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Local magnetic probes of superconductors

Simon J. Bending

School of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK

[Received 28 July 1997; revised 7 July 1998; accepted 14 July 1998]

Abstract

Investigations of the magnetic properties of high temperature superconductors(HTSs) have revealed the existence of striking newvortex phenomena due, in part,to their strong crystalline anisotropy, very short coherence lengths and the muchlarger thermal energies available at high temperatures. Some of these phenomena,for example vortex lattice `melting’, pose serious problems for technologicalapplications of the most anisotropic HTS materials and a fuller understandingof them is of considerable importance. The most direct information regardingvortex structures and dynamics is obtained through local measurement of themagnetic ® eld within or at the surface of a superconducting sample. A detailedreview of such local magnetic probes is presented here including Lorentzmicroscopy, magnetic force microscopy, Bitter decoration, scanning Hall probemicroscopy, magneto-optical imaging, and scanning superconducting quantuminterference device microscopy. In each case the principles underpinning thetechnique are described together with the factors that limit the magnetic ® eldand the spatial and temporal resolution. A range of examples will be given,emphasizing applications in the area of HTSs. In addition the ways in which theexisting techniques can be expected to develop over the next few years will bediscussed and new approaches that seem likely to be successful described.

Contents page

1. Introduction 4501.1. Comparison of the main imaging techniques 451

2. Magnetic ¯ ux structures in superconductors 4532.1. Type I materials 4542.2. Type II materials 454

2.2.1. Vortex structures 4542.2.2. Vortex dynamics 459

2.3. New phenomena in high-temperature superconductors 4603. Imaging of vortices by electron microscopy 462

3.1. Theory of phase contrast in electron microscopy 4623.2. Lorentz microscopy 4653.3. Electron holography 473

4. Magnetic force microscopy 4774.1. Theory of magnetic force microscopy of superconductors 4804.2. Magnetic force microscope design 4824.3. Results of magnetic force microscope imaging of vortices 483

5. The Bitter decoration technique 4865.1. Principles of Bitter decoration 4875.2. Examples of the use of Bitter patterning in superconductors 488

6. Scanning Hall probe microscopy 4926.1. Semiconductor heterostructure Hall probes 4936.2. Hall e ect and resolution in a heterostructure Hall probe 495

Advances in Physics, 1999, Vol. 48, No. 4, 449± 535

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6.3. Scanning Hall probe microscope design 4986.4. Examples of scanning Hall probe microscopy in superconductors 500

6.4.1. High spatial resolution 5006.4.2. High temporal resolution 505

7. Magneto-optical imaging 5087.1. Theoretical principles of magneto-optical imaging 5087.2. Examples of magneto-optical imaging in superconductors 510

7.2.1. Magneto-optical imaging with europium chalcogenides 5117.2.2. Magneto-optical imaging with yttrium iron garnet ® lms 511

7.3. High-speed magneto-optical imaging 5158. Scanning superconducting quantum interference device microscopy 516

8.1. Theory of superconducting quantum interference deviceoperation 516

8.2. Operation of the superconducting quantum interference devicein a ¯ ux locked loop 519

8.3. The state of the art in scanning superconducting quantuminterference device microscopy 519

8.4. Examples of scanning superconducting quantum interferencedevice microscopy images 521

9. Future perspectives and conclusions 5269.1. Future perspectives 526

9.1.1. Electron microscopy 5289.1.2. Magnetic force microscopy 5289.1.3. Bitter decoration 5299.1.4. Scanning Hall probe microscopy 5299.1.5. Magneto-optical imaging 5309.1.6. Scanning superconducting quantum interference device

microscopy 5309.2. Conclusions 531

References 531

1. Introduction

Over a decade after the discovery of high-temperature superconductivity itcontinues to present the condensed-matter physics community with major intellec-tual challenges. The mechanism leading to superconductivity aside, the rich magneticphenomena observed in these high-temperature superconductors (HTSs) have led toa dramatic renewal of interest in the mixed state of type II superconductors. To someextent this has involved the rediscovery of understanding acquired in the investiga-tion of low-temperature superconductors but has also frequently led to thediscarding of conventional theories and to the development of new lines of inquiry.In addition to its obvious academic fascination such work has great technologicalimportance for the development of superconducting materials since the motion ofvortices in the presence of a transport current and during ¯ ux creep causes aninduced voltage drop and a breakdown of the zero-resistance state. Thus theusefulness of a superconductor is only as good as one’s ability to control thep̀inning’ of vortices at ® xed positions within a sample.

A great deal of information concerning the magnetic properties of super-conductors can be gained from bulk measurements including magnetization, trans-port and heat capacity; yet it is virtually impossible to interpret such data fullywithout a microscopic picture of ¯ ux structures and dynamics. It is the purpose ofthis review, therefore, to describe the current capabilities of those techniques whichcan be used to image directly vortices in superconductors. As the title of this review

S. J. Bending450

implies, attention will be con® ned to those methods which can (or at least have thepotential to) resolve individual vortices and which are sensitive to their magnetic ® eldsdirectly. Hence neutron scattering (Cubitt et al. 1993), muon spin rotation (Lee et al.1993) and scanning tunnelling microscopy (STM) (Maggio-Aprile et al. 1995) lieoutside the scope of this work.

1.1. Comparison of the main imaging techniquesFigure 1 shows a diagrammatic plan of the current state of the art in magnetic

® eld sensitivity and spatial resolution for the six techniques considered here, namelyelectron microscopy, magnetic force microscopy (MFM), Bitter decoration, Scan-ning Hall probe microscopy (SHPM), magneto-optical (MO) imaging and scanningsuperconducting quantum interference device (SQUID) microscopy. A measure-ment bandwidth of 1Hz has been assumed (except for the s̀tatic’ case of Bitterdecoration). What is immediately evident from a plot of this type is the trade-obetween ® eld sensitivity and spatial resolution. This is well illustrated by the limitingcases of Lorentz microscopy (high spatial resolution) and scanning SQUIDmicroscopy (high ® eld resolution), while SHPM provides a compromise betweenthese two. The diagonal lines running across the ® gure represent the equivalent ¯ uxsensitivity Bmin l2min expressed in fractions of a superconducting ¯ ux quantumU 0 = h/2e and it is interesting that many of the techniques lie in the range(10 4-10 6) U 0, although for a variety of di erent reasons. The notable exceptionto this rule is MO imaging which has signi® cantly worse ¯ ux resolution but isnevertheless an important technique owing to its very high intrinsic temporalresolution.

Figure 1. Diagram comparing the magnetic ® eld sensitivity and spatial resolution ofelectron microscopy, MFM, Bitter decoration, SHPM, MO imaging and scanningSQUID microscopy.

Local magnetic probes of superconductors 451

Figure 2 shows a similar diagram where the time to capture one image frameis plotted against spatial resolution. Since in many cases the limit on scanning speedis set by signal-to-noise ratios, the optimized data points in this ® gure generallydo not correspond to those of ® gure 1. It is evident from this plot that thetemporal resolution of MO imaging far exceeds all the other techniques althoughLorentz microscopy can be performed at video rates with much higher spatialresolution. High scanning rates have, however, not been a priority in the develop-ment of many of these techniques and there is considerable scope for improvementsin this area.

As a roadmap to future sections it is probably useful to summarize here thestrengths and weaknesses of each of the six techniques indicated in ® gures 1 and 2.

A discussion of electron microscopy requires one to make a distinction betweenLorentz microscopy and electron holography. The former is an excellent techniquefor establishing the location of a vortex with very high spatial resolution (about10nm) and modest sensitivity (about 1mTHz 1/2). Moreover, since the outputrequires no post-processing, high-speed imaging in excess of video rates is possible.Electron holography is a more quantitative technique which allows one to study theinternal structure of vortices with similar spatial resolution and sensitivity butrequires considerable post-processing to reconstruct the image which inevitablyslows down image acquisition. Both techniques su er from the need for substantialsample preparation since very thin sections a few tens of nanometres thick arerequired to achieve adequate electron transmission. Consequently the possibleintroduction of artefacts and the in¯ uence of sample dimensions on the meas-urements are important considerations.

MFM has not been widely used in the ® eld of superconductivity despite its highspatial resolution (about 50nm) owing to its relatively poor sensitivity. The magnetic

Figure 2. Diagram comparing the image acquisition time and spatial resolution for ® ve ofthe techniques described in ® gure 1.

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tip used can alsobe highly invasiveand great experimental care must be taken duringimaging.

Bitter decoration is a mature technique for establishing the positions of vorticeswith relatively high spatial resolution (about 80nm) but has poor sensitivity andyields very little quantitative information about vortex structures. Furthermore ithas virtually no dynamic bandwidth in as much as the sample surface must becleaned after each decoration before another experiment can be performed.

SHPM is a niche technique which provides a unique compromise between spatialresolution (about 200nm) and sensitivity (about 100nTHz 1/2), making itparticularly well adapted for investigating vortices in superconductors. Video rateimaging is likely to become possible in the near future.

MO imaging is also a mature technology which has rather modest spatialresolution (about 1 m m) and sensitivity limited by the available MO materials andthe need to bring them into intimate contact with the surface of the superconductor.The strength of this technique is in high-speed imaging where modern pulsed lasershave made it possible to capture images at rates of 10ns frame 1 with much fasteracquisition a real possibility. MO imaging is therefore the only technique which cangenuinely claim to be able to study vortex dynamics on su ciently short time scalesto resolve microscopic motion.

Scanning SQUID microscopy is the technique with the highest sensitivity (lessthan 100pTHz 1/2) while the spatial resolution (about 4 m m) is limited by currentmicrofabrication capabilities and seems certain to improve. Existing applicationsconsiderably underutilize available signal-to-noise ratios (SNRs) and it is probablethat scanning at video rates and beyond will be realized in the near future.

2. Magnetic ¯ ux structures in superconductors

Before launching into more detail it is important to explain what sort of ¯ uxstructures can be expected in these materials. Only the main points will be sketchedhere and the reader is referred to one of the many excellent reviews for more details(for example Huebener (1979)). There are two important length scales whichdetermine many of the properties of superconductors. The coherence length( x 1± 100nm) is a measure of how rapidly the order parameter (wavefunction)describing the superconducting state can vary, for example at the junction with anon-superconducting region. All superconducting samples can completely expelmagnetic ¯ ux at su ciently low ® elds (the Meissner e ect) except for a thin surfacelayer where screening currents ¯ ow. The penetration depth (¸ 50± 200nm) is ameasure of the depth of this surface layer where ® eld penetration occurs. Figure 3is a sketch of the superconducting electron density ns and the magnetic ® eld nearthe surface of a sample, showing how both quantities decay approximatelyexponentially in this region with the appropriate length scales. The reduction inns in this region represents an energy gain for the sample since the fully super-conducting state is the equilibrium state. Conversely the penetration of the magnetic® eld at the surface represents a reduction in energy over the full Meissner state(zero ¯ ux within the superconductor). The net interface energy a per unit area can bewritten approximately as

a 12 ¹0H2c ( x )̧, (2.1)

where Hc is the thermodynamic critical ® eld. Superconductors are divided into typesI and II according to whether the overall surface energy is positive ( x > )̧ or


negative ( x < )̧. A more considered analysis of this problem leads to the conclusionthat a material will be type I if the Ginzburg± Landau parameter ·= /̧ x < 1/21/2and type II otherwise.

2.1. Type I materialsIn type I materials the interface energy is positive, and hence the lowest-energy

state in a magnetic ® eld is normally the full Meissner state. Since total ¯ ux expulsionimposes a large energy penalty on the sample, superconductivity is quenched byrelatively low magnetic ® elds and these materials are not generally of great tech-nological interest. An H± T phase diagram for a typical type I superconductor isshown in ® gure 4(a). Demagnetization factors due to sample shape can lead to anintermediate state between the Meissner phase and the ¯ ux vortex phase. Anappreciation of this can be gained by examination of ® gure 5(a) which shows athin type I superconducting ® lm in a perpendicular applied magnetic ® eld. Clearly, inthe Meissner state shown there is a very strong concentration of ® eld at the sampleedges (Hedge =Happlied/ (1 n), where n is the shape-dependent demagnetizationfactor) and the critical ® eld will be exceeded here long before it is in the centre of thesample. If Hedge > Hc, the system can, in practice, always reduce its energy bybreaking up into alternating normal and superconducting strips as shown in ® gure5 (b), and this is called the intermediate state. Although it is, in principle, interestingto image these ìntermediate’-¯ ux distributions, they occur on rather coarse scalescompared with the diameter of a vortex and will not be discussed further here.

2.2. Type II materials2.2.1. Vortex structures

In type II materials the wall energy is negative, and above a small lower critical® eld Hc1 the system would like to create as much interface as possible. Since ¯ ux isquantized in units of U 0(=h/2e) in a superconductor, this is achieved by allowing¯ ux to enter the sample in the form of vortex lines, each containing a single ¯ ux

Figure 3. Sketch of pro® les of magnetic ® eld B and superconducting electron density ns neara superconducting± normal interface.

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quantum. The H± T phase diagram of a typical type II superconductor is shown in® gure 4 (b); note that superconductivity is only destroyed at the boundary labelledHc2, which can represent a very large ® eld at low temperatures. A vortex consists of anormal core with radius x surrounded by a sheath of screening supercurrentsextending out to a distance ¸ as sketched in ® gure 6. The interaction between twoadjacent vortices is repulsive owing to the Lorentz force exerted by the supercurrentof one vortex on the magnetic ¯ ux of the other. This leads to Abrikosov’s (1957)famous prediction that the equilibrium state of a perfect type II superconductorwould be one in which the vortices are arranged on an ordered lattice. In practice,however, all real materials contain microscopic defects and inhomogeneities.Provided that the dimensions of these are comparable with or larger than x thesystem can usually reduce the energy penalty associated with the normal core bysiting a vortex there. Consequently vortices become p̀inned’ at these (generally

Figure 4. H± T phase diagrams of (a) type I and (b) type II superconductors.


randomly distributed) centres, introducing disorder into the vortex lattice. Larkinand Ovchinikov (1979) were able to show that even weak pinning destroys the long-range order of the ¯ ux line lattice, and only short-range crystalline order shouldremain. Flux structures can nevertheless have quite long-range sixfold bondorientational order: a so-called hexatic phase. More recently it has been proposedthat random pinning might lead to the formation of a vortex glass phase with thevortices frozen into a ® xed pattern determined by the distribution of pinning sites(Fisher et al. 1991).

The magnetic ® eld distribution at a vortex will depend strongly on the geometryof the sample of interest. For a bulk sample the Clem (1975) model is a very usefuldescription whereby the order parameter inside the vortex core is obtained from avariational trial function and the spatial dependence of the magnetic ® eld is given by

B(r) =U 0

2p x̧ vK0((r2 + x 2v)1/2/ )̧

K1( x v/ )̧ , (2.2)

where x v is a variation core-radius parameter (approximately x ), and K0 and K1are modi® ed Bessel functions. At low ® elds (mean vortex spacing about(2U 0/31/2B)1/2 )̧ the total ® eld distribution for a ¯ ux line lattice can beapproximated by the superposition of the ® elds of individual vortices, althoughcorrections must be introduced at high ® elds due to vortex overlap. This isdemonstrated for the Clem model in ® gure 7 (a) for a hexagonal lattice

Figure 5. (a) Field lines around a thin ® lm in the Meissner state. (b) The intermediate statein a type I superconducting ® lm.

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( x v =40nm; ¸ = 80nm) in an applied ® eld of 10mT. Many imaging techniquesactually sample the magnetic ® eld at a ® xed distance above a superconductingsurface. Performing a Fourier transform of the Maxwell equations, one can showthat the perpendicular component of magnetic ® eld a height z above a surface isgiven by

Bz(r, z) =s

~B( s , 0) exp ( sj jz) exp (is .r) (2.3)

where s is one of the reciprocal-lattice vectors of the ¯ ux line lattice and the ® rst termin the summation is the appropriate Fourier component of the ® eld distribution atthe sample surface. We note then that the Fourier components decay exponentiallywith increasing distance from the surface with the lowest-order reciprocal-latticevectors being the most robust. As a rule of thumb the ® eld modulation is onlydetectable at heights somewhat less than the lateral distance between ¯ ux vortices.This is illustrated in ® gures 7 (b) and (c) for two di erent stand-o heights.

In addition to e ects due to pinning centres, the vortex distribution may bestrongly in¯ uenced by surface or geometrical barriers at sample surfaces which areimportant even for negligible bulk pinning. The surface barrier (Bean and Livingston1964) can be understood if one imagines trying to introduce a single ¯ ux line parallelto the planar face of a semi-in® nite sample when there are two competing energyterms to consider. The ® rst is the repulsive interaction of the vortex with surfacescreening currents, and the second is the attraction of the ¯ ux line to its image insidethe sample. As a consequence a potential barrier for ¯ ux entry forms at the surfaceeven for H > Hc1. Flux entry can, in fact, be kinetically delayed until a much larger® eld Hen, when the barrier disappears. Even for H > Hen the surface potential leads

Figure 6. Pro® les of (a) superconducting electron density ns, (b) magnetic ® eld B and (c)supercurrent density Js near a vortex core.


Figure 7. Magnetic ® eld pro® le at various heights z above an Abrikosov lattice of vorticesin an applied ® eld of 10mT.

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to a concentration of ¯ ux in the middle of the sample as illustrated in ® gure 8.Geometrical barriers (Zeldov et al. 1994) arise owing to the tendency for vortices tobecome bowed as they penetrate at sample edges. For the case of a rectangularplatelet cross-section the vortices initially round o the sharp corners of the samplewithout complete penetration. As a consequence the energy of a penetrating vortexincreases gradually from zero to a maximum of about e 0d where e 0 is the vortex lineenergy and d the sample thickness. This represents a robust thermodynamic barrierup to the equilibrium ® eld Heq, and a kinetic barrier for ® elds above this up to thepenetration ® eld Hp at which point vortices start to enter freely. The geometricbarrier also leads to concentration of ¯ ux in the centre of the sample at equilibriumfor ® elds in excess of Heq. The e ects of surface and geometric barriers areparticularly pronounced in samples with very low pinning, which is often the casein high quality single crystals of HTS materials.

2.2.2. Vortex dynamicsThe dynamic properties of vortices are of particular importance since their

motion signals the breakdown of the zero resistance state. If a uniform transportsupercurrent density J is passed through a superconductor, there is a Lorentz forceon any ¯ ux lines present given by

Figure 8. Vortex line energy as a function of distance x from the sample surface for threedi erent values of applied ® eld.


F=J U 0, (2.4)where U 0 is a vector along the length of the vortex with the magnitude of the ¯ uxquantum. Provided that this force is much less than characteristic pinning forces,then the vortices will not move. However, above a critical current density Jc, pinningforces can be overcome and vortices start to move freely through the sample. If onedescribes all the damping processes (e.g. eddy current damping in the normal core) interms of a scalar damping factor ,́ the induced voltage in the ¯ ux ¯ ow regime is welldescribed by a ® eld-dependent ¯ ux ¯ ow resistivity and the vortex velocities vu aregiven by

vu =U 0(J Jc)

´ . (2.5)

These ¯ ux ¯ ow velocities obviously vary considerably depending on themagnitude of the applied transport current but, with the possible exception of theMOtechnique, values are typically much larger than the current temporal resolutionof imaging systems.

Even if J < Jc, vortex motion can still occur because of thermally activateddepinning. This phenomenon is called ¯ ux creep and was ® rst described theoreticallyby Anderson and Kim (1964). In practice a vortex or bundle of vortices undergoes athermally activated hop between two adjacent pinning points. The activation energyis typically much larger than kT in conventional superconductors and the meancreep rate tends to be rather slow and lies well within the temporal resolution ofseveral imaging techniques except for T very close to Tc. In HTS materials ¯ ux creepcan, however, be very rapid even for temperatures substantially below the criticaltemperature.

Flux ¯ ow and ¯ ux creep occur in the presence of su ciently weak pinning and/ora su ciently large driving force, either due to an applied transport current or tomagnetic ® eld gradients within the superconductor. For example if the appliedmagnetic ® eld threading a ® eld-cooled sample is suddenly reduced to zero, thevortices ¯ ow towards (and out of) the surface until a remanent state is produced suchthat J < Jc everywhere. This remanent state will then continue to relax further bytemperature dependent ¯ ux creep mechanisms.

2.3. New phenomena in high-temperature superconductorsHTSs are extreme type II materials and distinguish themselves from conventional

materials in a number of respects . The coherence length is very short ( x 1nm) andthe energy penalty associated with adding a ¯ ux vortex is rather small. As aconsequence the superconducting state exists up to very large magnetic ® elds. Thepenetration depth, on the other hand, is relatively large (̧ 100± 200nm) with theresult that the repulsive interactions between vortices at high ® elds (proportional to1/ 2̧) become very weak. Since, as the name implies, high-Tc materials remainsuperconducting up to much higher temperatures (about 100K), the magnitude ofthermal ¯ uctuations can also be very large. Finally, the new materials can show verylarge crystalline anisotropy because superconductivity is associated with layers ofCu± O atoms in the a± b plane which are only weakly coupled in the perpendiculardirection. The crystal structures are approximately tetragonal, although ofteninclude a small orthorhombic distortion. Therefore, for many purposes the aniso-tropy can be quanti® ed in terms of an anisotropy parameter ( C = (mc/ma)1/2)which is a function of the diagonal e ective-mass tensors mc and ma for the charge

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carriers (fourfold symmetry has been assumed). C can vary from about 5± 7 forYBa2Cu3O7 d , (YBCO) (Dolan et al. 1989) to about 50± 200 for Bi2Sr2CaCu2O8+d(BSCCO) (Farrell et al. 1989) for the HTSs whereas it is close to unity forconventional superconductors.

If a magnetic ® eld is applied along the high symmetry c direction, the fact thatsupercurrents are largely con® ned to planes of Cu± O atoms which are much thinnerthan the layer spacing causes vortices to have a strong two-dimensional (2D)character. In fact in BSCCO a vortex is formally viewed as a stack of point orp̀ancake’ vortices which interact weakly through Josephson coupling. Since pancakevortices within the same layer repel each other while those in di erent layers attracteach other, a regular lattice of straight ¯ ux lines has the lowest energy but isextremely soft with respect to 2D ¯ uctuations. Indeed when the typical shear energyof the ¯ ux line lattice starts to exceed the energy due to short range (interplanar) tiltdeformations, pancake vortices can start to move independently of those above andbelow them. In the presence of pinning, the energy of the systemmay well be reducedif the ¯ ux line becomes highly distorted such that each pancake vortex is situated onthe nearest adjacent pinning site within its layer. Eventually above a decoupling ® eldB2d, thermal ¯ uctuations lead to the total loss of phase coherence between pancakevortices in adjacent layers and the system essentially becomes 2D (BlaÈ tter et al.1994).

Up to now we have only considered vortex properties with the ® eld appliedparallel to the c axis. If the ® eld instead lies at oblique tilt angles with respect to the caxis, it is possible to realize a surprising regime where the normally repulsiveinteraction between vortices becomes attractive. This is a consequence of thetendency for vortex supercurrents to be con® ned to the Cu± O planes with the resultthat one of the magnetic ® eld components within the vortex reverses sign. This, inturn, leads to an attractive well in the vortex± vortex interaction within the planecontaining the magnetic ® eld vector and the crystal c axis (Grishin et al. 1990).

If the ® eld is applied exactly parallel to the Cu± Oplanes the vortex cores prefer tolocate in the ǹormal’ spaces between the planes. These are called Josephson vorticessince the circulating currents giving rise to them have to cross the superconductingCu± O planes by Josephson tunnelling. At angles slightly away from the a± b plane itcan become energetically favourable for the ¯ ux lines to form staircase-likestructures composed of a combination of pancake and Josephson vortices whichl̀ock in’ to the spaces between Cu± O planes (Oussena et al. 1994).

Finally the pronounced elastic softness of vortices in HTSs, and high availabletemperatures can lead to very large e ects of thermal ¯ uctuations and even melting(Nelson et al. 1987). It is possible to apply a Lindemann criterion to the ¯ ux systemto show that the vortex lattice should melt into a vortex liquid whenBm(T)/Bc2(T) 10 4. Such a `melting’ line has been identi® ed on the basis ofneutron scattering (Cubitt et al. 1993), muon spin rotation (Lee et al. 1993) andmagnetization measurements (Pastoriza et al. 1994, Zeldov et al. 1995), and in highlyanisotropic materials such as BSCCOis situated well below the critical temperature.On the basis of abrupt jumps in Hall probe measurements of local induction inBSCCO (Zeldov et al. 1995) and sharp peaks in the heat capacity of YBCO at themelting line (Schilling et al. 1996) it is widely believed that this represents a ® rst-order phase transition.

As a consequence of the phenomena described above, the H± T phase diagramforHTSs is complex and remains controversial.


This introduction merely scratches the surface of the richness of vortex physics inHTS materials, and other phenomena will be described in more detail later in thisarticle as a deeper understanding is required. For excellent comprehensive reviews ofthe area the reader is referred to BlaÈ tter et al. (1994) and Brandt (1995). In thedescriptions of imaging techniques that follow, examples will be presented toillustrate the state of the art in instrumental performance. Since the HTSs have, ingeneral, set very demanding measurement criteria in terms of ® eld resolution andoperation temperature, the vast majority of examples have inevitably been drawnfrom this area.

3. Imaging of vortices by electron microscopy

The ability to image magnetic vortices in transmission electron microscopy(TEM) stems from the fact that the ¯ uxon magnetic ® elds induce phase shifts inthe incident electron wavefunctions. Consequently electron phase-sensitive tech-niques such as electron holography must be employed in order to resolve them.Such developments only became possible when suitable highly coherent sources ofelectrons became available in the late 1960s. The theory of holography is usuallydiscussed in terms of in® nitely coherent plane wave electron states, but in reality anelectron beam in a microscope is a train of incoherent wave packets. In order forclear holographic images to be observed the transverse extent of the wave packetmust be su cient to overlap all the spatial points in the object plane which onewishes to interfere, while its longitudinal extent must be at least as long as themaximum phase di erence between any two of these points. Of these twoindependent criteria the former is normally most stringent and limits the maximumnumber of observable interference fringes and hence the resolution. In practice thelow brightness of early thermionic emission cathodes made them unsuitable forholography and it was only after a practical ® eld-emission cathode was developed in1968 (Crewe et al. 1968) that such applications took o . Tonomura et al. (1979) wereable to perfect the design still further to the point where more than 3000 interferencefringes became observable in a 70keV microscope and cathodes have continued toimprove since.

3.1. Theory of phase contrast in electron microscopyGiven that a coherent electron source is available it is trivial to show that a

superconducting vortex acts as a phase shifting object. Consider the gedankenexperiment sketched in ® gure 9 (a) where an electron plane wave is incident normalto a ¯ ux line containing a single superconducting ¯ ux quantum ( 0 = h/2e). Thevector potential for this situation can be represented by an azimuthal vector aboutthe axis of the vortex:

^Aµ = ( 0/2p r) ^µ . Using the standard line integral expression

for the phase shift of an electron trajectory passing a distance y from the ¯ ux stringwe ® nd that

u = 1

(2 0)z= 1z=1

^Au ds =

1 1

ydz2(y2 + z2) sgn( y) =

p2

sgn( y), (3.1)

where we assume that the limits of the integral can be set to in® nity and the factor of2 in the denominator of the second term arises because the ¯ ux quantum for a singleelectron is twice that for a superconducting Cooper pair. Thus we see that two

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di erent electron trajectories which pass on opposite sides of the ¯ ux stringexperience a p phase di erence (® gure 9(b)).

The scattering geometry described above is not, however, the preferred geometrysince it yields only a one-dimensional (1D) perspective on the spatial distribution ofvortices. On the other hand the vortices clearly cannot lie along the optical axis sincethey will induce no electron phase shifts in this orientation and hence no imagecontrast. Usually the normal to the sample plane is inclined at an angle of around45ë to the incident electron beam, allowing the application of a horizontal magnetic® eld to control the vortex density.

A more realistic geometry for calculating phase shifts in this situation is sketchedin ® gure 10 and one must now account for both ® elds within the superconductor aswell as fringing ® elds outside. These shifts have been calculated by Bonevich et al.(1994a) assuming that the ® elds in the free space above and below the sample can bedescribed by two point magnetic poles of strength 0 where the vortex stringintersects the superconductor surface with a radial ® eld line distribution everywhereoutside the perfectly diamagnetic sample. This is a reasonable approximation in thevicinity of the vortex although clearly it does not satisfy the boundary condition thatfurther away all ® eld lines must originate and terminate at opposite sides of thesample space. These workers have shown that

Figure 9. (a) Sketch of an electron trajectory passing by a horizontal ¯ ux string containing asingle ¯ ux quantum. (b) Total change u in phase accumulated by electrons ondi erent trajectories.


u (x, y) = 12 tan 1 y

x + a tan 1 y

x a +p4

sgny

x a sgny

x + a

12

sin 1y sin ( a )

[y2 + x + a( )2]1/2+ sin 1 ysin ( a )

[y2 + x a( )2]1/2, (3.2)

where t is the thickness of the superconducting ® lm and a =[t sin ( a )]/2. A three-dimensional (3D) plot of the phase shift described by equation (3.2) is shown in® gure 11(a); note that the ¯ ux string can be identi® ed with a discontinuous phasechange over the 2a long projection of the vortex onto the x axis. Its peak value of p(middle two terms of equation (3.2)) is exactly the phase shift for trajectories eitherside of an in® nite ¯ ux string which is independent of tilt angle a . However, the phasedi erence between large positive and negative values of y is only 2a (last two termsof equation (3.2)) and originates entirely from the fringing ® elds outside the sample.The phase shift for a vortex of ® nite width can be obtained from a convolution ofequation (3.2) with a model of the ® eld at a vortex core. This has been done for a¯ uxon described by a cylinder of uniform magnetic ® eld and radius 33nm in ® gure11(b). As one might expect, the abrupt phase discontinuity is smeared out over alength scale of the twice the penetration depth ¸ and, surprisingly, the maximumphase shift is quickly reduced from p down towards the limiting value of 2a . Thus

Figure 10. Coordinate system used to calculate the phase shift due to a ¯ ux string threadinga thin superconducting sample. The specimen of thickness t is inclined at an angle ofa to the optic axis and the corresponding phase shift in the object plane is depicted inthe contour map below.

S. J. Bending464

the height and breadth of this phase change are a measure of the diameter and innerstructure respectively of a superconducting ¯ ux vortex.

The two most common imaging techniques for ¯ ux vortices are Lorentzmicroscopy and true electron holography and these will be discussed separately.

3.2. Lorentz microscopyThis is in many ways the most convenient mode of operation since vortices can be

viewed directly without the need for further image processing. A simple classicalpicture can be obtained by considering the action of the Lorentz force (FL = ev̂ ^B)on electrons which pass through a ¯ uxon. Their trajectories will be de¯ ected awayfrom the vortex in such a way that the electron ¯ ux is enhanced near one edge of thevortex and depleted near the other in a similar way to the Hall e ect in conductingsolids. Thus, in the scattering geometry of ® gure 12(a), a vortex can be recognized asa pair of adjacent dark and light stripes as sketched. In practice, however, aquantitative understanding of Lorentz microscopy can only be achieved within afull quantum-mechanical description.

Figure 11. Phase shift for a sample 60nm thick at a tilt angle of 45ë for (a) a ¯ ux string and(b) a cylinder of 33nm radius and of uniform magnetic ® eld containing a single ¯ uxquantum ( Ļ =33nm).


A schematic diagramof the 300keV electron microscope used by Tonomura andco-workers is shown in ® gure 13 where the location of the ® eld emission source,cooled sample stage, lenses and image are indicated. A simpli® ed diagram of thegeometry to perform Lorentz microscopy is shown in ® gure 14. Note that the samplemust be thinned down to about 70nm before TEMcan be performed and the normalto the plane is inclined at an angle of around 45ë to the incident electron beam.

The ability to image this phase shift in Lorentz microscopy can be understoodfrom the schematic diagramshown in ® gure 15 (Chapman 1984). We assume that theelectron beam can be represented by a z-propagating plane wave (/exp (ikz)) whilethe vortex structure, since it only modulates the electron phase, can be representedby a transmittance f (x, y) /exp[i u (x, y)]. Thus the electron beam transmittedthrough the specimen will take the approximate form expfi[kz + u (x, y)]g. At theback focal plane of the objective lens the electron disturbance may be described bythe Fourier transform of the specimen transmittance ~f (kx, ky) modi® ed by a transferfunction t(kx, ky) which accounts for e ects such as spherical aberration (present inall electron lenses) and image defocusing. The electron disturbance in the imageplane is returned by the following reverse Fourier transform:

I(x0, y 0) / ~f (kx, ky)t(kx, ky) exp[ i(kxx0 + kyy 0)]dkx dky2

. (3.3)

It is clear from equation (3.3) that if the transfer function is everywhere unity,I(x0, y0) would simply be / f (x, y)j j2/ exp[iu (x, y)]j j2 which is constant andthe image would contain no contrast. A non-uniform transfer function is thenessential for Lorentz microscopy and is normally achieved by defocusing theimage by an amount z. In this situation the transfer function becomes

Figure 12. Classical response of a beam of electrons incident on a horizontal ¯ ux line.

S. J. Bending466

t(kx, ky) = exp[i z (̧k2x + k2y)/4p ] and the image intensity is proportional to thefollowing convolution:

I(x0, y 0) / exp[iu (x, y)]exp ip [(x x0)2 + ( y y 0)2]

¸ zdxdy

2

. (3.4)

Ultimately an individual ¯ ux vortex can be recognized in an image by an ovalregion with adjacent halves of bright and dark intensity. The line dividing thesehalves is de® ned by the projection of the vortex segment on to the z =0 plane, whilethe sense of light and dark regions is reversed for upward- or downward-directedvortices.

It is the nature of Lorentz mode images of vortices that it is very di cult toextract quantitative information from them about the inner structure of a ¯ uxon.However, since they indicate the position and polarity of ¯ uxons in real time they areideal for making studies of vortex dynamics. The ® rst example of this was achievedby Harada et al. (1992). A high-purity thin ® lm of niobium (R(300K/R(10K) 20)

Figure 13. Diagram of an electron microscope designed for investigating vortices insuperconductors.


was chemically thinned to 70 20nm and positioned on a low-temperature stage at45ë to the incident 300keV electron beam. Vortices were then imaged by Lorentzmicroscopy with the objective lens switched o and using a second intermediate lensfor focusing. This con® guration is used so that the sample is not exposed to therelatively high magnetic ® elds generated by the objective lens but has the disadvan-tage that the magni® cation is somewhat reduced from its maximum value. Figure 16shows a Lorentz micrograph of the niobium ® lm in a 10mT applied ® eld at 4.5Kwith the image defocused by 10mm to achieve contrast. The positions of the vorticesare clearly marked by the black and white spots and note that the 45ë tilt angle leadsto a 1/21/2 compression of one of the axes as indicated by the length markers. Thedark arcs superimposed on the image represent Bragg re¯ ections at atomic planesinduced by bends in the ® lm, and can be ignored. Detailed examination of ® gure 16reveals that the vortex lattice is not quite perfect in this image because of the

Figure 14. Schematic diagram of the experimental set-up for performing Lorentzmicroscopy.

S. J. Bending468

in¯ uence of randomly distributed weak pinning sites in the sample such as interstitialor substitutional atoms. These workers observed, under the in¯ uence of highertemperatures and/or applied magnetic ® elds, a greater degree of order as pinningforces became less important. By far the most impressive achievement of this work,however, was the ability to observe vortex motion in real time. Using a television(TV) camera attached to the electron microscope it was possible to capture data at amaximum rate of 30 framess 1. In this way, discrete vortex hopping processesbetween nearby pinning sites could be visualized in real time. Figure 17 shows threestills from a videotape generated some seconds after a 15mT ® eld was suddenlyswitched o at 4.5K. Most vortices were seen to hop in the direction of decreasingvortex density along pre-determined routes de® ned by ¯ uctuations in ® lm thickness.Examples of a few hopping processes about 0.5 m m long are indicated in ® gure 17 bydotted and solid circles showing positions before and after moving. In addition,some vortices simply oscillated around ® xed centres with amplitudes of about 0.3 m mand frequencies of about 10s 1.

Only a year later Harada et al. (1993) were able to use real-time Lorentz imagingto investigate the possibility of vortex lattice melting in the Bi2Sr1.8CaCu2O8+d HTS.This presented a special challenge since the a± b plane penetration depth in thismaterial is almost an order of magnitude larger than in niobium. Consequently theclassical electron de¯ ection by a single vortex is proportionately smaller and muchlonger defocusing distances (up to 10cm) have to be used to obtain su cientcontrast. Figure 18 shows Lorentz micrographs of a BSCCO ® lm 150± 250nm thickwhich had been cleaved from the face of a large single crystal. The sample wasinclined at 45ë to the incident electrons and cooled to 4.5K at H =0. The applied

Figure 15. Image formation during Lorentz microscopy.


® eld was then raised to 2mT and the sample was imaged at four di erenttemperatures as shown. Above 40K, when pinning forces cease to be important,the vortices are seen to form a very regular hexagonal lattice. As the temperature wasincreased further, the contrast gradually diminishes, disappearing entirely at 76.5K.Independent magnetization measurements found the irreversibility temperature atthis applied ® eld to be 74 2K which is close to the point where contrast is lost.These workers pointed out, however, that this cannot be taken as unequivocalevidence for lattice melting since the increase in penetration depth with increasingtemperature as well as possible vibrations of the vortices about their equilibriumpositions could alone be su cient to destroy contrast.

Recently Matsuda et al. (1996) have been studying the ways in which vorticesinteract with arti® cially introduced defect arrays. These were produced by irradiatinga 100nm niobium ® lm with a 30keV focused gallium-ion beam (diameter, 20nm).Figure 19 shows micrographs of a region of the ® lm containing a 4 4 rectangularlattice with periodicity 3.3 m m. Each `defect’ corresponds to a pit of 40nm diametersurrounded by a 300nm region of entangled dislocations. At low ® elds (less than thecommensurability ® eld of the defect lattice) the vortices have been shown to try toorder themselves in a periodic way on to selected defects so as to minimize theirpotential energy (Harada et al. 1996). Figure 19 shows the opposite limit when the ® eldis very much greater than the commensurability ® eld. Here the sample was cooled to6K in an applied ® eld of 18mT and allowed to reach equilibrium. The positions ofboth vortices and ion-implanted regions can be resolved in these micrographs anddetailed examination reveals that the presence of defects prevents the ¯ uxons fromforming one coherent lattice. Rather they form hexagonally ordered domains of about

Figure 16. Lorentz micrograph of a niobium ® lm at 4.5K in a ® eld of 10mT. [Reprintedwith permission from Nature (Harada et al. 1992) Copyright 1992 MacmillanMagazines Limited.]

S. J. Bending470

Figure 17. Dynamics of vortices in a niobium ® lm (a) 170s, (b) 170.1s and (c) 171.4s after a15mT ® eld has been switched o at 4.5K. Dotted (solid) circles show vortexpositions before (after) hopping. [Reprinted with permission from Nature (Harada etal. 1992). Copyright 1992 Macmillan Magazines Limited.]

(a)

(b)

(c)

(d )

Figure 18. Lorentz micrographs of a BSCCO ® lm in a 2mT applied ® eld at (a) 4.5K, (b)20K, (c) 56K and (d ) 68K. [Reprinted from Harada et al. (1993). Copyright 1993 bythe American Physical Society.]


(a)

(b)

(c)

(d )

Figure 19. Video frames of regions of vortex lattice in a niobium ® lm at various times afterthe ® eld was suddenly reduced from 18 to 8.5mT at 6K: (a) t =0s; (b) t =0.27s; (c)t =0.43s; (d ) t = 0.80s. Implanted defects are located at the black discs and domainboundaries for the vortex lattice are indicated by dotted lines. [Reprinted withpermission from T. Matsuda, K. Harada, H. Kasai, O. Kamimura and A.Tonomura, 1996, Science, 271, 1393. Copyright 1996 American Association for theAdvancement of Science.]

S. J. Bending472

5 5 vortices which appear to be pinned at defects near where domain boundaries arelocated. As soon as ® gure 19(a) was recorded, the applied ® eld was suddenly reducedto 8.5mT. Initially the system does not respond; then suddenly avalanche-like ¯ owbegins along one of the domain boundaries lying between the dotted curves in ® gure19(b). A little later, motion starts at a second domain wall as shown in ® gure 19(c).Finally in ® gure 19(d) a new stable domain structure becomes established. Such datayield unique insights into the dynamic interactions between vortices and pinning sitesand are certain to advance our understanding in this area greatly.

3.3. Electron holographyWhile Lorentz microscopy is capable of providing powerful insights into vortex

dynamics, it yields little quantitative information about the dimensions and internalstructure of individual vortices. If more quantitative data of this type are required,the complementary technique of electron holography can be applied. The o -axisgeometry as sketched in ® gure 20 is most commonly used for performing holographysince this allows the conjugate image (always present in holograms) to be separatedfrom the reconstructed image. As the name implies, the sample occupies one half ofthe electron beam path while the other half remains undisturbed and forms thereference wave. The two beams must now be made to interfere and this can beachieved with an electron biprism. The latter is simply a very ® ne (of less than 1 m mdiameter) positively charged ® lament which is place horizontally through themicroscope optic axis, ¯ anked by two grounded plate electrodes on either side.Close to the ® lament the biprism approximates to a coaxial cable and the potentialdepends logarithmically on the radial distance from it. It is straightforward to showthat electrons passing either side of the ® lament experience a ® xed angular de¯ ectiontowards it proportional to the biprism voltage which is independent of their incidentposition. In this way the scattered and reference beams can be made to interferecontrollably and to generate fringes in the hologram plane.

It is clear from ® gure 20 that the two beams are inclined at a relative angle awhen they interfere. A simple theoretical way to picture this situation is to imagine areference plane wave of form u r /exp[ik(z a y)] (tilted at an angle a to the opticaxis) interfering with the spherical wave from a point object u o /(if/ r) exp (ikr)where f is a scattering amplitude. The intensity in the hologram plane a distance lfrom the object will be

I(x, y) = u o + u rj j2 1 + fl2

2fl

sink(x2 + y2)

2l+ ka y . (3.5)

When this pattern is exposed on to ® lm the amplitude transmittance for anincident reconstruction beam depends on a coe cient g which indicates the contrastof the ® lm (t = I g /2). If g = 2, then t can simply be replaced by the expression forI given above. In this situation, if we now illuminate the hologram with a referencebeam identical with that used to create it, the resultant transmitted amplitude in thehologram plane is

T(x, y) = exp (ikl) 1 + fl2

+i fl

expik(x2 + y2)

2l i f

lexp

ik(x2 + y2)2l

2ka y .

(3.6)The ® rst and second terms represent the transmitted plane wave, the third term

the reconstructed image and the fourth term the conjugate image. The propagation


direction of the latter is now inclined at 2a with respect to the reconstructed imageand hence is spatially separated. Note that it is not necessary for the reference beamto be an electron wave; indeed it never is. In practice, holograms are magni® ed in themicroscope so that they can be reconstructed with the much longer wavelength oflight. This is, in fact, extremely convenient since the techniques available tomanipulate light are much more ¯ exible than those for electron waves.

The mere ability to generate holographic images is, however, insu cient toguarantee observation of ¯ ux vortices. This is because the phase shift across a vortexis typically about p /2 (about /̧4), corresponding to only a quarter of the di erence

Figure 20. Schematic diagram of the experimental set-up for performing electronholography.

S. J. Bending474

between a pair of interference fringes. Consequently techniques for phase-di erenceampli® cation are essential to improve spatial resolution. This is invariably achievedoptically owing to the far greater ¯ exibility of optical components. The simplestway to double the phase di erence is to use the Mach± Zehnder interferometersketched in ® gure 21 to illuminate the hologram with two separate plane waveswhose angles are chosen so that the reconstructed image from one beam overlapsthe conjugate image from the other. Since the phases of the two overlappingbeams are reversed in sign, the ® nal image reveals a phase distribution which hasbeen ampli® ed by a factor of two. If further ampli® cation is required, this processcan be repeated several times. In practice it is often quicker to exploit the higherharmonic data from nonlinear holograms when g 6= 2. In this case t(x, y) canbe expanded in a power series:

t(x, y) =I g /2

1 g flsin

k(x2 + y2)2l

+ ka y + g ( g + 2) f2

2l2sin2

k(x2 + y2)2l

+ ka y +

+ ( 1)n g ( g + 2)( g + 4) . . . ( g + 2n) fn

n!lnsinn

k(x2 + y2)2l

+ ka y . (3.7)

Thus the nth term in the series has been phase ampli® ed by a factor of n, and thiscan be overlapped with its conjugate image as described above to give a total of 2nampli® cation. Phase ampli® cations of up to 32 times have been demonstrated with acombination of these techniques. Vortex images are routinely produced with 16times ampli® cation when the approximate p /2 phase shift across a vortex roughlycorresponds to the separation between four adjacent fringes.

Figure 22 shows one of the ® rst holographically reconstructed images of a thinniobium foil which had been cooled to 4.5K in a ® eld of 10mT (Bonevich et al.1993). The objective lens has, once again, been turned o to eliminate its magnetic® elds from the vicinity of the sample, and focusing was achieved with anintermediate lens. This limits the spatial resolution to about 30nm over a samplearea 4 m m wide. Comparison of the 16 times phase-ampli® ed contour map with asimultaneously recorded Lorentz image reveals that vortices can be identi® ed by theregions where about four contour lines are tightly clustered together (indicated by

Figure 21. Optical reconstruction system for phase-ampli® ed interference microscopy.


open circles). As discussed earlier the feature running almost diagonally across theimage is a consequence of a slight bend of the ® lm and can be ignored here. Todemonstrate that these holographic images contain quantitative information aboutthe vortex structure much smaller sample regions about 1.5 m m wide were studiedwith the objective lens turned on for higher spatial resolution (about 7nm) (Bonevichet al. 1994b). These holograms were then digitized and reconstructed numerically.Figure 23 shows the resulting 12 phase-ampli® ed contour plots of a single vortex atthree di erent temperatures. In each case the vortex induces a phase shift of aboutp /2 but over an increasingly larger distance as the penetration depth increases athigher temperatures. For a given phase shift an average vortex diameter could beassigned which increased from 150 4nm at 4.5K to 185 4nm at 7K and230 4nm at 8K. A quantitative comparison was made with two di erent radialmodels of a ¯ uxon, namely the London (1935) model and the Clem (1975) model:

Figure 22. Interference micrograph of the vortex lattice in a niobium ® lm (phase ampli® ed16 times) at 4.5K in an applied ® eld of 10mT. [Reprinted from Bonevich et al.(1993). Copyright 1993 by the American Physical Society.]

(a) (b) (c)

Figure 23. Interference micrograph of a single vortex in niobium at (a) 4.5K, (b) 7K and (c)8K (phase ampli® ed 12 times). [Reprinted from Bonevich et al. (1994b). Copyright1994 by the American Physical Society.]

S. J. Bending476

BLon(r) = 02p 2̧LK0

rĻ

,

BClem(r) = 02p x̧ vK0((r2 + x 2v)1/2/ Ļ)

K1( x v/ Ļ) , (3.8)

where Ļ is the London penetration depth, x v is a variational parameter to describethe normal vortex core, and K0(x) and K1(x) are modi® ed Bessel functions.Assuming a two-¯ uid temperature dependence of Ļ(T) = Ļ(0)[1 (T/ Tc)4]1/2 itwas found that the Clem model probably provides a slightly better description of thereconstructed images. It will, however, be appreciated that a considerable amount ofmodelling goes into the simulation of images such as ® gure 23, and inverting the datato produce the true ® eld pro® le at a vortex is a very ambitious task.

The holographic reconstructions shown here were all produced optically ordigitally some time after the original holograms were formed. Recently there hasbeen considerable progress with real-time holography. This can be achieved in one oftwo ways and both involve detecting an electron hologram with a TV camera. Themost direct approach is then to reconstruct the image using Fourier transform-basedalgorithms and a very-high-performance computer. Even with current state-of-the-art hardware, however, reconstruction still takes a few minutes to achieve and it isnot yet possible to produce images in real time. A better approach is to transfer thevideo signal from the charge-coupled device (CCD) camera to a liquid-crystal panelas shown in ® gure 24. Since this panel is itself a phase hologram, illuminating it witha laser produces an image in real time (Chen et al. 1993).

4. Magnetic force microscopy

The general principles of scanning force microscopy (SFM) are well known andrequire little introduction here. Its development dates from work by Binnig et al.(1986) who recognized that it is possible to use the photolithographic processingtechniques developed for the semiconductor industry to fabricate microscopiccantilevers with force constants smaller than the e ective spring constant of anatom bonded at the surface of a solid. Thus they were able to show that one canmechanically image solid surfaces without perturbing the atomic structure. Toestablish a rough order of magnitude of the parameters involved (Sarid 1991),the vibration frequency and mass of a typical atom are x 1013 rads 1 andm 10 25 kg respectively, yielding an approximate spring constant k x 2m10Nm 1. This should be compared with the force constant of the rectangular leverof length l, width w and thickness t sketched in ® gure 25.

k =Ewt3

4l3, (4.1)

where E is Young’s modulus for the material. Inserting typical values for siliconcantilevers (E =1.79 1011 Nm2, q =2330kgm3, l 100 m m, w 10 m m andt 0.6 m m) yields k =0.1Nm 1 which is clearly well within what is required. Inpractice there is an additional requirement that the resonant frequency of thecantilever be su ciently high that there is no danger of exciting oscillations duringrapid scanning. This condition con¯ icts somewhat with the requirement of a softspring constant but is nevertheless readily achievable. For the cantilever sketchedabove


x =t

2l2E

0.24q

1/2

, (4.2)

yielding an acceptable resonant frequency of 85kHz for the above parameters.There are several di erent SFM techniques, but they all have many factors

in common. In all cases the de¯ ection of a micromachined cantilever is used tomonitor electrostatic or, as in the case of interest here, magnetostatic forcesbetween a sample surface and sensor. There are a variety of ways to monitor

Figure 24. Schematic diagram of a real-time electron holography system: YAG, yttriumaluminium garnet; VCR, video cassette recorder.

S. J. Bending478

de¯ ections ranging from STM detection as in the original instrument of Binnig et al.(1986), through capacitive (Goeddenhenrich et al. 1990) and piezoresistive(Tortonese et al. 1993) sensing to optical detection. The latter can be either a beamde¯ ection detector (Meyer and Amer 1988) or an interferometer (McClelland et al.1987) for laser light re¯ ecting from the back surface of the cantilever possibly via anoptical ® bre.

There are two distinct modes of SFM operation. In the constant-force mode thecantilever is brought into c̀ontact’ (i.e. within range of interatomic forces) with thesurface and a feedback loop is employed to control sample± cantilever separationsuch that the de¯ ection (hence force) remains ® xed during scanning. In this way thesurface topography can be measured under a constant force. Alternatively thecantilever can be oscillated near its resonance frequency and the resonance amplitudeused to probe force gradients near the sample surface. Treating a free (far from asurface) cantilever as a damped oscillator with resonance frequence x 0 and qualityfactor Q, its frequency-dependent amplitude is well described by the classicalexpression

A( x ) = a0x 20

[( x 20 x 2)2 + x 2 x 20/Q2]1/2, (4.3)

where a0Q is the resonant amplitude. As the cantilever approaches a surface,sample± probe interaction leads to a shifted resonant frequency x 00 which is afunction of the local force gradient F 0. Provided that x 00 x 0 the oscillationamplitude at a ® xed frequency near resonance is a measure of the force gradientat the cantilever tip. It can be shown (Sarid 1991) that the optimum operationfrequency in this mode is just o the free resonance x 0 at a value x m when theamplitude has dropped to about 82% of its maximum value. For small forcegradients the change in resonance amplitude is now linearly proportional to d F 0:

d A( x m) = 2a0Q2

33/2kd F 0, (4.4)

where k is the cantilever spring constant. The sensitivity of this operation mode canbe very high for a sensor with a large Q.

Figure 25. Sketch of a rectangular cantilever.


4.1. Theory of magnetic force microscopy of superconductorsIn order to be sensitive to magnetostatic forces the scanning force microscope tip

must be made of ferromagnetic material, ideally a microscopic single-domainparticle with a high coercive ® eld. In practice tips have been realized either by ® neelectrochemical etching of ferromagnetic wires or by depositing thin magnetic ® lmson top of the sharp atomic force microscope tips on micromachined cantilevers.Consequently the magnetic domain structure is rarely well known and may not evenbe the same from one scan to another. For this reason a truly quantitativeunderstanding of MFM images represents a major theoretical challenge in its ownright and is beyond the scope of this article. In most treatments of MFM images ofsuperconductors (Hug et al. 1991, Reittu and Laiho 1992, Wadas et al. 1992) thesimplifying assumption is made that the tip is actually a cylindrical single domainparticle magnetized uniformly along its axis which is perpendicular to the samplesurface. Provided that the length L of the particle is much larger than its radius R,the tip can be approximated by a magnetic point charge m = p ¹0MR2 (where M isthe magnetization along the domain) sited at the apex of the tip nearest the sample.In this limit the interaction force is simply proportional to the magnetic ® eld at thetip apex:

Ftip mH (4.5)If L R is not satis® ed, the tip must be viewed as a magnetic charge dipole, in

which case the force is proportional to the magnetic ® eld gradient. This highlightsone of the major di culties associated with interpreting MFM images and anexcellent discussion of this point has been given in Schoenenberger and Alvarado(1990).

When imaging superconductors there are two distinct contributions to the force.The ® rst of these is the s̀o-called’ Meissner levitation force and the second is theforce at the tip due to ¯ ux vortices threading the sample.

The levitation force is the microscopic analogue of the force which supports amacroscopic permanent magnet above a superconductor in the Meissner state. Thishas been calculated by Hug et al. (1991) assuming that the London equation (4.6) isobeyed in the semi-in® nite superconducting half-space z < 0:

r2A 12̧ A = 0, (4.6)where A is the vector potential and ¸ is the magnetic ® eld penetration depth. Atz =0 this solution must be matched with that of the Maxwell equations in the semi-in® nite half-space z > 0 containing the magnetic force microscope tip.

r2A = ¹0 Jm, (4.7)where Jm is the magnetization current density of the tip. Within the magnetic pointcharge model they found the following expression for the force with the tip a height dabove the superconductor:

Fz(d) = m2

4p ¹010

(1/ 2̧ + x2)1/2 x(1/ 2̧ + x2)1/2 + x

exp( 2xd) xdx. (4.8)

As one would expect, the l̀evitation’ force depends on m2 since, to a ® rstapproximation, the force arises due to the interaction between the magnetic pointcharge and its image within the superconductor.

S. J. Bending480

Using the same approximations, Reittu and Laiho (1992) have solved the sameset of equations to calculate the force at the tip due to a single ¯ ux line threading thesuperconductor, where equation (4.6) now contains a term on the right-hand side toaccount for the normal vortex core. They found that the vertical force on the tip aheight d above the superconductor and a radial distance r from the vortex axis is

Fz(r, d) = m 02p ¹010

(1/ 2̧ + x2)1/2(1/ 2̧ + x2)1/2 + x

exp ( xd)J0(xr)xdx1 + (x̧ )2

, (4.9)

where J0 is the zeroth-order Bessel function. The lateral force of the tip can also bestraightforwardly calculated and is found to be

Fr(r, d) = m 02p ¹010

(1/ 2̧ + x2)1/2(1/ 2̧ + x2)1/2 + x

exp( xd)J1(xr)xdx1 + (x̧ )2

, (4.10)

where J1 is the ® rst-order Bessel function. Equations (4.8)± (4.10) are plotted in ® gure26 as a function of r for a typical tip height of 20nm, assuming, as in the experimentof Moser et al. (1995), that the tip has a radius of 100nm and is made fromFe51Al8Ni14Co24Cu3 with M = 10.5 105 A m 1 and that the superconductor isYBCOat a low temperature with ¸ = 165nm. Note that, with these parameters, thelevitation force (always repulsive) is still somewhat larger than the force due to avortex (attractive or repulsive depending on the orientation of the vortex) and thecontrast at a vortex depends strongly on its orientation (i.e. up or down). It isinteresting to compare the peak lateral force of about 100pN with typical pinning

Figure 26. Theoretical forces exerted on an idealized magnetic force microscope tip 20nmabove a YBCO thin ® lm as a function of radial displacement r from a vortex core.


forces for vortices in these superconducting ® lms. These have been measured for afew pinning sites in 0.35 m m YBCO thin ® lms with a micron-sized Hall probe(Stoddart 1994) where the following expression for the typical temperaturedependence of the pinning force fp(T) was found:

fp(T) 100pN 1 TTp2

, (4.11)

where Tp is approximately the critical temperature of the ® lm. We note then that thelateral force is comparable with or exceeds the pinning force at all temperatures. Thisprobably represents a rather pessimistic prediction, however, since the stoichiometryof the ® lm of Stoddart (1994) was not optimal (Tc = 82K) and the Hall probemeasurement tends to select those vortices sited at the weakest pinning sites. Inaddition the magnetic point charge model of the magnetic tip almost certainlyoverestimates the force experienced by a real magnetic force microscope tip.Nevertheless these estimates highlight the potential invasiveness of MFM, and greatexperimental caution must be employed when imaging superconductors.

4.2. Magnetic force microscope designThe best MFMimages of vortices in superconductors have been obtained to date

by GuÈ ntherodt’s group at the University of Basel. A sketch of the ® bre-optic-basedhead of the microscope is shown in ® gure 27 (Moser et al. 1993). It is constructed ona clever design involving two concentric piezoelectric scanner tubes. The outer tubesupports both the ® bre mount and the cantilever and is used to scan the latter withrespect to a ® xed sample. The inner tube clamps only the monomode optical ® breand allows it to be moved relative to the cantilever surface. The micromachinedsilicon cantilever which actually performs the imaging itself sits on a Cu± Be springwhich has coarse and ® ne adjusting screws to allow one to set the initial cantilever±® bre tip separation. The sample itself sits on a mechanical approach system whichbrings it into contact with the cantilever tip. Figure 28 shows the full opticaldetection system employed. A laser diode source is coupled into a monomodeoptical ® bre through a Faraday isolator. The latter prevents re¯ ected light fromcoupling back into the laser which gives rise to mode hopping; a major source ofintensity noise in non-isolated diodes. The light passes through a bidirectionalcoupler, leaves the ® bre at a cleaved end and is incident on the highly re¯ ective rearside of the silicon cantilever. The inset of ® gure 28 shows how the narrow air spacebetween the cleaved ® bre end and the cantilever forms an interferometer whoseperformance depends on the re¯ ection amplitudes at the two interfaces. The signalphotodiode measures the re¯ ected interference signal and is compared with thereference photodiode to correct for shifts in laser power.

The system of two concentric scanner tubes allows three distinct dc operationmodes. The variable-interferometer mode involves contact scanning with the feed-back system disengaged. The resulting variable ® bre tip± cantilever separation resultsin a varying interference signal which roughly represents a force map of the surface.Interpretation of such images is non-trivial since the interferometer response dependsroughly sinusoidally (i.e. nonlinearly and non-monotonically) on cantilever displace-ments. In variable-de¯ ection mode the feedback system is connected to the ® brepiezo in order to maintain a constant interferometer air gap. Now the feedbacksignal generates a force map of the surface. Finally, in the constant-force mode the

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feedback loop is connected to the scanning piezo in order to maintain a constantcantilever de¯ ection, allowing a constant force map of the topography to begenerated. If ac operation is required, the scanner piezo can be used to oscillatethe cantilever above the sample surface.

4.3. Results of magnetic force microscope imaging of vorticesFigure 29 shows the ® rst MFM image obtained of vortices in a high-Tc material

by Hug et al. (1994) using the instrument described above. The image shows a22 m m 22 m m region of a 300nm YBCO thin ® lm deposited by laser ablation on aSrTiO3(001) substrate. A micromachined silicon cantilever with an integrated atomicforce microscope tip was used which had an iron ® lm 25nm thick deposited on it.

To obtain this image the tip was retracted to about 2mm above the surface andthe sample cooled through Tc to 77K in a ® eld of about 0.1mT. The tip was thenbrought to within about 20nm of the surface and a force map generated. Inspection

Figure 27. Diagram of the head of a low-temperature magnetic force microscope.


of ® gure 29 reveals that this region of the sample contains 25 vortices, eachproducing a repulsive force of about 0.8pN. Note that this is nearly three ordersof magnitude smaller than our earlier theoretical prediction, a discrepancy that canalmost certainly be traced back to an unrealistic model of the magnetic tip usedthere. By varying the cooling ® eld and counting the number of vortices in the image,these workers were able to verify that each did indeed contain a single super-conducting ¯ ux quantum. Moser et al. (1995) have examined the di erent contrast ofattractive and repulsive vortices in detail. If a full-width-at-half-maximum criterion

Figure 28. Schematic diagram of a ® bre optic interferometer displacement sensor. The lowerinset shows a sketch illustrating how the cleaved end of the ® bre and the uppersurface of the cantilever form an interferometer.

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is used to estimate the vortex diameter, they observed that attractive vortices alwaysappear broader than repulsive vortices. They attributed this fact to the in¯ uence ofthe scanning tip itself and the presence of a large approximately constant Meissnerrepulsion superimposed on the signal.

Figure 30 shows how the magnetic force microscope tip can be used to modifythe ¯ ux structure locally in the superconducting ® lm described above. In this15 m m 15 m m image the sample was cooled through Tc in the stray ® eld of thetip. Once the system has stabilized at 77K the MFM images reveal that a smallbundle of eight to 12 vortices has nucleated underneath the tip. Manipulation ofvortices in this way allows the possibility to perform unique experiments, forexample studying the creep of a single isolated vortex bundle, but at the same timehighlights the potential invasiveness of the technique.

Overlooking any noise associated with the de¯ ection detection system, thermalexcitation of the cantilever imposes bounds on the minimum detectable force. Sincethe cantilever is essentially a simple harmonic oscillator with two degrees of freedom,the equipartition theorem can be used to show that the rms oscillation amplitude atfrequencies much less than the resonant frequency x 0 is

d z21/2

=4kBT f

Qk x 0

1/2

, (4.12)

where f is the measurement bandwidth and the other symbols have their usualde® nitions. Assuming that Q 100, k = 0.1Nm 1, x 0 = 85kHz and a workingbandwidth of 1kHz we ® nd a rms vibration amplitude of about 4 10 12 m at room

Figure 29. MFM image of vortices in a YBCO thin ® lm (the image size is 22 m m 22 m m).[Reprinted from Physica C, 235± 240, H. J. Hug, A. Moser, I. Parashikov, B. Stiefel,O. Fritz, H.-J. Guentherodt and H. Thomas, `Observation and manipulation ofvortices in a YBa2Cu3O7 thin ® lm with a low temperature magnetic forcemicroscope’, p. 2695. Copyright 1994, with permission from Elsevier Science.]


temperature. Multiplying this by the cantilever force constant we obtain anapproximate uncertainty in the measured force of about 0.4pN. Note that this isas much as half of the peak force at a vortex measured in the experiments of Hug etal. (1994), explaining the relatively poor SNR in these images. No vortex-resolvedMFMimages measured in ac mode have been published to date. No doubt these willappear in due course since the ac mode is capable of measuring force gradients withvery high SNRs in very high-Q systems.

In the interferometer based MFM of Moser et al. (1993) the additional noise ofthe de¯ ection detection system is smaller than the thermal cantilever excitation. Thedominant source of this extrinsic noise is the diode laser which has both intensity andphase ¯ uctuations associated with it. The former largely arise owing to randomspontaneous emission events as well as changes in the way that the intensity isdistributed between the modes of the laser cavity. The latter also result fromspontaneous emission as well as ¯ uctuations in the carrier population and areparticularly important when phase-sensitive detection is being used as in theinterferometer described here.

5. The Bitter decoration technique

The ® rst technique ever used to image magnetic ¯ ux distributions in super-conductors was the Bitter decoration method which was developed independently by

Figure 30. Magnetic-force-microscope tip-induced nucleation of a bundle of eight to 12vortices in a YBCO thin ® lm (the image size is 15 m m 15 m m). [Reprinted fromPhysica C, 235± 240, H. J. Hug, A. Moser, I. Parashikov, B. Stiefel, O. Fritz, H.-J.Guentherodt and H. Thomas, `Observation and manipulation of vortices in aYBa2Cu3O7 thin ® lm with a low temperature magnetic force microscope’, p. 2695.Copyright 1994, with permission from Elsevier Science.]

S. J. Bending486

Trauble and Essmann (1966) and Sarma and Moon (1967). The technique isroutinely used to image magnetic domains in ferromagnetic structures when a liquidsuspension of tiny magnetic particles is placed on top of the sample. The lowtemperatures encountered in superconductivity dictate a much more sophisticatedapproach to decoration.

5.1. Principles of Bitter decorationFigure 31 shows a sketch of a typical decoration system where the sample is

sitting on a sample holder which, in this case, is thermally anchored to a liquid-helium bath at 4.2K. A ferromagnetic ® lament (typically iron, nickel or cobalt) ismounted about 2.5± 3cm away from the sample with a small shield in between toprevent radiative heating of the sample. In the presence of a low pressure of heliumgas a su ciently large current is passed through the ® lament so that it vaporizes. Theexact pressure of background gas is usually in the range 0.06± 0.26mbar, dependingon the chamber temperature, and critically controls the size and concentration ofparticles formed. These should be less than 5± 10nm in diameter with low kineticenergies and yet must not adhere to one another on the way to the surface. Uponapproaching the inhomogeneous ® eld distribution near the superconductor surface aferromagnetic particle with magnetic moment ¹ experiences a force equal to

Figure 31. Schematic diagram of a typical Bitter decoration system.


F= ¹0 Ñ (¹·H). (5.1)Since the magnetic moment will rapidly align with the direction of the local

magnetic ® eld to minimize its energy this can be approximated by

F ¹0 ¹j jÑ Hj j (5.2)and, ignoring any other forces, we see that the particle will follow the maximum ® eldgradient radially in towards the centre of a vortex. In this way the particles are drawnto regions of highest ® eld, which are the ¯ ux vortex cores in type II superconductors.Once they reach the sample surface they adhere to it via van der Waals forces,allowing the superconductor to be warmed up to room temperature and examinedunder a scanning electron microscope without disturbing the pattern formed. If thevortex image is to be correlated with the sample microstructure, TEM sections mustbe prepared. With conventional superconductors such as niobium it su ces to thin asample before decoration (Herring 1976). This approach does not work well withHTSs, however, because of their tendency to overheat near thinned edges duringdecoration and/or damage introduced during ion milling. To overcome theseproblems, such samples are ion mill thinned after decoration with an additionalprotective carbon ® lm a few tens of nanometres thick deposited on the surface(Bagnall et al. 1995).

The Bitter decoration technique is limited to low ® elds (typically H < 10mT)such that the mean vortex spacing a is considerably greater than the magnetic ® eldpenetration depth ¸ in the sample. At higher ® elds the vortex tails begin to overlapand ® eld gradients decrease rapidly. For this reason most images are taken in the® eld-cooling mode whereby the sample is cooled slowly through Tc in a small appliedmagnetic ® eld. Owing to an increase in the strength of pinning forces as thetemperature is lowered, the vortex pattern formed during cooling will lock in atsome temperature T in between Tc and the chamber base temperature (usually4.2K). Therefore the pattern observed will not be the true equilibrium con® gurationfor the decoration temperature; this is of course true for nearly all measurementsmade on superconductors. Recent estimates (Gammel et al. 1992, Grigorieva et al.1993, Marchevsky et al. 1997) indicate that T may be as high as (0.8± 0.9)Tc. Thequality of the decoration is strongly dependent on the sample in question. The bestimages are to be expected in materials with the strongest ® eld gradient parallel to thesample surface, that is in samples where vortices have the smallest diameters andlargest peak ® elds. Broadly speaking this requires materials with short penetrationdepths ¸ and narrow cores (small coherence length x 0). This last criterion is roughlyequivalent to selecting those materials with the largest lower critical ® eld Hc1. Anexcellent review of Bitter decoration in type II superconductors has been given byGrigorieva (1994).

5.2. Examples of the use of Bitter patterning in superconductorsThe concept of pinning is vital to nearly all practical applications of super-

conductivity. The decoration technique has proved invaluable since it easily enablesone to correlate directly the sample defect structure with local displacements ofvortices from their equilibrium position. In this way, speci® c pinning centres can beidenti® ed and, on the basis of the size of the displacements, the strength of pinningforces can be estimated (Trauble and Essmann 1968). A particularly good exampleof this work is the observation of pinning at twin boundaries in YBCOsingle crystals

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which are known to have a small orthorhombic distortion of the lattice (a 6= b) atlow temperatures (Vinnikov et al. 1990a, b). As-grown YBCO crystals generallycontain two orthogonal systems of {110} twins which form to relieve internal stressesintroduced as samples are cooled through the tetragonal± orthorhombic phasetransition. Figure 32 shows an optical micrograph of a small region of twinnedYBCO taken with polarized light to indicate the positions of the twin boundaries(dark lines). The adjacent decoration image clearly illustrates how vortices areattracted to these boundaries, in fact the vortex density there is roughly twice thatin the surrounding monodomain regions. Patterns of this type allow one to estimatethe pinning potential Up at the twin boundary since the mean vortex spacing on ando the boundary re¯ ects the di erence in energy in the two locations. After energyminimization the following expression is found for Up:

Up =20

(8p )1/2¹0 2̧ab¸

1/2

exp ab¸ 32

av¸

1/2

exp av¸ , (5.3)

where ab is the mean vortex spacing at the boundary and av is the spacing in theneighbouring domain. Using this equation the pinning potential per unit length oftwin boundary in YBCOwas estimated to be Up = 3.4 10 13 Jm 1. Vinnikov et al.argued that their system is close to equilibrium for the temperature at which thesample was decorated (4.2K) and hence this is the temperature that corresponds totheir estimate.

Another area where Bitter decoration has proved to be a powerful technique is inthe investigation of the in¯ uence of crystalline anisotropy on the vortex lattice. It haslong been known that anisotropy in conventional superconductors can lead todistortions of the hexagonal ¯ ux line lattice, and even the formation of a squarelattice for certain orientations of the magnetic ® eld (Huebener 1979). The extremeanisotropy characteristic of many of the HTSs has, however, given rise to a range ofqualitatively new phenomena. As discussed earlier, superconductivity in the high-Tcmaterials is linked to Cu± O planes oriented perpendicular to the c axis of the unit

(a) (b)

Figure 32. (a) Twin layers (dark lines) in a partly detwinned YBCO single crystal as viewedin the optical microscope with polarized light. (b) Bitter decoration image of the sameregion of the sample. [Reproduced from Grigorieva (1994) by permission of IOPPublishing Limited.]


cell. If adjacent planes are relatively strongly coupled (e.g. YBCO), the vortexstructure can be described by the anisotropic Ginsburg± Landau theory in terms ofan e ective-mass tensor mi, j = Mi, j/ Mav (where Mi, j are the components of the masstensor and Mav is its angular average). Within this description the penetration lengthalso becomes a tensor quantity. For the special orientations considered here asu cient notation is a̧ ,b = m1/2b ¸ where a represents the direction of the appliedmagnetic ® eld and b the direction of ® eld decay (mb is the diagonal element of thee ective-mass tensor). Since the orthorhombic distortion in YBCO is quite small,this gives rise to a relatively weak anisotropy of a̧ , b in the a± b plane which is di cultto identify in the decoration patterns of individual vortices. However, the anisotropyof the penetration depth gives rise to anisotropic vortex± vortex interactions andcauses distortion of the ¯ ux line lattice. Such distortions can be measured fromscanning electron microscopy images of decorated samples with relative ease andallow a̧ , b to be estimated with precision, for example ç,b/ ç,a = 1.15 0.02 inYBCO single crystals (Dolan et al. 1989), implying an e ective mass ratio of the a tothe b directions of 1.32 0.4.

With the ® eld applied at appreciable tilt angles to the c axis, òut-of-plane’ e ectsof anisotropy are much more dramatic. Figure 33 shows a decoration image for a150nm YBCOthin ® lm with a 0.45mT ® eld applied parallel to the c axis (Grigorievaet al. 1994). A homogeneous but disordered distribution of vortices is observed inthis micrograph as one would expect for these strongly pinned ® lms. In contrast,® gure 34 shows a decoration image when a ® eld of 0.20mT is applied at an angle of30ë to the c axis of a similar YBCO ® lm. The anisotropic penetration depth is now

Figure 33. Bitter decoration image of a 150nm YBCO ® lm with a 0.45mT ® eld appliedparallel to the c axis. The inset shows a digital Fourier transform of the vortexpattern. [Reproduced from Grigorieva et al. (1994) by permission of IOP PublishingLimited.]

S. J. Bending490

directly visible in the pronounced elliptical shape of the vortices, even at thisrelatively shallow tilt angle. The vortex spacings are also highly anisotropic ascon® rmed by the Fourier transforms of the digitized vortex patterns which are addedas insets in ® gures 33 and 34.

The tendency for vortex supercurrents to be con® ned to the Cu± O planes hasrather remarkable consequences when the ® eld is applied at an oblique angle withrespect to the c axis. It has been predicted theoretically (Grishin et al. 1990) that oneof the magnetic ® eld components within the vortex reverses sign, leading to anattractive well in the vortex± vortex interaction within the plane containing themagnetic ® eld vector and the crystal c axis. As a consequence the vortex lattice willbe shortened along the direction of attraction. The resulting vortex chain state isillustrated in ® gure 35 (Gammel et al. 1992) for an untwinned YBCO crystaldecorated on the a± b face in a ® eld of 2.48mT applied at an angle of 70ë awayfrom the c axis. These workers were able to show that the vortex spacing within thechains was approximately constant at low ® elds while the structure merged smoothlyinto an ordered isotropic vortex lattice at high ® elds (¹0H > 2mT).

The properties of YBCO can all be quite well understood in terms of ananisotropic e ective mass tensor, but the same is not true of the extremelyanisotropic HTSs such as BSCCO. In this case the vortices are thought to be betterdescribed by stacks of 2D p̀ancake’ vortices interacting exclusively via the magnetic® eld and the weak Josephson coupling in the space between Cu± O planes. As aconsequence the in¯ uence of anisotropy on vortex structures is more dramatic asillustrated by the new vortex states observed in tilted ® elds in BSCCO. Figure 36

Figure 34. Bitter decoration image of a YBCO thin ® lm in a ® eld of 0.2mT applied at anangle of 30ë to the c axis. The inset shows a digital Fourier transform of the vortexpattern. [Reproduced from Grigorieva et al. (1994) by permission of IOP PublishingLimited.]


(Bolle et al. 1991) shows an image of a BSCCO crystal which has been decoratedin a ® eld of 3.5mT applied at an angle of 70ë away from the c axis. Vortex chainsare clearly visible in this image but, surprisingly, are embedded in regions ofordered isotropic ¯ ux line lattice. Such a complex vortex structure would have beenvery di cult to predict in advance and its origin has still not yet been entirelyexplained.

6. Scanning Hall probe microscopy

The use of Hall e ect sensors to image ¯ ux pro® les at the surface of super-conductors dates back well over 30 years. Typically thin evaporated ® lms ofthe semimetal bismuth (Broom and Rhoderick 1962, Co ey 1967, Goren andTinkham 1971, Brawner and Ong 1993), InAs (Weber and Riegler 1973) or a GaAsepitaxial ® lm (Tamegai et al. 1992) have been employed in this role and, althoughmost of these studies present results on a much coarser scale, spatial resolutionas high as 4 m m with magnetic ® eld sensitivity of 0.01mT was already achievedby Goren and Tinkham (1971).

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