A dvances in Physics, 1999, Vol. 48, No. 4, 449± 535
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 new vortex phenomena due, in part,
to their strong crystalline anisotropy, very short coherence lengths and the much
larger thermal energies available at high temperatures. Some of these phenomena,
for example vortex lattice `melting’, pose serious problems for technological
applications of the most anisotropic HTS materials and a fuller understanding
of them is of considerable importance. The most direct information regarding
vortex structures and dynamics is obtained through local measurement of the
magnetic ® eld within or at the surface of a superconducting sample. A detailed
review of such local magnetic probes is presented here including Lorentz
microscopy, magnetic force microscopy, Bitter decoration, scanning Hall probe
microscopy, magneto-optical imaging, and scanning superconducting quantum
interference device microscopy. In each case the principles underpinning the
technique are described together with the factors that limit the magnetic ® eld
and 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 the
existing techniques can be expected to develop over the next few years will be
discussed and new approaches that seem likely to be successful described.
Contents
1.
2.
3.
4.
5.
6.
Introduction
1.1. Comparison of the main imaging techniques
Magnetic ¯ ux structures in superconductors
2.1. Type I materials
2.2. Type II materials
2.2.1. Vortex structures
2.2.2. Vortex dynamics
2.3. New phenomena in high-temperature superconductors
Imaging of vortices by electron microscopy
3.1. Theory of phase contrast in electron microscopy
3.2. Lorentz microscopy
3.3. Electron holography
Magnetic force microscopy
4.1. Theory of magnetic force microscopy of superconductors
4.2. Magnetic force microscope design
4.3. Results of magnetic force microscope imaging of vortices
The Bitter decoration technique
5.1. Principles of Bitter decoration
5.2. Examples of the use of Bitter patterning in superconductors
Scanning Hall probe microscopy
6.1. Semiconductor heterostructure Hall probes
6.2. Hall e€ ect and resolution in a heterostructure Hall probe
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6.3. Scanning Hall probe microscope design
6.4. Examples of scanning Hall probe microscopy in superconductors
6.4.1. High spatial resolution
6.4.2. High temporal resolution
7. Magneto-optical imaging
7.1. Theoretical principles of magneto-optical imaging
7.2. Examples of magneto-optical imaging in superconductors
7.2.1. Magneto-optical imaging with europium chalcogenides
7.2.2. Magneto-optical imaging with yttrium iron garnet ® lms
7.3. High-speed magneto-optical imaging
8. Scanning superconducting quantum interference device microscopy
8.1. Theory of superconducting quantum interference device
operation
8.2. Operation of the superconducting quantum interference device
in a ¯ ux locked loop
8.3. The state of the art in scanning superconducting quantum
interference device microscopy
8.4. Examples of scanning superconducting quantum interference
device microscopy images
9. Future perspectives and conclusions
9.1. Future perspectives
9.1.1. Electron microscopy
9.1.2. Magnetic force microscopy
9.1.3. Bitter decoration
9.1.4. Scanning Hall probe microscopy
9.1.5. Magneto-optical imaging
9.1.6. Scanning superconducting quantum interference device
microscopy
9.2. Conclusions
References
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1. Introduction
Over a decade after the discovery of high-temperature superconductivity it
continues to present the condensed-matter physics community with major intellectual challenges. The mechanism leading to superconductivity aside, the rich magnetic
phenomena observed in these high-temperature superconductors (HTSs) have led to
a dramatic renewal of interest in the mixed state of type II superconductors. To some
extent this has involved the rediscovery of understanding acquired in the investigation of low-temperature superconductors but has also frequently led to the
discarding of conventional theories and to the development of new lines of inquiry.
In addition to its obvious academic fascination such work has great technological
importance for the development of superconducting materials since the motion of
vortices in the presence of a transport current and during ¯ ux creep causes an
induced voltage drop and a breakdown of the zero-resistance state. Thus the
usefulness of a superconductor is only as good as one’s ability to control the
`pinning’ of vortices at ® xed positions within a sample.
A great deal of information concerning the magnetic properties of superconductors can be gained from bulk measurements including magnetization, transport and heat capacity; yet it is virtually impossible to interpret such data fully
without a microscopic picture of ¯ ux structures and dynamics. It is the purpose of
this review, therefore, to describe the current capabilities of those techniques which
can be used to image directly vortices in superconductors. As the title of this review
L ocal magnetic probes of superconductors
451
implies, attention will be con® ned to those methods which can (or at least have the
potential to) resolve individual vortices and which are sensitive to their magnetic ® elds
directly. Hence neutron scattering (Cubitt et al. 1993), muon spin rotation (Lee et al.
1993) and scanning tunnelling microscopy (STM) (Maggio-Aprile et al. 1995) lie
outside the scope of this work.
1.1. Comparison of the main imaging techniques
Figure 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, namely
electron microscopy, magnetic force microscopy (MFM), Bitter decoration, Scanning Hall probe microscopy (SHPM), magneto-optical (MO) imaging and scanning
superconducting quantum interference device (SQUID) microscopy. A measurement bandwidth of 1Hz has been assumed (except for the `static’ case of Bitter
decoration). What is immediately evident from a plot of this type is the trade-o€
between ® eld sensitivity and spatial resolution. This is well illustrated by the limiting
cases of Lorentz microscopy (high spatial resolution) and scanning SQUID
microscopy (high ® eld resolution), while SHPM provides a compromise between
these two. The diagonal lines running across the ® gure represent the equivalent ¯ ux
2
sensitivity Bmin lmin
expressed in fractions of a superconducting ¯ ux quantum
U 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 exception
to this rule is MO imaging which has signi® cantly worse ¯ ux resolution but is
nevertheless an important technique owing to its very high intrinsic temporal
resolution.
Figure 1. Diagram comparing the magnetic ® eld sensitivity and spatial resolution of
electron microscopy, MFM, Bitter decoration, SHPM, MO imaging and scanning
SQUID microscopy.
452
S. J. Bending
Figure 2. Diagram comparing the image acquisition time and spatial resolution for ® ve of
the techniques described in ® gure 1.
Figure 2 shows a similar diagram where the time to capture one image frame
is plotted against spatial resolution. Since in many cases the limit on scanning speed
is set by signal-to-noise ratios, the optimized data points in this ® gure generally
do not correspond to those of ® gure 1. It is evident from this plot that the
temporal resolution of MO imaging far exceeds all the other techniques although
Lorentz microscopy can be performed at video rates with much higher spatial
resolution. High scanning rates have, however, not been a priority in the development of many of these techniques and there is considerable scope for improvements
in this area.
As a roadmap to future sections it is probably useful to summarize here the
strengths 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 between
Lorentz microscopy and electron holography. The former is an excellent technique
for establishing the location of a vortex with very high spatial resolution (about
10nm) and modest sensitivity (about 1 mT Hz­ 1/ 2 ). Moreover, since the output
requires 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 the
internal structure of vortices with similar spatial resolution and sensitivity but
requires considerable post-processing to reconstruct the image which inevitably
slows down image acquisition. Both techniques su€ er from the need for substantial
sample preparation since very thin sections a few tens of nanometres thick are
required to achieve adequate electron transmission. Consequently the possible
introduction of artefacts and the in¯ uence of sample dimensions on the measurements are important considerations.
MFM has not been widely used in the ® eld of superconductivity despite its high
spatial resolution (about 50 nm) owing to its relatively poor sensitivity. The magnetic
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453
tip used can also be highly invasive and great experimental care must be taken during
imaging.
Bitter decoration is a mature technique for establishing the positions of vortices
with relatively high spatial resolution (about 80 nm) but has poor sensitivity and
yields very little quantitative information about vortex structures. Furthermore it
has virtually no dynamic bandwidth in as much as the sample surface must be
cleaned after each decoration before another experiment can be performed.
SHPM is a niche technique which provides a unique compromise between spatial
resolution (about 200nm) and sensitivity (about 100nT Hz­ 1/ 2), making it
particularly well adapted for investigating vortices in superconductors. Video rate
imaging is likely to become possible in the near future.
MO imaging is also a mature technology which has rather modest spatial
resolution (about 1 m m) and sensitivity limited by the available MO materials and
the 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 lasers
have made it possible to capture images at rates of 10ns frame­ 1 with much faster
acquisition a real possibility. MO imaging is therefore the only technique which can
genuinely claim to be able to study vortex dynamics on su ciently short time scales
to resolve microscopic motion.
Scanning SQUID microscopy is the technique with the highest sensitivity (less
than 100pT Hz­ 1/ 2) while the spatial resolution (about 4 m m) is limited by current
microfabrication capabilities and seems certain to improve. Existing applications
considerably underutilize available signal-to-noise ratios (SNRs) and it is probable
that 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 ¯ ux
structures can be expected in these materials. Only the main points will be sketched
here 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 which
determine 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 a
non-superconducting region. All superconducting samples can completely expel
magnetic ¯ ux at su ciently low ® elds (the Meissner e€ ect) except for a thin surface
layer where screening currents ¯ ow. The penetration depth (¸ 50± 200nm) is a
measure of the depth of this surface layer where ® eld penetration occurs. Figure 3
is a sketch of the superconducting electron density ns and the magnetic ® eld near
the surface of a sample, showing how both quantities decay approximately
exponentially in this region with the appropriate length scales. The reduction in
ns in this region represents an energy gain for the sample since the fully superconducting 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 be
written approximately as
2
1¹
¸),
( 2.1)
a
2 0Hc ( x ­
where Hc is the thermodynamic critical ® eld. Superconductors are divided into types
I and II according to whether the overall surface energy is positive ( x > ¸) or
454
S. J. Bending
Figure 3. Sketch of pro® les of magnetic ® eld B and superconducting electron density ns near
a superconducting± normal interface.
negative ( x < ¸). A more considered analysis of this problem leads to the conclusion
that a material will be type I if the Ginzburg± Landau parameter · = ¸/ x < 1/ 21/ 2
and type II otherwise.
2.1. Type I materials
In 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 expulsion
imposes a large energy penalty on the sample, superconductivity is quenched by
relatively low magnetic ® elds and these materials are not generally of great technological interest. An H± T phase diagram for a typical type I superconductor is
shown in ® gure 4(a). Demagnetization factors due to sample shape can lead to an
intermediate state between the Meissner phase and the ¯ ux vortex phase. An
appreciation of this can be gained by examination of ® gure 5 (a) which shows a
thin type I superconducting ® lm in a perpendicular applied magnetic ® eld. Clearly, in
the Meissner state shown there is a very strong concentration of ® eld at the sample
edges (Hedge = Happlied/ (1 ­ n), where n is the shape-dependent demagnetization
factor) and the critical ® eld will be exceeded here long before it is in the centre of the
sample. If Hedge > Hc , the system can, in practice, always reduce its energy by
breaking up into alternating normal and superconducting strips as shown in ® gure
5 (b), and this is called the intermediate state. Although it is, in principle, interesting
to image these `intermediate’-¯ ux distributions, they occur on rather coarse scales
compared with the diameter of a vortex and will not be discussed further here.
2.2. Type II materials
2.2.1. 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 is
quantized 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
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455
Figure 4. H± T phase diagrams of (a) type I and (b) type II superconductors.
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 labelled
Hc2, which can represent a very large ® eld at low temperatures. A vortex consists of a
normal core with radius x surrounded by a sheath of screening supercurrents
extending out to a distance ¸ as sketched in ® gure 6. The interaction between two
adjacent vortices is repulsive owing to the Lorentz force exerted by the supercurrent
of 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 superconductor
would 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 the
system can usually reduce the energy penalty associated with the normal core by
siting a vortex there. Consequently vortices become `pinned’ at these (generally
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S. J. Bending
Figure 5. (a) Field lines around a thin ® lm in the Meissner state. (b) The intermediate state
in a type I superconducting ® lm.
randomly distributed) centres, introducing disorder into the vortex lattice. Larkin
and Ovchinikov (1979) were able to show that even weak pinning destroys the longrange order of the ¯ ux line lattice, and only short-range crystalline order should
remain. Flux structures can nevertheless have quite long-range sixfold bond
orientational order: a so-called hexatic phase. More recently it has been proposed
that random pinning might lead to the formation of a vortex glass phase with the
vortices 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 geometry
of the sample of interest. For a bulk sample the Clem (1975) model is a very useful
description whereby the order parameter inside the vortex core is obtained from a
variational trial function and the spatial dependence of the magnetic ® eld is given by
B(r) =
K0 ((r2 + x v2 )1/ 2/ ¸)
,
2p ¸x v
K1(x v /¸)
U 0
(2.2)
where x v is a variation core-radius parameter (approximately x ), and K0 and K1
are 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 be
approximated by the superposition of the ® elds of individual vortices, although
corrections must be introduced at high ® elds due to vortex overlap. This is
demonstrated for the Clem model in ® gure 7 (a) for a hexagonal lattice
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457
Figure 6. Pro® les of (a) superconducting electron density ns, (b) magnetic ® eld B and (c)
supercurrent density Js near a vortex core.
( x v = 40 nm; ¸ = 80nm) in an applied ® eld of 10 mT. Many imaging techniques
actually sample the magnetic ® eld at a ® xed distance above a superconducting
surface. Performing a Fourier transform of the Maxwell equations, one can show
that the perpendicular component of magnetic ® eld a height z above a surface is
given by
~
Bz (r, z) =
B(s , 0) exp (­ js jz) exp (is . r)
( 2.3)
s
where s is one of the reciprocal-lattice vectors of the ¯ ux line lattice and the ® rst term
in the summation is the appropriate Fourier component of the ® eld distribution at
the sample surface. We note then that the Fourier components decay exponentially
with increasing distance from the surface with the lowest-order reciprocal-lattice
vectors being the most robust. As a rule of thumb the ® eld modulation is only
detectable 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 be
strongly in¯ uenced by surface or geometrical barriers at sample surfaces which are
important even for negligible bulk pinning. The surface barrier (Bean and Livingston
1964) can be understood if one imagines trying to introduce a single ¯ ux line parallel
to the planar face of a semi-in® nite sample when there are two competing energy
terms to consider. The ® rst is the repulsive interaction of the vortex with surface
screening currents, and the second is the attraction of the ¯ ux line to its image inside
the sample. As a consequence a potential barrier for ¯ ux entry forms at the surface
even 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
458
S. J. Bending
Figure 7. Magnetic ® eld pro® le at various heights z above an Abrikosov lattice of vortices
in an applied ® eld of 10mT.
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459
Figure 8. Vortex line energy as a function of distance x from the sample surface for three
di€ erent values of applied ® eld.
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 to
become bowed as they penetrate at sample edges. For the case of a rectangular
platelet cross-section the vortices initially round o€ the sharp corners of the sample
without complete penetration. As a consequence the energy of a penetrating vortex
increases gradually from zero to a maximum of about e 0d where e 0 is the vortex line
energy and d the sample thickness. This represents a robust thermodynamic barrier
up to the equilibrium ® eld Heq, and a kinetic barrier for ® elds above this up to the
penetration ® eld Hp at which point vortices start to enter freely. The geometric
barrier also leads to concentration of ¯ ux in the centre of the sample at equilibrium
for ® elds in excess of Heq. The e€ ects of surface and geometric barriers are
particularly pronounced in samples with very low pinning, which is often the case
in high quality single crystals of HTS materials.
2.2.2. Vortex dynamics
The dynamic properties of vortices are of particular importance since their
motion signals the breakdown of the zero resistance state. If a uniform transport
supercurrent density J is passed through a superconductor, there is a Lorentz force
on any ¯ ux lines present given by
460
S. J. Bending
F=J
U
0,
(2.4)
where U 0 is a vector along the length of the vortex with the magnitude of the ¯ ux
quantum. Provided that this force is much less than characteristic pinning forces,
then the vortices will not move. However, above a critical current density Jc, pinning
forces can be overcome and vortices start to move freely through the sample. If one
describes all the damping processes (e.g. eddy current damping in the normal core) in
terms of a scalar damping factor ´, the induced voltage in the ¯ ux ¯ ow regime is well
described by a ® eld-dependent ¯ ux ¯ ow resistivity and the vortex velocities vu are
given by
U 0 ( J ­ Jc )
vu =
.
(2.5)
´
These ¯ ux ¯ ow velocities obviously vary considerably depending on the
magnitude of the applied transport current but, with the possible exception of the
MO technique, values are typically much larger than the current temporal resolution
of imaging systems.
Even if J < Jc , vortex motion can still occur because of thermally activated
depinning. This phenomenon is called ¯ ux creep and was ® rst described theoretically
by Anderson and Kim (1964). In practice a vortex or bundle of vortices undergoes a
thermally activated hop between two adjacent pinning points. The activation energy
is typically much larger than kT in conventional superconductors and the mean
creep rate tends to be rather slow and lies well within the temporal resolution of
several imaging techniques except for T very close to Tc. In HTS materials ¯ ux creep
can, however, be very rapid even for temperatures substantially below the critical
temperature.
Flux ¯ ow and ¯ ux creep occur in the presence of su ciently weak pinning and/or
a su ciently large driving force, either due to an applied transport current or to
magnetic ® eld gradients within the superconductor. For example if the applied
magnetic ® eld threading a ® eld-cooled sample is suddenly reduced to zero, the
vortices ¯ ow towards (and out of) the surface until a remanent state is produced such
that J < Jc everywhere. This remanent state will then continue to relax further by
temperature dependent ¯ ux creep mechanisms.
2.3. New phenomena in high-temperature superconductors
HTSs are extreme type II materials and distinguish themselves from conventional
materials in a number of respects . The coherence length is very short (x
1 nm) and
the energy penalty associated with adding a ¯ ux vortex is rather small. As a
consequence the superconducting state exists up to very large magnetic ® elds. The
penetration depth, on the other hand, is relatively large (¸ 100± 200nm) with the
result that the repulsive interactions between vortices at high ® elds (proportional to
1/¸2 ) become very weak. Since, as the name implies, high-Tc materials remain
superconducting up to much higher temperatures (about 100K), the magnitude of
thermal ¯ uctuations can also be very large. Finally, the new materials can show very
large crystalline anisotropy because superconductivity is associated with layers of
Cu± O atoms in the a± b plane which are only weakly coupled in the perpendicular
direction. The crystal structures are approximately tetragonal, although often
include a small orthorhombic distortion. Therefore, for many purposes the anisotropy 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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461
carriers (fourfold symmetry has been assumed). C can vary from about 5± 7 for
YBa2 Cu3 O7­ d , (YBCO) (Dolan et al. 1989) to about 50± 200 for Bi2 Sr2 CaCu2 O8+d
(BSCCO) (Farrell et al. 1989) for the HTSs whereas it is close to unity for
conventional superconductors.
If a magnetic ® eld is applied along the high symmetry c direction, the fact that
supercurrents are largely con® ned to planes of Cu± O atoms which are much thinner
than 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 or
`pancake’ vortices which interact weakly through Josephson coupling. Since pancake
vortices within the same layer repel each other while those in di€ erent layers attract
each other, a regular lattice of straight ¯ ux lines has the lowest energy but is
extremely soft with respect to 2D ¯ uctuations. Indeed when the typical shear energy
of the ¯ ux line lattice starts to exceed the energy due to short range (interplanar) tilt
deformations, pancake vortices can start to move independently of those above and
below them. In the presence of pinning, the energy of the system may well be reduced
if the ¯ ux line becomes highly distorted such that each pancake vortex is situated on
the nearest adjacent pinning site within its layer. Eventually above a decoupling ® eld
B2d , thermal ¯ uctuations lead to the total loss of phase coherence between pancake
vortices 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 applied
parallel to the c axis. If the ® eld instead lies at oblique tilt angles with respect to the c
axis, it is possible to realize a surprising regime where the normally repulsive
interaction between vortices becomes attractive. This is a consequence of the
tendency for vortex supercurrents to be con® ned to the Cu± O planes with the result
that one of the magnetic ® eld components within the vortex reverses sign. This, in
turn, leads to an attractive well in the vortex± vortex interaction within the plane
containing the magnetic ® eld vector and the crystal c axis (Grishin et al. 1990).
If the ® eld is applied exactly parallel to the Cu± O planes the vortex cores prefer to
locate in the `normal’ spaces between the planes. These are called Josephson vortices
since the circulating currents giving rise to them have to cross the superconducting
Cu± O planes by Josephson tunnelling. At angles slightly away from the a± b plane it
can become energetically favourable for the ¯ ux lines to form staircase-like
structures composed of a combination of pancake and Josephson vortices which
`lock in’ to the spaces between Cu± O planes (Oussena et al. 1994).
Finally the pronounced elastic softness of vortices in HTSs, and high available
temperatures 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 system
to show that the vortex lattice should melt into a vortex liquid when
Bm( T )/ Bc2 ( T ) 10­ 4 . Such a `melting’ line has been identi® ed on the basis of
neutron scattering (Cubitt et al. 1993), muon spin rotation (Lee et al. 1993) and
magnetization measurements (Pastoriza et al. 1994, Zeldov et al. 1995), and in highly
anisotropic materials such as BSCCO is situated well below the critical temperature.
On the basis of abrupt jumps in Hall probe measurements of local induction in
BSCCO (Zeldov et al. 1995) and sharp peaks in the heat capacity of YBCO at the
melting line (Schilling et al. 1996) it is widely believed that this represents a ® rstorder phase transition.
As a consequence of the phenomena described above, the H± T phase diagram for
HTSs is complex and remains controversial.
462
S. J. Bending
This introduction merely scratches the surface of the richness of vortex physics in
HTS materials, and other phenomena will be described in more detail later in this
article as a deeper understanding is required. For excellent comprehensive reviews of
the area the reader is referred to BlaÈ tter et al. (1994) and Brandt (1995). In the
descriptions of imaging techniques that follow, examples will be presented to
illustrate the state of the art in instrumental performance. Since the HTSs have, in
general, set very demanding measurement criteria in terms of ® eld resolution and
operation temperature, the vast majority of examples have inevitably been drawn
from 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 in
the incident electron wavefunctions. Consequently electron phase-sensitive techniques such as electron holography must be employed in order to resolve them.
Such developments only became possible when suitable highly coherent sources of
electrons became available in the late 1960s. The theory of holography is usually
discussed in terms of in® nitely coherent plane wave electron states, but in reality an
electron beam in a microscope is a train of incoherent wave packets. In order for
clear holographic images to be observed the transverse extent of the wave packet
must be su cient to overlap all the spatial points in the object plane which one
wishes to interfere, while its longitudinal extent must be at least as long as the
maximum phase di€ erence between any two of these points. Of these two
independent criteria the former is normally most stringent and limits the maximum
number of observable interference fringes and hence the resolution. In practice the
low brightness of early thermionic emission cathodes made them unsuitable for
holography and it was only after a practical ® eld-emission cathode was developed in
1968 (Crewe et al. 1968) that such applications took o€ . Tonomura et al. (1979) were
able to perfect the design still further to the point where more than 3000 interference
fringes became observable in a 70 keV microscope and cathodes have continued to
improve since.
3.1. Theory of phase contrast in electron microscopy
Given that a coherent electron source is available it is trivial to show that a
superconducting vortex acts as a phase shifting object. Consider the gedanken
experiment sketched in ® gure 9 (a) where an electron plane wave is incident normal
to a ¯ ux line containing a single superconducting ¯ ux quantum ( 0 = h/ 2e). The
vector potential for this^ situation can^ be represented by an azimuthal vector about
the 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 string
we ® nd that
z=­ 1
1 y dz
p
^
u = ­1
Au ds =
sgn ( y) = ­ sgn( y),
2
2
2
(2 0) z=1
­ 12(y + z )
(3.1)
where we assume that the limits of the integral can be set to in® nity and the factor of
2 in the denominator of the second term arises because the ¯ ux quantum for a single
electron is twice that for a superconducting Cooper pair. Thus we see that two
L ocal magnetic probes of superconductors
463
Figure 9. (a) Sketch of an electron trajectory passing by a horizontal ¯ ux string containing a
single ¯ ux quantum. (b) Total change u in phase accumulated by electrons on
di€ erent trajectories.
di€ erent electron trajectories which pass on opposite sides of the ¯ ux string
experience a p phase di€ erence (® gure 9 (b)).
The scattering geometry described above is not, however, the preferred geometry
since it yields only a one-dimensional (1D) perspective on the spatial distribution of
vortices. On the other hand the vortices clearly cannot lie along the optical axis since
they will induce no electron phase shifts in this orientation and hence no image
contrast. Usually the normal to the sample plane is inclined at an angle of around
45ë 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 sketched
in ® gure 10 and one must now account for both ® elds within the superconductor as
well 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 be
described by two point magnetic poles of strength
0 where the vortex string
intersects the superconductor surface with a radial ® eld line distribution everywhere
outside the perfectly diamagnetic sample. This is a reasonable approximation in the
vicinity of the vortex although clearly it does not satisfy the boundary condition that
further away all ® eld lines must originate and terminate at opposite sides of the
sample space. These workers have shown that
464
S. J. Bending
Figure 10. Coordinate system used to calculate the phase shift due to a ¯ ux string threading
a thin superconducting sample. The specimen of thickness t is inclined at an angle of
a to the optic axis and the corresponding phase shift in the object plane is depicted in
the contour map below.
y
y
u (x, y) = 1 tan­ 1
­ tan­ 1
2
x+ a
x­ a
­ 1 sin­ 1
2
[
y2
+
y sin (a )
+ sin­ 1
2 1/ 2
+ (x + a) ]
p
4
sgn
[
y2
y
x­ a
y
­ sgn
y sin ( a )
+ (x ­ a)2]1/ 2
x+a
,
(3.2)
where t is the thickness of the superconducting ® lm and a = [t sin (a )]/ 2. A threedimensional (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 phase
change 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 either
side of an in® nite ¯ ux string which is independent of tilt angle a . However, the phase
di€ erence between large positive and negative values of y is only 2a (last two terms
of 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 of
equation (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 33 nm in ® gure
11(b). As one might expect, the abrupt phase discontinuity is smeared out over a
length scale of the twice the penetration depth ¸ and, surprisingly, the maximum
phase shift is quickly reduced from p down towards the limiting value of 2a . Thus
L ocal magnetic probes of superconductors
465
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 ¯ ux
quantum (¸L = 33nm).
the height and breadth of this phase change are a measure of the diameter and inner
structure respectively of a superconducting ¯ ux vortex.
The two most common imaging techniques for ¯ ux vortices are Lorentz
microscopy and true electron holography and these will be discussed separately.
3.2. L orentz microscopy
This 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 classical
^
picture can be obtained by considering the action of the Lorentz force (FL = ev^ B)
on electrons which pass through a ¯ uxon. Their trajectories will be de¯ ected away
from the vortex in such a way that the electron ¯ ux is enhanced near one edge of the
vortex and depleted near the other in a similar way to the Hall e€ ect in conducting
solids. Thus, in the scattering geometry of ® gure 12 (a), a vortex can be recognized as
a pair of adjacent dark and light stripes as sketched. In practice, however, a
quantitative understanding of Lorentz microscopy can only be achieved within a
full quantum-mechanical description.
466
S. J. Bending
Figure 12. Classical response of a beam of electrons incident on a horizontal ¯ ux line.
A schematic diagram of the 300keV electron microscope used by Tonomura and
co-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 the
geometry to perform Lorentz microscopy is shown in ® gure 14. Note that the sample
must be thinned down to about 70 nm before TEM can be performed and the normal
to 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 understood
from the schematic diagram shown in ® gure 15 (Chapman 1984). We assume that the
electron beam can be represented by a z-propagating plane wave (/exp (ikz)) while
the vortex structure, since it only modulates the electron phase, can be represented
by a transmittance f (x, y) /exp [i u (x, y)]. Thus the electron beam transmitted
through the specimen will take the approximate form exp i[kz + u (x, y)] . At the
back focal plane of the objective lens the electron disturbance may be described by
~
the Fourier transform of the specimen transmittance f (kx, ky ) modi® ed by a transfer
function t(kx, ky ) which accounts for e€ ects such as spherical aberration (present in
all electron lenses) and image defocusing. The electron disturbance in the image
plane is returned by the following reverse Fourier transform:
f
I(x 0, y 0) /
~
g
2
f (kx, ky)t(kx, ky) exp [­ i(kxx 0 + ky y 0)] dkx dky .
(3.3)
It is clear from equation (3.3) that if the transfer function is everywhere unity,
I(x 0, y 0) would simply be /jf (x, y)j2/jexp [iu (x, y)]j2 which is constant and
the image would contain no contrast. A non-uniform transfer function is then
essential for Lorentz microscopy and is normally achieved by defocusing the
image by an amount z. In this situation the transfer function becomes
467
L ocal magnetic probes of superconductors
Figure 13. Diagram of an electron microscope designed for investigating vortices in
superconductors.
t(kx, ky ) = exp [i z ¸(k2x + k2y)/ 4p
following convolution:
I(x 0, y 0)
/
] and the image intensity is proportional to the
ip
exp [iu (x, y)] exp
[(x ­ x 0)2 + ( y ­ y 0)2]
¸ z
2
dx dy .
( 3.4)
Ultimately an individual ¯ ux vortex can be recognized in an image by an oval
region with adjacent halves of bright and dark intensity. The line dividing these
halves is de® ned by the projection of the vortex segment on to the z = 0 plane, while
the sense of light and dark regions is reversed for upward- or downward-directed
vortices.
It is the nature of Lorentz mode images of vortices that it is very di cult to
extract 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 are
ideal for making studies of vortex dynamics. The ® rst example of this was achieved
by Harada et al. (1992). A high-purity thin ® lm of niobium (R(300K/R(10 K) 20)
468
S. J. Bending
Figure 14. Schematic diagram of the experimental set-up for performing Lorentz
microscopy.
was chemically thinned to 70 20 nm and positioned on a low-temperature stage at
45ë to the incident 300keV electron beam. Vortices were then imaged by Lorentz
microscopy with the objective lens switched o€ and using a second intermediate lens
for focusing. This con® guration is used so that the sample is not exposed to the
relatively high magnetic ® elds generated by the objective lens but has the disadvantage that the magni® cation is somewhat reduced from its maximum value. Figure 16
shows a Lorentz micrograph of the niobium ® lm in a 10 mT applied ® eld at 4.5 K
with the image defocused by 10 mm to achieve contrast. The positions of the vortices
are clearly marked by the black and white spots and note that the 45ë tilt angle leads
to a 1/ 21/ 2 compression of one of the axes as indicated by the length markers. The
dark arcs superimposed on the image represent Bragg re¯ ections at atomic planes
induced by bends in the ® lm, and can be ignored. Detailed examination of ® gure 16
reveals that the vortex lattice is not quite perfect in this image because of the
L ocal magnetic probes of superconductors
469
Figure 15. Image formation during Lorentz microscopy.
in¯ uence of randomly distributed weak pinning sites in the sample such as interstitial
or substitutional atoms. These workers observed, under the in¯ uence of higher
temperatures and/or applied magnetic ® elds, a greater degree of order as pinning
forces 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 a
maximum rate of 30 frames s­ 1 . In this way, discrete vortex hopping processes
between nearby pinning sites could be visualized in real time. Figure 17 shows three
stills from a videotape generated some seconds after a 15 mT ® eld was suddenly
switched o€ at 4.5K. Most vortices were seen to hop in the direction of decreasing
vortex 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 by
dotted 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 m
and frequencies of about 10 s­ 1 .
Only a year later Harada et al. (1993) were able to use real-time Lorentz imaging
to investigate the possibility of vortex lattice melting in the Bi2 Sr1.8CaCu2 O8+d HTS.
This presented a special challenge since the a± b plane penetration depth in this
material is almost an order of magnitude larger than in niobium. Consequently the
classical electron de¯ ection by a single vortex is proportionately smaller and much
longer defocusing distances (up to 10cm) have to be used to obtain su cient
contrast. Figure 18 shows Lorentz micrographs of a BSCCO ® lm 150± 250nm thick
which had been cleaved from the face of a large single crystal. The sample was
inclined at 45ë to the incident electrons and cooled to 4.5 K at H = 0. The applied
470
S. J. Bending
Figure 16. Lorentz micrograph of a niobium ® lm at 4.5K in a ® eld of 10mT. [Reprinted
with permission from Nature (Harada et al. 1992) Copyright 1992 Macmillan
Magazines Limited.]
® eld was then raised to 2 mT and the sample was imaged at four di€ erent
temperatures as shown. Above 40 K, when pinning forces cease to be important,
the vortices are seen to form a very regular hexagonal lattice. As the temperature was
increased further, the contrast gradually diminishes, disappearing entirely at 76.5 K.
Independent magnetization measurements found the irreversibility temperature at
this applied ® eld to be 74 2 K which is close to the point where contrast is lost.
These workers pointed out, however, that this cannot be taken as unequivocal
evidence for lattice melting since the increase in penetration depth with increasing
temperature as well as possible vibrations of the vortices about their equilibrium
positions could alone be su cient to destroy contrast.
Recently Matsuda et al. (1996) have been studying the ways in which vortices
interact with arti® cially introduced defect arrays. These were produced by irradiating
a 100nm niobium ® lm with a 30 keV focused gallium-ion beam (diameter, 20 nm).
Figure 19 shows micrographs of a region of the ® lm containing a 4 4 rectangular
lattice with periodicity 3.3 m m. Each `defect’ corresponds to a pit of 40 nm diameter
surrounded by a 300nm region of entangled dislocations. At low ® elds (less than the
commensurability ® eld of the defect lattice) the vortices have been shown to try to
order themselves in a periodic way on to selected defects so as to minimize their
potential energy (Harada et al. 1996). Figure 19 shows the opposite limit when the ® eld
is very much greater than the commensurability ® eld. Here the sample was cooled to
6K in an applied ® eld of 18mT and allowed to reach equilibrium. The positions of
both vortices and ion-implanted regions can be resolved in these micrographs and
detailed examination reveals that the presence of defects prevents the ¯ uxons from
forming one coherent lattice. Rather they form hexagonally ordered domains of about
L ocal magnetic probes of superconductors
471
Figure 17. Dynamics of vortices in a niobium ® lm (a) 170s, (b) 170.1s and (c) 171.4s after a
15mT ® eld has been switched o€ at 4.5K. Dotted (solid) circles show vortex
positions before (after) hopping. [Reprinted with permission from Nature (Harada et
al. 1992). Copyright 1992 Macmillan Magazines Limited.]
(a)
(b)
(c)
(d )
Figure 18. Lorentz micrographs of a BSCCO ® lm in a 2 mT applied ® eld at (a) 4.5K, (b)
20K, (c) 56K and (d ) 68K. [Reprinted from Harada et al. (1993). Copyright 1993 by
the American Physical Society.]
472
S. J. Bending
(a)
(b)
(c)
(d )
Figure 19. Video frames of regions of vortex lattice in a niobium ® lm at various times after
the ® eld was suddenly reduced from 18 to 8.5mT at 6 K: (a) t = 0 s; (b) t = 0. 27s; (c)
t = 0.43s; (d) t = 0.80s. Implanted defects are located at the black discs and domain
boundaries for the vortex lattice are indicated by dotted lines. [Reprinted with
permission from T. Matsuda, K. Harada, H. Kasai, O. Kamimura and A.
Tonomura, 1996, Science, 271, 1393. Copyright 1996 American Association for the
Advancement of Science.]
L ocal magnetic probes of superconductors
473
5 5 vortices which appear to be pinned at defects near where domain boundaries are
located. As soon as ® gure 19(a) was recorded, the applied ® eld was suddenly reduced
to 8.5mT. Initially the system does not respond; then suddenly avalanche-like ¯ ow
begins along one of the domain boundaries lying between the dotted curves in ® gure
19(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 data
yield unique insights into the dynamic interactions between vortices and pinning sites
and are certain to advance our understanding in this area greatly.
3.3. Electron holography
While Lorentz microscopy is capable of providing powerful insights into vortex
dynamics, it yields little quantitative information about the dimensions and internal
structure of individual vortices. If more quantitative data of this type are required,
the complementary technique of electron holography can be applied. The o€ -axis
geometry as sketched in ® gure 20 is most commonly used for performing holography
since this allows the conjugate image (always present in holograms) to be separated
from the reconstructed image. As the name implies, the sample occupies one half of
the electron beam path while the other half remains undisturbed and forms the
reference wave. The two beams must now be made to interfere and this can be
achieved with an electron biprism. The latter is simply a very ® ne (of less than 1 m m
diameter) positively charged ® lament which is place horizontally through the
microscope optic axis, ¯ anked by two grounded plate electrodes on either side.
Close to the ® lament the biprism approximates to a coaxial cable and the potential
depends logarithmically on the radial distance from it. It is straightforward to show
that electrons passing either side of the ® lament experience a ® xed angular de¯ ection
towards it proportional to the biprism voltage which is independent of their incident
position. In this way the scattered and reference beams can be made to interfere
controllably and to generate fringes in the hologram plane.
It is clear from ® gure 20 that the two beams are inclined at a relative angle a
when they interfere. A simple theoretical way to picture this situation is to imagine a
reference plane wave of form u r /exp [ik(z ­ a y)] (tilted at an angle a to the optic
axis) 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 l
from the object will be
2
2
+ 2
f
­ 2f sin k( x y ) + ka y .
I(x, y) = ju o + u r j2 1 +
( 3.5)
l
l
2l
When this pattern is exposed on to ® lm the amplitude transmittance for an
incident reconstruction beam depends on a coe cient g which indicates the contrast
of the ® lm (t = I­ g / 2). If g = ­ 2, then t can simply be replaced by the expression for
I given above. In this situation, if we now illuminate the hologram with a reference
beam identical with that used to create it, the resultant transmitted amplitude in the
hologram plane is
T (x, y) = exp (ikl) 1 +
f
l
2
+
if
ik( x2 + y2)
exp
l
2l
2
2
­
­ i f exp ik(x + y ) ­ 2ka y .
l
2l
(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
474
S. J. Bending
Figure 20. Schematic diagram of the experimental set-up for performing electron
holography.
direction of the latter is now inclined at 2a with respect to the reconstructed image
and hence is spatially separated. Note that it is not necessary for the reference beam
to be an electron wave; indeed it never is. In practice, holograms are magni® ed in the
microscope so that they can be reconstructed with the much longer wavelength of
light. This is, in fact, extremely convenient since the techniques available to
manipulate light are much more ¯ exible than those for electron waves.
The mere ability to generate holographic images is, however, insu cient to
guarantee observation of ¯ ux vortices. This is because the phase shift across a vortex
is typically about p /2 (about ¸/4), corresponding to only a quarter of the di€ erence
L ocal magnetic probes of superconductors
475
Figure 21. Optical reconstruction system for phase-ampli® ed interference microscopy.
between a pair of interference fringes. Consequently techniques for phase-di€ erence
ampli® cation are essential to improve spatial resolution. This is invariably achieved
optically owing to the far greater ¯ exibility of optical components. The simplest
way to double the phase di€ erence is to use the Mach± Zehnder interferometer
sketched in ® gure 21 to illuminate the hologram with two separate plane waves
whose angles are chosen so that the reconstructed image from one beam overlaps
the conjugate image from the other. Since the phases of the two overlapping
beams are reversed in sign, the ® nal image reveals a phase distribution which has
been ampli® ed by a factor of two. If further ampli® cation is required, this process
can be repeated several times. In practice it is often quicker to exploit the higher
harmonic data from nonlinear holograms when g =
6 ­ 2. In this case t(x, y) can
be expanded in a power series:
t(x, y) = I­ g
/2
1­ g
f
k(x2 + y2 )
f2
k(x2 + y2)
+ ka y + g (g + 2) 2 sin2
+ ka y +
sin
l
2l
2l
2l
+ (­ 1)ng (g + 2)(g + 4) . . . (g + 2n)
f n n k(x2 + y2 )
+ ka y .
sin
n! ln
2l
(3.7)
Thus the nth term in the series has been phase ampli® ed by a factor of n, and this
can be overlapped with its conjugate image as described above to give a total of 2n
ampli® cation. Phase ampli® cations of up to 32 times have been demonstrated with a
combination of these techniques. Vortex images are routinely produced with 16
times ampli® cation when the approximate p /2 phase shift across a vortex roughly
corresponds to the separation between four adjacent fringes.
Figure 22 shows one of the ® rst holographically reconstructed images of a thin
niobium foil which had been cooled to 4.5 K 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 an
intermediate lens. This limits the spatial resolution to about 30nm over a sample
area 4 m m wide. Comparison of the 16 times phase-ampli® ed contour map with a
simultaneously recorded Lorentz image reveals that vortices can be identi® ed by the
regions where about four contour lines are tightly clustered together (indicated by
476
S. J. Bending
Figure 22. Interference micrograph of the vortex lattice in a niobium ® lm (phase ampli® ed
16 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) 7 K and (c)
8 K (phase ampli® ed 12 times). [Reprinted from Bonevich et al. (1994b). Copyright
1994 by the American Physical Society.]
open circles). As discussed earlier the feature running almost diagonally across the
image is a consequence of a slight bend of the ® lm and can be ignored here. To
demonstrate that these holographic images contain quantitative information about
the vortex structure much smaller sample regions about 1.5 m m wide were studied
with the objective lens turned on for higher spatial resolution (about 7 nm) (Bonevich
et 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 at
three di€ erent temperatures. In each case the vortex induces a phase shift of about
p /2 but over an increasingly larger distance as the penetration depth increases at
higher temperatures. For a given phase shift an average vortex diameter could be
assigned which increased from 150 4 nm at 4.5K to 185 4nm at 7 K and
230 4 nm at 8 K. A quantitative comparison was made with two di€ erent radial
models of a ¯ uxon, namely the London (1935) model and the Clem (1975) model:
L ocal magnetic probes of superconductors
BLon (r) =
BClem(r) =
0
2p ¸2L
477
r
K0 ¸ ,
L
K0((r2 + x v2 )1/ 2 /¸L)
,
2p ¸x v
K1(x v /¸L)
0
( 3.8)
where ¸L is the London penetration depth, x v is a variational parameter to describe
the normal vortex core, and K0(x) and K1 (x) are modi® ed Bessel functions.
Assuming a two-¯ uid temperature dependence of ¸L( T ) = ¸L(0)[1 ­ ( T / Tc )4]1/ 2 it
was found that the Clem model probably provides a slightly better description of the
reconstructed images. It will, however, be appreciated that a considerable amount of
modelling goes into the simulation of images such as ® gure 23, and inverting the data
to produce the true ® eld pro® le at a vortex is a very ambitious task.
The holographic reconstructions shown here were all produced optically or
digitally some time after the original holograms were formed. Recently there has
been considerable progress with real-time holography. This can be achieved in one of
two ways and both involve detecting an electron hologram with a TV camera. The
most direct approach is then to reconstruct the image using Fourier transform-based
algorithms and a very-high-performance computer. Even with current state-of-theart hardware, however, reconstruction still takes a few minutes to achieve and it is
not yet possible to produce images in real time. A better approach is to transfer the
video signal from the charge-coupled device (CCD) camera to a liquid-crystal panel
as shown in ® gure 24. Since this panel is itself a phase hologram, illuminating it with
a 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 and
require little introduction here. Its development dates from work by Binnig et al.
(1986) who recognized that it is possible to use the photolithographic processing
techniques developed for the semiconductor industry to fabricate microscopic
cantilevers with force constants smaller than the e€ ective spring constant of an
atom bonded at the surface of a solid. Thus they were able to show that one can
mechanically image solid surfaces without perturbing the atomic structure. To
establish 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 and
­
25
m 10 kg respectively, yielding an approximate spring constant k x 2 m
10Nm­ 1 . This should be compared with the force constant of the rectangular lever
of length l, width w and thickness t sketched in ® gure 25.
Ewt3
,
( 4.1)
4l3
where E is Young’s modulus for the material. Inserting typical values for silicon
cantilevers (E = 1.79 1011 Nm2 , q = 2330kg m3 , l 100 m m, w 10 m m and
t 0.6 m m) yields k = 0.1 Nm­ 1 which is clearly well within what is required. In
practice there is an additional requirement that the resonant frequency of the
cantilever be su ciently high that there is no danger of exciting oscillations during
rapid scanning. This condition con¯ icts somewhat with the requirement of a soft
spring constant but is nevertheless readily achievable. For the cantilever sketched
above
k=
478
S. J. Bending
Figure 24. Schematic diagram of a real-time electron holography system: YAG, yttrium
aluminium garnet; VCR, video cassette recorder.
x
t
E
= 2l2 0. 24q
1/ 2
,
(4.2)
yielding an acceptable resonant frequency of 85 kHz 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 to
monitor electrostatic or, as in the case of interest here, magnetostatic forces
between a sample surface and sensor. There are a variety of ways to monitor
L ocal magnetic probes of superconductors
479
Figure 25. Sketch of a rectangular cantilever.
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 beam
de¯ 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 an
optical ® bre.
There are two distinct modes of SFM operation. In the constant-force mode the
cantilever is brought into `contact’ (i.e. within range of interatomic forces) with the
surface and a feedback loop is employed to control sample± cantilever separation
such that the de¯ ection (hence force) remains ® xed during scanning. In this way the
surface topography can be measured under a constant force. Alternatively the
cantilever can be oscillated near its resonance frequency and the resonance amplitude
used to probe force gradients near the sample surface. Treating a free (far from a
surface) cantilever as a damped oscillator with resonance frequence x 0 and quality
factor Q, its frequency-dependent amplitude is well described by the classical
expression
a0 x 20
A(x ) = 2
,
( 4.3)
[(x 0 ­ x 2)2 + x 2x 20/ Q2]1/ 2
where a0 Q is the resonant amplitude. As the cantilever approaches a surface,
sample± probe interaction leads to a shifted resonant frequency x 00 which is a
function of the local force gradient F 0. Provided that x 00 x 0 the oscillation
amplitude at a ® xed frequency near resonance is a measure of the force gradient
at the cantilever tip. It can be shown (Sarid 1991) that the optimum operation
frequency in this mode is just o€ the free resonance x 0 at a value x m when the
amplitude has dropped to about 82% of its maximum value. For small force
gradients the change in resonance amplitude is now linearly proportional to d F 0:
d A(x
2
m
0Q
d
) = 2a
3
3 / 2k
F 0,
( 4.4)
where k is the cantilever spring constant. The sensitivity of this operation mode can
be very high for a sensor with a large Q.
480
S. J. Bending
4.1. Theory of magnetic force microscopy of superconductors
In order to be sensitive to magnetostatic forces the scanning force microscope tip
must be made of ferromagnetic material, ideally a microscopic single-domain
particle with a high coercive ® eld. In practice tips have been realized either by ® ne
electrochemical etching of ferromagnetic wires or by depositing thin magnetic ® lms
on top of the sharp atomic force microscope tips on micromachined cantilevers.
Consequently the magnetic domain structure is rarely well known and may not even
be the same from one scan to another. For this reason a truly quantitative
understanding of MFM images represents a major theoretical challenge in its own
right and is beyond the scope of this article. In most treatments of MFM images of
superconductors (Hug et al. 1991, Reittu and Laiho 1992, Wadas et al. 1992) the
simplifying assumption is made that the tip is actually a cylindrical single domain
particle magnetized uniformly along its axis which is perpendicular to the sample
surface. 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 ¹0 MR2 (where M is
the 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 the
tip 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 highlights
one of the major di culties associated with interpreting MFM images and an
excellent 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 `so-called’ Meissner levitation force and the second is the
force at the tip due to ¯ ux vortices threading the sample.
The levitation force is the microscopic analogue of the force which supports a
macroscopic permanent magnet above a superconductor in the Meissner state. This
has been calculated by Hug et al. (1991) assuming that the London equation (4.6) is
obeyed in the semi-in® nite superconducting half-space z < 0:
rA ­
2
1
A 0
¸2 = ,
(4.6)
where A is the vector potential and ¸ is the magnetic ® eld penetration depth. At
z = 0 this solution must be matched with that of the Maxwell equations in the semiin® nite half-space z > 0 containing the magnetic force microscope tip.
rA = ­ ¹0 Jm,
2
(4.7)
where Jm is the magnetization current density of the tip. Within the magnetic point
charge model they found the following expression for the force with the tip a height d
above the superconductor:
Fz (d ) =
m2 1(1/¸2 + x2 )1/ 2 ­ x
exp(­ 2xd ) x dx.
4p ¹0 0 (1/¸2 + x2 )1/ 2 + x
(4.8)
As one would expect, the `levitation’ force depends on m2 since, to a ® rst
approximation, the force arises due to the interaction between the magnetic point
charge and its image within the superconductor.
L ocal magnetic probes of superconductors
481
Figure 26. Theoretical forces exerted on an idealized magnetic force microscope tip 20nm
above a YBCO thin ® lm as a function of radial displacement r from a vortex core.
Using the same approximations, Reittu and Laiho (1992) have solved the same
set of equations to calculate the force at the tip due to a single ¯ ux line threading the
superconductor, where equation (4.6) now contains a term on the right-hand side to
account for the normal vortex core. They found that the vertical force on the tip a
height d above the superconductor and a radial distance r from the vortex axis is
Fz (r, d ) =
m 0 1 (1/¸2 + x2 )1/ 2 exp (­ xd )J0(xr) xdx
,
2p ¹0 0 (1/ ¸2 + x2 )1/ 2 + x
1 + (x¸)2
( 4.9)
where J0 is the zeroth-order Bessel function. The lateral force of the tip can also be
straightforwardly calculated and is found to be
Fr(r, d ) =
m 0 1 (1/ ¸2 + x2)1/ 2 exp (­ xd )J1 (xr)x dx
,
2p ¹0 0 (1/ ¸2 + x2 )1/ 2 + x
1 + ( x¸)2
(4.10)
where J1 is the ® rst-order Bessel function. Equations (4.8)± (4.10) are plotted in ® gure
26 as a function of r for a typical tip height of 20 nm, assuming, as in the experiment
of Moser et al. (1995), that the tip has a radius of 100nm and is made from
Fe51 Al8 Ni14 Co24 Cu3 with M = 10. 5 105 A m­ 1 and that the superconductor is
YBCO at a low temperature with ¸ = 165nm. Note that, with these parameters, the
levitation force (always repulsive) is still somewhat larger than the force due to a
vortex (attractive or repulsive depending on the orientation of the vortex) and the
contrast at a vortex depends strongly on its orientation (i.e. up or down). It is
interesting to compare the peak lateral force of about 100pN with typical pinning
482
S. J. Bending
forces for vortices in these superconducting ® lms. These have been measured for a
few 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 temperature
dependence of the pinning force f p( T ) was found:
fp( T )
100pN 1 ­
T
Tp
2
,
(4.11)
where Tp is approximately the critical temperature of the ® lm. We note then that the
lateral force is comparable with or exceeds the pinning force at all temperatures. This
probably represents a rather pessimistic prediction, however, since the stoichiometry
of the ® lm of Stoddart (1994) was not optimal ( Tc = 82K) and the Hall probe
measurement tends to select those vortices sited at the weakest pinning sites. In
addition the magnetic point charge model of the magnetic tip almost certainly
overestimates the force experienced by a real magnetic force microscope tip.
Nevertheless these estimates highlight the potential invasiveness of MFM, and great
experimental caution must be employed when imaging superconductors.
4.2. Magnetic force microscope design
The best MFM images 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-based
head of the microscope is shown in ® gure 27 (Moser et al. 1993). It is constructed on
a clever design involving two concentric piezoelectric scanner tubes. The outer tube
supports both the ® bre mount and the cantilever and is used to scan the latter with
respect to a ® xed sample. The inner tube clamps only the monomode optical ® bre
and allows it to be moved relative to the cantilever surface. The micromachined
silicon cantilever which actually performs the imaging itself sits on a Cu± Be spring
which 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 which
brings it into contact with the cantilever tip. Figure 28 shows the full optical
detection system employed. A laser diode source is coupled into a monomode
optical ® bre through a Faraday isolator. The latter prevents re¯ ected light from
coupling back into the laser which gives rise to mode hopping; a major source of
intensity noise in non-isolated diodes. The light passes through a bidirectional
coupler, leaves the ® bre at a cleaved end and is incident on the highly re¯ ective rear
side of the silicon cantilever. The inset of ® gure 28 shows how the narrow air space
between the cleaved ® bre end and the cantilever forms an interferometer whose
performance depends on the re¯ ection amplitudes at the two interfaces. The signal
photodiode measures the re¯ ected interference signal and is compared with the
reference photodiode to correct for shifts in laser power.
The system of two concentric scanner tubes allows three distinct dc operation
modes. The variable-interferometer mode involves contact scanning with the feedback system disengaged. The resulting variable ® bre tip± cantilever separation results
in a varying interference signal which roughly represents a force map of the surface.
Interpretation of such images is non-trivial since the interferometer response depends
roughly sinusoidally (i.e. nonlinearly and non-monotonically) on cantilever displacements. In variable-de¯ ection mode the feedback system is connected to the ® bre
piezo in order to maintain a constant interferometer air gap. Now the feedback
signal generates a force map of the surface. Finally, in the constant-force mode the
L ocal magnetic probes of superconductors
483
Figure 27. Diagram of the head of a low-temperature magnetic force microscope.
feedback loop is connected to the scanning piezo in order to maintain a constant
cantilever de¯ ection, allowing a constant force map of the topography to be
generated. If ac operation is required, the scanner piezo can be used to oscillate
the cantilever above the sample surface.
4.3. Results of magnetic force microscope imaging of vortices
Figure 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 a
22 m m 22 m m region of a 300nm YBCO thin ® lm deposited by laser ablation on a
SrTiO3 (001) substrate. A micromachined silicon cantilever with an integrated atomic
force microscope tip was used which had an iron ® lm 25nm thick deposited on it.
To obtain this image the tip was retracted to about 2 mm above the surface and
the sample cooled through Tc to 77K in a ® eld of about 0.1 mT. The tip was then
brought to within about 20nm of the surface and a force map generated. Inspection
484
S. J. Bending
Figure 28. Schematic diagram of a ® bre optic interferometer displacement sensor. The lower
inset shows a sketch illustrating how the cleaved end of the ® bre and the upper
surface of the cantilever form an interferometer.
of ® gure 29 reveals that this region of the sample contains 25 vortices, each
producing a repulsive force of about 0.8pN. Note that this is nearly three orders
of magnitude smaller than our earlier theoretical prediction, a discrepancy that can
almost certainly be traced back to an unrealistic model of the magnetic tip used
there. 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 superconducting ¯ ux quantum. Moser et al. (1995) have examined the di€ erent contrast of
attractive and repulsive vortices in detail. If a full-width-at-half-maximum criterion
L ocal magnetic probes of superconductors
485
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 of
vortices in a YBa2Cu3O7 thin ® lm with a low temperature magnetic force
microscope’, p. 2695. Copyright 1994, with permission from Elsevier Science.]
is used to estimate the vortex diameter, they observed that attractive vortices always
appear broader than repulsive vortices. They attributed this fact to the in¯ uence of
the scanning tip itself and the presence of a large approximately constant Meissner
repulsion superimposed on the signal.
Figure 30 shows how the magnetic force microscope tip can be used to modify
the ¯ ux structure locally in the superconducting ® lm described above. In this
15 m m 15 m m image the sample was cooled through Tc in the stray ® eld of the
tip. Once the system has stabilized at 77 K the MFM images reveal that a small
bundle of eight to 12 vortices has nucleated underneath the tip. Manipulation of
vortices in this way allows the possibility to perform unique experiments, for
example studying the creep of a single isolated vortex bundle, but at the same time
highlights the potential invasiveness of the technique.
Overlooking any noise associated with the de¯ ection detection system, thermal
excitation of the cantilever imposes bounds on the minimum detectable force. Since
the 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 at
frequencies much less than the resonant frequency x 0 is
d z2
1/ 2
=
4kB T
Qkx
f
0
1/ 2
,
(4.12)
where f is the measurement bandwidth and the other symbols have their usual
de® nitions. Assuming that Q 100, k = 0.1 Nm­ 1 , x 0 = 85 kHz and a working
bandwidth of 1 kHz we ® nd a rms vibration amplitude of about 4 10­ 12 m at room
486
S. J. Bending
Figure 30. Magnetic-force-microscope tip-induced nucleation of a bundle of eight to 12
vortices in a YBCO thin ® lm (the image size is 15 m m 15 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 of vortices in a
YBa2 Cu3O7 thin ® lm with a low temperature magnetic force microscope’, p. 2695.
Copyright 1994, with permission from Elsevier Science.]
temperature. Multiplying this by the cantilever force constant we obtain an
approximate uncertainty in the measured force of about 0.4pN. Note that this is
as much as half of the peak force at a vortex measured in the experiments of Hug et
al. (1994), explaining the relatively poor SNR in these images. No vortex-resolved
MFM images measured in ac mode have been published to date. No doubt these will
appear in due course since the ac mode is capable of measuring force gradients with
very high SNRs in very high-Q systems.
In the interferometer based MFM of Moser et al. (1993) the additional noise of
the de¯ ection detection system is smaller than the thermal cantilever excitation. The
dominant source of this extrinsic noise is the diode laser which has both intensity and
phase ¯ uctuations associated with it. The former largely arise owing to random
spontaneous emission events as well as changes in the way that the intensity is
distributed between the modes of the laser cavity. The latter also result from
spontaneous emission as well as ¯ uctuations in the carrier population and are
particularly important when phase-sensitive detection is being used as in the
interferometer described here.
5. The Bitter decoration technique
The ® rst technique ever used to image magnetic ¯ ux distributions in superconductors was the Bitter decoration method which was developed independently by
L ocal magnetic probes of superconductors
487
Trauble and Essmann (1966) and Sarma and Moon (1967). The technique is
routinely used to image magnetic domains in ferromagnetic structures when a liquid
suspension of tiny magnetic particles is placed on top of the sample. The low
temperatures encountered in superconductivity dictate a much more sophisticated
approach to decoration.
5.1. Principles of Bitter decoration
Figure 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 liquidhelium bath at 4.2 K. A ferromagnetic ® lament (typically iron, nickel or cobalt) is
mounted about 2.5± 3cm away from the sample with a small shield in between to
prevent radiative heating of the sample. In the presence of a low pressure of helium
gas a su ciently large current is passed through the ® lament so that it vaporizes. The
exact pressure of background gas is usually in the range 0.06± 0.26 mbar, depending
on the chamber temperature, and critically controls the size and concentration of
particles formed. These should be less than 5± 10 nm in diameter with low kinetic
energies and yet must not adhere to one another on the way to the surface. Upon
approaching the inhomogeneous ® eld distribution near the superconductor surface a
ferromagnetic particle with magnetic moment ¹ experiences a force equal to
Figure 31. Schematic diagram of a typical Bitter decoration system.
488
S. J. Bending
·
F = ¹0Ñ (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 j¹jÑ jHj
(5.2)
and, ignoring any other forces, we see that the particle will follow the maximum ® eld
gradient radially in towards the centre of a vortex. In this way the particles are drawn
to 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 examined
under a scanning electron microscope without disturbing the pattern formed. If the
vortex image is to be correlated with the sample microstructure, TEM sections must
be prepared. With conventional superconductors such as niobium it su ces to thin a
sample before decoration (Herring 1976). This approach does not work well with
HTSs, however, because of their tendency to overheat near thinned edges during
decoration and/or damage introduced during ion milling. To overcome these
problems, such samples are ion mill thinned after decoration with an additional
protective 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 ® eld
penetration depth ¸ in the sample. At higher ® elds the vortex tails begin to overlap
and ® 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 applied
magnetic ® eld. Owing to an increase in the strength of pinning forces as the
temperature is lowered, the vortex pattern formed during cooling will lock in at
some temperature T in between Tc and the chamber base temperature (usually
4.2 K). Therefore the pattern observed will not be the true equilibrium con® guration
for the decoration temperature; this is of course true for nearly all measurements
made 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 . The
quality of the decoration is strongly dependent on the sample in question. The best
images are to be expected in materials with the strongest ® eld gradient parallel to the
sample surface, that is in samples where vortices have the smallest diameters and
largest peak ® elds. Broadly speaking this requires materials with short penetration
depths ¸ and narrow cores (small coherence length x 0 ). This last criterion is roughly
equivalent to selecting those materials with the largest lower critical ® eld Hc1 . An
excellent review of Bitter decoration in type II superconductors has been given by
Grigorieva (1994).
5.2. Examples of the use of Bitter patterning in superconductors
The concept of pinning is vital to nearly all practical applications of superconductivity. The decoration technique has proved invaluable since it easily enables
one to correlate directly the sample defect structure with local displacements of
vortices from their equilibrium position. In this way, speci® c pinning centres can be
identi® ed and, on the basis of the size of the displacements, the strength of pinning
forces can be estimated (Trauble and Essmann 1968). A particularly good example
of this work is the observation of pinning at twin boundaries in YBCOsingle crystals
489
L ocal magnetic probes of superconductors
(a)
(b)
Figure 32. (a) Twin layers (dark lines) in a partly detwinned YBCO single crystal as viewed
in the optical microscope with polarized light. (b) Bitter decoration image of the same
region of the sample. [Reproduced from Grigorieva (1994) by permission of IOP
Publishing Limited.]
which are known to have a small orthorhombic distortion of the lattice (a 6= b) at
low temperatures (Vinnikov et al. 1990a, b). As-grown YBCO crystals generally
contain two orthogonal systems of {110} twins which form to relieve internal stresses
introduced as samples are cooled through the tetragonal± orthorhombic phase
transition. Figure 32 shows an optical micrograph of a small region of twinned
YBCO taken with polarized light to indicate the positions of the twin boundaries
(dark lines). The adjacent decoration image clearly illustrates how vortices are
attracted to these boundaries, in fact the vortex density there is roughly twice that
in the surrounding monodomain regions. Patterns of this type allow one to estimate
the pinning potential Up at the twin boundary since the mean vortex spacing on and
o€ the boundary re¯ ects the di€ erence in energy in the two locations. After energy
minimization the following expression is found for Up:
Up =
(8p
)
2
0
1/ 2
¹0¸2
ab
¸
1/ 2
exp
­ a¸b ­ 3 a¸v
2
1/ 2
exp
­ a¸v ,
( 5.3)
where ab is the mean vortex spacing at the boundary and av is the spacing in the
neighbouring domain. Using this equation the pinning potential per unit length of
twin boundary in YBCOwas estimated to be Up = 3.4 10­ 13 J m­ 1 . Vinnikov et al.
argued that their system is close to equilibrium for the temperature at which the
sample was decorated (4.2 K) and hence this is the temperature that corresponds to
their estimate.
Another area where Bitter decoration has proved to be a powerful technique is in
the investigation of the in¯ uence of crystalline anisotropy on the vortex lattice. It has
long been known that anisotropy in conventional superconductors can lead to
distortions of the hexagonal ¯ ux line lattice, and even the formation of a square
lattice for certain orientations of the magnetic ® eld (Huebener 1979). The extreme
anisotropy characteristic of many of the HTSs has, however, given rise to a range of
qualitatively new phenomena. As discussed earlier, superconductivity in the high-Tc
materials is linked to Cu± O planes oriented perpendicular to the c axis of the unit
490
S. J. Bending
Figure 33. Bitter decoration image of a 150nm YBCO ® lm with a 0.45mT ® eld applied
parallel to the c axis. The inset shows a digital Fourier transform of the vortex
pattern. [Reproduced from Grigorieva et al. (1994) by permission of IOP Publishing
Limited.]
cell. If adjacent planes are relatively strongly coupled (e.g. YBCO), the vortex
structure can be described by the anisotropic Ginsburg± Landau theory in terms of
an e€ ective-mass tensor mi, j = Mi, j / Mav (where Mi, j are the components of the mass
tensor and Mav is its angular average). Within this description the penetration length
also becomes a tensor quantity. For the special orientations considered here a
1/ 2
su cient notation is ¸a ,b = mb ¸ where a represents the direction of the applied
magnetic ® eld and b the direction of ® eld decay (mb is the diagonal element of the
e€ ective-mass tensor). Since the orthorhombic distortion in YBCO is quite small,
this gives rise to a relatively weak anisotropy of ¸a ,b in the a± b plane which is di cult
to identify in the decoration patterns of individual vortices. However, the anisotropy
of the penetration depth gives rise to anisotropic vortex± vortex interactions and
causes distortion of the ¯ ux line lattice. Such distortions can be measured from
scanning electron microscopy images of decorated samples with relative ease and
allow ¸a ,b to be estimated with precision, for example ¸c,b / ¸c,a = 1.15 0.02 in
YBCO single crystals (Dolan et al. 1989), implying an e€ ective mass ratio of the a to
the b directions of 1. 32 0.4.
With the ® eld applied at appreciable tilt angles to the c axis, `out-of-plane’ e€ ects
of anisotropy are much more dramatic. Figure 33 shows a decoration image for a
150nm YBCOthin ® lm with a 0.45 mT ® eld applied parallel to the c axis (Grigorieva
et al. 1994). A homogeneous but disordered distribution of vortices is observed in
this micrograph as one would expect for these strongly pinned ® lms. In contrast,
® gure 34 shows a decoration image when a ® eld of 0.20 mT is applied at an angle of
30ë to the c axis of a similar YBCO ® lm. The anisotropic penetration depth is now
L ocal magnetic probes of superconductors
491
Figure 34. Bitter decoration image of a YBCO thin ® lm in a ® eld of 0.2mT applied at an
angle of 30ë to the c axis. The inset shows a digital Fourier transform of the vortex
pattern. [Reproduced from Grigorieva et al. (1994) by permission of IOP Publishing
Limited.]
directly visible in the pronounced elliptical shape of the vortices, even at this
relatively shallow tilt angle. The vortex spacings are also highly anisotropic as
con® rmed by the Fourier transforms of the digitized vortex patterns which are added
as insets in ® gures 33 and 34.
The tendency for vortex supercurrents to be con® ned to the Cu± O planes has
rather remarkable consequences when the ® eld is applied at an oblique angle with
respect to the c axis. It has been predicted theoretically (Grishin et al. 1990) that one
of the magnetic ® eld components within the vortex reverses sign, leading to an
attractive well in the vortex± vortex interaction within the plane containing the
magnetic ® eld vector and the crystal c axis. As a consequence the vortex lattice will
be shortened along the direction of attraction. The resulting vortex chain state is
illustrated in ® gure 35 (Gammel et al. 1992) for an untwinned YBCO crystal
decorated on the a± b face in a ® eld of 2.48mT applied at an angle of 70ë away
from the c axis. These workers were able to show that the vortex spacing within the
chains was approximately constant at low ® elds while the structure merged smoothly
into an ordered isotropic vortex lattice at high ® elds ( ¹0H > 2 mT).
The properties of YBCO can all be quite well understood in terms of an
anisotropic e€ ective mass tensor, but the same is not true of the extremely
anisotropic HTSs such as BSCCO. In this case the vortices are thought to be better
described by stacks of 2D `pancake’ vortices interacting exclusively via the magnetic
® eld and the weak Josephson coupling in the space between Cu± O planes. As a
consequence the in¯ uence of anisotropy on vortex structures is more dramatic as
illustrated by the new vortex states observed in tilted ® elds in BSCCO. Figure 36
492
S. J. Bending
Figure 35. Bitter decoration image of vortex chains in an untwinned YBCO single crystal in
a ® eld of 2.48mT applied at 70ë away from the crystal c axis. [Reprinted from
Gammel et al. (1992). Copyright 1992 by the American Physical Society.]
(Bolle et al. 1991) shows an image of a BSCCO crystal which has been decorated
in a ® eld of 3.5 mT applied at an angle of 70ë away from the c axis. Vortex chains
are clearly visible in this image but, surprisingly, are embedded in regions of
ordered isotropic ¯ ux line lattice. Such a complex vortex structure would have been
very di cult to predict in advance and its origin has still not yet been entirely
explained.
6. Scanning Hall probe microscopy
The use of Hall e€ ect sensors to image ¯ ux pro® les at the surface of superconductors dates back well over 30 years. Typically thin evaporated ® lms of
the semimetal bismuth (Broom and Rhoderick 1962, Co€ ey 1967, Goren and
Tinkham 1971, Brawner and Ong 1993), InAs (Weber and Riegler 1973) or a GaAs
epitaxial ® lm (Tamegai et al. 1992) have been employed in this role and, although
most of these studies present results on a much coarser scale, spatial resolution
as high as 4 m m with magnetic ® eld sensitivity of 0.01 mT was already achieved
by Goren and Tinkham (1971). Using micrometer-based scanning stages these
experiments gave key insights into the critical state of a range of superconducting materials. Further development of these instruments was, until
recently, largely hampered by the inevitable trade-o€ between spatial and ® eld
resolution (discussed later) and the lack of convenient automated precision scanning
systems.
L ocal magnetic probes of superconductors
493
Figure 36. Bitter decoration image of the vortex pattern in a BSCCO single crystal in a ® eld
of 3.5mT applied at 70ë away from the crystal c axis. [Reprinted from Bolle et al.
(1991). Copyright 1991 by the American Physical Society.]
The invention of the modulation doped semiconductor heterostucture (Dingle et
al. 1978) has subsequently revolutionized the ® eld. These epitaxial structures contain
2D layers of electrons with carrier mobilities which can exceed 100m2 V­ 1 s­ 1 at low
temperatures, far greater than that found in any naturally occurring compound. As a
consequence it is now possible to combine very high ® eld sensitivity with submicron
spatial resolution. In addition, developments in automation and mechanical scanning using either stepping-motor-driven stages or piezoelectric tubes have made
accurate Hall probe positioning quite routine.
6.1. Semiconductor heterostructure Hall probes
Currently all state-of-the-art systems employ GaAs/Al0.3 Ga0.7 As heterostructure
Hall probes which were ® rst used for investigating superconductors independently
by Geim (1989) and Bending et al. (1990a, b). Figure 37 shows a sketch of the
conduction-band edge through a typical heterostructure which is normally grown by
molecular-beam epitaxy on a semi-insulating GaAs substrate. Since the lattice
constants of GaAs and Al0.3Ga0.7As are almost identical, the entire structure can
be viewed as a coherent single crystal with modulated conduction- and valence-band
edges owing to the much larger bandgap in Al0.3Ga0.7As. Note that, while the silicon
dopants are actually placed within one of the Al0.3Ga0.7As layers, the electrons
(2DEG) become trapped in a V-shaped potential well at the interface between slabs
of Al0.3Ga0.7As and GaAs, an undoped `spacer’ layer away. This spatial separation
494
S. J. Bending
Figure 37. Layer structure of a GaAs/Al0.3 Ga0.7As heterostructure (top). Sketch of the
corresponding conduction-band edge perpendicular to the layers showing the location
of the two-dimensional electron gas (2DEG) (bottom).
between donors and electrons is the secret of modulation doping and virtually
eliminates ionized impurity scattering which strongly limits the low temperature
mobility of bulk doped semiconductors. Thus, the conductivity of the system is very
high at low temperatures, although the gains at room temperature, when the
mobility is largely dominated by phonon scattering, are only modest.
The fact that the electrons are con® ned to a very narrow layer about 10 nm wide
has the additional bene® t that the Hall probe samples the ® eld distribution at a well
de® ned height above the superconductor. Moreover, since the electrons typically lie
50± 100nm below the wafer surface they can, in principle, be brought into very close
proximity with the sample to achieve high spatial resolution. Finally the 2D electron
concentrations in such structures tend to be low, giving rise to a large Hall coe cient
and high ® eld sensitivity.
L ocal magnetic probes of superconductors
495
The ® rst applications of GaAs heterostructure Hall sensors were as static linear
arrays which could be brought into ® xed intimate contact with a superconductor
surface. In this way induction pro® les along a line could be generated with modest
spatial resolution as a function of applied ® eld and temperature. Such experiments
include an investigation of vortex bundle dynamics in lead ® lms (Stoddart et al.
1993), a study of geometrical barriers in high-Tc single crystals (Zeldov et al. 1994)
and an identi® cation of the melting line in BSCCO single crystals (Zeldov et al.
1995). Local induction measurements using linear Hall probe arrays have now
become very common and this list is intended to be illustrative rather than
exhaustive.
6.2. Hall e€ ect and resolution in a heterostructure Hall probe
The Hall e€ ect in bulk semiconductors will certainly be familiar to the reader.
The derivation of the Hall e€ ect in a 2D probe traditionally follows by analogy
where the system is now described by the usual carrier mobility ¹ and a 2D carrier
concentration n2D. Figure 38 shows a sketch of an idealized Hall probe based around
the intersection of a pair of current and voltage leads in a uniform applied magnetic
® eld perpendicular to the plane of the Hall probe. Shaded regions represent Ohmic
contacts, which are assumed to de® ne equipotentials.
Provided that the current leads are very much longer than the width of the
voltage leads, the Lorentz force can be equated to the force on the electron due to the
Hall ® eld Ey once a dynamic equilibrium has been established:
Figure 38. Sketch of an idealized 2D Hall probe.
496
S. J. Bending
(6.1)
­ evdB = ­ eEy,
where vd is the carrier drift velocity. Using the known relationship for the 2D current
density ( J2D = ­ n2Devd) the Hall coe cient RH becomes
­1
Ey
= RH = n e ,
J2DB
2D
(6.2)
and we see that it does not depend on any of the dimensions of the probe. In reality,
equation (6.1) is not strictly valid for a 2D sample since it is impossible to distribute
the line charges at the edges of the probe to get a uniform Hall electric ® eld. In
practice, however, equation (6.2) provides a reasonable ® rst-order approximation to
the behaviour of a sensor. Typical Hall coe cients for a GaAs/Al0.3 Ga0. 7As
heterostructure Hall probe are 3 mT­ 1 and vary by as little as 50% in the
temperature range 5± 300K, changing by only a few per cent below 100K.
For T < 100K the main noise component of such probes up to a critical dc bias
current Imax is the Johnson noise of the Hall voltage contacts ( V J = (4kB TRv f )1/ 2)
where Rv is the two-terminal voltage probe resistance and f is the measurement
bandwidth. Consequently for IHall < Imax the SNR of a sensor with voltage leads of
width w and total length l is given by
SNR =
IHall RHB
(4kB TRv f )1/ 2
¹
n2D
1/ 2
w
4kB Tel
1/ 2
f
IHall B.
(6.3)
Consequently, if all other factors remain constant, the ratio of carrier mobility to
carrier concentration provides a good ® gure of merit for the sensor. In practice,
however, the maximum current bias imposes an equally severe constraint. Once Imax
is exceeded, the low-frequency output becomes dominated by random telegraph-like
1/ f noise, presumably due to the trapping and emission of `hot’ electrons at deep
donor impurities in the Al0.3 Ga0.7As layer. Hall probes based on the intersection of
two wires 1 m m wide can sustain currents of 4 m A at 300K and up to 60 m A below
77K without increasing the noise signi® cantly and typical values of Rv are 60 k ,
3 k and 1.5k at 300K, 77 K and 4.2 K respectively (Oral et al. 1996). The
magnetic ® eld resolution of such Hall probes is measured to be about
4 10­ 6 T Hz­ 1/ 2 at 300K and about 3 10­ 8 THz­ 1/ 2 below 77 K (when it is
limited in this case by the noise of the room temperature pre-ampli® er). The active
electronic area of these sensors is reduced to about 0.8 m m 0.8 m m by edge depletion
because of pinning at surface states, yielding a ¯ ux resolution of about
1 10­ 5 U 0 Hz­ 1/ 2 at 77 K which is comparable with the best scanning SQUID
systems operated at 4.2 K (Tsuei et al. 1994). Even below Imax there is a small 1/ f
component to the output at 77 K with a corner frequency of about 10Hz and the
frequency response of the sensor is ¯ at up to about 10 kHz and then drops o€ at
about 10 dBdecade­ 1 .
The Hall voltage is inevitably superimposed on an `o€ set’ voltage which
notionally arises due to slight misalignments of the opposing contacts. In reality
this `o€ set’ occurs because of microscopic inhomogeneities in the heterostructure
and, since it represents an Ohmic longitudinal voltage drop, it has the same strong
temperature dependence as the carrier mobility. In practice this o€ set must be
electronically subtracted every time that a new temperature is established, with the
consequence that the SHPM is not good at measuring absolute values of the local
magnetic induction. It does, however, yield a very precise linear measure of relative
497
L ocal magnetic probes of superconductors
changes in induction and the ability to build high quality images is in no way
compromised.
As the spatial resolution is increased but the total lead length is kept constant, the
SNR of a probe falls as the square root of the Hall contact wire width (equation
(6.3)). In practice this reduction can be minimized by ¯ uting the leads, but a
reduction in Imax and an observed increase in 1/ f noise in submicron devices
signi® cantly degrade the sensitivity. In addition the o€ set voltage is found to increase
rapidly in submicron sensors, considerably complicating electronic compensation.
Consequently the present state of the art in spatial resolution is about 0.2 m m (Oral
et al. 1998) which is still much larger than the limits imposed by submicron
lithography.
Since the spatial resolution is comparable with the length scales of interest in type
II superconductors, there is appreciable instrumental broadening of images and it is
important to account for this if quantitative estimates of, for example, the
penetration depth are required. This problem has been examined in two limits. In
very-high-carrier-mobility sensors at low temperatures (below 4.2 K) one is in the
ballistic limit if the electron mean free path exceeds typical sensor dimensions.
Peeters et al. (1997) have shown that the active area of the Hall probe is well
described by the square intersection of current and voltage leads in this limit. Most
experiments, however, are performed at much higher temperatures when the mean
free path is considerably smaller than the sensor dimensions. In this di€ usive limit
the Hall e€ ect is described by the simultaneous solution of
E+q
H
and
(x, y)J
( 6.4)
z = s ­ 1J
·J = 0
( 6.5)
¶ Ex ¶ Ey
+
= f 1(x, y)Ex + f 2(x, y)Ey,
¶ x
¶ y
( 6.6)
Ñ
where q H(x, y) = B(x, y)/ n2De is the spatially dependent Hall resistivity and s the 2D
conductivity. Eliminating current density J we ® nd that the system is described by a
2D Poisson equation containing `charge density’ terms which are proportional to
magnetic ® eld gradients within the sensor:
where
1
f 1(2) ( x, y) = ­ 2
+q
s
2
H
2q
H
¶ q H
+ (­ )s (s ­ 2 ­ q
¶ x( y)
2
H
) ¶ ¶ y(Hx)
q
.
( 6.7)
Equation (6.6) has been solved numerically for a realistic Hall probe geometry
(Bending and Oral 1997) using a simple analytic exponential approximation for the
® eld near a vortex core: B(x, y) = ( 0 / 2p ¸2 ) exp ­ [(x ­ x0 )2 + ( y ­ y0 )2]1/ 2 /¸ ,
where x0 , y0 specify the origin of the vortex and ¸ is a notional penetration depth
which de® nes the scale of the ® eld inhomogeneity. It was found that a local charge
dipole is created in the Hall probe at the position of the vortex with a complex
associated current distribution. If ¸ is much smaller than typical Hall probe
dimensions, solutions of equation (6.6) can be used to de® ne a response function
describing the Hall response for a local bundle of ¯ ux at an arbitrary position in
the sensor. Figure 39 shows a 3D representation of this as a function of x and y
for a Hall probe 1 m m wide. Note that the response extends well outside the
square intersection of voltage and current leads. In contrast with the ballistic
f
g
498
S. J. Bending
Figure 39. Response function for a 2D Hall probe with leads 1 m m wide as a function of the
position of a highly localized vortex within it.
limit it is found that the active Hall probe area is about twice that of the geometric
intersection of current and voltage wires. This is easy to understand in terms of the
® nite currents that clearly must ¯ ow in the voltage leads near the point where they
join the current contacts. More recently an approximate analytic response function
which agrees well with this numerical model has been calculated for this Hall probe
geometry using conformal mappings in the complex plane (Thiaville et al. 1997). If a
precise model of the expected ® eld pro® le at a vortex is available, it has been shown
(Bending and Oral 1997) that this can be numerically convolved with the Hall probe
response function to obtain an excellent ® t to measured SHPM pro® les, allowing
quantitative estimates of the penetration depth to be obtained.
6.3. Scanning Hall probe microscope design
Figure 40 shows a schematic layout of a state-of-the-art scanning Hall probe
microscope (Oral et al. 1996). The Hall probe is mounted on the piezoelectric
scanner tube of a commercial low-temperature scanning tunnelling microscope with
a stick± slip coarse approach mechanism and is tilted 1± 2ë with respect to the sample
plane. Figure 41 is a scanning electron micrograph of a typical Hall probe of 0.8 m m
resolution. The active Hall sensor is patterned about 13 m m away from the corner of
a deep mesa etch which is coated with gold to act as an integrated tunnel tip, and the
L ocal magnetic probes of superconductors
499
Figure 40. Sketch of a typical scanning Hall probe microscope: PC, personal computer.
relative tilt angle between sensor and sample ensures that this is the closest point to
the sample surface. The sample is ® rst approached until tunnelling is established, and
then the Hall probe is scanned across the surface to measure the magnetic ® eld and
surface topography simultaneously as sketched in ® gure 42(a) (STM-tracking
scanning Hall probe microscope). Alternatively the sample can be retracted a
fraction of a micron and the Hall probe scanned much more rapidly with a slightly
lower spatial resolution (¯ ying scanning Hall probe microscope (® gure 42 (b))).
Before commencing `¯ ying’ SHPM it is usual to measure the scanner plane and
sample tilt angle with the scanning tunnelling microscope tip so that this can be
electronically compensated during scanning. This is the preferred mode of SHPM
operation since it is fast and avoids the risk of a `head crash’ often encountered
during the much slower scans with STM tracking.
The piezoelectric coe cient of the scanner tube falls as the temperature
is lowered limiting the scan area of a typical system to about 25 m m 25 m m at
77K (Oral et al. 1996). The microscope is placed in a cryostat containing a
1.5± 300K variable-temperature insert and a 7 T superconducting magnet which
is all mounted on a double-stage vibration isolation system to eliminate external
disturbances.
500
S. J. Bending
Figure 41. Electron micrograph of a scanner Hall probe with 0.8 m m spatial resolution.
6.4. Examples of scanning Hall probe microscopy in superconductors
6.4.1. High spatial resolution
The ® rst scanning Hall probe microscope with a submicron Hall probe was
developed by Chang et al. (1992a, b) at AT&T. While this system had excellent
spatial resolution (about 0.35 m m) the ® eld resolution of 3.6 10­ 5 T Hz­ 1/ 2 was
rather poor and demanded very slow image acquisition. Simple improvements in
Hall probe design (Oral et al. 1996) rapidly led to improvements in sensitivity of
several orders of magnitude, allowing high-resolution images to be captured at about
1 line s­ 1 with a measurement bandwidth of 1 kHz. Figure 43 (a) shows a collage of
such images which have been obtained on the same region of a high-quality YBCO
thin ® lm (d = 0.35 m m; Jc (77 K) = 1.4 106 A cm­ 2 and Tc = 90.8 K (dc magnetization)) after ® eld cooling at 0.1 mT to various temperatures (Oral et al. 1997b).
Note that the scanning range becomes smaller at low temperatures owing to a
reduction in the piezoelectric coe cient of the scanner tube. As one expects, the
vortex diameter decreases and contrast improves at low temperatures owing to a
reduction in the superconducting penetration depth. Careful examination of line
scans through these pinned vortices reveals a surprisingly large variation in vortex
diameter and peak ® eld as one crosses from left to right.
In ® gure 43 (a) the active Hall probe area is not much smaller than the measured
vortex diameters, and the response function described earlier has been used to
account for this. For a thin superconducting ® lm it can be shown that the ¯ ux
distribution a height z above the sample surface is given by (Chang et al. 1992b)
Bz (r, ¸, z) =
U 0
2p ¸2
1J0(g r) exp [­ g ( jzj ­ d/ 2)]dg
,
0
kg (coth (kg d / 2) + kg /g
)
(6.8)
L ocal magnetic probes of superconductors
501
Figure 42. Schematic diagram illustrating (a) the STM tracking and (b) the ¯ ying modes of
SHPM.
where kg = (g 2 + 1/ ¸2 )1/ 2 , z is measured from the centre of the ® lm of thickness d
and ¸ is the bulk penetration depth. This expression has been convoluted with the
numerical Hall probe response function and ® tted to line scans through the centre of
the numbered vortices to establish a value of ¸ for each vortex. Figure 43 (b) shows
the results of this ® tting procedure for the vortices numbered 1± 5 in the 60 K image
measured in 5K steps from 10 to 80 K (see ® gure caption for symbol de® nition). The
excellent quality of the ® t is illustrated in the inset of the ® gure at 10K. The
calculated Hall probe response function is shown beneath these data and even at the
502
S. J. Bending
Figure 43. (a) SHPM images of vortices in a YBCO thin ® lm after ® eld cooling to various
temperatures at 0.1mT. (b) Penetration depth as a function of temperature measured
for the individual vortices labelled 1± 5 on the displayed image ((r ), 1; (d ), 2; ( j ), 3;
(m ), 4; ( . ), 5). The inset shows a ® t to one of the vortex cross-sections at 10K. The
response function of the Hall probe is plotted beneath this ® t. [Reproduced from Oral
et al. (1997b) with the permission of IOP Publishing Limited.]
lowest measurement temperature it is appreciably narrower than the measured crosssection. Note that for a given vortex there is a modest scatter in the measured
penetration depth as a function of temperature, which is presumably introduced
during the extensive calibration procedures which have to be repeated at each new
measurement temperature. Most signi® cantly, however, at any given temperature,
L ocal magnetic probes of superconductors
503
when such errors are expected to a€ ect all data in the same fashion, we observe a
wide distribution in values of ¸ (e.g. ¸ 0.23± 0.34 m m at T = 35 K). There is a clear
correlation between vortex position and penetration depth which becomes systematically smaller as one moves from the left to the right of the image. Such an e€ ect
would be an obvious artefact of a ® nite tilt angle between the sample surface and
scanner plane. However, any tilt angles have been measured beforehand and
electronically compensated during scanning with great precision, so this is unlikely
to be the origin. It seems more likely that the distribution is the result of local
¯ uctuations in oxygen concentration in the sample which are known to have a strong
in¯ uence on the penetration depth (Fuchs et al. 1996). Such strong inhomogeneity
over length scales as short as tens of microns is rather surprising and yet has been a
recurring theme running through all SHPM studies of HTSs at Bath to date (cf.
vortex stripes in highly ordered BSCCO single crystals at low ® elds (Oral et al.
1997a)).
Recent SHPM work has focused on studies of the vortex `melting’ transition in
BSCCO single crystals. As the name implies, it is now widely accepted that the
ordered hexagonal vortex solid undergoes a phase transition into an uncorrelated
vortex liquid along a well de® ned phase boundary in the H± T plane. One of the
signatures of the transition is a sharp jump in the local magnetic induction as a
`liquid± solid’ boundary passes beneath a microscopic Hall probe (Zeldov et al.
1995). This, combined with the direct measurement of a very sharp peak in the
speci® c heat of YBCO at the analogous phase boundary (Schilling et al. 1996),
indicates strongly that `melting’ occurs via a ® rst order thermodynamic phase
transition. The `melting’ line of an as-grown BSCCO crystal spans the ® eld region
0± 40 mT, and it was necessary to develop a higher-resolution Hall probe in order to
explore an appreciable fraction of this boundary. Electron-beam lithography was
used to pattern a new generation of heterostructure probes which was identical to
that of ® gure 41 except that the Hall sensor was now de® ned at the intersection of
two wires 0.4 m m wide. Accounting for edge depletion, the expected spatial resolution
of these sensors is about 0.2 m m, and this was con® rmed by imaging the 0.25 m m long
bits on an ultrahigh-density hard disc. The sensitivity of the electron-beam patterned
probes was somewhat reduced over the ® rst generation (about 3 10­ 7 T Hz­ 1/ 2 at
77K) but it was now possible to image discrete vortices up to several milliteslas.
Figure 44(a) shows a family of vortex images at various points along a ® eld cut at
85K in a high quality BSCCO single crystal (Oral et al. 1998). A reasonably well
ordered sixfold symmetric structure is evident at 1 mT, 1.5mT, 2.0 mT and with
some di culty at 2.25mT. At all ® elds above this (for example 2.5 mT), discrete
vortices could not be resolved, suggesting a loss of static order. This was consistent
with the parallel observation that the melting line occurs at 2.3 mT at 85K on the
basis of a jump in local induction as a function of applied ® eld. The vortex lattice is
seen to undergo surprisingly large rotations as the ® eld is increased; this e€ ect is
attributed to the incommensurability of the hexagonal vortex solid and the
rectangular surface barriers of the sample which con® ne it. Figure 44(b) plots the
average peak-to-valley vortex `corrugations’ along the unit vectors of each of the
images in ® gure 44 (a) as a function of applied ® eld. Clearly this quantity drops
abruptly and discontinuously to zero at the melting line indicated by the vertical
dotted line, consistent with what one would expect for a ® rst-order transition. For
comparison the corrugation predicted by the Clem (1975) variational model
assuming no ¯ uctuation broadening or melting is also plotted on this ® gure (solid
504
S. J. Bending
Figure 44. (a) Family of 85K SHPM images of vortices in a BSCCO single crystal at
applied ® elds of 1 mT (greyscale spans about 0.042mT), 1.5mT (greyscale spans
about 0.031mT), 2mT (greyscale spans about 0.021mT), 2.25mT (greyscale spans
about 0.013mT) and 2.5mT (greyscale spans about 0.003mT) (Hm = 2.3 mT). (b)
Plot of mean peak-to-valley vortex `corrugation’ as a function of the applied ® eld
(d ). Also shown for comparison is the prediction of the Clem variational model
(
). [Reprinted from Oral et al. (1998). Copyright 1998 by the American Physical
Society.]
line) and shows the positive curvature characteristic of the increasing overlap of
exponential vortex tails. The strong negative curvature of the measured `corrugation’
data as a function of applied ® eld highlights the growing importance of 2D
¯ uctuations (wavier lines) as the melting line is approached from below.
L ocal magnetic probes of superconductors
505
Figure 45. SHPM image of ¯ ux vortices in a niobium strip 100 m m wide after cooling to
6.3K in a ® eld of 0.034mT (Field 1997).
For many applications it is by no means essential to image vortices with high
resolution, but su cient merely to establish their positions within much larger area
scans. Siegel et al. (1995) have designed a scanner head based on laminar piezo
benders capable of imaging with high spatial resolution over ® elds as large as
275 m m 275 m m at 4.2 K. Figure 45 shows large-area scanning Hall probe images
with 1 m m resolution of a 100 m m wide strip of niobium ® lm which has been ® eld
cooled to 6.3 K in 0.034mT. The dark points in this image are discrete vortices and
do not form an ordered hexagonal lattice owing to the large vortex separations and
the high density of pinning sites in the ® lm. Nevertheless the distribution of vortices
is quite uniform throughout the image. In ® gure 46 a snapshot of ¯ ux penetration
into the same niobium strip is shown in a regime where single vortex resolution has
been lost after the applied ® eld was increased to 33 mT from zero at 4.5 K. Of
particular interest here is the way in which ¯ ux penetrates in feathery `® ngers’, which
is a phenomenon ascribed to long-range interactions between the ends of the vortices
(Field 1997).
6.4.2. High temporal resolution
The excellent SNR of heterostructure Hall probes, combined with their very
rapid intrinsic response times, allows the possibility to image with high temporal
resolution. In practice the scanning rate is limited by the need to remain below the
resonance frequency of the piezo scanner tube (about 1 kHz). Increasing the
measurement bandwidth to 10 kHz, quasireal-time SHPM with single vortex
resolution has recently been demonstrated (Oral et al. 1996). Figure 47(a) shows
506
S. J. Bending
Figure 46. SHPM image showing ® nger-like penetration of ¯ ux into the niobium strip of
® gure 45 after ramping the ® eld from zero to 33mT at 4.5K (Field 1997).
an example of this for ¯ ux penetration into the same YBCO thin ® lm discussed
earlier with the scanner head about 2 mm from one edge of the 5 mm 5 mm sample.
The ® lm is ® rst zero ® eld cooled to 85 K and the applied ® eld then gradually cycled
to +1.65 mT, decreased to ­ 1.65 mT and ® nally brought back to +1.65 mT while
continuously imaging the 25 m m 25 m m (128 128 pixel) scan area at about
1 frames­ 1 . Figure 47 (b) shows a B± H loop taken with the Hall probe backed o€
about 4.2 m m from the surface to smear out microscopic ¯ ux inhomogeneities. The
images (i)± (vi) in ® gure 47 (a) represent ¯ ux snapshots corresponding to the indicated
positions on this hysteresis loop. Image (i) shows the virgin state of the superconductor with no ¯ ux evident. As the external ® eld increases, the ¯ ux is clearly seen
to enter from the right in the form of localized bundles (labelled in image (iv)).
Bundle 1 in image (ii) ® rst grows to a few microns in diameter and then a large
number of vortices suddenly jump to position 2 (image (iii)). Bundle 2 then grows in
turn and there is a second jump to position 3 sometime later (image (iv)). Two
isolated vortices (white arrows) and an antivortex (black arrow) can also clearly be
resolved in this image. Upon ® eld reversal, bundles of antivortices are seen to enter
along the same route, leading eventually to vortex± antivortex annihilation and the
formation of antivortex bundles (images (v) and (vi)). Careful analysis of the frames
between those shown here reveals that the ¯ ux bundles typically contain three to ten
individual vortices at this temperature, while the paths of the isolated vortices can
also be tracked on an approximately 1s time scale.
It is, in fact, not necessary to achieve such high scanning rates to observe
interesting dynamic phenomena in high-Tc materials. Figure 48 shows three images
L ocal magnetic probes of superconductors
507
Figure 47. (a) Snapshots of ¯ ux penetration into a YBCO thin ® lm at 85K with the Hall
probe positioned 0.67 m m above the sample. (b) A hysteresis curve obtained with the
Hall probe static and about 4.2 m m above the sample. Labels refer to the positions on
the hysteresis curve to which the six images correspond. [Reprinted from Oral et al.
(1996). Copyright 1996 by the American Physical Society.]
from a region near the centre of a BSCCO single crystal after the applied ® eld was
suddenly (t = 0 s) increased from 0 to 0.8 mT at 77 K (Oral et al. 1998). Each scan
lasted 45s and the microscope had been set to map the same area repeatedly. The
three images in ® gures 48 (a)± (c) represent di€ erent delay times spanning 45± 315s
and, although there must be some movement within each frame, substantial changes
can be seen to occur in the vortex structures on the scale of minutes. As a guide to the
eye a triangular mesh has been superimposed on the data to indicate approximately
508
S. J. Bending
Figure 48. Real-time SHPM images of vortices in a BSCCO single crystal (a) about 45s, (b)
about 180s and (c) about 315s after the ® eld was suddenly increased from 0 to
0.8mT at 77K (image sizes, about 7 m m 5.6 m m, greyscale spans about 0.255mT).
[Reprinted from Oral et al. (1996). Copyright 1996 by the American Physical Society.]
the locations of the vortex centres. This makes it quite clear that successive images
are strongly distorted by shear waves, and vortex vacancies are visible in the lower
left corner of ® gure 48 (a) and upper right corner of ® gure 48(c). This highlights the
softness of the ¯ ux line lattice and weakness of the pinning in these single crystals
even at temperatures substantially below the critical temperature ( Tc = 90.5 K).
7. Magneto-optical imaging
7.1. Theoretical principles of magneto-optical imaging
Another way to image the stray magnetic ® elds at the surfaces of superconductors is to bring a magneto-optically active ® lm into intimate contact with
the sample and to examine it under linearly polarized light. The contrast achieved
between regions of di€ erent magnetic ® eld is a consequence of the Faraday rotation
of the polarization of the incident light. A qualitative understanding of the Faraday
e€ ect can be obtained by considering an elastically bound carrier in a medium which
is driven into a circular orbit by the rotating electric ® eld of a circularly polarized
wave passing through it. If a magnetic ® eld is now applied perpendicular to this
orbit, there will be an additional radial Lorentz force on the carrier. Depending on
the handedness of the light, this force can act either inwards or outwards, yielding
two di€ erent values of the polarization and hence index of refraction. As a
consequence there will be a di€ erence in propagation velocity (and a phase shift)
between left- and right-handed polarizations. For small angles this Faraday rotation
d is given by
d = VdH,
(7.1)
where V is the Verdet constant for the medium of thickness d, and H is the applied
® eld. Viewing linearly polarized light as a superposition of two circularly polarized
beams of opposite handedness (using complex notation),
509
L ocal magnetic probes of superconductors
Figure 49. Sketch of the position of the magneto-optically active layer (MOL) and
superconducting sample for (a) high-resolution imaging with europium chalcogenide
® lms and (b) imaging with garnets.
+
Elin(z) = Ecir
(z) + Ecir­ ( z)
E0
= 2 [exp (­ ikz) + exp (ikz)],
( 7.2)
which accumulate equal and opposite phase changes of d after traversing a medium
of thickness d, we ® nd that the phase of the linearly polarized ray has been rotated by
d upon emergence.
E0
Elin(d) =
exp [­ i(kd ­ d )] + exp [i(kd + d )]
2
f
= E0 cos (kd) exp (id )
g
( 7.3)
This rotation can then be visualized by analysing the transmitted light with a
crossed polarizer.
In practice the measurement geometry is as shown in ® gure 49. The MO layer can
be deposited either directly on top of the superconductor or on to a separate
transparent substrate which is ¯ ipped to bring the ® lm into intimate contact with the
sample. Since HTSs are neither transparent nor highly re¯ ecting, a thin mirror layer
of aluminium is usually deposited between the sample and the ® lm to re¯ ect the
linearly polarized light back to an analyser on the same side as the source. This has
the advantage that the light passes through the MO ® lm twice, doubling the rotation
angle. Since MO ® lms have a high absorption coe cient b , the intensity of the
re¯ ected light after passing through a crossed polarizer will be
I = I0 exp (­ 2b d ) sin2 (2VdH)
4I0 V 2 d2 H2 exp (­ 2b d)
( 7.4)
and the maximum signal is given when d 1/ b . In practice this may not be the most
important criterion and optimum contrast can often be achieved when the analyser is
set slightly away from the complete extinction position and the thickness of the ® lm
chosen so that unrotated light re¯ ected from the top of the MO ® lm destructively
interferes with light re¯ ected from the aluminium mirror in ® eld-free regions.
510
S. J. Bending
Figure 50. Diagram of a typical experimental set-up for performing MO imaging on
superconductors.
MO measurement systems can vary considerably and ® gure 50 shows a
particular con® guration due to Moser et al. (1989). The sample sits on the
temperature controlled cold stage of a helium ¯ ow cryostat in an evacuated
chamber at the centre of a normal solenoid whose axis is perpendicular to the
superconductor surface. The sample is illuminated from above through a
polarizing microscope, with which the re¯ ected light is also collected and analysed.
The image can be viewed directly or captured with a CCD camera for further image
processing.
7.2. Examples of magneto-optical imaging of superconductors
The types of experiments that can be performed using MO imaging are largely
dictated by the choice of materials used. These commonly fall into two categories,
europium chalcogenides or yttrium iron garnets (YIGs), and a good review of the
area has been given by Koblischka and Wijngaarden (1995).
L ocal magnetic probes of superconductors
511
7.2.1. Magneto-optical imaging with europium chalcogenides
Until recently most MO imaging has been performed with thin ® lms of europium
chalcogenides owing to their large Verdet constants. Alloys of EuS and EuF2 have
frequently been employed where the paramagnetic ¯ uoride is added to suppress
ferromagnetism in the sulphide below 16K. Because of the di culties associated
with depositing EuS and EuF2 in controlled proportions, more recently singlecomponent EuSe ® lms have been used which are paramagnetic down to 4.6K. At
low temperatures the Verdet constant of EuSe ( V (4. 2 K) 0.1ë mT­ 1 m m­ 1 ) is very
high, although it varies somewhat with ® lm morphology, allowing a magnetic ® eld
resolution of about 1 mT with the spatial resolution limited to about 0.5 m m by the
optical microscope up to saturation ® elds of several teslas. In principle this should
allow the observation of isolated ¯ ux vortices in low applied ® elds, although the
present author is not aware that such images have ever been achieved to date. The
main drawback with the use of europium chalcogenides, particularly for applications
in high-temperature superconductivity, is that the Verdet constant falls rapidly with
increasing temperature, and imaging is only possible below 20 K. For this reason,
other MO materials have been developed recently to extend the temperature range to
the much higher temperatures of current interest.
MO imaging is extremely well suited to correlating ¯ ux pro® les with microstructural information. Figure 51 (Koblishka 1992) shows a set of images from a
twinned YBCO bicrystal which is divided across the middle by a grain boundary. In
this case a 250nm EuSe layer has been electron-beam evaporated directly onto the
sample following an intermediate 100nm aluminium mirror layer. Figure 51 (a)
displays a polarized light micrograph of the sample showing the twin structure and a
clear grain boundary running horizontally slightly above the centre of the sample.
In ® gure 51(b) a MO image is shown in a magnetic ® eld of 273mT, applied after
zero ® eld cooling to 5 K. Note how ¯ ux enters easily along the grain boundary
(white regions) as well as up twin boundaries, forming pronounced `¯ ux ® ngers’.
Figure 51(c) displays a MO image of the remanent state after the applied ® eld has
been removed. The ¯ ux ® ngers at twin boundaries remain, revealing strong pinning
there. In addition, negative ¯ ux has now entered along the grain boundary as
revealed by the dark stripes there.
It is straightforward to capture images of the type shown in ® gure 51 digitally
and to invert them to obtain true ¯ ux pro® les. In samples with a suitably regular
geometry, measured ¯ ux gradients can be directly related to critical current densities
Jc and pinning forces f p. For example for a slab which is in® nite in the y and z
directions with ® nite thickness along x and ® eld applied along z, Friedel et al. (1963)
have shown that
1 ¶ Bz
Jc = ¹
( 7.5)
0 ¶ x
and
f p (x) = Jc Bz (x)
( 7.6)
For a more general discussion of this inversion problem see Roth et al. (1989)
and Paishitski et al. (1997).
7.2.2. Magneto-optical imaging with yttrium iron garnet ® lms
Owing to the demanding growth conditions for high-quality garnet ® lms these
materials are usually deposited on separate substrates and ¯ ipped into intimate
512
S. J. Bending
Figure 51. Flux penetration into a twinned YBCO bicrystal. (a) optical micrograph with
polarized light clearly revealing the existence of a horizontal grain boundary near the
middle of the sample. (b) MO image at 5 K and an applied ® eld of 273mT. (c)
remanent state after the ® eld has been reduced to zero at 5 K. [Reproduced from
Koblischka and Wijngaarden (1995) with the permission of IOP Publishing Limited.]
contact with the sample under study. Films can be chosen with either perpendicular
or in-plane anisotropy and the MO technique is now sensitive to the magnetization
component along the light propagation direction. The former have characteristic
labyrinth domains of up and down magnetization perpendicular to the sample plane
and were originally developed for bubble memory applications. In this case changes
in the domain structure (i.e. growth of one domain orientation and shrinkage of the
other) act as an indicator of ¯ ux pro® les and ® lms are often doped with bismuth to
increase contrast. Spatial resolution is, therefore, limited by the characteristic
domain widths (about 5 m m). Films with in-plane anisotropy, on the other hand,
allow a direct observation of ¯ ux patterns as the magnetization vector is rotated out
of the plane of the ® lm under the in¯ uence of the magnetic ® eld distribution. In this
L ocal magnetic probes of superconductors
513
Figure 52. MO image of a Tl± Ba± Ca± Cu± O single crystal in the Meissner state at 76K with
an applied ® eld of 7.2mT. The sample is about 600 m m wide at its largest point.
[Reprinted from Indenbom et al. (1990) with permission from Elsevier Science.]
case the spatial resolution is again limited to about 4 m m by the thickness of the YIG
layer plus the gap between sample and ® lm.
The major advantage of YIG ® lms is that their MO response is good all the way
up to their Curie temperatures (about 800K), making them much better suited to
studying HTSs. In addition the sensitivity of YIG materials (about 10 m T) is far
superior to that of the europium chalcogenides but at the price of considerably lower
saturation ® elds (50± 200mT).
Figure 52 shows an example of the visualization of magnetic ® eld screening in a
Tl± Ba± Ca± Cu± O single crystal using a bismuth-doped YIG ® lm with perpendicular
anisotropy (Indenbom et al. 1990). The measurement was performed in a ® eld of
7.2 mT applied after zero-® eld cooling to 76 K and the mirror layer was deliberately
omitted so that the outline of the crystal is apparent (dark shadow). Note how the
labyrinth domain pattern is much denser in the screened region above the sample
and rather sparse elsewhere where one domain orientation (black contrast) has
virtually been eliminated.
Finally the use of YIG ® lms with in-plane anisotropy is demonstrated in ® gure 53
(Schuster et al. 1994). These images summarize the results of an interesting experiment whereby the central portion of a BSCCO single crystal (bright region in ® gure
53(a)) has been masked by an absorber while the edge regions were irradiated with
high-energy heavy ions. Such radiation is known to produce columnar defects right
through the crystal which act as very strong pinning sites and increase the critical
current by a factor of 20± 50. Figures 53 (b)± ( f ) show MO images after zero-® eld
cooling to 50 K as progressively higher ® elds are applied (see caption). In these
images, bright areas correspond to regions of high ¯ ux density while dark areas
re¯ ect the Meissner phase. Initially vortices are seen to penetrate the sample from the
surface, moving deeper into the sample at the middle of each edge where screening
currents and stray ® elds are largest. In ® gure 53 (c) the ¯ ux front just reaches the
514
S. J. Bending
Figure 53. (a) Optical micrograph of a BSCCO single crystal showing the dark outer
irradiated region. (b)± ( f ) MO images at 50K after the applied ® eld has been
increased to (b) 85mT, (c) 107mT, (d ) 128mT, (e) 150mT and ( f ) 171mT. (g) Flux
density pro® les from (b)± ( f ) taken along the line indicated by the two arrows in ( f )
(a.u., arbitrary units). [Reprinted from Schuster et al. (1994). Copyright 1994 by the
American Physical Society.]
unirradiated area and ¯ ux suddenly starts to appear in the centre of the sample. The
penetration of ¯ ux at the edges now slows down and magnetization becomes
dominated by growth of the ¯ ux `dome’ in the centre. Clearly any vortices appearing
in the centre must have crossed the Meissner region to get there and are driven by
screening currents which exceed jc in the low-pinning unirradiated region. Figure
53(g) shows vertical line scans across the di€ erent images taken at the point
L ocal magnetic probes of superconductors
515
indicated by the two arrows in ® gure 53 ( f ). These workers found excellent
agreement between these traces and theoretical calculations of the ® eld pro® les for
a sample of this type apart from the suppression of very sharp cusps at the sample
edges where the YIG ® lm had become saturated.
7.3. High-speed magneto-optical imaging
The great strength of MO imaging lies in the extremely high potential rate of
image acquisition. Over 30 years ago, Goodman and Wertheimer (1965) attached a
high speed cine camera to the microscope of their MO system in order to study the
kinetics of ¯ ux jumps in niobium discs. They were able to capture an image every
103 m s and recorded ¯ ux jumps with velocities as high as 30 ms­ 1. More recently
Leiderer et al. (1993) have been able to achieve 10 ns temporal resolution by
adopting a pump-and-probe approach. A 7 ns laser pulse was split into two parts,
one of which was focused onto the back side of a superconducting ® lm. As a
consequence the sample was locally raised above its critical temperature, creating a
nucleation site for a ¯ ux instability. After frequency shifting, the other part of the
pulse was passed down the MO optical path and used to illuminate a EuS layer at a
well de® ned time delay after the instability had been nucleated. The resulting
snapshot could then be recorded with a video camera. Figure 54 shows an example
Figure 54. Time evolution of the instability in the ¯ ux distribution induced by a laser pulse
in a 4mm 4 mm section of a YBCO thin ® lm: (a) Before the laser pulse, (b) 56ns
after the pulse; (c) the ® nal ¯ ux distribution. [Reprinted from Leiderer et al. 1993.
Copyright 1993 by the American Physical Society.]
516
S. J. Bending
of this technique for a YBCO ® lm 300nm thick which had been zero-® eld cooled to
1.8 K after which a ® eld of 25 mT was applied. Figure 54 (a) shows the ¯ ux
distribution over a 4 mm 4mm region of the sample before the laser pulse and
most of the sample is free of ¯ ux (dark). Figure 54(b) shows the situation 56ns after
the pulse while ® gure 54 (c) displays the eventual ® nal ¯ ux distribution revealing a
strong branching character. Leiderer et al. came to the conclusion that there are two
steps associated with the nucleated instabilities. The ® rst takes place on a time scale
less than 10 ns during which ¯ ux is homogeneously redistributed over part of the
sample. The second involves penetration of ¯ ux from outside the sample in the form
of branches which propagate at a speed of about 50 kms­ 1 .
8. Scanning superconducting quantum interference device microscopy
It is widely recognized that the SQUID is the most sensitive magnetic ® eld
sensing element known to man. Historically applications have focused on areas such
as biomedical or military remote sensing which require extremely high magnetic ® eld
sensitivity combined with rather coarse spatial resolution. The basic technology to
build scanning SQUID systems has, however, long been available and it is perhaps
surprising that, with one exception (Rogers 1983), the ® rst prototypes (Minami et al.
1992, Black et al. 1993, 1995, Ma et al. 1993, Matthai et al. 1993, Vu et al. 1993,
Kirtley et al. 1995a) have only been developed in the last few years as part of the
current explosion in scanning probe techniques.
8.1. Theory of superconducting quantum interference device operation
At present, SQUIDs based on low critical temperature materials such as niobium
dominate the scanning area since device fabrication and integration tends to be more
straightforward and a broad knowledge base already exists. For this reason we
con® ne our attention to low-Tc devices, although high-Tc materials seem certain to
become more prevalent in the near future. The theory of operation of SQUIDs has
been well described in a number of excellent texts (for example Rose-Innes and
Rhoderick (1978) and Clarke (1990)) and only the most relevant details will be given
here. The basic SQUID building block is the Josephson junction formed at a weak
link between two macroscopically large superconducting electrodes. A weak link can
be realized in a variety of ways, for example by a physical constriction, a grain
boundary, a point contact or a tunnel junction. The latter is the element of choice for
low-Tc SQUID microscopes when it is usually formed between two parallel metal
electrodes separated by an oxide tunnel barrier. In the most advanced devices,
niobium ( Tc = 9. 2K) is employed in both electrodes. Native niobium oxides are,
however, very unstable and `arti® cial’ tunnel barriers consisting of very thin layers of
oxidized aluminium are generally employed. The current± voltage characteristics of
such junctions are strongly hysteretic as shown in ® gure 55 (a) and not at all suitable
for dc SQUID applications. To eliminate this, a resistive shunt is normally connected
in parallel with the tunnel junction, leading to the reversible characteristic shown in
® gure 55 (b).
Superconductivity arises owing to the formation of Cooper pairs of electrons and
can be described in terms of a macroscopic two-component electron-pair wavefunction or order parameter w (r) = jw j exp (iu ). This allows the current density to be
evaluated as
L ocal magnetic probes of superconductors
517
Figure 55. Current± voltage characteristics of (a) a hysteretic and (b) a resistively shunted
Josephson junction.
J(r) = ­
hen
w
4im
(r) Ñ ^ w (r) ­ w (r) Ñ ^ w (r) ,
( 8.1)
where n represents the density of electrons. Thus, if we assume that the magnitude of
the order parameter is constant throughout a piece of superconductor ( jw j2 is
proportional to the fraction of electrons that are superconducting and would not
normally depend on position), equation (8.1) implies that a supercurrent is associated with a gradient in the phase u in the direction of ¯ ow. This picture can be
generalized for the case of a weak link where one can assume that the order
parameter in the macroscopic electrodes on either side has the same magnitude
but a di€ erent phase ( w 1 = jw 0 jexp (iu 1); w r = jw 0 jexp (iu r)). In this case, equation
(8.1) predicts a small but ® nite supercurrent for the weak link which depends on the
di€ erence in phase between the two sides:
I = Ic sin ( u
r
­ u 1) .
( 8.2)
The constant Ic denotes the maximum or critical supercurrent for the weak link in
question and equation (8.2) describes the dc Josephson e€ ect.
A dc SQUID is constructed by connecting two Josephson junctions in series with
a loop made of macroscopically wide superconducting leads as shown in ® gure 56. If
there is no applied magnetic ® eld, the phase di€ erences across both junctions will be
identical and the two supercurrents will simply add, giving double the currentcarrying capacity. In the presence of an applied ® eld, however, this will no longer be
the case and there will be an additional Aharonov± Bohm phase shift around the
SQUID loop as discussed earlier in the context of electron holography:
518
S. J. Bending
Figure 56. Sketch of a SQUID.
·
^
^
u = A
dl =
^
S
·
B dS = 2p
^
a
0
,
(8.3)
where a is the ¯ ux enclosed within the SQUID loop and 0 = h/ 2e is the
superconducting ¯ ux quantum. Since this additional phase term enters in a circular
sense, it breaks the symmetry of the problem and leads to interference e€ ects
between the two junctions. Provided that the weak link critical current is small
( LIc
0, where L is the SQUID loop inductance), then the screening of the applied
magnetic ® eld can be neglected and the following result is obtained for the net critical
current of the SQUID:
IcSQUID = 2Ic cos p
a
0
.
(8.4)
Thus we see that the critical current of the SQUID oscillates with period
B = 0/ A, where A is the area of the loop. Scanning dc SQUIDs typically have
pick-up loops of diameter about 10 m m, an oscillation period of about 2 10­ 5 T
and a fundamental magnetic ® eld sensitivity many orders of magnitude smaller than
L ocal magnetic probes of superconductors
519
this. The ultimate resolution of the device is limited by Nyquist noise currents
associated with the two junction shunt resistors. An excellent discussion of this has
been given by Clarke (1990), who showed that an optimized device under best
operation conditions has a minimum in the (white) ¯ ux noise density given
approximately by
16kB TL
Su ( f )
,
( 8.5)
R
where L is the SQUID inductance and R is the resistance of one of the shunts. At low
frequencies (below 1 Hz) the system becomes dominated by 1/ f noise which has two
main sources. The ® rst arises from critical current ¯ uctuations in the junctions
associated with electron traps in the oxide barriers. The second is due to the motion
of ¯ ux lines trapped in the main body of the SQUID which, for all practical
purposes, behaves as if an external ¯ ux noise source were applied to the SQUID.
There is an important di€ erence between the two types since the ® rst can be
eliminated by an appropriate modulation scheme while the second cannot. Nevertheless state-of-the-art devices operated in high-frequency ¯ ux-locked loops exhibit
¯ ux noises quite close to their intrinsic values and are almost operating with ideal
resolution. In practice, SNRs are rarely the primary concern in scanning SQUID
systems when they are used to image superconductors owing to the relatively large
magnetic ® elds present (compared with applications in biomagnetism for example).
In many of the images shown here this ratio can be as high as 103 ± 104 .
8.2. Operation of the superconducting quantum interference device in a ¯ ux locked
loop
In nearly all practical applications the SQUID is incorporated into a feedback
circuit and used as a null detector for magnetic ¯ ux. This requires the fabrication of
an additional input coil which couples into the SQUID loop as sketched schematically in ® gure 57. The SQUID is biased with a constant current (greater than 2Ic0 )
and the output voltage monitored as the input coil is then used to apply an
oscillating ¯ ux ( f i 100kHz) to the SQUID with a peak-to-peak amplitude of
u 0/ 2. If there is no ambient dc magnetic ® eld present, the SQUID output will be
proportional to a recti® ed version of the input signal and contains only the frequency
2f i . Consequently, if this signal is passed (normally via a cooled transformer) to a
lock-in ampli® er referenced to the fundamental frequency, its output will be zero. If,
on the other hand, the ambient dc magnetic ® eld is non-zero, the SQUID output will
contain a large component at the fundamental frequency which will be detected by
the lock-in ampli® er. In practice the output of the lock-in ampli® er is used in a
negative feedback loop to null the background dc ® eld exactly. This is achieved by
passing the signal through an integrator which drives an additional dc current into
the input coil in parallel with the 100kHz oscillator via a large series resistor Rf . The
feedback voltage across Rf can be monitored and is an extremely sensitive measure
of the ambient ® eld near the SQUID.
8.3. The state of the art in scanning superconducting quantum interference device
microscopy
Recent developments have been based exclusively around dc SQUIDs operated
in ¯ ux locked loops. The trend with time has been towards smaller pick-up loops
(higher spatial resolution) and higher ¯ ux sensitivity, and without wishing to
diminish the e€ orts of other workers in the ® eld we shall con® ne ourselves to a
520
S. J. Bending
Figure 57. Schematic diagram of a SQUID operating in a ¯ ux-locked loop.
detailed discussion of an instrument developed by Kirtley et al. (1995b) which
currently represents the state of the art. The sensors for this instrument are
fabricated by a sophisticated planarized all-refractory technology for low-Tc superconductivity (Ketchen et al. 1991) process which allows an entire integrated SQUID
magnetometer to be built up starting from a Nb± AlOx± Nb trilayer deposited on a
silicon wafer. A planar surface is recovered after initial patterning using SiOx in® ll,
allowing both optical and electron beam lithography to be performed on the same
chip to de® ne coarse (greater than 1 m m) and submicron features respectively. Figure
58 shows a schematic diagram of one of these integrated SQUIDs. The octagonal
niobium pick-up loop of 4 m m diameter (with 0.8 m m linewidth) on the right of the
® gure is connected to the niobium SQUID `washer’ via a 20 m m section of coplanar
lead structure attached to a low-inductance strip-line, 1.2 mm long. The `washer’
itself contains a square hole 10 m m wide and is attached to the two 1 m m tunnel
junctions and associated platinum or Pt± Rh resistive shunts to its left. Flux
modulation is achieved through a single turn coil which is de® ned on top of the
`washer’. The ® gures of merit of the SQUIDs fabricated in this way are most
impressive and the typical ¯ ux noise of about 2 10­ 6u 0 Hz­ 1/ 2 corresponds to a
magnetic ® eld noise of about 4 10­ 10 THz­ 1/ 2 at 4.2 K. Data capture rates are
generally in the range 1± 10 Hz, allowing the resolution of variations in ® elds as small
as 1 nT.
In operation the SQUID is mounted on a cantilever fabricated from a 13 m m
brass shim which is ® xed to the end of a piezoelectric scanner tube at an angle of
about 20ë to the tube face. The sample on its mount is normally mechanically
scanned with respect to the sensor with one corner of the SQUID chip actually riding
along its surface. In order to optimize spatial resolution, care is taken that the pickup loop lies within one diameter of the active corner and hence is typically a few
micrometres above the sample. The sample on its mount approaches the sensor on a
linear actuator with a di€ erential micrometer. Scanning is achieved by pivoting the
sample mount with a stepper motor driven x± y stage. The pivot ring is positioned
close to the end of the sample mount in order to obtain a factor-of-seven reduction in
scanning range at the sample. This mechanical system allows a 400 m m scan range
L ocal magnetic probes of superconductors
521
Figure 58. Sketch of a scanning SQUID assembly. The lower inset shows an expanded
schematic view of the integrated SQUID and pick-up loop.
with sub-micron resolution. If even ® ner resolution is required, the piezoelectric tube
can be used to scan the SQUID with respect to the sample with nanometre
resolution. All the data presented here are taken with both the sample and SQUID
at 4.2 K.
8.4. Examples of scanning superconducting quantum interference device microscopy
images
A valuable use of the instrument arises from its ability to diagnose problems with
other superconducting devices. Figure 59 shows a family of four images of a high-Tc
YBCO`washer’ SQUID which has a scratch running across it from the top left-hand
corner to the centre of the right-hand side. It was known beforehand that this
particular device had a particularly large hysteresis in its response as a function of
applied ® eld. This can arise due to the trapping and subsequent motion of vortices in
the superconducting ® lm and is clearly undesirable. All the images in the ® gure were
measured at 4.2 K but after di€ erent ® eld and temperature cycles. In ® gure 59 (a) the
SQUID was cooled in a very low ® eld (about 0.2 m T) after which 18 (black) vortices
have become trapped in the YBCO washer. Figure 59 (b) was taken after the ® eld
was cycled to 0.06 mT and back again. Note that a single ¯ ux bundle is trapped in
the upper right-hand corner of the washer where the scratch crosses it. More ¯ ux
522
S. J. Bending
Figure 59. Image of vortices in a YBCO thin-® lm edge-junction washer SQUID with a
scratch running from top left to centre right. Images were recorded after (a) ® eld
cooling at a very low ® eld and cycling to (b) 0.06mT and (c) 0.22mT at 4.2K. (d )
Image at 4.2K after cycling to 0.24mT at 77K (Kirtley et al. 1995b). [Copyright 1995
International Business Machines Corporation. Reprinted with permission of IBM
Journal of Research and Development, Vol. 39, No. 6.]
becomes trapped along the scratch as the ® eld is cycled to progressively higher values
as shown in ® gure 59(c). Finally in ® gure 59(d) the sample is cycled to 0.24 mT at
77K before cooling to 4.2 K for imaging. We now ® nd vortices trapped at the inside
corners of the square `washer’ hole, illustrating the fact that vortices are ® rst trapped
at these points where the magnetic ® eld strengths are highest.
The di€ use `tails’ clearly visible on the vortices trapped in the SQUID washer
arise due to the ® nite area of the coplanar leads connecting the pick-up loop to the
rest of the SQUID. Hence, if a vortex lies in the gap between these leads, it also
contributes to the signal. Turning the problem around, the highly localized nature of
these vortices suggests a convenient way to measure the response of the pick-up loop
assembly. Figure 60(a) shows an image of one of these vortices with the correctly
oriented pick-up loop superimposed on it. Clearly the shape of the image is dictated
by the geometry of the loop and its leads. In ® gures 59 and 60 the size of the pick-up
loop was actually 10 m m which is over a hundred times larger than the magnetic
L ocal magnetic probes of superconductors
523
Figure 60. (a) Scanning SQUID image of a single vortex in a YBCO thin ® lm with the pickup loop superimposed (left). (b) Line scans (
) and theoretical ® ts in the
directions indicated assuming that the vortex contains a single ¯ ux quantum (
)
(right) (Kirtley et al. 1995b). [Copyright 1995 International Business Machines
Corporation. Reprinted with permission of IBM Journal of Research and
Development, Vol. 39, No. 6.]
penetration depth in YBCO thin ® lms at these temperatures (¸(4.2 K) 150nm).
Since the height of the loop above the sample is always comparable with its
dimensions, we can approximate the ® eld due to an isolated vortex by
B(r)
r.
( 8.6)
2p r3
It is now straightforward to integrate the ¯ ux through a loop (assumed circular
with radius r0) at a height h directly above the vortex:
loop
=
0
1­
0
h/ r0
[1 + (h/ r0)2]1/ 2
( 8.7)
Consequently we might expect that, if h r0 , about 0.3 0 would be coupled into
the SQUID. Figure 60(b) shows two experimental line scans (full circles) in the
directions indicated on ® gure 60(a). Note that the electronic SNR for these images is
about 105 and the apparent irregularities in these scans must arise from surface
irregularities or tip± surface interactions as the cantilever is scanned in `contact’
mode. The solid curves represent numerical integration of equation (8.7) for the
actual pick-up loop geometry with the height above the sample ® tted to h = 8 m m.
The excellent agreement between data and ® ts suggests that there is a good
quantitative understanding of the scanning SQUID images.
The most remarkable success of this scanning SQUID system has been its use to
investigate the symmetry of the pairing state in HTSs. In the introduction to this
review some of the key issues in contemporary superconductivity were discussed.
Magnetic imaging systems tend to address those of immediate technological import-
524
S. J. Bending
ance relating to the impact of vortex dynamics on the ability to sustain supercurrents
or the performance of superconducting devices. It is, however, also possible to probe
issues as fundamental as the mechanism of superconductivity in carefully designed
experiments. The best known of these is an ingenious series of investigations of grain
boundary junctions fabricated on tricrystal substrates designed by Tsuei et al. The
theoretical background to this work is quite complex and only an outline will be
given here; for more details the reader is directed to an excellent review by Tsuei et al.
(1995) and references therein.
We know that superconductivity in the cuprate materials is also associated with
electron pairing and, as discussed earlier in the context of SQUIDs, can be described
by a pair wavefunction or order parameter W . This can be written in the following
separable form:
W ( r, R) = u ( r) w ( R) ,
(8.8)
where w (R) describes the centre-of-mass motion of the pair and u (r) the relative
motion of the two electrons. The former is related to the macroscopic aspects of
superconductivity and is invoked to describe phenomena such as ¯ ux quantization
and the Meissner e€ ect. The latter contains the microscopic information about the
pairing state and is not easily accessible experimentally. However, in the Bardeen±
Cooper± Schriefter theory the order parameter in momentum space is related to the
gap function (k) in the following way
¢ k
u ~ (k) = u (r) exp (ik r) d3r = ( ) ,
(8.9)
E(k)
·
where E(k) is the quasiparticle energy. Since ¢(k) and u (r) transform in the same
way under the symmetry group operations of the host crystal lattice, the symmetry of
the gap function re¯ ects that of the pair wavefunction. Consequently a systematic
study of ¢(k) using quasiparticle or pair tunnelling across a Josephson junction or
photoemission spectroscopy should reveal the pairing symmetry of the superconducting electrodes. In practice, such measurements are rarely conclusive and it
is the IBM tricrystal experiment which currently provides the most compelling data.
It is believed that the electron pair state is s wave in the majority of low-Tc
materials, that is the order parameter is of one sign and has no zero nodes. This does
not, however, preclude the possibility of anisotropy; indeed it is known from
quasiparticle tunnelling studies of superconducting elements such as niobium and
lead that the gap function in these materials is weakly anisotropic. In contrast,
superconductivity in high-temperature materials is associated with fourfold symmetric Cu± O planes within crystals with tetragonal symmetry (if we ignore small
orthorhombic distortions in some materials). If we adopt the premise that the gap
function ¢k could be described by any of the 1D even-parity irreducible representations for a tetragonal lattice with D4h symmetry (Scalapino 1995) the d-wave G +
3
representation is a likely candidate with basis function ¢k /¢0 (cos kx ­ cos ky ).
This is usually referred to as the dx2­ y2 pair state and looks a promising choice,
although higher-angular-momentum states cannot be excluded. dx2­ y2 is characterized by four lobes with opposite signs and zero nodes along the diagonals ( y = x )
as sketched in ® gure 61. This raises the novel possibility that a Josephson junction
could be formed between two lobes of opposite sign in the two electrodes. Such a
structure has been named a p junction on account of the phase shift at the interface.
If a ring is now constructed which contains an integer number N of such junctions,
L ocal magnetic probes of superconductors
525
Figure 61. Schematic diagram of constant contours of the gap function in momentum space
(note the reversed sign of adjacent lobes).
two di€ erent situations can arise. If N is even, the system has the option of ¯ ipping
the polarity of the order parameter to eliminate all the p junctions. Even if it does not
achieve this, there will always be an even number of such junctions remaining and
the net accumulated phase around the ring will be a trivial integer multiple of 2p and
will not impact on the required single-valuedness of the wavefunction. If, on the
other hand, N is odd, then it will never be possible to reorient the wavefunctions to
eliminate all p junctions, and there will always be an odd number of them remaining.
The accumulated phase around the ring will now be an odd multiple of p , and the
system must compensate by spontaneously generating a half-¯ ux-quantum ( 0/ 2)
within the ring to preserve the single-valuedness of the wavefunction. If one examines
the energetics of this problem, this basic conclusion can be con® rmed provided that
Ic L
0 for the ring and the cost of creating the spontaneous half-¯ ux quantum is
well compensated by the gain in the Josephson energy (Sigrist and Rice 1992).
The prediction of the spontaneous generation of a ¯ ux of
0/ 2 in zero applied
® eld is one that Tsuei et al. set out to investigate. Since grain boundaries between
di€ erently oriented regions of high-Tc superconducting ® lm act as Josephson
junctions, they represent an ideal building block for such studies. Furthermore the
® lm orientation can be controlled by locking it to that of a SrTiO3 (100) substrate
during epitaxial growth. There is no way to design a ring structure containing a
single p junction with such an approach, but Tsuei et al. were able to invent a clever
scheme based on a tricrystal substrate to create a ring with three such junctions. To
achieve this, three pieces of SrTiO3 substrate were cut at the desired angles, polished,
reassembled and fused with the crystallographic orientations shown in ® gure 62. An
epitaxial ® lm of YBCO was then laser ablated on top of the substrate and patterned
into four strategically placed rings (inner diameter, 48 m m; wire width, 10 m m) only
one of which, around the tricrystal point, contained an odd number of junctions as
indicated in ® gure 62. It was con® rmed that a Josephson junction is formed at all
places where the rings cross a join in the substrate and that LIc 100 0
0/ 2 at
4.2 K as required.
Figure 63 shows a scanning SQUID microscope image of the sample obtained
with a pick-up loop of 10 m m diameter inclined at an angle of about 20ë to the sample
526
S. J. Bending
Figure 62. Schematic diagram of the tricrystal YBCO ring samples used to observe halfinteger ¯ ux quanta: GB, grain boundary.
surface after cooling to 4.2 K in a ® eld of less than 0.5 m T. The positions of all four
rings are clearly visible (the outer control rings can be seen because of slight changes
in the inductance of the SQUID sensor when the pick-up loop lies directly above the
superconducting ® lm); yet only the central ring with three junctions appears to
contain any ¯ ux. The fraction of the ring ¯ ux threading the sensor when it is placed
at a position (x, y) in the scanner plane is given by M(x, y)/ L, where M(x, y) is the
mutual inductance between pick-up loop and ring at that position and L is the ring
self-inductance ( M(x, y)/ L 0. 02 at its maximum). M(x, y) has been calculated
numerically for the octagonal pick-up loop geometry and used to generate
theoretical line scans in the directions indicated on the diagram assuming that the
ring is threaded by a ¯ ux of 0 / 2. These theoretical traces (solid curves) are
compared with the data (dotted curves) at the bottom of the ® gure and the
agreement is clearly outstanding. The asymmetry in the scans arises, in part, from
the unshielded section of the coplanar leads and also because the pick-up loop is
inclined with respect to the surface. Subsequent experiments with di€ erent sample
geometries and materials systems have lent further support to these results and have
provided one of the strongest pieces of evidence to date for d-wave pairing in the
HTS materials.
9. Future perspectives and conclusions
9.1. Future perspectives
The majority of the imaging techniques described in this review have developed
considerably over the last 10 years and there is no reason to suppose that they will
L ocal magnetic probes of superconductors
527
Figure 63. (a) Scanning SQUID image of the sample sketched in ® gure 62 after cooling in a
® eld of less than 0.5 m T (top). (b) Line scans (
) through the central threejunction ring along with ® ts to the data assuming it contains half of a ¯ ux quantum
(
) (bottom) (Kirtley et al. 1995b). [Copyright 1995 International Business
Machines Corporation. Reprinted with permission of IBM Journal of Research and
Development, Vol. 39, No. 6.]
not continue to do so for at least another decade. The discovery of high-temperature
superconductivity has contributed, in part, to renewed interest in imaging the
microscopic ¯ ux distributions in superconductors, although it should be remembered
that these techniques can equally well be applied to studies of ferromagnetic
528
S. J. Bending
materials. Indeed considerable e€ ort is currently being devoted to investigations of
the domain structure of thin ferromagnetic ® lms, multilayers and nanostructures
owing to their substantial technological potential for sensing and data storage. Since
the length scales of interest in ferromagnetism are somewhat shorter than in
superconductivity, such work seems certain to lead to a drive for improved spatial
resolution.
The following section will attempt to identify likely future developments for each
given technique. Such comments are inevitably highly speculative but will hopefully
be of some value to the reader nevertheless.
9.1.1. Electron microscopy
The current state of the art in both Lorentz microscopy and electron holography
is already most impressive, and future developments seem likely to be qualitative
rather than quantitative. Although electron holography is, in principle, a more
quantitative technique than Lorentz microscopy, for most practical purposes it is
su cient to identify the location of a vortex rather than to examine its internal
structure. For this reason the fact that Lorentz microscopy produces images directly
without the need for post-processing seems to give it a signi® cant advantage. One
area where developments seem sure to be made is in the speed of image acquisition.
Video rates are already being achieved and should be considerably exceeded if
brighter electron sources and more sensitive image capture media become available.
In addition it may become possible to study rather thicker samples and to minimize
the potential risk of damage during thinning procedures. Finally advances in the
electronic and/or optical reconstruction of interference images seems certain to speed
this process up by many orders of magnitude over the next decade and rapid
quantitative holographic imaging may yet prove a serious rival for Lorentz
microscopy.
9.1.2. Magnetic force microscopy
MFM has not been widely used in the ® eld of superconductivity, although it has
now become a standard research tool for ferromagnetic materials. The reasons for
this are two-fold; it can be highly invasive if great experimental care is not taken and,
as we have seen in section 4, the SNR is currently very low (2:1) at the single vortex
level.
The ® rst of these issues would be improved (although not solved) by using a noncontact ac mode of microscopy. This was recently attempted by Yuan et al. (1996)
using a vibrating piezoresistive silicon cantilever whose resonant frequency was
tracked with a phase-locked loop. The tip of the atomic force microscope cantilever
had been coated with 16nm of iron to make it magnetic and was vibrated with an
amplitude of 20nm at an average height of 120nm. These workers observed features
in their images indicative of the presence of asymmetric ¯ ux structures, although it
appears that they do not yet have su cient resolution to observe individual ¯ ux
vortices. It seems likely, however, that results of this type cannot be far away.
The problem of poor SNR seems certain to be related to the non-ideal domain
structure of the magnetic force microscope tip. This is illustrated by the fact that the
forces predicted for single-domain particles are many orders of magnitude larger
than those actually measured with the (polycrystalline) thin ® lms forming real
magnetic tips. Recently Kent et al. (1993) have demonstrated that it is possible to
produce single-domain ferromagnetic pillars in a controlled way by electrodeposition
L ocal magnetic probes of superconductors
529
from a chemical vapour precursor using a scanning tunnelling microscope tip.
Ferromagnetic ® laments with diameter as small as 7 nm and aspect ratios in excess
of 20 have been grown on silicon substrates and it should be possible to create such a
structure on the tip of a silicon cantilever. It would be interesting to perform MFM
with such a structure, although whether the particle would be robust enough to
survive during imaging remains to be seen. Furthermore great care would still have
to be taken to avoid perturbing the ¯ ux pro® le of the sample with the stray ® elds of
the tip.
9.1.3. Bitter decoration
Bitter decoration is a mature technique and it is di cult to foresee any major
developments in the near future. The control of background helium pressure during
evaporation in order to optimize the magnetic particle size and kinetic energy is one
area where work continues. Recently Blum et al. (1992) have developed a rather
di€ erent decoration apparatus from that described in section 5 where the sample is
screened from the evaporating ® lament by a massive copper block. In this
con® guration, particles are transported by laminar helium gas ¯ ow through a narrow
conduit where the sample sits. In this way the particle ¯ ow rate can be much more
precisely controlled.
9.1.4. Scanning Hall probe microscopy
The ® eld sensitivity of state-of-the-art SHPM instruments is currently limited by
pre-ampli® er noise at temperatures below 77 K. In principle the reduction in Johnson
noise and increase in electron mobility at 4.2 K would allow an approximately
tenfold increase in sensitivity if pre-ampli® er noise could be reduced to these levels
(e.g. a cooled pre-ampli® er stage). This would increase the ¯ ux sensitivity of a typical
system to less than 10­ 6U 0 which, to the best of our knowledge, would out-perform
any other current imaging system.
The state-of-the-art spatial resolution of SHPM (about 0.2 m m) is still rather
coarse and limits the study of discrete vortices to ® elds below about 10 mT.
Experience suggests that the GaAs/Al0.3 Ga0.7 As heterostructure Hall probe is
already approaching its operation limits at this resolution if an acceptable magnetic
® eld sensitivity is to be retained. Considerable bene® ts are to be expected, however,
by turning to new materials systems where the carriers reside in layers of higher
mobility. Two strong candidates are InAs and InSb with electron mobilities which
are three and ten times larger respectively than that of GaAs at room temperature.
The use of these materials should lead to signi® cant reductions in the Johnson noise
and an additional reduction in 1/ f noise can be expected for other reasons. Also a
decrease in or absence of surface depletion at patterned edges should make
fabrication somewhat easier and it may be possible to achieve a spatial resolution
as high as 50 nm in this way.
The Hall e€ ect is a very convenient basis for an imaging system since it is linear
and highly local. Nevertheless there are other magnetotransport phenomena which
could form the basis of detection schemes with considerably superior magnetic ® eld
sensitivity, for example giant magnetoresistance in magnetic superlattices (Baibich et
al. 1988) and giant magnetoimpedance in amorphous alloys (Panina et al. 1994).
Both these materials have the drawback that the response is not so local but it
should, nevertheless, be possible to realize high resolution sensors based on narrow
conducting constrictions. The nonlinear magnetic ® eld response is a signi® cant
530
S. J. Bending
complication for analogue imaging techniques although it may be possible to `bias’
sensors magnetically into an approximately linear regime depending on the speci® c
application. Imaging systems based on such principles seem likely to appear over the
next few years.
9.1.5. Magneto-optical imaging
This, once again, is a rather mature technology which seems unlikely to undergo
dramatic development. The present capabilities are largely limited by the available
MO materials; one can either work at high spatial resolution but poor ® eld
resolution (EuSe) or high ® eld resolution but poor spatial resolution (YIG). A
new material which could be evaporated as a thin ® lm directly on the surface of a
sample and used in a wide range of temperatures (4± 100K) with high ® eld sensitivity
would give an instant boost to the area.
In the absence of a new material one wonders whether the technique of scanning
near-® eld optical microscopy could be modi® ed to improve the spatial resolution of
MO imaging. A thin MO ® lm could be deposited directly onto the tip of a tapered
optical ® bre and scanned across the surface of a superconducting sample in the usual
way. This should allow substantial gains in spatial resolution although it may not be
practical in reality since it is di cult to preserve linear polarization along optical
® bres. Even with existing capabilities it should be possible theoretically to resolve
individual vortices with EuSe at low magnetic ® elds.
The current state of the art rate of MO image acquisition (10 ns frame­ 1 ), while
impressive, in no way represents the limit of what can be generated in short optical
pulses. Since it is clear that important vortex dynamic processes take place on much
faster time scales, it seems likely that there will be a push to improve this resolution
still further. We speculate that time resolution as short as 100fs may be possible
using currently available femtosecond optical pulses and very sensitive CCD cameras
to record images.
9.1.6. Scanning superconducting quantum interference device microscopy
It is likely that this technique will undergo considerable development in the next few
years. The spatial resolution can still be improved by patterning smaller pick-up loops.
Based on a (conservative) minimum wire width of 100nm and a 1:10 aspect ratio
between this and the loop diameter, a resolution of 1 m m should be achievable. It is then
possible to arti® cially increase spatial resolution further by integrating a nanometresized soft magnetic `needle’ into the SQUID loop which channels surface ® elds into the
sensor as demonstrated recently in a collaboration between Forschungszentrum JuÈ lich
and The University of SaarbruÈ cken (Forschungszentrum JuÈ lich 1997).
The requirement that the probe must be very close to the sample surface during
imaging dictates that the SQUID and superconductor must be almost at the same
temperature. The current predominant use of niobium SQUIDs clearly limits the
working temperature range to less than 9.2 K. It may, however, become possible to
use high-Tc SQUIDs to perform similar measurements at higher temperatures and in
several cases their 77 K noise ® gures have been shown to be comparable with low-Tc
SQUIDs at 4.2K (Zhang et al. 1994, Lee et al. 1995, Ludwig et al. 1995). Simple
high-Tc scanning systems already exist but seem unlikely to achieve a spatial
resolution comparable with existing low-Tc systems for some years. This will require
the integration of a submicron high-Tc pick-up loop with all the associated
fabrication pitfalls.
L ocal magnetic probes of superconductors
531
Scanning SQUID microscopes are currently operated at very low bandwidths
(1± 10 Hz). This is not due to poor SNRs which are more than adequate and allow
high-speed operation on a scale of a few picoseconds under some circumstances (for
example Tuckerman (1980)). The problem lies, instead, with the comparatively crude
mechanical system which is used to achieve relative motion between sample and
sensor. In the speci® c system described here, the SQUID is mounted on a brass shim
`cantilever’ and the corner of the chip holding the pick-up loop is dragged along the
sample surface in direct contact. Scanning must inevitably be slow to prevent the
cantilever bouncing or resonating and leads to appreciable wear at the contact point
and hence a slow drift in probe± sample separation with time. As an alternative it
should be possible to integrate an entire SQUID onto an atomic force microscope
cantilever with the pick-up loop close to or even around the atomic force microscope
tip. In this way a very small controllable force can be maintained and surface
topography imaged at the same time as the magnetic ¯ ux distribution. However (in
the absence of a new concept, e.g. a dense array of SQUID pick-up loops), even with
a redesigned support system the need to scan the SQUID (or the sample)
mechanically seems likely to limit imaging to video rates.
9.2. Conclusions
In conclusion the six main techniques for imaging vortices in superconductors
with near-single-vortex resolution are reviewed. The principles underpinning their
operation are described as well as factors limiting the spatial, magnetic ® eld and
temporal resolution. In general there is an approximately reciprocal relationship
between ® eld sensitivity and spatial resolution and many of the techniques have
comparable ¯ ux resolution lying in the range (10­ 4 - 10­ 6)U 0 . Within this constraint,
imaging systems are already available to satisfy nearly all potential requirements at
low temperatures (4.2 K), although not necessarily at the higher temperatures which
are now of interest in the context of high-Tc superconductivity (about 77 K). The
only technique which can hope to access the short time scales associated with vortex
dynamics is MO imaging. This has already been demonstrated at speeds of
10ns frame­ 1 , and there is considerable scope for improvement, perhaps by as much
as two orders of magnitude.
The capabilities of the instruments are illustrated by a variety of recent images of
superconducting materials. The diverse range of physical problems which can be
investigated in this way highlights the key role played by such local magnetic probes
in our understanding of superconductors. This has never been more so than in the
struggle to understand HTSs where many rich new vortex phenomena have been
discovered.
Speculations have been made as regards future instrumental developments. These
seem likely to lead to more overlap between the operating regimes of di€ erent
techniques rather than dramatic new capabilities although an order-of-magnitude
improvement in overall ¯ ux resolution does seem likely.
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