions can also be fabricated by

design, using for example a ramp-edge YBCO/Nb junctiontechnology [183]. An example is shown in figure 14, whichshows scanning SQUID microscope images of two facettedjunctions [184]. In figure 14(a) the corner junctions areconnected by superconducting material. In this case there isa strong interaction between the half-flux quantum Josephsonvortices upon cooling, so that they have a strong tendency toalign with alternating signs of the circulating supercurrentsand spontaneous magnetizations. In figure 14(b) the corner

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Figure 15. (a) Schematic of YBCO–Nb two-junction rings fabricated to phase sensitively determine the momentum dependence of thein-plane gap in YBCO. (b) SQUID microscope image, taken with a square pickup loop 8 µm on a side, of a square area 150 µm on a sidecentered on one of the rings, cooled and imaged in zero field at 4.2 K. (c) The outer circle of images, taken after cooling in zero field, has afull-scale variation of 0.04 �0 flux through the SQUID: the inner ring, taken after the sample was cooled in a field of 0.2 µT, has a full-scalevariation of 0.09 �0. The rings cooled in zero field had either 0 or �0/2 of flux in them; the rings cooled in 0.2 µT had �0/2, �0 or 2�0 offlux, as labelled. Figure reprinted with permission from [187]. Copyright 2006 Macmillan Publishing Ltd.

junctions have been isolated from each other by etchingchannels between them. In this case the half-flux quantumJosephson vortices all have the same alignment, even whencooled in a relatively small field.

Lombardi et al [185] and Smilde et al [186] havemapped out the in-plane momentum dependence of thesuperconducting gap amplitude of YBCO by measuring thecritical current as a function of junction normal angle fora series of biepitaxial YBCO grain boundary junctions,and YBCO/Nb ramp junctions, respectively. The latterexperiments, performed in part on untwinned epitaxial YBCOfilms, were able to determine the anisotropy of the gapbetween the a and b in-plane crystalline directions. However,these experiments were only sensitive to the amplitude ofthe superconducting order parameter, not its phase. Phase-sensitive measurements of the momentum dependence of thein-plane gap in YBCO were performed by Kirtley et al [187]by imaging the spontaneously generated magnetic flux in aseries of YBCO–Nb two-junction rings, with one junctionangle normal relative to the YBCO crystalline axes heldfixed from ring to ring, while the other junction normalangle varied. Some results from this study are shown infigure 15. The rings either had n�0 (integer flux quantization)or (n + 1/2)�0 (half-integer flux quantization) of flux in them,n an integer. The transition between integer and half-integerflux quantization happened at junction angles slightly differentfrom 45◦ + n × 90◦, as would be expected for a pure dx2−y2

superconductor, because of the difference in gap amplitudesbetween the crystalline a and b phases. In addition, carefulintegration of the total flux in the rings showed that anyimaginary component to the order parameter, if present, mustbe small.

3.1.4. Ring arrays. Arrays of superconducting rings areof interest as a physical analog of a spin: for conventionalsuperconducting rings the states with shielding supercurrentflowing clockwise or counter-clockwise are degenerate ata magnetic flux bias of �0/2 (half of a flux quantumthreading each ring). Two neighboring rings can interactmagnetically, making them a physical analog of the Ising spin.Frustration can be introduced into the system by choosingthe arrangement of the rings: square and hexagonal latticesare geometrically unfrustrated, while triangular and Kagomelattices are frustrated.

Davidovic et al used scanning Hall bar microscopy tostudy two-dimensional arrays of closely spaced Nb rings(see figure 16) [144, 145]. In these experiments Davidovicet al found that it was possible to find small islands ofantiferromagnetically ordered ‘spins’, but no perfect ordering,even in arrays with no geometrical frustration. One sourceof disorder in arrays of conventional superconducting rings isvariation in area from ring to ring from lithographic variations.Conventional rings must be flux biased to near �0/2 flux withhigh precision (see figure 16), and area variations lead to fluxvariations.

The same technology used for the pairing symmetry testsof figure 15 was used to form two-dimensional arrays ofπ -rings, superconducting rings with an intrinsic phase shiftof π [184]. An example of SQUID microscope imaging ofseveral π -ring arrays is shown in figure 17 [188]. π -rings havetwo degenerate states at zero flux bias, therefore eliminatingthe need for precise flux biasing of the rings, and therebyeliminating slight differences in the sizes of the rings as asource of disorder [188]. Although π -ring arrays showed morenegative values of the bond-order σ , corresponding to greater

15


Figure 16. A sequence of field-cooled images of a honeycomb lattice of 1 µm diameter hexagonal Nb thin-film rings, taken in increasingapplied fluxes near an applied flux per ring of �0/2. The flux increases from left to right starting at the upper left corner with 0.4913�0 andending in the lower right corner at 0.5066�0. Figure reprinted with permission from [145]. Copyright 1997 by the American PhysicalSociety.

antiferromagnetic ordering, than arrays of conventional rings,perfect ordering was never observed in either conventional orπ -ring arrays, possibly because of the large number of nearlydegenerate states in this system [189].

3.1.5. Tests of the interlayer tunneling model. One candidatemechanism for superconductivity at high temperatures in thecuprate perovskite superconductors is the interlayer tunnelingmodel, in which the superconductivity results from anincreased coupling between the layers in the superconductingstate [190–194]. An essential test of this theory is thestrength of the interlayer Josephson tunneling in layeredsuperconductors. For the interlayer tunneling model tosucceed, the interlayer coupling in the superconducting

state must be sufficiently strong to account for the largecondensation energy of the cuprate superconductors. Thebest materials for testing this requirement are Tl2Ba2CuO6+δ

(Tl-2201) and HgBa2CuO4+δ (Hg-1201), which have highcritical temperatures (Tc ≈ 90 K) and a single copper-oxideplane per unit cell. Scanning SQUID microscope images weremade of interlayer Josephson vortices emerging from the a−c

face of Tl-2201 [195] (figure 18) and Hg-1201 [196]. Theextent of these vortices in the a-axis direction is set by theinterlayer penetration depth λc. Some spreading of the vortexfields occurs as the crystal surface is approached [197], but notenough to affect the qualitative conclusions from these studies:the experimentally determined interlayer penetration depth λc

is an order of magnitude larger than required by the interlayertunneling model. These conclusions are in agreement with

16


Figure 17. SQUID microscopy images of four electricallydisconnected arrays of π -rings with 11.5 µm nearest neighbordistances. These images were taken at 4.2 K with a 4 µm diameterpickup loop after cooling in nominally zero field at 1–10 mK s−1.The bond-order σ is a measure of the antiferromagnetic order in thearrays. The geometrically unfrustrated square and honeycombarrays could have perfect antiferromagnetic correlation, whichwould correspond to σ = −1. The minimum possible bond-orderfor the frustrated Kagome and triangle lattices is σ = −1/3. Figurereprinted with permission from [188]. Copyright 2005 by theAmerican Physical Society.

measurements of the Josephson plasma frequency in thesematerials [198].

3.1.6. Pearl vortices. Vortices in thin superconductorssatisfying d � λL, where d is the thickness and λL isthe London penetration depth, first described by Pearl [199],have several interesting attributes. The field strengths Hz

perpendicular to the films diverge as 1/r at distances r � �,where � = 2λ2

L/d is the Pearl length, in Pearl vortices,whereas in Abrikosov (bulk) vortices the fields diverge asln(r/λL) [200]. Since in the Pearl vortex much of thevortex energy is associated with the fields outside of thesuperconductor, the interaction potential Vint(r) between Pearlvortices has a long range component Vint ∼ �/r for r �

[199], unlike Abrikosov vortices, which have only short rangeinteractions. Although it was originally believed that theinteraction between Pearl vortices Vint ∼ ln(�/r) for r � �

leads to a Berezinskii–Kosterlitz–Thouless (BKT) transitionwhich is cut off due to screening on a scale � [111], Kogan[201] reports that the BKT transition could not happen inthin superconducting films of any size on insulating substratesbecause of boundary conditions at the film edges that turnthe interaction into a near exponential decay. Grigorenkoet al, using SHM, observed domains of commensurate vortex

Figure 18. (a) Scanning SQUID microscope image of an interlayerJosephson vortex emerging from the ac face of a single crystal of thesingle-layer cuprate superconductor Tl2Ba2CuO6+δ , imaged at 4.2 Kwith a square pickup loop 8.2 µm on a side. (b) A two-dimensionalfit to the vortex in (a) with λc, the penetration depth perpendicular tothe CuO2 planes, equal to 18 µm and z0, the spacing between theSQUID pickup loop and the sample surface, equal to 3.6 µm.(c) The difference between the data and the fit. (d) Cross-sectionsperpendicular and parallel to the long axis of the vortex, offset forclarity. Figure reprinted with permission from [195]. Copyright1998 AAAS.

patterns near rational fractional matching fields of a periodicpinning array in a thin-film superconductor, which could onlyform when the vortex–vortex interactions are long range [202].

Figure 19 shows scanning SQUID microscope images ofPearl vortices in ultra-thin [Ba0.9Nd0.1CuO2+x]m/[CaCuO2]n(CBCO) high-temperature superconductor films. Pearlvortices can be magnetically imaged in such thin films,although the Pearl length can be as large as 1 mm, because ofthe strong 1/r divergence of the Pearl vortex magnetic fieldsat the vortex core: the thin-film limit for the two-dimensionalFourier transform of the z-component of the field from anisolated vortex trapped in a thin film is given by [199, 203]

Bz(k, z) = �0e−kz

1 + k�, (25)

where z is the height above the film, k =√

k2x + k2

y , � =2λ2

ab/d, λab is the in-plane penetration depth, d is the filmthickness and �0 = h/2e.

The open circles in figures 19(b) and (d) are cross-sectionsthrough the data of figures 19(a) and (c). The solid lines are

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Figure 19. SQUID microscope image and cross-sectional data along the positions indicated by the dashed line in (a), (c) of Pearl vorticestrapped in two artificially layered (Ba0.9Nd0.1CuO2+x)m/(CaCuO2)n superconducting films. The SQUID pickup loops were a square 7.5 µmon a side (a),(b), and an octagon 4 µm on a side (c),(d) (schematics superimposed on image). The open symbols in (b),(d) are thecross-sectional data; the solid lines in (b), (d) are fits. Figure reprinted with permission from [299]. Copyright 2004 by the AmericanPhysical Society.

fits to the data using equation (25), using the Pearl length �

as a fitting parameter. The in-plane London penetration depthλab of these thin films can be inferred from � if d is known.

3.1.7. Mesoscopic superconductors. Vortices in super-conductors with dimensions comparable to the super-conducting coherence length ξ have been predicted to nucleatespontaneously in a number of interesting configurations, suchas shells [204], giant vortices [205] and antivortices, so asto preserve the symmetry of the sample [206, 207]. Vortextrapping patterns in mesoscopic superconducting samples havebeen studied using Bitter decoration [208], scanning SQUIDmicroscopy [209], ballistic Hall magnetometry [91], transport[210] and Hall bar microscopy. An example of Hall barmicroscopy results is shown in figure 20. Nishio et al [211]report results in agreement with Ginzburg–Landau calculationsfor temperatures considerably lower than the superconductingtransition temperature.

3.1.8. Sr2RuO4. Sr2RuO4, with the same crystal structureas La2CuO4, the parent compound for the first class

of cuprate perovskite compounds shown to exhibit high-temperature superconductivity [2], was discovered in 1994to be superconducting at about 1 K [6]. It is thought tobe a px ± ipy-wave, triplet pairing superconductor withbroken time-reversal symmetry. One of the consequencesof this pairing symmetry state is spontaneously generatedsupercurrents, and consequent large magnetic fields at surfacesand boundaries between px + ipy and px − ipy domains [212].Although there is evidence for characteristic magnetic fieldsof 0.5 G in Sr2RuO4 from muon spin resonance experiments[213], and evidence for broken time-reversal symmetry atthe surface of Sr2RuO4 has been obtained using Sagnacinterferometry [18], no evidence for spontaneously generatedmagnetic fields at the surfaces of Sr2RuO4 samples has beenfound in scanning Hall bar and scanning SQUID experiments,with much higher sensitivity than the magnetic fields originallypredicted [214–216]. A possible explanation for this failure toobserve spontaneous magnetization directly is that the px ±ipy

domains are small, so that the fields from closely spaceddomain boundaries nearly cancel each other. The experimentalinformation on px ± ipy domain sizes is not consistent, with

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Figure 20. Scanning Hall probe microscope images of a4 µm × 4 µm square Pb film at applied fields of (a) 4.6 G, (c) 6.6 G,(e) 8.6 G and (g) 10.6 G. ((b), (d), ( f ) and (h)). The results of 2Dmonopole fits to (a), (c), (e) and (g), respectively. The broken linesdenote the shape of the square. The locations of the vortices areshown as white points. Figure reprinted with permission from [211].Copyright 2008 by the American Physical Society.

the first phase-sensitive pairing symmetry experiments onSr2RuO4 being consistent with domain sizes of order 1 mm[217], while later phase-sensitive results are consistent withdomain sizes 1 µm or less [218], and the Sagnac interferometrybeing consistent with domain sizes intermediate betweenthese two sizes [18]. Recently Raghu et al have suggestedthat the superconductivity in Sr2RuO4 arises primarily inquasi-one-dimensional bands, and is therefore not expectedto generate experimentally detectable edge currents [219].

Dolocan et al [220, 221] have magnetically imaged singlecrystals of Sr2RuO4 using a scanning µSQUID mountedon a tuning fork in a dilution refrigerator. This allowedcombined magnetic and topographic information on a lengthscale of 1 µm, at temperatures below 400 mK. They foundcoalescence of vortices and the formation of flux domains(see figure 21). The formation of lines of vortices in a fieldtilted relative to the crystalline c-axis [222] can be attributedin this case to the large anisotropy in Sr2RuO4, resulting inan attractive interaction between vortices in a plane definedby the c-axis and the magnetic field direction [221]. Thevortex coalescence observed when the applied magnetic fieldis parallel to the c-axis may be due to weak intrinsic pinning atdomain walls, and the existence of a mechanism for bringing

vortices together that overcomes the conventional repulsivevortex–vortex interaction.

3.1.9. Magnetic superconductors. Magnetic order andsuperconductivity are competing orders since the Meissnerstate excludes a magnetic field from the bulk andsuperconductivity is destroyed at sufficiently high fields.However, superconductivity and magnetism can coexist[223–226] if the orientation of the local magnetic momentsvaries on a length scale shorter than the superconductingpenetration depth λ, or if the field generated by themagnetization is carried by a so-called spontaneous vortexlattice [227]. The former order has been observed inErRh4B4 and HoMo6S8 [228, 229], and indirect evidence forthe latter has been reported in UCoGe [230]. Several Hallbar studies have been performed on artificial ferromagnetic-superconducting hybrid structures [231–234]. figure 22[235] shows scanning Hall bar microscopy measurementsof ErNi2B2C, which has a superconducting Tc of ≈11 K,becomes antiferromagnetic below TN ≈ 6 K, and exhibitsweak ferromagnetism below TWFM ≈ 2.3 K [236]. Bluhm et al[235] found that in this material the superconducting vorticesspontaneously rearranged to pin on twin boundaries uponcooling through the antiferromagnetic transition temperature,and that a weak random magnetic signal appears in theferromagnetic phase.

3.2. Fluxoid dynamics

3.2.1. Limits on spin-charge separation. One explanation forthe peculiar normal state properties and high superconductingtransition temperatures of the high-Tc cuprate perovskites isthat there exists a new state of matter in which the elementaryexcitations are not electron-like, as in conventional metals, butrather the carriers ‘fractionalize’ into ‘spinons’, chargeless spin1/2 fermions, and ‘chargons’, bosons with charge +e [237].Senthil and Fisher proposed a model for such a separationin two dimensions with sharp experimental tests [238]: inconventional superconductors a Cooper pair with charge 2e

circling around a vortex with h/2e of total magnetic fluxexperiences a phase shift of 2π . However, a chargon wouldpick up only a phase shift of π . In order for the chargon wavefunction to remain single-valued, it must be accompanied bya ‘vison’ that provides an additional π phase shift. In theSenthil-Fisher test, a superconducting ring with flux trappedin it is warmed through the superconducting transition. If thenumber of vortices in the ring is odd, then it will also containa vison that could persist above Tc. If the ring is then recooledin zero field before the vison escapes, it will spontaneouslygenerate a single flux quantum with random sign. On theother hand, if the number of vortices in the ring is even, nosuch ‘vortex memory’ effect would exist. Bonn et al [239](figure 23) tested for this effect in highly underdoped singlecrystals of YBa2Cu3O6+x . These crystals have a very sharpbut low transition temperature favorable for such a test. Theyfound no evidence for a vortex memory effect, and were ableto put stringent upper limits on the energy associated with avison. Some models in which superconductivity results from

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Figure 21. Scanning SQUID microscope images of flux domains in Sr2RuO4 at T = 0.36 K after field cooling at various fields. In all cases,the magnetic field amplitude applied along the c-axis (H⊥) was kept constant at 2 G while the in-plane field (Hab) was (a) 0 G, (b) 5 G,(c) 10 G, (d) 50 G. The imaging area is 31 µm × 17 µm. The field scale in G is shown on the right; dark regions are superconductingvortex-free regions. Figure reprinted with permission from [220]. Copyright 2005 by the American Physical Society.

Figure 22. Scanning Hall bar image of a single-crystal sample of ErNi2B2C. After cooling below Tc in a weak field (2.4 Oe in (a), (b), (d),(e), 18 Oe in (c)), a vortex distribution consistent with random pinning is observed on the ab face ((a) 7.3 K). Upon reducing the temperaturebelow TN, the vortices spontaneously organize along twin domain walls along the [1 1 0] ((b) 5.3 K) or [1 1 0] ((c) 4.2 K) direction. Thepattern gradually disappears as the temperature is raised again ((d) 5.7 K, (e) 6.0 K, different cycle than (a) and (b)). Figure reprinted withpermission from [235]. Copyright 2006 by the American Physical Society.

spin-charge separation predicts h/e fluxoids in materials withlow superfluid density [240–242]. Wynn et al [243], usingscanning SQUID and Hall bar microscopies, found no evidenceforh/e vortices in strongly underdoped YBa2Cu3O6+x crystals,also placing limits on spin-charge separation in these materials.

3.2.2. Superconducting rings. As we have seen in previoussections, the dynamics of fluxoids in a ring configuration areof interest as a model for the Ising spin system [144, 145, 188]section 3.1.4, and for placing limits on spin-charge separationin underdoped cuprate superconductors [239] section 3.2.1.Kirtley et al studied fluxoid dynamics in photolithographicallypatterned thin-film rings of the underdoped high-temperature

superconductor Bi2Sr2CaCu2O8+δ using a scanning SQUIDmicroscope, and concluded that their results could beunderstood in terms of thermally activated nucleation of a Pearlvortex in, and transport of the Pearl vortex across, the ringwall [244].

Fluxoid dynamics in superconducting rings may alsoprovide clues to physics on an entirely different scale: the earlydevelopment of the universe. Immediately after the Big Banga fundamental symmetry related the particles that mediatedthe electromagnetic, strong and weak forces. However,as the universe cooled down this symmetry was broken insuch a way that, for example, the W and Z bosons, whichmediate the weak interaction, have mass, while photons, which

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Figure 23. Cross-sections and images of single (bottom rightimage, lower cross-section) and double (middle right image, highercross section) flux quanta trapped in a 50µm × 50 µm square ofsingle-crystal YBa2Cu3O6.35 (Tc = 6.0 K) with a 10 µm hole drilledusing a focussed ion beam (top right image). When the sample washeated to 5.6 K for 1 s and recooled to 2 K, these flux quanta escapedwith no sign of the vortex memory associated with visons. Figurereprinted with permission from [239]. Copyright 2001 MacmillanPublishing Ltd.

mediate the electromagnetic force, do not. Kibble proposed anintuitive picture of this symmetry breaking of the early universe[245, 246], with the underlying idea that causality governedthe number of defects because a finite time delay is needed forinformation to be transferred between different regions of asystem. Zurek pointed out that Kibble’s ideas could be testedby studying vortices in superfluids and superconductors [247].As a superconductor or superfluid cools through its transitiontemperature different regions can nucleate into a state withthe same order parameter amplitude ψ but different phasesφ. This will produce ‘kinks’ or ‘defects’ where φ wouldeventually (at low temperature) change by 2π . These kinks canbe removed by annihilation with anti-kinks if the coherencelength ξ is long enough and the cooling rate is sufficientlyslow. However, as the system cools this annihilation processproceeds less rapidly, until at some point the kinks ‘freeze-in’ because part of the system cannot communicate with otherparts quickly enough. In a superconducting ring a phase shiftof 2π around the ring corresponds to a fluxoid trapped in thering, so that the process of forming and freezing in kinks isalso called spontaneous fluxoid generation. Kibble and Zurekshowed that causality implies that the probability P of finding aspontaneous fluxoid in the ring (when cooled in zero externalfield) depends on the cooling rate τ−1

Q as P ∝ (τQ/τ0)−σ ,

where τ0 is a characteristic time and σ depends on howthe correlation length ξ and the relaxation time τ vary withtemperature close to the transition. For a superconducting ringand assuming a mean-field approximation σ = 1/4.

Tests of the Kibble–Zurek prediction have been madeby looking for vortices in superfluid 4He [248, 249] and3He [250, 251], in superconducting films [15, 252], andsuperconducting rings interrupted by Josephson junctions[253–255]. However, verification of the ideas of Kibble andZurek using arguably the simplest system, superconductingrings, was not made until recently. Kirtley et al [256]

Figure 24. SQUID microscope images of a 12 × 12 array of 20 µminside diameter, 30 µm outside diameter thin-film rings of Mo3Si,cooled in zero field through the superconducting Tc at various rates.The resulting ring vortex numbers were consistent with thermallyactivated spontaneous fluxoid formation. Figure reprinted withpermission from [256]. Copyright 2003 by the American PhysicalSociety.

(figure 24) used a SQUID microscope to image an array ofrings after repeated cooling in various magnetic fields andcooling rates to determine the probability of spontaneousfluxoid generation. They showed that a second mechanismprevailed, in which the final density of fluxoids depended on abalance between thermal generation and the relaxation rates offluxoids [257–259]. They argued that the thermal fluctuationmechanism is complementary to the ‘causal’ mechanismand should be considered in attempts to understand phasetransitions, both in the laboratory and in the early universe.The Kirtley et al experiments [256] were done on rings withparameters which favored the thermal activation mechanismover the Kibble–Zurek (causal) mechanism. Recently Monacoet al [260] have studied superconducting rings with parametersmore favorable to the causal mechanism, using a geometry inwhich the flux state of a single ring can be manipulated andsensed rapidly, and find results consistent with the Kibble–Zurek prediction, taking into account the fact that the ring’scircumference was much smaller than the coherence length atthe temperature at which the fluctuations were frozen in.

3.2.3. High-Tc grain boundaries. Grain boundaries in thecuprate high-Tc superconductors are widely used for devicesand fundamental studies, and govern the transport propertiesof superconductors with technological applications [178]. It istherefore important to understand the mechanism of dissipationin transport across grain boundaries. Transport across grain

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Figure 25. Scanning Hall bar measurements of vortex noise in a 24◦

symmetric YBCO grain boundary (GB). (a) Sketch of the sample.The bridge width is 50 µm. (b), (c) Magnetic image of the GB area,(b) at 10 K and applied current I = 29 mA and (c) at 10 K afterthermal cycling to 70 K with I = 0, showing individual vortices (redonline) and antivortices (blue online). The regions imaged areoutlined by the square dashed line (orange online) in (a). (d), (e)The Hall probe signals versus time at 10 K for different currentsapplied to the GB measured at two positions marked in figure 1(b).The switching events are vortices passing under the probe. Thecircles (red online) indicate three-level switching events. Figurereprinted with permission from [261]. Copyright 2009 by theAmerican Institute of Physics.

boundaries is usually studied with a voltage threshold on theorder of nanovolts. At these voltages millions of vorticestraverse the boundary per second. Much smaller voltages canbe measured by magnetically imaging the vortices in the grainboundaries. Kalisky et al [261] performed such measurementson grain boundaries produced by epitaxial growth of thecuprate high-Tc superconductor YBa2Cu3O7−δ on bicrystals ofSrTiO3 using a large scanning area Hall bar microscope [84].They observed (figure 25) telegraph noise in the Hall barsignal when the sensor was directly above the grain boundary,which they attributed to the motion of vortices. They inferredthe voltage across the grain boundary from the frequencyof the telegraph noise, with voltage sensitivity as small as2 × 10−16 V. The dependence of the inferred grain boundaryvoltage on current followed either an exponential V ∝ emI/Ic

or a power law V ∝ In dependence with high values ofm or n. These results were qualitatively different fromgrain boundary transport measurements at higher voltages

Figure 26. SQUID signal (left axis) and ring current (right axis) asa function of applied flux �a for two aluminum rings, both withthickness d = 60 nm and annulus width w = 110 nm. Thefluctuation theory (dashed, red online) was fit to the data (noisysolid, blue online) through the dependence of Iring on �a andtemperature. (a) to (c) ring radius R = 0.35 µm, fittedTc(�a = 0) = 1.247 K, and γ = 0.075. The parameter γcharacterizes the size of the ring. The smooth solid line (greenonline) is the theoretical mean-field response for T = 1.22 K andshows the characteristic Little–Parks line shape, in which the ring isnot superconducting near �a = �0/2. The excess persistent currentin this region indicates the large fluctuations in the Little–Parksregime. (d ) R = 2 µm, fitted Tc(�a = 0) = 1.252 K and γ = 13.Figure reprinted with permission from [265]. Copyright 2007 byAAAS.

[178, 262, 263], and could not be fully explained using existingmodels.

3.3. Local susceptibility measurements

3.3.1. Superconducting fluctuations. Experimental knowl-edge of superconducting fluctuations in one dimensionhas largely been derived from transport measurements[264], which require electrical contacts and an externallyapplied current. The SQUID susceptometer, along witha ring geometry for the superconductor, allows the studyof fluctuation effects, without contacts, in isolated, quasione-dimensional rings in the temperature range wherethe circumference is comparable to the temperature-dependent Ginzburg–Landau coherence length ξ(T ). In suchmeasurements, an example of which is shown in figure 26[265], a magnetic field is applied to the ring using a single-turn field coil, co-planar and concentric with the pickup loop,both of which are integrated into the SQUID sensor [72]. Theresponse of the ring to this applied flux is measured by thepickup loop. The SQUID sensor is in a carefully balancedgradiometer configuration, such that it is insensitive to the

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Figure 27. Local susceptibility image in underdoped Ba(Fe1−xCox)2As2, indicating increased diamagnetic shielding on twin boundaries.(a) Local diamagnetic susceptibility, at T = 17 K, of the ab face of a sample with x = 0.051 and Tc = 18.25 K, showing stripes of enhanceddiamagnetic response (white). In addition there is a mottled background associated with local Tc variations that becomes more pronouncedas T → Tc. Overlay: sketch of the scanning SQUID sensor. The size of the pickup loop sets the spatial resolution of the susceptibilityimages. (b) and (c) Images of the same region at (b) T = 17.5 K and (c) at T = 18.5 K show that the stripes disappear above Tc. Atopographic feature (scratch) appears in (b) and (c). (d) Crystal structure of a unit cell of the parent compound BaFe2As2. (e) Top view ofthe FeAs layer with tetragonal symmetry, and ( f ) an exaggerated view of the orthorhombic distortion that occurs at low temperatures.(g) Schematic of a possible arrangement of spins across a twin boundary in the antiferromagnetic state. Figure reprinted with permissionfrom [285]. Copyright 2010 by the American Physical Society.

fields applied by the field coil [73]. Nevertheless, furtherbackground subtraction, by comparing the mutual inductancebetween the field coil and SQUID with the pickup loop closeto the ring with that with the pickup loop at a distance, isrequired to measure the response field due to the ring, whichcan be 7 orders of magnitude smaller than the applied field.A scanned sensor allows multiple rings to be measured in asingle cooldown. The experimental results agree with a fullnumeric solution of thermal fluctuations in a Ginzburg–Landauframework that includes non-Gaussian effects [266, 267] for allof the rings for which calculations were numerically tractable.This is in contrast to previous work on a single Al ring [268],which disagreed strongly with theory.

Similar techniques have allowed the measurement ofspontaneous persistent currents in normal metal rings [269].

3.3.2. Stripes. The pnictide class of superconductors [8]have the highest critical temperatures (57 K) observed for anon-cuprate superconductor, multiband Fermi surfaces, and atcertain doping levels a paramagnetic to antiferromagnetic aswell as a tetragonal to orthorhombic transition above the super-conducting transition temperature. In Ba(Fe1−xCox)2As2

(Co doped Ba-122) and other members of the 122 family

(AFe2As2 with A = Ca, Sr, and Ba), doping causes thespin-density-wave transition temperature and the structuraltransition temperature to decrease [270, 271] falling to zero ator near the doping where the highest Tc occurs, suggestingthe importance of lattice changes in determining transportproperties. Both experiment [272–274] and theory [275]have suggested a close relationship between structural andmagnetic properties, leading some authors to describe thelattice and spin-density-wave transition by a single orderparameter [276, 277]. In addition, structural strain appears toplay a significant role in the superconductivity. For example, inthe 122 compounds, small amounts of non-hydrostatic pressurecan induce superconductivity [278–282]. In addition, there isevidence that the structural perfection of the Fe-As tetrahedronis important for the high critical temperatures observed in theFe pnictides [283, 284].

Kalisky et al [285] have shown that Co doped Ba-122 hasstripes of enhanced susceptibility below the superconductingtransition temperature (see figure 27). These stripes areresolution limited using a SQUID pickup loop with an effectivediameter of 4 µm, and are believed to be associated withboundaries between twins- crystallites with their in-plane a, b

axes rotated by 90◦. The amplitude of the susceptibility

23


Figure 28. (a) Geometry of a SQUID susceptometer. (b) Mutual inductance between the field coil and the pickup loop, as a function of thespacing z between the SQUID substrate and a thin CBCO film. The symbols are data, taken with various alternating currents through thefield coil. The solid line is modeling using equation (25), with � = 11.8 µm. (b) Comparison of the Pearl length � for a number of CBCOsamples using fitting of SQUID magnetometry images of vortices (vertical axis) versus susceptibility measurements (horizontal axis).Figure reprinted with permission from [299]. Copyright 2004 by the American Physical Society.

stripes becomes larger as the superconducting transitiontemperature is approached from below. Since the susceptibilitysignal for a homogeneous superconductor becomes largeras the penetration depth becomes shorter and the superfluiddensity becomes larger, it seems reasonable to associatethe susceptibility stripes with shorter penetration depths andenhanced superfluid densities on the twin boundaries. Twinboundaries have been associated with enhanced superfluiddensity in conventional superconductors [286], but theinfluence of twin planes on superconductivity in, for example,the cuprates is less clear [287–291].

It is difficult to calculate analytically the effect of a planeof reduced penetration depth (enhanced superfluid density) onthe susceptibility observed in scanning SQUID measurements.Kirtley et al [292] have done a finite-element calculation inthe appropriate geometry. Because of the uncertainty in thewidth of the region with enhanced superfluid density, it is alsodifficult to estimate the size of the enhancement in the Cooperpair density from the scanning susceptibility measurements.Nevertheless, Kirtley et al estimate an enhancement in thetwo-dimensional Cooper pair density in the range between1019 and 1020 m−2. For comparison, the two-dimensionalelectron liquid [293] at the LaAlO3–SiTiO3 interface, whichexhibits superconductivity at about 0.2 K [294], has a carrierconcentration of about 1017 m−2. The temperature dependenceof the stripe amplitude can be best fit by assuming thatthe twin boundary has a different critical temperature thanthe bulk, although no stripes were observed above thebulk Tc.

Kalisky et al also observed that vortices tended to avoidpinning on the twin boundaries, and that they were difficultto drag over the twin boundaries using either a SQUIDsusceptometer sensor or a magnetic force microscope tip [292].The enhancement of supercurrent density, as well as the barrierto vortex motion presented by the twin boundaries, providestwo mechanisms for enhanced critical currents in twinnedsamples. Prozorov et al [295] have noted an enhancementof the critical current of slightly underdoped single crystals ofBa(Fe1−xCox)2As2, which they associate with twinning.

3.3.3. Spinlike susceptibility. There appears to be acomponent of 1/f α (α ∼ 1) noise in superconductingdevices, which is apparently due to magnetic flux noise, and isremarkably universal [296, 297]. Recently Koch, DiVincenzoand Clarke (KDC) have proposed that this noise is due tounpaired, metastable spin states [44]. The proposed spin statesare bistable, with a broad distribution of activation energies forchanging their spin orientation. When the spin state changesit changes the amount of magnetic flux coupling into thesuperconducting device. Each contributes a Lorentzian witha particular cutoff frequency to the total noise; adding upa large number of Lorentzians with a broad distribution ofcutoff frequencies results in noise with 1/f distribution [43].KDC estimate that a density of unpaired spin states of about5 × 1017 m−2 is required to explain universal 1/f noise.

Bluhm et al [298] have found a signal in scanningSQUID susceptometry measurements at low temperatures in anumber of materials, both metals and insulators, even including

24


Figure 29. (a), (b) Optical micrographs of two single crystals of the pnictide superconductor LaFePO. (c) Susceptibility scan of #1 atT = 0.4 K. (d) Change in heff = h + λ between 0.4 and 3 K over sample #2. (e), ( f ) Maps of local Tc over the same areas as in (c) and (d).The crosses indicate points where �λ(T ) data were collected. (g) �λ of samples #1 and #2. The dashed lines are fits between0.7 K < T < 1.6 K. (h) Black lines are possible superfluid densities for LaFePO sample #1, point 2, with different λ(0). Shaded area:superfluid density of YBa2Cu3O6.99 from [306, 307]. The width of the shaded area reflects uncertainty in λ(0). Figure reprinted withpermission from [304]. Copyright 2009 by the American Physical Society.

Au, that has a paramagnetic response with a temperaturedependence consistent with unpaired spins. This susceptibilityhas a component that is out of phase with the applied field,implying that it could contribute to 1/f -like magnetic noise.The density of these spin states is estimated to be in the range of1017 m−2, consistent with the KDC estimate. This implies thatscanning SQUID susceptometry could be used as a diagnosticfor determining which materials and processes contribute moststrongly to 1/f noise in superconducting devices.

3.4. Penetration depths

Penetration depths have been inferred from fitting the magneticimages of superconducting vortices for a number of years[32, 79, 80, 299–302]. An alternate means is to measure themutual inductance between the SQUID pickup loop and a fieldcoil integrated into the sensor in SSM [72] (figure 28). Thetwo-dimensional Fourier transform of the z-component of theresponse field in the pickup loop, with a current I in a circularring of radius R oriented parallel to, and a height z above ahomogeneous bulk superconductor with penetration depth λ is

given by [303]

Bz(k) = −πµ0IR

(q − k

q + k

)J1(kR)e−2kz, (26)

where q =√

k2 + 1/λ2 and J1 is the order 1 Bessel functionof the first kind. Similarly the response field for a thin-filmsuperconducting sample with Pearl length � is given by [303]

Bz(k) = −πµ0IRJ1(kR)e−2kz

1 + k�. (27)

To a good approximation, if z λ the magnetic fields resultingfrom the screening of the field coil fields by the sample actas if they are due to an image coil spaced by a distance2heff = 2(z + λ) from the real coil, where z is the physicalspacing between the sample surface and the field coil. ASQUID sensor with a single circular field coil of radius R anda co-planar, coaxial circular pickup loop of radius a orientedparallel to a homogeneous bulk sample surface will have amutual inductance [304]

M = µ0

2πa2

(1

R− R2

(R2 + 4h2eff)

3/2

). (28)

25


The solid line in figure 28(b) is obtained by numericallyintegrating the 2D Fourier transform of equation (27) over thearea of the pickup loop, for various values of z, and fit to thedata by varying �. Figure 28(c) compares the values obtainedfor the Pearl lengths for a number of the CBCO samples usingmagnetometry and susceptometry methods. The two methodsagree within experimental error over the range of Pearl lengthspresent. Since the fitting to the Pearl vortex images was doneassuming each vortex has �0 of total flux threading through it,rather than a fractional value [305], this agreement means thatthe superconducting layers are sufficiently strongly Josephson-coupled that it is energetically favorable for the vortex fluxto thread vertically through the superconducting layers, asopposed to escaping between the layers.

Figure 29 shows results from a study using scanningSQUID susceptometry of the penetration depth of the pnictidesuperconductor LaFePO [304]. This study used the techniqueof measuring the mutual inductance between the SQUIDfield coil and pickup loop as a function of spacing z, andthen repeating the measurement at fixed z while varying thetemperature. Since the mutual inductance is a function of z + λ,the temperature dependence of λ can be inferred from suchmeasurements, even without a detailed model for the mutualinductance. Spatially resolved susceptibility measurements(figures 29(c)–( f)) showed that the effective height dependedon the topography of the sample, with scratches, bumpsand pits effecting the measurements strongly. However, themeasured penetration depth and the temperature dependenceof the penetration depth was reproducible for positions morethan 10 µm from surface irregularities. In this case thepenetration depth had a linear temperature dependence atlow temperatures, similar to the cuprate high-temperaturesuperconductors, indicative of well formed nodes in the energygap. It should be noted that in this geometry it is λab, thein-plane penetration depth, that controls the mutual inductance.

Perhaps the most promising application of MFM tothe study of superconductivity is for high-spatial resolution,absolute measurements of penetration depths. There are twoways of inferring superconducting penetration depths fromMFM measurements. The first is, as mentioned above, to fitMFM vortex images to a model with the penetration depth asa fitting parameter [301, 302]. While Nazaretski et al modeledtheir tip numerically [301], Luan et al used an analyticalapproach which is easy to fit to experimental data and notstrongly affected by differences between tips [302]. A secondmethod is to consider the interaction between the magneticmaterial in the tip and its image in the superconducting materialdue to Meissner screening of the tip fields. Following Xuet al [308], Lu et al [309] wrote the force due to the interactionbetween the magnetic tip, approximated as a magnetic dipole,and its image in the superconductor as

F(z) = 3µ0m20

32π [z + λ(T )]4, (29)

where m0 is the tip’s magnetic moment and z is the physicaldistance between tip and sample. They fit their measured MFMforce derivative curves to the z λ approximation

dF(z)

dz= − 3µ0m

20

8π [z + λ(T )]5(30)

Figure 30. Normalized superfluid densityρs(T )/ρs(0) ≡ λab(0)2/λab(T )2 versus T for the pnictidesuperconductor Ba(Fe0.95Co0.05)2As2. �λab(T ) is determined fromMFM (squares) and SSS (diamonds) experiments by measuring thechange in the diamagnetic response at fixed height. These values areoffset to match the absolute value of λab(T ) by fitting the MFM datato a truncated cone model (circles). The green solid line shows a fitto a two-band s-wave model. The width of the dashed band reflectsthe uncertainty in λab(0). Inset: �λab(T ) versus T at low T . Blackdashed line: one gap s-wave model with a = 1.5 and �0 = 1.95Tc.Magenta dashed line: �λ(T ) = cT 2.2 (c = 0.14 nm K2.2). Figurereprinted with permission from [302]. Copyright 2010 by theAmerican Physical Society.

to obtain the temperature dependence of the penetration depthλ

for a single crystal of YBCO. Luan et al [302] used a truncatedcone tip model to write

∂Fz(z, T )

∂z− ∂Fz(z, T )

∂z

∣∣∣∣z=∞

= A

{1

z + λab(T )+

h0

[z + λab(T )]2+

h20

2[z + λab(T )]3

}(31)

for the difference between the force derivative at a givenheight z and that at large distances, where A, which dependson the tip shape and coating, is determined from fittingexperimental data at low temperatures, λab(T ) is the in-planecomponent of the penetration depth and h0 is the truncationheight of the tip. Model independent values for �λ(T )

were obtained by mapping a change in λ to a change inz since they are equivalent. Absolute values for λ(T )

were obtained by fitting to equation (31). Figure 30 showsexperimental results for the superfluid density ρs of thepnictide superconductor Ba(Fe0.95Co0.05)2As2 obtained byMFM and SSM measurements. The �λ(T ) results from MFMmeasurements were later confirmed by a bulk technique onsimilar samples [310]. The advantage of MFM is that both theabsolute value and high sensitivity relative values of λ can beobtained in the same measurement, so that superfluid densityis obtained over the full temperature range and from it the gapstructure of the order parameter. Luan et al found a two full-gap model model fit the data well, consistent with the s+− orderparameter most frequently discussed in the pnictides. Luan

26


Figure 31. MFM imaging and manipulation of individual vortices in YBCO at T = 22.3 K. (a), (b) Schematic drawings of an MFM tip(triangle) that attracts a vortex (thick lines) in a sample with randomly distributed pinning sites (dots): at large heights (a) the force betweentip and vortex is too weak to move the vortex; at small heights (b) the vortex moves right and then left as the tip rasters over it. (c) MFMimage taken at z = 420 nm (maximum applied lateral force F lat

max ≈ 6 nN). (d) z = 170 nm (F latmax ≈ 12 nm). Inset: scan taken at a

comparable height at T = 5.2 K. (e) Line cut through the data in (c) along the dashed line, showing the signal from a stationary vortex (solidcurve with peak to the left, blue online). Overlapping it is a line cut from the reverse scan (solid curve with peak to the right, green online).( f ) Line cut through the data in (d) along the dashed line, showing a typical signal from a dragged vortex. Figure reprinted by permissionfrom [312]. Copyright 2008 Macmillan Publishers Ltd.

et al also found that ρs was uniform on the sub-micrometerscale despite highly disordered vortex pinning.

3.5. Manipulation of individual vortices

While there is a vast literature on the properties ofsuperconducting vortices [111], there has been relatively littlework on manipulation of individual vortices by scanningmagnetic probes. Such manipulation, performed with MFM[112, 116, 119, 311–313] or SSM [314], can directly measure

the interaction of a moving vortex with the local disorderpotential [312]. Keay et al used MFM to observe sequentialvortex hopping between sites in an array of artificial pinningcenters [313]. An example of another such MFM studyis shown in Figure 31 [312]. Here vortices in a highquality, detwinned sample of YBa2Cu3O6.991 are fixed at lowtemperatures and high scan heights, but at higher temperaturesand lower scan heights the tops of the vortices can be draggedby an attractive magnetic interaction with the tip. The vorticesare dragged along the fast scan direction until a combination

27


of pinning and elastic forces overcomes the attractive tip–vortex force. The displacement of the vortex in the slowscan direction is much longer than in the fast scan direction,demonstrating a ‘vortex wiggling’ effect: alternating transverseforces markedly enhance vortex dragging. Vortex motion inboth the slow and fast scan directions is stochastic. From suchmeasurements Auslaender et al [312] were able to infer themaximum lateral dragging force before the top of the vorticeswere depinned. They also found a marked in-plane anisotropyin the distance w the top of the vortex was dragged by thetip in the fast scan direction, larger than could be accountedfor by the in-plane anisotropy of the vortex core and pointpinning. They speculated that a likely source for extra pinninganisotropy is nanoscale clustering of oxygen vacancies alongthe Cu–O chains in YBCO.

4. Conclusions

In summary, there are now many types of scanning magneticmicroscopies, each with their own strengths and weaknesses.Of the three covered here, SSM is the most sensitive for lowresolution applications, but requires a cooled sensor. MFM hasthe highest spatial resolution, but also applies the largest fieldsto the sample. SHM can be used over a broader temperaturerange than SSM. A major thrust in development has beentoward higher spatial resolution, which requires smallersensors scanned closer to the sample. The scaling of resolutionand sensitivity with sensor size and spacing depends both onthe technique and the source of field. Many of the applicationsdiscussed here have involved the imaging of magnetic flux inspecial geometries, unconventional superconductors or both.Others have involved time dependence, through, e.g., multiplecooldowns, cooldowns at varying cooling rates or noisemeasurements. Some exciting new developments have beenscanning susceptibility, which allows the measurement of localpenetration depths, spins and superconducting fluctuationswith unprecedented sensitivity and resolution; MFM imagingof Meissner force derivatives, which allow spatially resolvedabsolute measurements of penetration depths; and controlledmanipulation of superconducting vortices. I expect to seemany other exciting developments in the future.

Acknowledgments

I would like to thank Lan Luan, Beena Kalisky, Cliff Hicks,Danny Hykel and Klaus Hasselbach for carefully readingthis review. Any errors are mine, not theirs. I would alsolike to acknowledge support from the NSF under Grant No.PHY-0425897, by the German Humboldt Foundation, and bythe French NanoSciences Fondation, during this work.

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