Rep. Prog. Phys. 57 (1994) 651-741. PFinted in the UK
Low temperature scanning electron microscopy of
superconducting thin films and Josephson junctions
Rudolf Gross? and Dieter Koellet
7 Physikalixhes Institut, Lehrstuhl Enperimentalphysik 11, Universitat Tiibingen, Morgenstelle 14, D-72076
Tiibingen, Federal Republic of Germany
f Department of Physics, University of California, Berkeley and Materials Sciences Division, Lawrence
Berkeley Laboratory, Berkeley, CA 94720, USA
Abstract
By extending scanning electron microscopy to the temperature regime of liquid helium
and nitrogen a powerful technique for the imaging of the local properties of superconducting thin films and Josephson junctions is obtained. Low temperature scanning
electron microscopy (LTSEM) allows one both to investigate interesting physical phenomena in superconducting thin film samples with a spatial resolution of about 1 p m
and to perform a functional test of superconducting devices and circuits at their operation temperature. We discuss the technical and physical background of the LTSEM
imaging technique including the electron optical and cryogenic requirements, the interaction of the electron beam with the superconducting sample, the dynamics of the
electron beam induced non-equilibrium state, and the electron beam induced signal.
The origin of spatial structures in superconducting thin films and Josephson junctions
and their spatially resolved analysis by LTSEM is reviewed. The use of LTSEM in the
functional test of both low- and high-temperature superconducting thin films, devices,
and circuits is summarized.
This review was received in its present form in December 1993
00344885/94/070651 t9B59.50 0 1994 IOP Publishing Ltd
65 I
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R Gross and D Koelle
Contents
I. Introduction
2. Principle of low temperature scanning electron microscopy
2.1. Basic elements and electron beam parameters
2.2. Cryogenic requirements
3. interaction of the beam electrons with the superconductor
3.1. Beam electron range and thermalization time
3.2. Electron beam induced non-equilibrium state
4. Response of a superconductor to electron beam irradiation
4.1. General response theory
4.2. Response at low temperatures
4.3. Response at higher temperatures
5. The signal
5.1. General aspects
5.2. Measuring techniques
6. LTSEM study of superconducting films
6. I. One-dimensional case
6.2. Two-dimensional case
6.3. Three-dimensional case
6.4. LTSEM study of passive thin-film devices
6.5. Imaging of hotspots in superconducting film
7. LTSEM study of superconducting tunnel junctions and circuits
7.1. Basic equations
7.2. Pair tunnelling current density
7.3. Quasiparticle tunnelling current density
7.4. Arrays of superconducting tunnel junctions
8. LTSEM study of superconducting weak links
8. I. Basic equations
8.2. Pair current density
8.3. Quasiparficle current density
8.4. Disordered arrays of weak links
Acknowledgments
References
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Low remnperature scanning electron microscopy
653
1. Introduction
Scanning electron microscopy (SEM) represents a powerful tool for studying spatial
structures in condensed matter [I]. Over the last decades this method has found wide
application not only in materials science but also in biology and medicine. Moreover,
electron beam testing has become an indispensable aid for failure analysis of semiconducting integrated circuits. The principle of this imaging technique is simple. A focused
electron beam is scanned over the surface of an object and simultaneously an appropriate response signal (for example, the emission of secondary electrons) generated by the
electron beam irradiation is recorded. A two-dimensional image of a specific sample
property (the surface topography in the case of detecting the secondary electrons) is
then obtained by synchronously displaying this signal on a video screen. The success
and the widespread application of scanning electron microscopy is mainly based on
two properties of this technique. Firstly, the technology for the generation and manipulation of a sharply focused electron beam is highly developed. At present, an electron
beam appears to be the most important probe for scanning microscopy. Secondly, there
is a large variety of different response signals generated by the electron beam irradiation.
Depending on the actually detected signal, different properties of the scanned object
can be imaged. Beyond the most commonly used secondary and reflected electrons,
excitations generated by the electron beam irradiation in the scanned object, such as
charge carriers, phonons, photons etc, can be used for the imaging process.
Whereas SEM has become a standard technique for investigating objects kept at room
temperature, less work has been done with respect to low temperature applications. Here
by low temperatures we understand the temperatwe range from the boiling temperature
of liquid nitrogen (77 K) down to below the boiling temperature of liquid helium
(4.2 K). This temperature range is of particular interest for the application of SEM to
the investigation of superconductors. Starting about 15 years ago, low temperature
scanning electron microscopy (LTSEM) has been developed as a tool for the spatially
resolved investigation of superconducting thin films and circuits. In particular, the
cryogenic requirements [2-4], a general response theory [SI, and a theoretical description of the dynamics of the electron beam induced non-equilibrium processes in superconductors [6] have been developed. Up to now, in many experiments LTSEM has been
demonstrated to be a powerful tool for studying not only superconductors but also the
low temperature properties of semiconductors and insulators [7,8]. Some of the recently
demonstrated applications of LTSEM are the imaging of:
the quasiparticle and Josephson current density, and the energy gap distribution
in superconducting tunnel junctions [9-161;
trapped magnetic flux quanta, self-resonant and RF-induced states in single Josephson junctions, arrays of Josephson junctions, and superconducting microwave circuits
[ 1 1-28];
the spatial distribution of the critical current density in high temperature superconducting films and grain boundary junctions [20-491;
the anisotropic phonon propagation in single crystals [50-55];
*defects in single crystals by phonon tomography [56-621;
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R Gross and D Koelle
dissipative structures generated during the low temperatures avalanche break
down of semiconductors [63-661;
the response of superconducting tunnel junction detectors to short electron beam
pulses allowing the spatially resolved testing of superconducting x-ray spectrometers
[67-771.
The recent discovery of the high temperature superconductors has initiated a large
amount of research aimed at the understanding of the fundamental properties of these
new superconducting materials. We have proposed LTSEM as an analytical technique
for investigating the local superconducting properties of high-T, superconductors with
high spatial resolution [29-341. In recent experiments we have demonstrated that LTSEM
can be used to image the spatial distribution of the critical current density in polycrystalline [29, 31, 341 and epitaxial [32,33] high-T. films or to study the transport properties
across single grain boundaries in these films 135-37, 44-46]. Meanwhile, the LTSEM
measuring technique is frequently used for analysing the high temperature superconductors [38-471. In contrast to usual electric transport measurement yielding only the global
values of the superconducting sample properties, by LTSEM the local superconducting
properties such as the local resistive transition or the critical current density J, can be
measured with a spatial resolution of about 1 pm, For the high-T, superconductors
information on the local material properties is important for the further improvement
of these materials. Beyond the already demonstrated measurement of the local critical
current density and resistive transition, it should be possible to apply LTSEM to the
study offlux pinning and flux flour. Furthermore, LTSEM has already been demonstrated
to be useful for the testing of high temperature superconducling devices and circuits
[47-491.
It is interesting to compare LTSEM to other scanning techniques such as laser scanning microscopy [78-921 or scanning tunnelling microscopy [93-1001, which recently
have been successfully extended to the low temperature regime. Beyond the large variety
of different response signals the basic advantages of LTSEM surely are the small beam
diameter, which is achievable at a large working distance, the highly developed techniques for the manipulation of a focused electron beam, and the magnitude of its
probing depth. For laser beam scanning the achievable beam diameter is much larger
resulting in a worse spatial resolution. However, laser scanning microscopy is less
expensive and does not require a vacuum system for the laser beam. The probing depth
of the electron beam depends on the beam voltage and the sample material and typically
ranges between 0.1 and IO pm. This magnitude of the probing depth, which can easily
be adjusted by varying the beam voltage, is suitable for the study of thin film samples
and devices. The probing depth of the laser beam is strongly material dependent and
typically is only some nm for metals. Dielectric materials, in contrast, may also be
transparent for the laser light. In general, the principle of operation of laser and electron
beam scanning microscopy is quite similar and the specificadvantages of each technique
will determine which technique will be preferred for what kind ofapplication. Scanning
tunnelling microscopy is unrivaled in terms of spatial resolution. However, its probing
depth is restricted to only a few atomic layers allowing only the investigation of surface
properties of solids.
The number of applications of LTSEM has increased continuously within the last
decade. With the recent discovery of the high temperature superconductors, which often
show spatially inhomogenous properties, the need for a spatially resolving measuring
technique for the analysis of superconducting thin films and circuits became even more
Low temperature scanning elecrron microscopy
655
clear. In this article we want to review the application of LTSEM to the investigation of
thin-film superconductors and Josephson junction devices. In particular, we will discuss
the electron optical and cryogenic requirements for LTSEM (section 2) and the interaction of the electron beam with the superconducting material (section 3). In section 4
we discuss the temporal and spatial evolution of the beam induced non-equilibrium
state in superconductors. The response of the superconducting sample to the local
electron beam perturbation and the relevant time and length scales are derived. The
electron beam induced response signal is analysed in section 5. In sections 6 and 7 we
show how LTSEM can be applied to the investigation of the local superconducting
properties of thin films and Josephson junctions, respectively, as well as to the study
of more complex superconducting circuits. Here, the relationship between the measured
response signals and the local sample properties is derived. Finally, in section 8 we
discuss the application of LTSEM to the study of superconducting weak links.
2. Principle of low temperature scanning electron microscopy
2.1. Basic elements and electron beam paranierers
The basic elements of LTSEM are shown in figure 1. The sample, for example a substrate
carrying a superconducting film or Josephson junction, is mounted on a temperature
-V
BEAM BLANKING UNIT
,'HI
DEFLECTION UNIT
Figure 1. Principle of low temperature scanning electron microscopy of superconducting
thin-film devices and circuits.
controlled low-temperature stage 12-41 of a scanning electron microscope in such a way
that its surface can directly be scanned by the electron beam. The back of the sample is
mounted in good thermal contact with a thermal reservoir kept at the desired operating
temperature. During the scanning process the sample is perturbed locally and the beam
induced response signal is recorded as a function of the beam coordinates ( x , y ) on the
sample surface. The large number of interesting applications of LTSEM is related to the
large variety of possible response signals. In most LTSEM studies of superconducting
samples the global change of an electric quantity such as the sample current or voltage
is measured. The obtained images usually are referred to as electron beam induced
current (EBIC) or voltage (EBIV)images similar to those in studies of semiconductors
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R Gross and D Koelle
by means of SEM [I]. As will be shown below, the response signal, i.e., the change of
the global quantity is related to the local sample properties at the position of the focused
electron beam. Therefore, by displaying this response signal synchronously on a video
screen, a two-dimensional image of the superconducting sample properties is obtained.
Ofcourse, similar to rmm temperature scanningelectron microscopy beyond the electrical response signals, the backscattered (BE)and secondary electrons (SE), the Auger
electrons (AE),or the x-rays generated by the electron beam irradiation can be used as
response signals. By this, information on the film topology and its chemical composition
is obtained. Using a computer controlled image processing system, the different signals
can be recorded simultaneously and compared directly. In many cases a conclusive
interpretation of the sample behaviour is only possible by the correlation of the different
images, which are obtained by using different response signals and, hence, contain
different kind of information.
The beam diameter achievable with commercial scanning electron microscopes usually is of the order of IO0 A. Clearly, the ultimate resolution of LTSEM will be determined
by the diameter of the primary electron beam. However, for the study of the local
properties of thin film superconductors the spatial resolution will be determined by the
spreading of the beam induced perturbation in the specimen, i.e., by the spreading of
the sample region contributing to the recorded EBIC or EBIV signal. Therefore, in most
cases the spatial resolution will be much worse than expected from the value of the
beam diameter. We will show in section 4 that the spatial resolution depends on both
the sample parameters, such as the specificheat or thermal conductivity, and the electron
beam parameters, such as the energy of the beam electrons or the beam power, Typically,
the spatial resolution is in the pm range.
The beam parameters of commercial SEMS typically 1-40 keV beam voltage and
1 pA to 1 p A beam current yielding about 1 nW to 40 mW beam power. The scanning
speed usually can be varied over a wide range from more than several seconds to less
than IO0 ps per line. If the response signal is very small, signal averaging methods have
to be used to increase the signal to noise ratio. Typically, the recording time of complete
LTSEM mic (EBIV) images ranges between some seconds and some minutes. To improve
the signal to noise ratio, the electron beam often is periodically modulated using a
beam blanking unit and the signal is detected by a phase sensitive detector (lock-in
amplifier). The beam blanking unit can also be used to generate short electron beam
pulses for studying transient phenomena. Commercial beam blanking units allow the
generation of short electron beam pulses having a duration of only several ps. Note
that at a beam current of -10 pA and a pulse duration of -10 ns each electron beam
pulse in average only contains a single electron. Such pulses are used for the study of
the energy resolution of superconducting tunnel junction detectors [73, 75, 771.
2.2. Cryogenic reguirements
The study of both low and high-T, superconducting samples below their critical temperature requires operation temperatures in the range from well below liquid helium
temperature up to more than 120 K. This temperature range can be accessed by using
liquid helium (4.2 K ) or liquid nitrogen (77 K) as cooling liquids. The low temperature
stage used in our set-up is formed by a conventional 'He bath cryostat located outside
of the specimen chamber of the SEM. The large cylindrical liquid helium tanks is surrounded by a liquid nitrogen tank for precooling and thermal shielding. The large
helium tank serves as a reservoir for a small tank, which is located inside of the specimen
Low temperature scanning electron microscopy
657
chamber of the SEM [ 2 , 3 ] .The small tank is mounted on a liquid nitrogen temperature
place and surrounded by a gold plated copper radiation shield, which is kept at about
77 K. The liquid nitrogen plate, in turn, is mounted on a movable room temperature
x-y stage. We note that in designing this part of the cryostage special care has to be
taken to avoid possible vibrations reducing the spatial resolution. The large and small
helium tank are connected via flexible stainless steel bellows. This allows the motion
of the small tank in x and y direction by about f15 mm. In this way the sample
can be positioned exactly as with a usual room temperature sample stage. If sample
temperatures only above 77 K are required, the helium cryostat also can be filled with
liquid nitrogen. The thermal insulation between the liquid helium, liquid nitrogen and
room temperature parts of the cryostage is obtained by stainless steel tubes and the
vacuum provided by the vacuum system of the SEM.
Figure 2. Cross-sectional view of the small liquid helium tank including the sample mounting for the operation k f o w (a) and above 4.2 K (b). 1: sample; 2: sample holder; 3:
clamping screw; 4: copper ring for wire heat sink; 5: thermal shield; 6 : LHe-tank; 7:
clamping ring; 8: indium seal; 9: LHe tubes; 10:temperature Scnsor; I 1 :heater; 12: copper
brock; 13: nylon disk; 14: lid for liquid helium tank.
Figure 2 shows a cross-sectional view of the small helium tank including the sample
mounting for the temperature range below and above 4.2 K. For the temperature regime
below 4.2 K (figure 2(a)) the most useful set-up i s the arrangement where one side of
the sample is in direct contact with the liquid helium bath whereas the opposite side
can be directly scanned by the electron beam. The temperature of the helium bath can
be reduced down to about 1.5 K by pumping the gas above the liquid. For this arrangement the sample material directly separates the liquid helium bath from the vacuum of
the specimen chamber. That is, the sample must be mechanically strong enough and
must be shaped as a disk of certain diameter (about 20 and 60 mm for different types
of sample holders). Since often this is not the case, the sample is usually mounted on
a stable substrate material with high heat conductivity (e.g. single crystalline sapphire)
658
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R Gross and D Koelle
using a suitable low temperature adhesive. We note that the sample mounting is in
particular useful for the investigation of superconducting films deposited directly on
top of a proper substrate material. As indicated in figure 2(a) the sample is fixed by a
clamping screw which also compresses the indium seal between the sample and the
stainless steel sample holder. The sample holder acts as the top plate of the tank. It is
fixed by a clamping ring and a sealed also by indium.
Figure 2(h) shows the cryogenic set-up and the sample mounting for operation
temperatures above 4.2 K [3]. Whereas the sample is coupled closely to a gold plated
copper block acting as a heat sink, this block is coupled only weakly to the helium
bath in order to reduce the consumption of the cooling liquid for operation temperatures
well above its boiling temperature. The temperature of the copper block is measured
by suitable temperature sensors (carbon-glass resistor, Si diode, platinum resistor, etc)
and kept constant by a temperature controller regulating the power of the bifilarly
wound manganin heater. I n this way the temperature can be varied between about 4.2
and 150 K and stabilized within a few m K [3]. Details on the temperature regulation,
the thermal time constants, and the consumption of the cooling liquid can be found
in [3].
For many applications it is advantageous that the low temperature stage can be
attached and removed from the SEM easily. This allows a quick sample change and
an intermittent operation of the microscope at room temperature for conventional
applications. A simple sample mounting and exchange of the cryostage is achieved by
attaching the whole cryostage to the hinged door of the specimen chamber without any
further mechanical supporting elements. We note that the weight of the low temperature
stages is similar to that of the standard room temperature sample stage. After warming
up the cryostage and venting the vacuum of the specimen chamber one simply can open
the door of the specimen chamber in order to change the sample, A photograph of the
opened specimen chamber showing those parts of the cryostage located inside the
specimen chamber is depicted in figure 3. The 77 K radiation shield is removed so that
the small liquid helium tank is visible (configuration of figure 2 ( a ) ) , The figure also
shows the liquid nitrogen base plate, which is mounted on the room temperature positioning stage by four thermally insulating stainless steel tubes.
The electrical current and voltage leads attached to the top of the sample are thermally anchored close to the sample in order to avoid sample heating effects. For some
applications (e.g. for the investigation of superconducting tunnel junctions and circuits)
it is important to carefully shield the sample against any ambient magnetic fields such
as the Earth’s magnetic field or the stray fields of the final lens. A n effective magnetic
shielding of the sample is obtained by using magnetically soft materials such as
cryoperm-I0 (trademark of the Vakuumschmelze GmbH, Germany) and superconducting shields [3, 131. On the other hand, for some experiments a variable magnetic field
is required. This is achieved by a small superconducting coil placed in the small liquid
helium tank. Moreover, it is possible to irradiate the sample with microwaves during
LTSEM experiments. This is particularly interesting for experiments with Josephson
junctions and circuits. The microwave is guided to the vicinity of the sample using a
waveguide (E-band, i.e. 60-90 GHz) [4]. The curved front end of the waveguide can
be seen in figure 3.
We note that a cryostage based on a liquid helium bath cryostat is most effective
in terms of applicability and cooling power over a large temperature regime. The mechanical vibrations of the cryostage are small due to the mechanically calm cooling
medium. For higher operation temperatures ( T > 4 . 2 K) simpler and less expensive cold
Low temperature scanning electron microscopy
659
Figure 2. Opcned specimeii chiimher of thc ('amScan J ~ D Vmicroscops sliowing thc small
liquid helium tank with the sample holder installed. The liquid nitrogcn temperature radiation shield is removed. The wliole cryostage is attached lo the hinged door of the chamber.
stages are often adequate and commercially available [ 1011. For these stages, which
mostly use continuously flowing cold helium gas as the cooling medium, the sample
usually is mounted on a cold finger extending into the specimen chamber of the SEM.
However, these stages often are worse with respect to temperature stability and mechanical vibrations.
3. Interaction of the beam electrons with the superconductor
3.1. Beam electron range and thermalization time
To evaluate the important time and length scales of the specimen response tu the
electron beam irradiation, the electron beam range R and the thermalization time rb
for the beam electrons to reach the thermal energy of the target atoms has to be
estimated. From electron stopping theory [ I , 1021 the thermalization time t bcan be
estimated as
rb=
(m4/2)*
31rNZe4vo'
(1)
Here N is the number of the target atoms per unit volume, Z is the number of electrons
per target atom, e is the elementary charge, m is the electron mass, and u0 is the velocity
660
R Gross and D Koelle
of the beam electrons. For typical beam energies (EoG40 keV) the stopping time is
smaller than about IO-” s both for low- and high-T, superconductors.
Along their stopping path in a solid the beam electrons transfer their energy by a
large number of scattering events. Assuming that these scattering events are isotropic,
the beam electrons are thermalized within a hemisphere. The radius of this hemisphere
is given by the beam electron range and its centre by the coordinates of the beam focus
on the sample surface. The beam electron range was found to be proportional to about
the 1.5 power of the beam energy Eoand inversely proportional to the mass density p
of the absorbing medium. The experimentally obtained values for the beam electron
range were found to follow the empirical dimensional formula [ I ]
Here R is obtained in p m when Eo is in units of keV and p in units of g ~ m - With
~.
the values of the mass densities of the different superconducting materials an electron
beam range in the pm range is obtained for beam energies up to about 40 keV. Figure
4 shows the calculated dependence of the beam electron range R on the beam energy
for differenl low- and high-T, superconductors and for some relevant substrate materials
LIZ
f
v
LIZ
Beam Energy
( keV )
Figure4. Beam electron range R versus beam energy of high-T, (a)and low-T. superconductors ( 6 ) as well as of some relevant substrate materials (e).
Low temperature scanning electron microscopy
661
in the range up to 40 keV. For materials with low mass density a low beam energy has
to be used to keep R small. Note that equation (2) is an empirical formula giving only
approximate values. The actual value of the electron beam range in some materials can
deviate by more than 10% from the value given by the above expression. From electronstopping theory the actual path length travelled by the electron in the solid can be
estimated. This total path length L is given by [5]
Since the path of the electron in the stopping medium is tortuous, the length L is always
considerably larger than the average electron beam range.
The evaluation of the characteristic time and length scale for the energy loss of the
primary electrons due to their interaction with the target atoms shows that the beam
electrons are thermalized within a time scale of typically less than IO-” s. The typical
length scale for the spreading of the primary electrons during their stopping process is
typically in the p n range. During their stopping process the beam electrons generate
different kinds of excitations in the stopping medium. In the case of a superconductor
these excitations mainly will be quasiparticles and phonons. In the next section the
relaxation and diffusion processes of these excitations will be discussed.
3.2. Eleclroon beam induced non-equilibrium slate
The non-equilibrium behaviour of conventional BCS superconductors exposed to an
external perturbation such as irradiation with light, microwaves, heat pulses, or electrons has been studied both experimentally and theoretically [6,103, 1041. The combined configuration of Cooper pairs, quasiparticles, and phonons in a superconductor
under non-equilibrium conditions represents a complicated system. In general, the nonequilibrium distribution of quasiparticles and phonons is obtained by solving the Boltzmann equations consistently with the BCS gap equation. However, this is complicated
even for the well known BCS superconductors, since the detailed energy dependence of
the different scattering rates is not precisely known. For the high-T, superconductors
our knowledge on the nature of the excitations and their scattering rates is small.
Therefore, we can say little on the detailed nature of the non-equilibrium state generated
by an external Perturbation such as a focused electron beam.
The electron beam irradiation of conventional BCS superconductors has been treated
in detail in [ 5 , 6 , 1041. There, the treatment of the non-equilibrium state was simplified
by dividing the relaxation process of the generated excitations in three different time
and length scales as shown schematically in figure 5. During the first, very short time
scale (regime I, <IO-” s) the high energy beam electrons (EkeV) are stopped in the
superconducting material by Coulomb interaction with the target atoms. The characteristic time is the thermalization time q,.During this stopping process high energy excitations (-eV) are generated. Within a second, also rather short time scale (regime 11,
s), these excitations relax down to energies of the order of the gap energy of
the superconductor (EmeV). Here, the characteristic times are the electron-electron
scattering time ,z and the electron-phonon scattering time rep. In this regime a large
number of phonons is generated. Finally, within a third time scale the energy deposited
by theexternal perturbation isremoved by transfer to the heat sink (regime 111, >IO-* s).
Here, the characteristic times are the quasiparticle recombination time rR, the phonon
pairbreaking time zg, the quasiparticle scattering time zs, and the phonon escape time
662
R Gross and D Koelle
characterislic length ( pm )
10-2
bo-3
10
100
10-1
101
'
Pb
,
,
m
COODipl POWS
w Phonaor
102
Beom Electron
x
100
c
E,,
.---
,
characteristic time f sec)
Figure 5. Characteristic time and length scales for the relaxation of tlie electron beam
induced non-equilibrium state in a superconductor. Regime I : stopping of the beam clectmm by Coulomb interaction. Regime 11: relaxation of the high cnergy exitations by
electron-electron and electron-phonon scattering. Regime Ill: relaxation of tlie low energy
excitations by quasiparticle recombination and the transfer of phonons to the heal sink
(phonon escape process).
r Y ,i.e., the time the phonons require to escape to the substrate. These times are listed
in table I for some superconducting materials together with the thermalization time of
the beam electrons. The third time scale is usually much longer than the first and
second ones. This is in particular the case at very low temperatures, where some of the
characteristic times as for example the quasiparticle recombination time can become
very long.
Each characteristic time scale can be associated with a characteristic length scale
that is given by the spatial spreading of the beam induced non-equilibrium state within
these time scales. Within the first time scale the beam perturbation is restricted to a
volume having the diameter of the beam electron range R. The diffusion length of the
high energy excitations within the second time scale is usually short and can be neglected
compared to the diffusion length of the low energy excitations within the third time
scale. In most LTSEM experiments the spatial and temporal resolution is not sufficient
to investigate the non-equilibrium state within regimes I and 11. Usually one is interested
TaMc 1. Thermalization lime rb of 10 kcV beam electrons, quasiparticle recombination
time IR and phonon pairbreaking time I . at T/Tc=0.5 [IOS], phonon escape time 1, and
phonon trapping coefficient I + T J T ~for different low-T, superconducting Alms. The
phonon escape time rI is calculated for a 100nm thick film on a sapphire (SrTiO, for
YECO) substrate using the phonon transmission coefficientsi r o n [I%).
Tht data for
YBaiCu,O,.a are taken from [106-112].
Material
rb(xIo-"s)
Pb
In
0.41
Sn
Nb
AI
YECO
0.58
0.59
0.49
I .4
0.73
rR
( X I I P S )ra(xIO-vs)
0.19
0.79
2.3
0.034
0.17
0.15
0.0042
0.24
438
<0.1
0.11
<0.001
r,(XIO-qS)
12.9
18.4
2. I
1.2
0.40
<0.01
l+r,/re
38 I
110
21
288
2.6
-10
Low teinperature scanning electron microscopy
663
only in the longest time and largest length scale (regime HI), since these scales detennine
the sample response and the spatial resolution in the LTSEM experiment. Hence, in the
following we do not consider the details of the very fast processes as the beam electron
stopping and the relaxation of the high energy excitations. We restrict our discussion
to the relaxation and diffusion processes of the excess excitations having already energies
of the order of the gap energy. The scattering rates of these excitations are, at least
for the conventional superconductors, well known [IOS]. As shown in section 4, the
characteristic time and length scales of the electron beam induced non-equilibrium state
are obtained by solving the combined diffusion equations for the quasiparticles and the
phonons [6, 1041.
The electron beam irradiation of high-T, superconductors can be treated in a similar
way. The characteristic time and length scales for the beam electron stopping process
are about the same as those of conventional superconductors. The relaxation time of
the high energy excitations down to energies of the order of the gap energy is similar
or even shorter than that of conventional superconductors because of the higher energy
gap values and the usually higher operation temperatures of the high-T, materials. For
YBaZCu30,.a this time is less than I ps [106]. To estimate the important time and
length scales, which are relevant for the LTSEM imaging process, again only the coupled
diffusion process of the low energy excitations has to be considered using the characteristic times known so far [106-1121.
A purely thermal treatment of the electron beam perturbation of a superconductor
often represents a good approximation, if the different excitations get into thermal
equilibrium within a short time scale [5,6, 1041. In fhis case the non-equilibrium distribution of the excitations in regime Ill can be approximated by a thermal distribution
with an elevated temperature at every time and position during their diffusion process.
However, if the interaction of the different excitations is weak, they can diffuse completely decoupled and, even if their energy distributions can be described by thermal
distributions, they are probably characterized by diRerent elevated temperatures 161.
For example, this is the case for conventional superconductors a t temperatures well
below the critical femperature, where the quasiparticle recombination time T~ and the
quasiparticle scattering time T~ are much longer than the phonon pair breaking time
T~ and the phonon scattering time T~~~ [ IOS]. The detailed conditions under which a
thermal or a non-thermal treatment of the electron beam induced-equilibrium state is
appropriate are given in [6].
For high-T, materials the different scattering times are not well known up to now.
However, at the higher operating temperatures of these materials the scattering times
of the different excitations in general are shorter resulting in a rapid thennalization
[106-1121. Therefore, for these materials, at least at temperatures well above liquid
helium temperature, a purely thermal treatment of the electron beam perturbation is
appropriate. That is, one simply treats the electron beam as a local heat source and
assumes that the generated excitations have a thermal distribution. In this case their
coupled diffusion is described by the heat diffusion equation. To solve the heat diffusion
equation the thermal conductivity and the specific heat of the high-T, materials have to
be known. These thermal parameters are known for most high-T, materials [I 13-1261.
4. Response of a superconductor to electron beam irradiation
In this section the response of a superconducting thin-film sample to focused electron
beam irradiation is discussed. The task is to derive expressions for the characteristic
664
R Gross and D Koelle
decay time and decay length of the electron beam induced response of the superwnducting film. The value of the spatial decay length is important, since it usually determines
the spatial resolution of LTSEM for the investigation of superconducting thin films and
circuits. The decay time determines the electron beam modulation frequency and the
scanning speed below which transient effects do not have to be taken into account. As
will be shown below, transient effects can be used to dynamically localize the beam
perturbation and, hence, to improve the spatial resolution.
The discussion given in this section to a large extent also can be applied to describe
the response of a superconductor to focused laser irradiation (laser scanning microscopy). Beyond the different size of the beam spot the main difference between focused
electron and laser beam irradiation is the different energy distribution of the primary
excitations. Whereas the beam electrons typically have an energy in the keV regime the
laser photons have an energy in the eV regime. That is, by electron beam irradiation
usually excitations with higher energy are generated. However, with respect to the
imaging techniques only the non-equilibrium state in regime I11 (see figure S), i.e., after
the relaxation of the high energy excitations, is relevant, since one is interested only in
phenomena occurring on a time scale longer than the temporal resolution of these
techniques (typically > I ns). In regime I11 the electron and laser beam induced nonequilibrium state is quite similar due to the fast decay of the high energy excitations.
4.1, General response theory
The initial effect of the electron beam can be characterized by an appropriate perturbation P(x, y , z, f). Here (x, y ) is the coordinate point of the beam focus on the specimen
surface, z is the coordinate perpendicular to the surface, and f is the time. For a well
focused electron beam, P(x, y , z , I ) will be peaked around the coordinates ( x o ( f ) ,y o ( f ) )
of the beam spot at the specimen surface and in a depth io(f)from the surface where
the maximum power deposition occurs. With a beam resolution function ~ ( xy , z),
which has unit volume integral
s
dxdydz a(x,y,z)=I
(4)
the pertubation can be expressed as
p(X, Y , A I ) = ~ o ( 0 4 x - . ~ d O )(Y-Yo(O),
,
(z-20(0)1
(5)
where Po(f) is the integral beam perturbation at the time f. The radius of the perturbation is determined by the beam resolution function c and should be as small as possible.
In our case this radius is given by the beam electron range R, which determines the
volume, where the beam energy is initially deposited.
The specimen responds to the electron beam perturbation in a number of ways. In
general, the response can be characterized by a specific response function F(x, y , z, I).
If the response F is proportional to the perturbation P, it can be calculated using a
linear response function G as
s s
F(x, y , Z , I ) = dx’ dy’ dz’ d t G(x, y, z, x‘, y‘, z’, f - t‘)P(x’,y’, 2, t‘)
(6)
where for causality reasons G = 0 for f‘- f GO. If P(x, y , z, I ) varies slowly in time, the
response Fwill follow P quasistatically. In this case the static linear response function
665
Low temper.ature scanning electron microscopy
G(x, y , z, x’, y’, 2’) can be used and the time integration in equation (6) can be omitted.
A quasistatic response theory can he applied, if the modulation frequencyfof the
electron beam is small compared to the inverse of the characteristic decay time r of the
generated non-equilibrium state and if the beam scanning speed U, is small compared
to A/z. Here A is the characteristic decay length of the electron beam induced nonequilibrium state. As will be shown below, r typically is shorter than 1 ps nd A less
than IO pm. Hence, the conditions for the applicability of a quasistatic resp nse theory
a r e 2 x f < l / r - l MHzand v,<A/r=lOOcm
Whereas thescanningspe disusually
much smaller than 100 cms-’, the beam modulation frequency can be made onsiderably
higher than l / r by state of the art beam blanking units. In the fo1lowing;analysis any
transient effects due to the scanning speed of the electron beam are negle ted and the
response is calculated for a fixed beam position. The transient effects due to the electron
beam modulation, however, are taken into account. Moreover, it is shown that they
can be used to improve the spatial resolution of the LTSEM imaging technique [ 5 , 6 ] .
The perturbation P(x, y , z , I ) usually can be approximated by a simple functional
form. If the initial perturbation volume is small compared to the length scale A of the
generated non-equilibrium state, P can be approximated by a Dirac delta function. In
other cases P can be modelled by a Gauss or exponential function. A further possibility
is to assume that the beam power is homogeneously absorbed inside of a hemisphere
having the diameter of the beam electron range. However, considering only the qualitative behaviour of the electron beam induced response, the knowledge of the detailed
functional form of the beam perturbation is not important. Only for a quantitative
analysis P ( x , y, z, t ) should be known in detail.
High temperature superconductors are highly anisotropic materials. Therefore, the
response function G will be anisotropic, that is, it will depend on the crystallographic
orientation of the irradiated superconducting material. Note that with an anisotropic
response function G, a completely isotropic perturbation P will cause an anisotropic
response F.
In the following we will estimate the response of superconducting thin-film samples
to focused electron beam irradiation for the low and high temperature regimes. Here,
low and high temperature regimes denote the temperature range where the thermal
boundary resistance between the superconducting film and the substrate material is
large or negligibly small, respectively. At low temperatures (4.2 K) the major obstacle
for the heat flow to the heat sink is the thermal boundary resistance between the film
and the substrate cansed by the acoustic mismatch between the different materials [ 1271.
The heat transfer coefficient a between superconducting films and single crystalline
substrates is small and ranges between about 0.1 and IOW cm-’ K-’at 7=4.2 K [ 1281.
In contrast, the transfer across the superconducting film typically is by more than
two orders of magnitude larger. For example, with the thermal conductivity of PbIn,
~ ~ ( 4K)-5
. 2 x IO-’W c K 2K-’[ 1041, we have a heat transfer of - 1 O ’ W cm-’ K-’
across a 500 nm thick film. Therefore, at low temperatures the solid-solid thermal
boundary resistance is dominant as compared to the thermal resistance across the film.
In this case, any temperature gradient across the superconducting film can be neglected
compared to that across the interface between the film and the substrate. At higher
temperatures (77 K) we usually have the opposite situation. At this temperature the
heat transfer coefficient between metal films and single crystalline substrates typically
is larger than IO3 Wcm-’K-’ [113-117]. The heat transfer across a 500 nm thick
YBa2Cu307-6 fdm, in contrast, is obtained to about IO3 Wcm-’K-’ using
~ ~ ( K)
7 7- 5 x lO-’W
K-‘[ I 18-1251. That is, in most cases the thermal boundary
s C ’ .
iy‘
F
666
R Gross and D Koelle
resistance can be neglected compared to the thermal resistance across the superconducting film. I n this case, the temperature gradient across the interface between the superconducting film and the substrate can be neglected compared to that in the film or in the
substrate material. The two temperature regimes typically overlap between about IO K
and 50 K depending on the detailed thermal properties of the superconducting film and
the substrate material.
4.2. Respo~iseat loiv lempcmrures
In the low temperature regime we can make the following assumptions. Firstly, the
substrate is modelled as isothermal at the bath temperature due to the low thermal
impedance o f most substrate materials relative to the film-substrate boundary resistance. Secondly, adiabatic boundary conditions at the top side of the film are assumed,
since the film surface is exposed to vacuum. Thirdly, variations of the electron beam
induced non-equilibrium state in r-direction, i.e. perpendicular to the film plane, are
neglected allowing a two-dimensional analysis. Since the typical thickness d of the
investigated films is of the order of the beam electron range, the film in good approximation can be assumed to be perturbed homogeneously across its thickness. Finally, the
electron beam is modelled by a pointlike perturbation i.e. by a Dirac delta function.
The spatial spreading of the electron beam induced perturbation in a superconducting film in the low temperature regime is schematically shown in figure 6. Using a
e-BEAM
(01
e-BEAM
(bl
I
I
I
I
A
I
I
I
DIFFUSION REGION
,
/
\
A
L
QUASIPARTICLES
m COOPER PAIRS
-PHONOUS
A
Figure 6. Spreading of the electron beam induced pcrturbation in a superconducting film
for the case of a dcscriplion by the heat diffusion equation ( 0 ) and the coupled diffusion
equations of quasiparticles and phonons (b).
thermal treatment (figure 6(a)) the heat diffusion in the film is determined by the
thermal conductivity K~ and the heat capacity CF of the film. The heat transfer to the
substrate is determined by the heat transfer coeficient a , Using a non-thermal treatment
(figure 6(b))the coupled diffusion of the generated excess excitations (quasiparticles and
phonons) has to be considered. Here, the most important processes are the quasiparticle
recombination (process I), the phonon pairbreaking (process 3), and the phonon escape
process (process 2) with the rates l i r R , I / r B and l / r T ,respectively. Depending on the
ratio of the different rates, the diffusion of the different excitations can vary from a
completely decoupled to a closely coupled diffusion [6, 1041. In the following, only small
perturbations will be considered. In this case the different parameters characterizing the
diffusion process can be assumed to be constant and equal to the thermal equilibrium
values. For large perturbations the temperature dependence of the thermal parameters
or the dependence of the scattering times on the excess density of the excitations has
to be taken into account [6, 1041. This results in non-linear diffusion equations.
Low temperature scantzhg electron microscopy
667
With the assumptions discussed above the response of the superconducting film to
the electron beam irradiation is obtained by solving the heat diffusion equation (thermal
treatment) or the coupled diffusion equations of the generated excitations (non-thermal
treatment). The sharply focused electron beam modulated at a frequency 0 / 2 x can be
modelled by
)~is
where6 is the Diracdelta function and p ( / ) = ~ ( x - ~ , ( t ) ) ~ + ( y - y ~ ( r ) thedistance
from the beam focus in the film plane. For the calculation of the response F t h e electron
beam is assumed to have a fixed position. The response for a slowly moving beam is
obtained by moving the response calculated for the fixed beam accordingly.
For the perturbation according to equation (7), both for a thermal and a nonthermal treatment the response of the superconducting film is given by [ 5 , 61
Here, KO is the modified Bessel function of zero order.
If a thermal treatment is appropriate, the response F(p, t ) corresponds to a temperature rise 6T(p, [) and we have [S,61
T=-
C,d
a
Here, A is the thermal healing length and z is the thermal response time; K F , CF
and d a r e the thermal conductivity, the specific heat capacity, and the thickness of the
superconducting film, respectively.
If a non-thermal treatment is appropriate, the response F ( p , t ) of the superconducting film corresponds to an excess quasiparticle density SNq(p,/) and we have 16,1041
Here, Iq and Iphare the quasiparticle and phonon generation rates due to the electron
beam irradiation, D, is the quasiparticle diffusion constant, A is the quasiparticle diffusion length, and 7 the effective decay time of the excess quasiparticle density.
668
R Gross and D Koelle
(i) low fieqequency limit (wrcc I )
In the low frequency limit the response is given by
that is, the response is following the perturbation quasistatically without any phase
shift. The spatial decay of the response is given by the Bessel function, which decays
cc exp(-p/A) for p > A and diverges logarithmically for small p . The divergence results
from the assumption of a pointlike electron beam perturbation. Practically, it is smeared
out over the radius of the perturbation given by the beam electron range R.
(ii) high frequency h i i t (an>l)
In the high frequency limit the response is composed of a
p((P,I )
where the
Fdc(P?
DC component
/) +
/)
DC
and an
AC
component,
(16)
is given by Wo/2Ko(p/A) and the AC component is given by
In contrast to the DC component, the AC Component extends to a frequency dependent
distance
= A/&
(18)
from the point of the beam focus. The AC response has a phaseshift of lr/4 relative to
the perturbation and a frequency dependent amplitude. The AC response is dynamically
localized near the perturbation. This can be used to improve the spatial resolution of
the LTSEM imaging technique. Detecting only the AC signal by a synchronous detection
technique, information for a smaller sample area and, hence, a better spatial resolution
is obtained [S, 6, 104, 1291.
The conditions for which a thermal or a non-thermal treatment has to be used for
the description of the electron beam induced non-equilibrium state are discussed in
detail in 161 and [104]. Typically, for most superconductingfilm/substrate combinations
the decay length A ranges between about I and IOpm and the decay time r between
about IO" and lob's [6,104]. In experiments a spatial resolution of about 1 pm could
be obtained in most cases using high frequency beam modulation techniques
CfGSOMHz) [104, 1291.
The expressions given above have been found to describe well the response of thin
films of conventional BCS superconductors perturbed by a focused electron beam. They
also should be applicable for high-T, films. However, the detailed nature of the excitations in the high-T, materials and the different characteristic times are not well known.
As discussed above, a thermal treatment o f the response represents a good approximation for high-T. films. For YBa2Cu307-s, we have ~ ~ ( 4K). 2 1 mW cm-* K-' and
C ~ ( 4 . K)13
2
mJ crK3 K-' 1118-1251. The heat transfer coefficient to the frequency
used substrate materials such as SrTiO,, LaA103, MgO, NdGa03 etc is expected to be
of the order of I W cm-2 K-' at 4.2 K [ I 13-1 171. Then, for a 500 nm thick film we
obtain A - 3 p m and z z 150 ns. Similar values are obtained for other high-T, materials.
That is, in the low temperature regime the spatial resolution of LTSEM for the investigation of high-T, films will be in the pm regime. Since the thermal response time is quite
Low teinperature scanning electron microscopy
669
short, beam modulation frequencies of several MHz are required to improve the spatial
resolution.
The anisotropy of the high-Tcmaterials has been neglected so far. For YBaaCu30,-s
we have K , ~ x ~[126]
K ~ and, hence, Aabiby2Ac, that is, the thermal healing length is
different along the ab-plane and the c-axis direction of the material. Hence for an
YBaCuO film with the c-axis parallel to the plane of the film the pointlike electron
beam perturbation results in an elliptical temperature profile. In contrast, for a film
with the c axis perpendicular to the film surface a circular profile is obtained. For the
case of a non-thermal treatment a similar result is expected, since the diffusion
coeficients of the quasiparticles and phonons should be anisotropic.
4.3. Response at higher teinperatures
According to acoustic mismatch theory the thermal boundary resistance between the
superconducting film and the substrate increases proportional to about T 3for temperatures well below the Debye temperature. Therefore, at higher temperatures (above about
IO to 50 K) the thermal boundary resistance between the superconducting film and the
substrate becomes small compared to the thermal resistance of the film. Hence, in the
following discussion the solid-solid thermal boundary resistance will be neglected for
simplicity. We note that this temperature regime mainly applies to high-T, films.
The non-thermal treatment of the electron beam perturbation of a superconducting
film given above for the low temperature case in general can be extended to the higher
temperature regime. However, at higher temperatures the quasiparticle-phonon and
phonon-phonon scattering rates, which could be neglected at low temperatures, become
important. Furthermore, the phonon escape rate becomes by several orders of magnitude higher. Due to the enhanced phonon escape rate, gradients of the quasiparticle
and phonon density in z-direction have to be taken into account allowing no longer a
two-dimensional analysis [ 1041. Therefore, the solution of the coupled diffusion equations of quasiparticles and phonons is complicated and the derivation of expressions
for A and r more difficult than in the low temperature regime. However, due to the
enhanced scattering rates the generated excitations thermalize very fast and the thermal
treatment of the non-equilibrium state represents a good approximation in the higher
temperature regime. Therefore, in the following we only give a thermal treatment of
the sample response.
The perturbation of a thick ( d > R ) and a thin ( d < R ) superconducting film by a
focused electron beam of diameter a is shown in figure 7. Using a thermal treatment
of the beam perturbation, the response of the superconducting film is obtained by
solving the heat diffusion equation
Here K F , S is the thermal conductivity and CF,s the specific heat of the film and the
substrate, respectively. A ( x , y , z, t ) is the energy supplied by the electron beam per unit
time and volume. Since the film surface is exposed to vacuum, adiabatic boundary
conditions at the top side of the film have to be assumed. In most LTSEM experiments
the temperature rise caused by the electron beam irradiation is small. Therefore, in the
following the temperature dependence of K F , S and CF.Sare neglected solving the heat
diffusion equation. Taking into account the temperature dependences of K and C results
610
R Gross and D Koelle
Figure 1. Electron beam irradiation of a thick ( 0 ) and a thin (6) superconducting film. The
beam power is assumed to be deposited homogeneously in a hemisphere of diameter R.
in a highly non-linear thermal diffusion equation [57]. Furthermore, with a<<R the
finite diameter of the electron beam can be neglected.
We first consider the solution of the heat diffusion equation for a very thin (dc<R)
o r a very thick (d>>R ) film. For these cases the temperature field generated by the beam
irradiation is determined only by the thermal parameters of the film or the substrate,
respectively. Again, the electron beam is modelled by a pointlike perturbation modulated periodically in time as
Po
1
p(r,f)=-(I+e'~'))--(r(f))
2
r(t)
(20)
where r(f) = J ( x - x o ( f ) ) 2 + ( y - y o ( l ) ) 2 + ( z - z o ( f ) ) 2 is the distance from the beam focus
on the film surface (zo=O). As in the previous section the scanning speed of the electron
beam shall be slow enough so that transient effects related to the scanning process can
be neglected. Furthermore, an isotropic thermal conductivity is assumed. With these
assumptions the temperature field is obtained as
F(r, f)
Po
= __
{ I +exp[ -kr+ i(wr-kr)]}
Z~KFS~
(21)
with
k = I/ I , =
&
.
(22)
Here, DF.s= K ~ , ~ /isCthe~ thermal
. ~
diffusivity of the substrate or the film, respectively.
For the case of a thick film (figure 7 ( a ) ) and a thin film (figure 7(h)), we have K = I C
and K = K~ in equations (21) and (22), respectively. For the very thin film (d<cR) the
temperature field is determined by the thermal properties of the substrate. Due to the
small thermal boundary resistance, the film temperature is given by the temperature of
the adiabatic surface of the substrate.
For d - R a simple solution of equation (19) is obtained only for K ~ Y K ~ N I CIn.
this case the temperature field is given by equation (21) independent of the film thickness. For d - R and K ~ # Kthe~ solution of equation (19) becomes more complicated.
A detailed treatment of this case is given elsewhere [130]. I n most cases the thermal
~
Low teniperature scanning electron microscopy
67 I
conductivities of the substrate and the superconducting film are of the same order of
magnitude and the assumption K ~ K~
= represents a good approximation for a qualitative analysis. However, for a quantitative evaluation of the electron beam induced
perturbation a more detailed analysis is required [131-134].
The solution of the heat diffusion equation represents a highly damped thermal
wave. It is attenuated to I/e after propagating a single reduced wavelength I,, which
is usually referred to as the thermal diffusion length. For YBa2Cu,0,-d, I, is about
1.5pml.Jat T=77 K, that is, for a typical beam modulation frequency of
10 kHz we have I, 15pm. Note, that similar to the low temperature case the oscillating
part of F(r, 1 ) is dynamically localized near the beam focus at high modulation frequencies. Again, this can be used to improve the spatial resolution of the LTSEM imaging
technique, if only the oscillating part of the signal is measured by a synchronous
detector.
F(r, t ) diverges for small r due to the assumption of a pointlike electron beam
perturbation. This divergence disappears taking into account the finite volume for the
energy deposition of the electron beam. Assuming that the beam energy is deposited
homogeneously in a hemisphere with diameter R (see figure S), the solution of the heat
diffusion equation is obtained as
-
(
,","I
F(r,t)=F, 1 - 7
F(r, t ) =-Fo R [ 1 +e'"']
3r
[l+e'"']
for r I 4R / 2
for I r l > R/2
123)
(24)
with
for beam modulation frequencies o r < < l . Here T = R ' / ~ & , ~is the time required to
obtain a steady state response inside the hemisphere of diameter R after switching on
the beam perturbation. If the beam is modulated at a frequency ~ < < I / T , the response
can follow the perturbation quasistatically. The thermal diffusion length I, is large
compared to R in this case and the spatial decay of F(r, t ) is mainly determined by the
I / r term. For a YBaCu,07-a film, z = @/4DF lypically ranges between IO and 100 ns
in the temperature range between 10 and 90 K. That is, for commonly used beam
modulation frequencies in the kHz regime a quasistatic response is obtained.
Figure 8 shows the normalized temperature field F(r) for different beam voltages.
The curves have been calculated according to equations (23) and (24) for a
YBa,Cu,07-s film on SrTiO,. The spatial decay length of F(r), that is, the width of
the electron beam induced temperature field is mainly determined by the beam electron
range R, which, in turn, is determined by the beam voltage. Since the width of the
temperature profile determines the spatial resolution of the imaging fechnique, an
improved spatial resolution can be obtained by using lower beam voltages. For a beam
voltage of 10 keV a spatial resolution of about I p m is expected for the investigation
of YBa2Cu307-bfilms. This agrees well with the experimentally observed spatial resolution limit [29-341.
The absolute magnitude Fo of the beam induced temperature rise can be varied over
a wide range by varying the beam power P. Note that Fa will vary with temperature
R Gross and D Koelle
672
-a
-4
0
r
4
8
(”
Figure 8. Normalized response lunction F(r) in a Y B a D ~ , 0 7 - s61m for diRerent beam
voltages.
for a constant beam power, since K depends on temperature. The dependence of Fo on
K and P has been shown in detail in [34]. Typically, with the available beam power of
commercial SEMS Fo can be varied from well below 1 mK up to several K.
Finally, we point out that in the case of a non-thermal treatment the diffusion
equation for the quasiparticles and phonons in a semi-infinite superconductor has the
same form as equation (19). Hence, if the diffusion of the quasiparticles and phonons
is completely decoupled the resulting excess quasiparticle and phonon profile generated
by a pointlike electron beam perturbation is given by equation (21) replacing & / 2 n ~ ~ , ~ r
by Iq,.ph/2~Dqp.phr
and equations (23)-(25) with FO= 3~qp,ph/2nDqp,p~R.
Here, Iqp.phis
the quasiparticle/phonon injection rate due to the electron beam irradiation and Dqp.ph
is the quasiparticle/phonon diffusion constant.
5. The signal
5.1. General aspects
In the previous sections we have discussed the perturbation P and the resulting response
F of the superconducting material. In this section we analyse the signal S and its
relationship to the local properties of the superconducting sample at the location of
the beam spot. If the signal is proportional to the response F o f the sample, the resulting
signal for a slowly moving electron beam is given by
r
J
S(x, y, f)= dx’ dy‘ d:’
.”
J
dt’ H(x, y,
2,
x’, y’, z‘,
f-
f‘)I:(x’, y’, z‘, t’)
(26)
where If is the linear response function. By causality, H=O for f ’ - t < O . With N = O
for a beam position outside of the sample area the integration in equation (26) extends
only over the sample area. If I: changes sufficiently slowly in time, the signal responds
instantaneously to Fand the time integration in equation (26) can be omitted. Expressions for the LTSEM signal obtained in the study of superconducting thin films, Josephsoil junctions, and weak links are derived in sections 6, 7 and 8 . In this section only
general aspects and the signal detection methods are discussed.
Low temperature scanning electron nricroscopy
613
For a perfectly homogeneous sample the linear response function Hwill be spatially
homogeneous and S ( x , y , t ) will not depend on the beam position. Note that a spatially
varying signal for a perfectly homogeneous sample can be caused by a spatially varying
response F. Spatial variations of F can be introduced by spatial variations of the
absorbed beam power due to a varying electron backscattering coefficient or spatial
variations of the thermal sample parameters. The homogeneity of the electron backscattering coefficient can directly be controlled by measuring the spatial variation of the
intensity of the backscattered electrons (BEimaging). In contrast, the homogeneity of
the thermal parameters is difficult to measure. However, they are expected to have
good spatial homogeneity for high quality superconducting films and interfaces to the
substrate. In the following ana1ysis.a spatially homogeneous response F is assumed.
For superconducting samples the dependence of the signal on F can be highly nonlinear, since the response F can carry the superconductor across the phase boundary
between the superconducting and the normal conducting state. In this case, the resistivity of the specimen changes in a highly non-linear way. The dependence of the signal
on the response can then no longer be expressed in terms of a linear response function.
In most LTSEM experiments only small perturbations are used so that the electron beam
can be considered as a passive probe. For sufficiently small perturbations equation (26)
is always applicable. A specific problem related to superconducting samples is the fact
that there can be non-local signal contributions beyond the usual local signal contribution resulting from the change of the local sample properties at the position of the
electron beam. The importance of the non-local signal contributions has been shown
in our work on superconducting tunnel junctions [ I ] , 13, 151. The non-local signal
contributions are caused by the macroscopic phase coherence in superconducting specimens. In the presence of non-local signal Contributions the interpretation of the
measured signa! can be quite complex.
5.2. Measuring techniques
Beyond the standard SEM imaging methods such as secondary or backscattered electron
imaging in LTSEM experiments various other imaging methods are used. In the LTSEM
study of superconducting samples the most commonly used technique is the EBIC and
EBIV method which is described in detail below. We briefly discuss also the voltage
contrast method and some special measuring techniques.
5.2.1. EEIV and EBIC inethod. In the EBIV (EBIC)method the electron beam induced
change of the global voltage (current) across the current (voltage) biased sample is
measured as a function of the coordinates of the beam focus . Figure 9 shows the
typical sample geometry used for the investigation of superconducting films by LTSEM.
The superconducting film is deposited on a proper substrate and patterned into a
microbridge using standard microfabrication techniques. Current and voltage leads can
easily be attached to the film for electrical measurements. During the L r s m experiments, the surface of the film is scanned directly by the focused electron beam whereas
the bottom side of the substrate is in direct contact with the cooling reservoir. In most
cases superconducting specimens have low electrical resistance and the external circuit
represents a current source. Then, the measured signal is the electron beam induced
674
R Gross and D Koelle
ELECTRON BEAM
Figure 9. Typical sample geometry used for the investigation of superconducting films by
LTSSEM.
change 6 V of the sample voltage
(EBIV
imaging) given as
6 v(X, J’)
= 6Rd.G Y ) ~ B .
(27)
Here, (I,J?) is the coordinate point of the beam focus on the sample surface, IB the
applied bias current, and 6R, the beam induced change of the film resistance. If the
external circuit represents a voltage source, the measured signal will be the beam induced
change of the sample current
provided that 6R,<<RR.Here B’C is the applied bias voltage and R, the total sample
resistance. Usually, the electron beam perturbation is periodically modulated resulting
i i i a periodically modulated response signal, which is detected phase sensitively by a
lock-in amplifier. This technique allows the detection of voltage (current) signals as
small as only a few nV (PA). A two-dimensional voltage ( 6 V ( X ,p)) or current ( 6 I ( x ,p))
image of the sample is obtained by scanning the sample and measuring the electron
beam induced change o f the sample voltage or current simultaneously. Equations (27)
and (28) show that a finite signal is obtained only if the electron beam perturbation
causes a non-vanishing change of the sample resistance. For a superconducting sample
this is the case if the sample already is in the resistive state without electron beam
irradiation or if it is switched into the resistive state by the beam perturbation. That
is, the bias point (sample current and temperature) has to be chosen such that the
sample resides in or close to the resistive state.
A rough estimate of the magnitude of the signal for a spatially homogeneous sample
gives
Here, W is the width, L the length, and d the thickness of the investigated thin-film
sample, ,C
, is the sample volume perturbed by the electron beam. The estimate of the
expected signal shows that the sample dimensions should not be much larger than the
perturbed sample volume. In practice, typically C,J WLd>
is required. That is,
with C,,,-1 to I O j”,
the sample volume should be smaller than about
mm3.
Low temperature scanning electron microscopjr
675
Therefore, the LTSEM imaging technique is restricted to small volume samples such as
patterned thin film samples. The study of bulk samples is difficult. Note, that for the
above estimate a spatially homogeneous sample has been assumed. For an inhomogeneous sample the volume WLd has to be replaced by the volume actually contributing to
the sample resistance. This volume can be much smaller than the whole sample volume.
Figure IO. Sketch of the sample geometly used for the invesligalion of superconducting
tunnel junctions by LTSEM.
Figure I O shows the typical experimental configuration used for the LTSEM imaging
of planar type superconducting tunnel junctions. The tunnel junction is formed by two
superconducting electrodes SI and S2 of widths Wl and W2and thicknesses 4 and dz
respectively. The two electrodes are separated from each other by a thin, electrically
insulating tunnelling barrier. The thin-film structure is deposited on a proper substrate.
During the LTSEM experiments, the thin film structure is scanned directly with the
electron beam. Usually, the entire resistance of a superconducting tunnel junction is
related to the tunelling barrier, that is, the junction electrodes do not contribute to the
resistance. Then, for a current or voltage biased junction the electron beam induced
voltage or current change is given by equation (27) and (28), respectively, replacing R,
and 6R, by Rtunand 6R,,. . Here, R,"" is the tunnelling resistance and 6R,.. its electron
beam induced change. The magnitude of the electron beam induced signal is estimated
to be
Here, A,, is the perturbed and A,,= Wl W, the tunnelling area of the junction. In
order to be able to detect the electron beam signal usually APr/A,""2 IO-' is required.
That is, with Ap,-I to IOpmZ the tunnel junction area should not be larger than
about 1 mm2.
Figure I I(a) schematically shows the measuring set-up used for the recording of
the LTSEM voltage images. The sample is biased at a constant current and irradiated
by a periodically modulated electron beam. The electron beam induced voltage signal
is amplified by a differential amplifier and detected by a Lock-in amplifier synchronized
by the beam blankirig unit. The output voltage of the lock-in amplifier is used to control
the brightness of the SEM video screen and stored by the image processing system. We
note that this method only can be applied, if the superconducting sample is in or close
to the resistive state under the operative conditions (temperature, bias current). No
676
R Gross and D KoeNe
11
Ref. In
(beamblankingunit)
Figure 11. Experimental set-up used lor the measurement of the spatial variation of the
quasiparticle ( a ) and pair lunnelling current density ( b ) .
beam induced voltage signal can be detected, if the sample stays in the zero voltage
(superconducting) state upon electron beam irradiation.
In a different measuring method (figure 1 I(b)), which in the following is referred
to as maximum critical current detection (MCCD) [13, 1951, the electron beam induced
change of the critical current, 61c, of a superconducting film or tunnel junction is
measured. In this technique the sample current is increased at a constant rate with the
electron beam irradiating the sample at the position ( x , y ) . Then, the current value at
which the sample voltage exceeds a certain threshold value, V,', is detected and stored
using a voltage comparator and a sample&hold unit. The voltage comparator triggers
the sample&hold unit and the switch connecting the periodic sweeper and the current
source, whenever the sample voltage exceeds the threshold value. Afterwards the periodic sweeper and the sample current are reset to zero. This measuring cycle is repeated
periodically at a rate of up to 10,000 measurements/s. Subtracting the critical current
without electron beam irradiation from the value obtained with electron beam irradiation, the electron beam induced change 6I.(x, y) is obtained. A two-dimensional critical
current image 6I.(x,y) is obtained by scanning the electron beam across the sample
and measuring FI. permanently. The magnitude of SJ,(x,y) is proportional to Cp,/
U'd for thin film samples and AFr/A,"" for tunnel junctions, respectively.
By the electron beam irradiation a finite electical current is injected locally into the
superconducting film or tunnel junction at the beam position, This current typically
ranges between 1 pA and 1 nA and is by several orders of magnitude smaller than the
sample current in most experiments. To estimate the local current density generated by
the beam current, the beam current is assumed to be injected homogeneously into a
hemisphere having a diameter equal to the beam electron range R. With R- 1 pm, the
~~.~.
~~
Low temperature scanning electron microscopy
677
current density through the surface of the hemisphere can be estimated to be smaller
than 1 A ~ m - Since
~ . this current density is small compared to the typical bias current
density generated by the external circuit, the effect of the current injection by the
electron beam usually can be neglected.
In the following sections the electron beam induced voltage (current) signal will be
estimated for superconducting films (section 6), Josephson tunnel junctions (section 7),
and high temperature superconducting weak links (section 8 ) of various geometries.
Typical experimental result of the LTSEM study of such samples are shown. An important feature will be the interpretation of the measured LTSEM signal. Dependent on the
different experimental parameters such as bias current, sample temperature, or electron
beam power we will discuss what kind of information on the local sample properties such
as the critical current density, critical temperature, resistivity, tunnelling conductivity etc
is contained in the detected signal. In our analysis, we will restrict ourselves to the case
that the response F(r, t) can be considered as a purely thermal response and that its
magnitude and characteristic decay length is spatially homogeneous. In most cases the
results of the thermal treatment qualitatively agree with those of a non-thermal treatment. Therefore, the thermal treatment usually represents a good approximation and
is sufficient as long as no quantitative analysis is required. The external circuit is assumed
to represent a current source. The derivation of expressions for the voltage biased case
is straightforward.
5.2.2. Voltage conlrust method. The method of voltage contrast in a SEM makes it
possible to quantitatively visualize the local distribution of static or low frequency
voltage differences on conducting lines of a circuit [135-142]. The method is based on
the influence of the local sample voltage on the secondary electron image. Above regions
with positive voltage, the resulting electric fields attract a portion of the secondary
electrons back to the sample surface. With increasing voltage more and more secondary
electrons are attracted. That is, those regions with more positive voltage appear darker
in the secondary electron image. In this way the voltage distribution in conducting
circuits can be imaged. Of course, the voltage contrast method can be applied also at low
temperatures. Recently, it has been used to study CMOS circuits and high temperature
superconducting films at liquid nitrogen temperature [143, 1441.
The voltage changes which can be detected by the voltage contrast method Iypically
are well above 100 mV [135]. This voltage resolution is sufficient for the analysis of
semiconducting circuits. However, in superconducting circuits, the voltage differences
are usually less than the gap voltage, i.e., typically well below IO mV. Therefore, in
most cases the voltage resolution of the voltage contrast method is not sufficient for
the study of superconducting circuits and we do not discuss this method in more detail.
In contrast, in the EBJV method described above, electron beam induced changes of the
sample voltage as small as a few nV can be detected. Comparing both methods one
should keep in mind that their principle is quite different making their direct comparison
difficult. Whereas in the voltage contrast method the distribution of static voltage
differences in the sample is directly imaged, in the EBIV method the sample is perturbed
locally and the induced change of the global sample voltage is detected.
5.23. OIher methods Beyond the EBIV/EBIC and the voltage contrast method there are
various other LTSEM imaging techniques yielding spatially resolved information on
superconducting thin-films and circuits. A particularly interesting method that recently
has been applied to the study of arrays of superconducting tunnel junctions [216] is
678
R Gross and D Koelle
the ‘microwave’ imaging. In this method the change of the microwave power emitted
by the Josephson junction array is recorded as a function of the coordinates of the
beam focus on the array. As a microwave detector a single Josephson junction coupled
capacitively to the array can be used. A two-dimensional ‘microwave’ image is obtained
by scanning the array and measuring the change of the emitted microwave power
simultaneously. This imaging technique can be applied to get information on the phase
locking of the individual Josephson junctions within the array [216]. A further application of this method is the study of standing wave patterns in superconducting microwave
circuits [26].
6.
LTSEM
study of superconducting films
The critical current density of a superconductor is defined as the highest current density
the superconductor can support without measurable resistancc, There are three mechanisms controlling the critical current density in superconducting materials: Depairing,
depinning and decoupling. In this section we will consider samples containing no socalled weak links. Here, weak link stands for the whole variety of structures resulting
in a weak coupling of the superconducting order parameter [145-147]. The critical
current density of a superconducting sample containing no weak links is determined
either by the depairing critical current density [ 148, 1491 or the depinning critical current
density [150, 1511. The depairing critical current density is reached, if the velocity of
the Cooper pairs exceeds a critical value. The depairing critical current density for a
clean superconductor at T=OK is given by Jf’=n*e*fi/nm*&,, where n*, e* and nz*
are, respectively, the density, the charge and the effective mass of the Cooper pairs, and
tothe coherence length. A similar expression can be inferred from the London theory
by calculating the current density at which the kinetic energy of the charge carriers
equals the condensation energy. Jn the London model we have Jfp=Bc(T)/p,dL(T),
where B, is the thermodynamic critical field and AL the London penetration depth [ 149,
1511. The depairing limit of the critical current density usually is reached only in extreme
cases [I5ll. For type I1 superconductors in the mixed state [I511 a finite resistance is
obtained already at current densities well below the depairing critical current density.
The resistance is caused by the motion of magnetic flux lines in the sample which are
generated either by an external magnetic field or the self-field of the sample current,
The flux lines start to move, if the Lorentz force FL= J x QoB/B per unit length acting
on the flux lines exceeds the pinning force Fp. I n this case, the critical current density
is determined by depinning. Note that in high temperature superconductors grain
boundaries can act as weak links (for recent reviews see 11521 and [153]). Therefore,
the critical current density of polycrystalline films containing a high density of grain
boundaries is no longer given by the depinning but by the usually much smaller decoupling critical current density [148, 1501. A high-T, film containing a high density of
grain boundaries should be treated as an array of coupled Josephson junctions (see
sections 7.4 and 8.4).
In this section we consider type I1 superconducting films for which the critical
current density is limited by depinning. If such a film is biased at or above its critical
current value, the current will be distributed over the entire cross-sectional area of the
sample so that the voltage drop along the direction ofcurrent flow is minimized. Sample
regions with stronger pinning sites will carry a higher current density and vice versa.
Low temperature scanning electron microscopy
679
Hence, it can be assumed that the critical current of some cross-sectional area perpendicular to the current direction is reached, if every part of this cross-section is carrying its
critical current density. This is in contrast to superconducting samples containing weak
links as discussed in sections 7 and 8.
6.I . One-dimensional case
If the width and the thickness of the investigated sample is small compared to the
characteristic decay length of the electron beam induced temperature field 6T(r, I ) , the
sample can be treated as a one-dimensional object. In this case the electron beam
induced change of the sample resistance is given by
~ R , ( xt,) =
s
dt’-
i d j
dx [P(x,t - t ’ , Tb+ST(x, I , ) , J B ) - P ( x ,
t-l’,
T~,JB)]
(31)
where Tb is the sample temperature without electron beam perturbation and & = I B /
Wd is the bias current density. In most experiments the scanning speed of the electron
beam and its modulation frequency are small and the time integration in equation (31)
can be omitted. Then, SR,(x, t ) has the same time dependence as ST(x, t ) .
For a superconducting film 6Rs will be zero as long as the sample stays in the zero
resistance state with the electron beam perturbation switched on. Increasing either the
temperature or the bias current density, a signal vr,ill be obtained for T> T:(&) or
JB>J,(T). Thesamplepositionswhere T>T:(x) orJB>JE(x)arereferred toascritical
regions. Only these critical regions yield a measurable signal. As will be shown below,
the local critical temperature or critical current density of a distinct sample region can
be obtained by varying the temperature or the bias current and by measuring at what
value of T or JB a signal is detected for the first time. The ‘critical’ temperature T:
obtained in this way may differ from the mean field transition temperature T.. In
particular, for the high temperature superconductors T: can be considerably smaller
than T, due to the presence of thermally activated processes such as thermally activated
phase slippage [154], flux flow, or flux creep [155, 1561.
6.I . I . Measurement of the local resistive transition. For small perturbations equation
(3 1 ) can be simplified to
For a current biased sample the measured signal is 6 V = l ~ 6 R ,that
, is, the static linear
response function I 1 appearing in equation (26) is J B [ a p ( x ) / a T ] .
Assuming that the temperature derivative of the resistivity is about constant within
the perturbed sample region and approximating the integral over the temperature profile
by 6T0(l)h,the electron beam induced voltage signal for a current biased sample is
obtained to
Here, 6To(x.t ) is the electron beam induced maximum temperature rise (equation (25))
and A the characteristic width of the electron beam induced temperature field. Usually,
6To is constant for all beam positions, i.e. 6To is independent of x .
680
R Gross and D Koeile
-1'
87
'
*
aa
'
'
a9
'
'
90
'
'
91
'
'
92
'
I
93
T (K)
Figure IZ Electric resistivily p ( q ) (solid lines) and eleclron beam induced voltage signal
~ V ( . T ~(broken
.)
lines) versus lemperature for three difTerent sample positions I,wilh different critical temperatures 7'&,).
According to equation (33), S V ( s ) is proportional to ap(.r, JB)/2Tfor small perturbation. Therefore, by measuring the temperature dependence of SV(x) we obtain
Sp(x, T);aT. The local resistive transition p(x, T)can be obtained by integration. The
maximum of the 6 V ( x , T) curve corresponds lo the temperature T:(x), at which the
local p(x, T ) curve has its steepest slope. This is shown schematically in figure 12,
where typical p ( x , . T ) curves of three sample positions xi with different values of T:
(x,) are shown together with the resulting voltage signal 6V(xi). For small bias current
density T:(xj) is close to the thermodynamic transition temperature T,. That is, for a
fixed electron beam position x, the local critical temperature at the coordinate point x,
can be determined by recording the temperature dependence of 6 V ( x j ) .The distribution
of the critical temperature along the superconducting film is obtained by scanning
the electron beam along the sample and recording the beam induced voltage signal
simultaneously. The highest signal is detected at those positions xi where Tb%T:(x,).
By recording several voltage images at different temperatures the distribution of the
critical temperature along the film is obtained.
Figure 13 shows the temperature dependence of the electron beam induced voltage
signal S V ( x ) recorded at five different positions along a c-axis oriented, epitaxial
YBa2Cu307-bfilm (L=180pm, W = IOpm, d=200nm). It is clearly visible that the
local 6 V(x, T ) curves are shifted relative to each other to a maximum shift of about
200 mK, whereas the width of the curves does not vary significantly. This gives clear
evidence that the different sample regions have different 'critical' temperatures T:. The
curves were recorded using an electron beam power that causes a maximum temperature
rise 6To of about 100 mK. Note that the magnitude of 6To directly determines the
temperature resolution of the measurement.
According to equation (33) the voltage signal increases proportional to the bias
current density. However, increasing the bias current may result in a considerable
broadening of the local resistive transition curves and, hence, of the SV(x, T)curves.
In particular, this is the case for high-T, films due to the presence of dissipative processes
such as thermally activated flux flow and creep [ 155, 1561. That is, for large bias currents
the zero resistance critical temperature is determined by the onset ofdissipative processes
and may be much lower than the thermodynamic critical temperature.
In the case of a linear response function the spatial resolution for the imaging of
the local resistive transition is determined by the characteristic decay length A of the
Low terizperalure scanning electron microscopy
681
five locations on a
cmP ( a ) and J,=
electron beam induced temperature field and does not depend on the absolute magnitude
of 6T.Note, however, that for a non-linear response function H , the spatial resolution
also depends on the magnitude of 6T. For illustration, we have calculated the 6 V ( x )
curves according to equation (31) using p(x, T ) = p ( T ) G ( x - x , ) . For the temperature
dependence of p a linear dependence (linear response) and a step-like dependence (nonlinear response) is assumed. The temperature field was taken according to equations
(23) and (24). The result is shown in figure 14. For the linear p ( T ) dependence the
same normalized 6 V ( x ) curve is obtained for all beam powers and its width is equal
to the width of the temperature field ST(x). For the step-like p ( T ) dependence, in
contrast, the width of the normalized 6 V ( x ) curves and, hence, the spatial resolution
strongly depends on the beam power. Figure 14 clearly demonstrates the reduction of
the spatial resolution with increasing beam power. The effect of the beam power on
0.0 '
-8
I
'
-4
x--xi
0
(!Jm)
4
a
Figure 14. Normalired voltage signal bV(r)/6V(.r,)calculated according to equation (31)
using p(x, T ) = p ( T ) G ( x - x l ) . For p ( T ) a linear dependence (linear response function,
solid line) and a step-like dependence (nonlinear response function, broken lines) was used.
682
R Gross and D Koelle
the spatial resolution for the investigation of YBa2Cu,07-6-films by
shown recently [31, 341.
LTSEM
has been
6.1.2. Measurefttentofthc local critical current density, For operating temperatures well
below T,(ST,<<T. - T,)the general condition for the detection of a non-vanishing signal
(6R,>O) directly follows from equation (31) to
J~>Jc(xc)-SJc(x,).
(34)
For smaller bias currents the superconducting film stays in the zero resistance slate
both with and without electron beam irradiation resulting in 6R,=0.
For small perturbation the electron beam induced change of the sample resistance
can be expressed as
Here, small perturbation means that the election beam change of the critical current
SJ,(x) is small compared to J,(x). For a current biased sample the measured signal is
6 V = I B 6 R , , and the static linear response function H of equation (26) is J B [ a p ( x ) /
a~,(x)i[ a ~ , ( x ) i a ~ i .
In most cases one can assume that ap/aJ, and 8JJZTare constant in the perturbed
sample region. Then, approximating the integral over the temperature profile by ASTO,
we obtain
In figure 15 examples of typical p ( x ) and ap(x)/aJ,(x) versus Jcurves for three different
sample positions x, with different critical current density values Jc(x2) are shown.
According to equation (36) the voltage signal is directly proportional to c?p/aJ,. Figure
15 shows that for small perturbation dp(xj)/aJc(x,)is zero for J B < J c ( x # ) .Therefore,
increasing the bias current density in small steps and recording the electron beam
induced voltage signal simultaneously, a signal at the position x, will be detected for
the first time at JB=Jc(x,).In this way the local critical current density value at the
201
-5
12
'
I
14
'
I
16
'
l
i
18
Y
I
20
J ( arb. units )
Figure 15. Electric resistivity p ( x , ) (solid lines) and 8p(x,)/aJc(x,)(broken lines) versus
currenf density for three &Rerent sample positions x, with different critical current density
values JC(xt).
Low temperature scanning electron nricroscopj,
683
position J,can be determined. The spatial distribution of the critical current density
along a superconducting film usually is obtained by recording a series of voltage images
for increasing bias current values. For a distinct current value only those sample regions
where J n > J , ( x ) will yield a measurable signal. Of course such measurement can be
repeated at different sample temperatures to obtain J,(x, T ) . Note that the magnitude
of the measured voltage signal depends on the value 8.JC/8T.In most cases, JJ,/dT#O
for all temperatures T i T, allowing the measurement of J,(x) for all temperatures
below T,. For example, the temperature dependence of the critical current density of
epitaxial YBa2Cn107-dfilms is almost linear.
For large perturbations the electron beam induced signal has to be calculated according to equation (31). Since SJ,(x) can be large, a signal is detected already forJnc<J,(x).
Furthermore, for temperatures close to T,, we can have 6To?Tc-Tb and hence
SJ,-J,. This shows that it is difficult to determine the local value of the critical current
density using high beam power and operating temperatures close to T,. For small
perturbation the spatial resolution for the imaging of J&) is determined by the characteristic width A of the temperature field 6 T ( x ) . In the same way as discussed above,
for large perturbation the spatial resolution also depends on the absolute magnitude
of the beam induced temperature increment, if the response function is non-linear.
Figure 16. LTSEM voltage images showing the spatial variation ofthe critical current density
along a quasi-one-dimensional YBaiCulOl-s microbridge at T=77 K and zero applied
magnelic field. Bright regions correspond to sample regions where the critical current density
is exceeded at the respective bias current density. Bias current density: (a) J s =
3 . 5 ~ 1 0 ~ A c m - ~( h, ) J e = 4 . 0 x I O 6 A c m ~ ~ ' ,(c)
J B = 4 . 5 x I f 1 6 A c n ~ ' , ( d ) JB=
5.0 Y I f16 A cm .a. Thc electron beam parameten are 26 kV voltage. 0.2 nA current, and
20 kHz modulation frequency (from [321).
Figure 16 shows LTSEM voltage images showing the distribution of the critical
current density along a 1 0 p m wide YBaZCu107-6 line. The thickness of the c-axis
oriented YBa2Cui0,-a film was 100 nm. The different voltage images were recorded
at T=77 K for different values of the bias current density ranging between 3.5 and
684
R Gross and D Koelfe
5 x IO6 A cm-2. Bright regions correspond to those regions yielding a high electron
beam induced voltage signal. According t o the above discussion the images clearly show
an inhomogeneous distribution of the critical current density along the narrow line. At
the lowest bias current density, which corresponds to the critical current density
obtained by a standard four probe transport measurement, the critical current density
is exceeded only in a few sample regions. Only at the highest bias current density is the
critical current density reached in almost all sample regions. Measurements at even
higher bias current densities are prevented by strong Joule heating effects.
6.2. Two-dimensional case
A thin film sample with width and length, which is large, and thickness, which is small
compared to the characteristic decay length A of the sample response F, can be considered a s a two-dimensional sample. In this case the superconducting film can be assumed
to be perturbed homogeneously across the whole film thickness and variations of the
sample properties in z-direction can be negtected. For a two-dimensional sample it is
difficult to give expressions for the electron beam induced change of the sample resistance similar to the one-dimensional case. In contrast to the one-dimensional case, the
current density in the perturbed region does not need t o stay constant even for a current
biased sample. Only the integral of the current density over a complete cross-sectional
area has to be constant. That is, the current in the perturbed region can decrease, if
this decrease is compensated by an increase of the current density in the unperturbed
part of the cross-sectional area, and vice versa. Furthermore, the current will no longer
flow exactly parallel to the x-direction (as in the one-dimensional case) for a sample
Figure 17. Sketch or the sample configuration for the investigation of a two-dimensional
superconducting film by LTSFM.
having a spatially inhomogeneous resistivity. As indicated in figure 17, a cross-sectional
area perpendicular to the current flow can have a complicated shape for a spatially
inhomogeneous sample.
Perturbing the sample at some coordinate point ( x , y ) along the cross-sectional area
A i , which is about parallel to the y-direction in this case, the local resistivity p will
Low temperature scanning elecrron microscopy
685
change to p(x,y)+6p(x,y) and the local current density J(x,y) will change to
J(x,y ) + 6J(x,y ) . For small electron beam perturbation the change of the local electric
field 6E(x,y, 1) is then given by
6 W , J.’, 1) = J(x, Y ) 6 P ( J ,Y, 0 +P(& Y ) W X , y, 0
(37)
if only quantities first order in 6T are retained. For slow scanning speed, the electric
field can be calculated quasistatically, that is, to good approximation V x 6E=0 and
V.6J=O.
We first will consider a sample with a spatially homogeneous resistivity p(x,y ) such
that J ( x , y ) and the electric field E(x, y ) are spatially homogeneous. However, the
derivative dp(x, y)/dT shall be inhomogeneous. For a current biased sample we have
lowdySJ(x,y ) = 0.
(38)
Here, the integration has to be done along the cross-sectional area A iperpendicular to
the current flow. The voltage change 6 V ( t ) is then obtained to
6V(x,y,l)U--
low[
dy
dx JGpfx,y , t ) .
(39)
Similarly, for a voltage biased sample we have
JoL
dx ~ E ( xy,) = o
and the electron beam induced change of the sample current is obtained to
6 r ( x ,y , I ) U -
loLjo”
1
L
dx
dy
6p(x, y , I ) .
For small perturbations we have 6p(x,y, t)--.[I?p(x,y)/I?T]6T(x,
J’. I ) and the static
response function of equation (26) is [ap(x,y)/aT]J/W and [ap(x, y)/aT]Jd/pL for
the current and voltage biased situation, respectively. The integral over the temperature
profile ST(x, y , t ) approximately gives 6To(r)A2,where A is the characteristic decay
length of the temperature field and 6To its maximum value. With these approximations
we obtain
for the case of a current and a voltage biased sample, respectively. In both cases, the
signal is proportional to the electron beam induced temperature rise 6To, to the perturbed sample area -A2, the temperature derivative of the local resistivity, and to the
current density (electric field) for the current (voltage) biased case. In most experiments
the electron beam induced temperature increment 6To and the decay length A are
independent of the beam position. Therefore, variations of the detected signal are related
to variations of the temperature derivative of the local resistivity allowing to image the
spatial distribution of this quantity by LTSEM.
686
R Gross and D Kueile
For a spatially inhomogeneous resistivity the current density and the electric field
will be inhomogeneous. In this case, it is more difficult to derive expressions for the
electron beam induced signal. However, it can be shown that equation (42) and (43)
are approximately valid as long as the spatial variations of the resistivity are small
[157].
In the one-dimensional case the local current density J(x) was constant and equal
to the bias current density. In the two-dimensional case, in contrast, only the integral
of the current density over a complete cross-sectional area A, perpendicular to the
current flow is fixed. The local current density J(n,y) can vary considerably along A,
and cannot be determined by just dividing the bias current fBthrough the cross-sectional
area A , . This makes the interpretation of the detected signal more complicated. In the
same way as for the one-dimensional case, for a two-dimensional superconducting film
no signal will be detected as long as the sample stays in the zero resistance state after
the electron beam perturbation is switched on. Increasing, for example, the bias current,
a signal at a beam position (x, y ) is obtained, only if IBis larger than the critical current
of the irradiated cross-sectional area. For smaller bias current values the electron beam
induced reduction of the critical current in the perturbed region can be redistributed
over the remainder cross-sectional area. Hence, for small perturbation the first signal
is obtained for In>lc(T)or Tb> Tc(fB),increasing either the bias current or the temperature. That is, the criticul regions are complete cross-sectional areas, i.e., they are twodimensional.
6.2.1. .Meusurcnenr uJST(x, y,J.We first discuss whether it is possible to measure the
variation of the critical temperature along a single, isolated cross-sectional area A,
biased at an infinitesimal current. Let us assume that the critical temperature T.(J>)
along some cross-sectional area varies between T,, and Tc2.For a sample temperature
T,, <Tb<Tcz, the resistivity is highly inhomogeneous. The current will flow only in
those parts of the cross-section where T,(y)> T,, i.e. where p(y)=O. The current
density in the regions with lower critical temperature is zero and, according to equation
(37), these regions yield no electron beam induced signal, since both J ( y ) and 6 J ( y )
are zero. Increasing the temperature, a voltage signal will be detected for the first
time, when the sample temperature becomes equal to the highest value of the critical
temperature along thecross-sectional area. Since ap(y)/aTand, hence, 6 p ( y ) i ( a p ( y ) /
aT)6Tois maximum at Tbz.Tc(y),a large signal is expected only for those parts of the
cross-section with the highest critical temperature. That is, it is possible to image these
parts of the film. However, it is impossible to obtain the distribution of the critical
temperature along a complete cross-sectional area.
Up to now we have considered only a single, isolated cross-sectional area. The
actual current density distribution in a distinct cross-sectional area, however, depends
on the current density distribution in the adjacent cross-sections. For example, we have
J(x, y ) = O for a sample region, which is surrounded by regions having Tc<Tb,even if
this region has a critical temperature higher than Tb.Then, according to equation (37)
no signal is expected for this region since J(x, y ) = O and 6J(x, y ) =O. Therefore, using a
small bias current, only the superconducting percolation path along the inhomogeneous
sample yields a signal and can be imaged by the LTSEM technique. Nevertheless, the
imaging of the superconducting percolation path and its correlation to chemical and
structural sample properties represents an interesting application of LTSEM.
6.2,2. Memurertrent o f J c ( x , y). We will now discuss how the current density along a
single isolated cross-sectional area A, can be measured. We will assume for the moment
Low temperarure scanning elecrron microscopy
681
that the sample temperature is well below the minimum value of the critical temperature
along the cross-section and that the electron beam perturbation is small. As discussed
above, the critical regions are now complete cross-sectional areas. In analogy to the
one-dimensional case, the condition for the appearance of an electron beam induced
voltage signal is
fe>1c(x2)- 8fdXJ.
(44)
Here
is the critical current of the cross-sectional area A; without electron beam irradiation
and
is the change of the critical current caused by the electron beam irradiation at the
position (xj,y). Note that for l e = l c ( x ; ) the current density at every coordinate point
y along the cross-sectional area A , is equal to the local critical current density value.
Therefore, at this bias current an electron beam induced signal will be detected along
the whole cross-sectional area. With increasing bias current the different cross-sectional
areas with different critical current values successively generate a signal when the bias
current approaches their critical current value. Therefore, increasing the bias current
in small steps and scanning the sample after each step, the critical current values of the
different cross-sectional areas can be determined. This is completely analogous to the
one-dimensional case.
A typical example for the imaging of the critical cross-sectional areas of a c-axis
oriented, epitaxial YBa2Cu,07-a films is shown in figure 18. The spatial dimensions of
the film are L=500pm, W=70pm, d=60nm. The different voltage images are
recorded for different bias current values IE>fcmin
at T = 83 K. Here, ICmi,
is the critical
current value of the weakest cross-sectional area of the superconducting line. This
value corresponds to the critical current value determined by an electric transport
measurement. The bright areas in figure 18 correspond to those cross-sectional areas
which have a critical current value smaller than the applied bias current. It is evident
that more and more cross-sectional areas become resistive with increasing bias current.
A series of voltage images as shown in figure I S allows the determination of the critical
current distribution along the superconducting film. For the investigated film thecritical
current values of different cross-sections differ by more than a factor of four. The
voltage images in figure I S demonstrate that the weak cross-sectional areas are not
necessarily perpendicular to the superconducting line. As discussed in detail in 1321, for
the sample of figure I S the weak cross-sectional areas arise from scratches in the SrTiO,
substrate due to an improper polishing process. These scratches prevent the epitaxial
growth of the YBa2Cu30-i-s film resulting in a local reduction of the critical current
density.
Figure 19 shows the critical cross-sectional areas of a heteroepitaxially grown
YBa2Cu,0-i-s/Ndl,aCeo,l~Cu0,
superlattice consisting of 6 alternating layers of c-axis
oriented YBa2Cu,0,-s and Ndl,s3Ceo.17Cu0,(d=40 nm) [I%]. The dimensions of the
investigated microbridge patterned into this superlattice by Ar ion beam etching are
688
R Gross and D Koelle
Figure 18. Voltage images of a 7Opm wide superconducting line in an epitaxial
YRaiCulOl-s film recorded at T=83 K and fR/fm,m=1.8 (a), 2.7 ( b ) , 3.5 ( e ) and 4.0 ( d ) .
The film extends horizontally beyond the field of view. The arrows mark the film boundaries.
The electron beam parameters are I O kV voltage, 1 nA current, and I O kHz modulation
frequency.
L=200 pm and W=40pm. Figure 19 clearly shows that the weak cross-sectional areas
(dark regions) are not necessarily perpendicular to the geometrical strncture of the
microbridge and can have a wavy shape. For this sample their detailed shape is determined by the microstructure (defects, precipitates etc) of the sample and not by substrate
defects as in figure 18 [I%]. We note that for the bias currents used in figure 19 the
magnetic flux lines are moving only along the weak cross-sectional areas. Only for these
areas the critical current is exceeded. In the bright regions in between the weakest crosssectional area the critical current is not yet reached and the flux line lattice is still fixed.
That is, by LTSEM one can image the channels along which the magnetic flux lines move
when the bias current is increased above the critical current of the microbridge. This
situation is completely analogous to the flow of charge carriers along current filaments
in an insulating sample when the critical voltage is exceeded [63-661.
The magnitude of the voltage signal can vary considerably along a distinct crosssectional area Ai. In the following we discuss the variation of the electron beam induced
voltage signal 6 V(xi, y ) along a cross-sectional area A; with a spatially varying critical
current density J c ( x j , y )as shown schematically in figure 20. Perturbing the sample at
the position ( x j , y ) causes a change o f the critical current S l , ( x j , y ) according to
equation (46). Assuming that aJ,/aT is about constant in the perturbed sample
Low temperature scanning electron microscopy
689
Figure 19. Two-dimensional voltage images of a 4 0 g m wide line in a Ndl.slCe,,,,CuV,/
Y R ~ K u , O , -superlattice
~
recorded at T=34 ( n ) , 60 ( h ) , and 77K ( e ) . Dark and bright
areas correspond to regions yielding large and small voltage signal, respectively.The elemon
beam parameters are 10 kV voltage, 1.2 nA current and 10 kHz modulation frequency.
..
Figure 20, Schematic sketch of the current density distribution .I&
j) along
,, the crosssectional area A i together with the expected voltage signal S V ( x , , y).
690
R Gross and D Koelle
region and approximating the integral over the temperature profile by A6Ta the electron
beam induced change of the critical current of the cross-sectional area Ai is obtained
to
Sl,(x,, Y ) can be measured directly using the MCCD technique. For 6 1 c ( x j , y ) < < L ( x t )
the resulting voltage signal for a current biased sample is given by
Ifd, A and GToareindependent of the beam position, thevoltage signal directly displays
the variation of the temperature derivative of the local critical current density. Usually,
the aJc(xj,y)/a7'cc J c ( x i , y )and, hence, 6 V ( x j ,U) cc J,(xj, y).That is, thevariation ofthe
voltage signal along the cross-sectionalarea A, is directly proportional to the variation of
the critical current density as shown in figure 20. Hence, LTSEM does not only allow
the determination of the critical current values of the complete cross-sectional areas as
discussed above, but also the measurement of the variation of the critical current density
along the cross-sectional areas.
Above it was assumed that the film thickness d is constant and that the critical
current density does not vary in the z-direction. If the effectivefilm thickness is reduced
at some position y along Ai (for example by precipitates), this region will yield a smaller
signal, since 61,(xi,y) is smaller due to the reduced film thickness. For example, sample
regions yielding a smaller signal along the weak cross-sectional areas in figure I9 could
be correlated with the presence of precipitates. In the same way, regions where the
critical current density is reduced over some part of the film thickness will yield a
smaller signal. A partly reduced critical current density can for example be caused by
growth defects in epitaxial high-l: films at the film-substrate' interface. Such regions
appear as weak regions and can be imaged by LTSEM 1321 as shown in figure 18. The
imaging of weak cross-sectional areas is highly interesting for the analysis of complex
superconducting circuits. Firstly, it provides a test of the homogeneity of the superconducting films forming the circuit. Secondly, it allows us to test the influence of different
fabrication processes such as the patterning of the films or chemical treatments on the
film quality. Finally, it allows us to examine the influence of specific circuit elements
on the overall current carrying capacity. Such elements are, for example, insulated
crossovers of two superconducting lines or superconducting contacts between two lines
through a window in an insulating layer.
In the above discussion a single, isolated cross-sectional area has been considered.
It was assumed that every part of the cross-section is biased at its critical current density
for IB=Ic(x,). In a real sample, however, the current density along a distinct crosssectional area also may be determined by the current density in the adjacent crosssectional areas making the interpretation of the electron beam voltage signal more
complicated. The assumption of an isolated cross-sectional area is always applicable,
if the considered cross-sectional area has a smaller critical current density than the
adjacent ones. The arguments given in section 6.1.2 with respect to the use of larger
perturbations or temperatures near the critical temperature are valid also for the twodimensional case. Of course, in the same way as for the one-dimensional case the spatial
resolution is determined by the width of the temperature field.
Low temperature scanning electron microscopy
691
6.3. Three-diinensional case
A thin film sample with width, length and thickness large compared to the characteristic
decay length of the sample response F represents a three-dimensional sample. The
discussion given for the two-dimensional case is directly applicable also for the threedimensional case. The integrations in section 6.2 have to be performed also in zdirection. The electron beam induced voltage signal will be by a factor A / d smaller
than in the two-dimensional case. The factor A / d gives the fraction of the sample in
z-direction perturbed by the electron beam.
6.4.
LTSEM
study of passive thin-film devices
Above we have shown how LTSEM can be used for the imaging of inhomogeneities in
simply shaped superconducting films. It is quite natural to use LTSEM also for the
spatially resolved electrical characterization of more complex thin film structures (e.g.
insulating cross-overs, contacts between different superconducting layers in multilevel
structures etc) and of passive thin film devices (transmission lines, flux transformers,
bolometers, etc). Here, LTSEM can be used for a non-destructive spatially resolved
device test. LTSEM can provide important information on damaging effects due to the
patterning process, the presence and position of electrical shorts between t&'o superconducting layers separated by an insulating layer, or on those parts of the device limiting
its critical current. In general, this information is difficult to obtain by standard electrical
characterization methods.
As a typical example figure 21 shows a LTSEM voltage image of a seven-turn
YBa2Cu307-s input coil fabricated by subsequent deposition and patterning of three
expitaxial layers in the sequence YBa2Cu,O7-&rTiO3-Y Ba2Cu,07-s. The image was
recorded at T=88 K using a constant bias current. The resistive transition of the input
coil showed a foot structure extending down to below 88 K, i.e., at the measuring
temperature the input coil still showed a small resistance. Our LTSEM analysis clearly
showed that this residual resistance is only caused by those parts of the spiral .where
the YBCO lines run across the edges of the SrTiO, layer, which is marked by the bright
line in figure 21. These parts yield vanishingly small voltage signal and appear as dark
regions in the LTSEM voltage image. The SrTiO, layer provides the insulation between
the YBCO cross-under (indicated by the dark rectangular area crossing the lines of the
spiral) and the multi-turn spiral. Figure 21 also shows that the measured voltage signal
decreases from the left to the right hand side. This is most likely caused by a slight
increase of the critical temperature in this direction. For a more detailed discussion of
figure 21 see [47].
6.5. Imaging of hotspots in superconducting films
In a superconducting thin film the electric current can flow without energy dissipation
as long as the current density does not exceed the critical current density. If the critical
current density in a superconducting strip is exceeded at a particular cross-sectional
area, energy is dissipated resulting in an increase of the local film temperature. The
magnitude of the temperature rise is determined by the amount of dissipated energy
and by the efficiency of heat removal, i.e., by the thermal conductivity K of the thin
film material and the heat transfer coefficient a between the film and the substrate
material. Due to the increase of the film temperature the local critical current density
692
R Gross and D Koe/le
Figure 21. LTSEM voltage m a g e o f a S W C I I - ~ U Fpiral
~ ~ ~ input coil rccorded a1 T-XX K and
a bias current of I mA. Dark and bright areas correspond to region yielding small and
large voltage signal, respectively. The dark rectangular structure on the left hand side of
the image is caused b y the YBCO cross-under providing electrical contact to the inner cnd
of the spiral. The electron beam parameters are 25 kV voltage. 1.8 nA current, and 10 kHz
modulation frequency (from [47]).
is reduced further resulting in a further increase of the sample temperature. In this way
the energy dissipation eventually becomes large enough such that a so-called self-heating
hotspot is formed. The hotspot represents a stable temperature structure consisting of
a domain of length / where the temperature is elevated above the critical temperature
of the superconducting material. At the boundaries of the hotspot the temperature
passes i", and approaches the bath temperature Tb well outside the hotspot. The width
of the transition region is determined by the thermal healing length of the superconducting film. If the power supplied by the external circuit is kept constant the size of the
hotspot is growing until the energy input equals the energy transfer to the substrate,
which approximately is proportional to the length I of the hotspot.
The stable temperature structure of a self-heating hotspot is a typical example of
the dissipative structures formed in the non-equilibrium state of open systems. As
pointed out by Landauer cf a/ [159, 1601 the temperature structure associated with a
hotspot results from a S-shaped temperature dependence of the resistance. This Sshaped resistance versus temperature curve is particularly pronounced in superconducting materials, where the resistance rises rapidly from zero to a finite value at the critical
temperature. A detailed analysis of the heat balance equation for describing a hotspot
in a thin-film superconductor was given by Skocpol et a/ [161]. Recent reviews of the
subject of hotspots in superconducting films are given in [162-1631.
In the following we will give a qualitative discussion of the imaging of hotspots by
LTSEM. Here we will assume that the additional perturbation of the electron beam is
small, i.e., the electron beam is considered as a passive probe that does not influence
Low zemperature scanning electron microscopy
693
the system strongly. The electron beam is scanned across the sample and, depending
on the biasing conditions, the electron beam induced change of the sample current or
voltage is recorded as a function of the beam coordinates. For simplicity we will consider
a quasi one-dimensional superconducting line as discussed in section 6.1. For twodimensional samples the discussion of section 6.2 can be applied. As shown in section
6.1.1 the electron beam induced signal is about proportional to ap(x)/aT. Hence, if
the bath temperature is well below the critical temperature of the superconducting film
a significant signal is expected only for the edges of the hotspot where the temperature
profile of the hotspot passes T,. Inside of the hotspot the film temperature is above T,.
At this temperature ap(x)/JTusually is small resulting in a small signal. Furthermore,
dp(x)/aTis zero well outside the hotspot if the applied current density JB is well below
the critical current density at the bath temperature. Due to the sharp increase of the
resistivity at T= T,, ap(x)/aTis maximum T=TC.Hence, if the electron beam induced
perturbation is small (Uo<<
T,- Tb), we expect a peak in the signal to appear at the
boundaries of the hotspot. The width of the signal peak is determined by the spatial
decay length A of the beam perturbation and is expected to be about 2A. A detailed
treatment of the origin of the electron beam induced signal based on the analysis of
the heat balance equation can be found elsewhere [51, 1641.
Typical experimental results of the imaging of hotspots by LTSEM can be found in
[50,51] and [164-1661. Using LTSEM the formation and the growth of hotspots in
superconducting microbridges with increasing power input could be observed directly.
The experimental results agree well with the theoretical considerations including the
effects due to high-frequency beam modulation [ 165, 1661. Fitting the experimental
data to the model predictions the thermal parameters of the thin film sample such as
the thermal conductivity and the specific heat of the film material, and the heat transfer
coefficient between the film and the substrate could be derived.
7.
LTSEM
study of superconducting tunnel junctions and circuits
In section 6 we have shown how LTSEM can be used for the study of superconducting
samples for which the critical current density was determined by depinning. In the
following two sections the spatially resolved investigation of weakly coupled superconducting systems is discussed. In contrast to the samples considered in section 6, for
such samples decoupling limits the critical current density. We consider both single
Josephson junctions or weak links and complex networks of weak links contained for
example in polycrystalline high-T, films. Superconducting tunnel junctions and the
various types of weak links are of particular interest for many cryoelectric applications
such as superconducting quantum interference devices (sQu~D.),microwave detectors,
fast switching elements etc [145-1471. The spatially resolved study of the current transport across individual Josephson junctions or weak links as well as the investigation of
complex circuits formed by these elements is highly important, since it provides valuable
information on their local superconducting properties.
In the preceding section we have discussed the application of LTSEM to the imaging
of the superconducting properties of samples containing no weak links. For these
samples it could be assumed that every part of a cross-sectional area A, is carrying its
critical current density, if the sample current is equal to the critical current across Ai.
The critical current density was determined either by the depairing or depinning current
density of the superconducting film. For weakly coupled superconductors the local
694
R Gross and D KoeIle
critical current density is determined by the local coupling strength between the two
superconducting electrodes forming the weak link and the local phase difference of the
superconducting order parameter [ 145, 1471. In the following we show that LTSEM can
provide important information on spatial structures in superconducting weak links.
These structures can have various origins such as a spatially inhomogeneous coupling
between the superconducting electrodes, inhomogeneities of the superconducting electrode material, or a spatially inhomogeneous phase difference due to an applied magnetic field or trapped vortices. Beyond the strong physical interest in the direct
observation of such structures, LTSEM experiments are highly important for the spatially
resolved analysis of cryoelectronic devices. As we will show below, LTSEM represents a
powerful tool for superconducting circuit testing.
7.I . Basic equations
7.I . I . Zero voltage state, Let us consider a superconducting tunnel junction as sketched
in figure IO. As first predicted by Josephson [167], Cooper pairs can tunnel across the
thin insulating barrier separating the superconducting electrodes resulting in an electric
current flow at zero voltage. The Josephson current density that can flow without
electrical resistance is given by
Jh,
B) = Jh,
J.) sin 6 (n,Y ) .
(49)
Here, J&, y) is the local critical current density and 4 (x, y) the local difference between
the phase of the superconducting wavefunctions in both electrodes. We note that for
the different types of weak links the phase difference function in general is not sinusoidal
but some arbitrary 2n periodic function, For a superconducting tunnel junction formed
by two identical homogeneous superconductors the critical current density is given by
the Ambegaokar-Baratoff (AB) expression [ 1681
where A ( T ) is the temperature dependent energy gap, e the elementary charge, R, = I /cn
the tunnelling resistance times the junction area in the normal state, k B the Boltzmann’s
constant, and f a numerical factor for strong coupling superconductors. The critical
current density of the various types of weak links is given by expressions differing more
or less from the A B expression [145, 1461. The phase difference function $(.u,y) is
determined by the magnetic flux density B(n, y ) penetrating the tunnelling barrier. For
a tunnel junction with d , , d2>AL,we have
v$(.,,.)=q(x,Y)
(Ex21
with
(51)
Here, d(x, y ) = 2 A ~ ( n , y )is the effective magnetic thickness of the tunnelling barrier, li
is Planck’s constant divided by 2n, and i and Bare the unit vectors in the z-direction
and in the direction of the local flux density, respectively. ,ILis the London penetration
depth of the electrode material. The magnetic flux density B(x,y) is composed of the
magnetic field BJ generated by the tunnelling current and of the magnetic field BE
Low teniperature scanning electron microscopy
695
applied to the junction by external sources. With B = BJ+Be and taking into account
that BJ is determined by Js(x,y ) and, hence, by 4 ( x , y ) yields the well known stationary
sine-Gordon equation [ 145, 1461
where AJ(x,y)is the Josephson penetration depth, which is given by
Here, p o is the vacuum permeability. The Josephson penetration depth represents the
characteristic magnetic screening length of a superconducting tunnel junction.
In the zero-voltage state the phase difference +(x, y ) obeys the time independent
sine-Gordon equation and can be calculated using appropriate boundary conditions.
The boundary condition of equation (52) is given by the normal component of Vb(x, y )
along the boundary of the tunnel junction area. It can be expressed by the current
Z,(x, y ) per unit length flowing into the tunnel junction area as
A detailed discussion of the sine-Gordon equation and its boundary conditions can be
found in most textbooks on the Josephson effect [145-147].
(a) SmaN funneljunctions ( W , , Wz<dJ). For small tunnel junctions the self-field o f
the Josephson current can be neglected resulting in a spatially constant phase difference
function in the absence of an external magnetic field. Applying an extemal field parallel
to the barrier the modulation of the maximum Josephson current
Is=
JJx,Y ) dx dy
(55)
follows the well known Fraunhofer diffraction pattern
if J,(x, y ) is spatially homogeneous. Here, @ is the magnetic flux threading the junction
and cD,=h/Ze js the magnetic flux quantum. Equation (56) shows that the pair tunnelling current can be suppressed by applying an external magnetic field parallel to the
barrier. This is used in LTSEM experiments where the pair tunnelling current is
disturbing.
( h ) Large tunnel junctions ( W ,or W2>dJ).For large Josephson junctions the selffield of the tunnelling current has to be taken into account. This results in a spatially
varying phase difference function and, hence, in a spatially varying Josephson current
density even in the absence of an external magnetic field. If one dimension of the
junction is small compared to A, (one-dimensional geometry), Js(x,y ) can be calculated
analytically [ 145, 146, 1691. For a two-dimensional geometry ( W , , W2>>A,),# ( x , y )
is obtained by integrating equation (52) numerically. However, the determination of
696
R Gross and D Koelle
the boundary condition (equation (54)) represents a critical point for the calculation
of the phase difference function and the Josephson current density. In most cases,
the boundary condition cannot be determined exactly because ii. is influenced by the
complicated geometrical configuration of the junction electrodes and the current feeding
lines, as well as by the unknown tunnelling current density itself. Without knowing the
exact boundary conditions it is difficult to calculate J%(x,y) and the modulation of the
maximum Josephson current f, by an external magnetic field [ 170-1741. In some cases
the problem of the unknown boundary conditions can be circumvented by calculating
J&y) using a numerical iteration procedure [12, 14, 151. I n any case, the direct imaging of J,(x, y ) by LTSEM is highly interesting, since this allows a critical test of the
numerical methods.
7,/.2. Finite uoltage state. For a constant voltage V#O between the superconducting
electrodes, the phase difference 4 varies in time according to the Josephson equation
11671
resulting in an oscillating pair tunnelling current
( a ) SmaN tunneljunctions ( Wl, W2<AJ),We first consider small Josephson junctions
in the absence of an external magnetic field. In this case the phase difference can be
assumed to be spatially homogeneous and the time average of the pair tunnelling current
vanishes, if the voltage across the junction i s kept constant. In addition to the pair
tunnelling current the normal excitations or quasiparticles can tunnel across the barrier
11751. The quasiparticle tunnelling results in a finite tunnelling current if the voltage
between the junction electrodes is non-zero. For an ideal tunnel junction formed by
twO superconducting electrodes with energy gaps A, and A2 the quasiparticle tunnelling
current is given by [ 145, 1461
Here, R. is the normal resistance times junction area, E the quasiparticle energy and
f(E, T)the Fermi distribution function at energy E and temperature T. The voltage
dependence of Jq, is obtained by integrating equation (59). The temperature dependence of Jqp is determined by the temperature dependence of the energy gap and the
Fermi distribution function. The different expressions for the voltage and temperature
dependence of the quasiparticle current density of the various types of weak links are
not discussed here. They are discussed in detail in 1176-1801.
Note that the time average of the pair tunnelling current is zero only if the voltage
across the tunnel junction is kept constant. In this case the current-voltage characteristic
(IVC) for V>O is given by the quasiparticle tunnelling characteristic. However, due to
Low temperature scanning electron microscopy
691
the small impedence of the tunnel junctions a voltage bias usually is difficult to establish
and many experiments are performed with a current biased junction. For a current
biased junction only the time average (V(t)>,of the junction voltage is constant. The
time dependent junction voltage results in a non-sinusoidal oscillation of the pair current
with a non-vanishing time average. In this case the total tunnelling current is larger
than the quasiparticle tunnelling current due to the finite contribution of the pair
current. This is especially the case for overdamped Josephson junctions having a
McCumber parameter [ 145, 1461 &=2eJcR,?C,/fi< 1. The various types of weak links
discussed in section 8 often represent such overdamped Josephson junctions. Superconducting tunnel junctions, in contrast, usually are strongly underdamped (&>> 1) due
to their relatively large specific capacitance C, and J,R. products. We note that only
for small underdamped Josephson junctions at zero external magnetic field the time
average of the pair current is about zero for the current biased situation. That is, the
IVC is about equal to the quasiparticle tunnelling characteristic at V>O [181, 1821.
(b) Large runnel junctions ( W , or W, > AI). For large Josephson junctions the spatial
distribution and the temporal evolution of the phase difference is determined by the
unperturbed, time dependent sine-Gordon equation (SCE) [ 145, 1461
if dissipative effects can be neglected. Here, E=c(.c0/C,d)'!* is the Swilhart velocity
1145, 1461; c and c0 are, respectively, the velocity of light and the permeability in
vacuum. Equation (60) describes the propagation of electromagnetic waves in a nonlinear, dispersive medium. Due to the non-linearity of the SGE there exist interesting
dynamic excitations in Josephson junctions. Solutions of equation (60) are, for example,
solitons, i.e. particle-like excitations that propagate without dispersion between the
superconducting electrodes forming a Josephson transmission line [ 145, 146, 1831. These
solitons are usually referred to as fluxons or Josephson vortices.
The SGE has to be solved under appropriate boundary conditions. Here, in the same
way as for the stationary case the determination of the boundary conditions represents
a severe problem. The boundary conditions have a significant impact on the dynamic
behaviour of large Josephson junctions. They are determined by the junction size, the
external magnetic field, and the electrical current applied to the junction [ 145, 1461. In
general one can distinguish between 'weak' (qA,< 1) and 'strong' (qA,> 1) boundary
conditions, where q is given by equation (51). At zero external magnetic field the flux
density parallel to the junction barrier is determined only by the self-field of the tunnelling current and the notation weak and strong boundary condition corresponds to large
and small tunnel junctions. At a finite external magnetic field the condition qA,> 1
corresponds to the case where the Josephson vortices inside the junction overlap. In
this case the dynamics of the Josephson junction usually is discussed in terms of nonlinear microwave interactions [ 1451. For qAL< I, the non-linearity of the sine-Gordon
system compresses both the supercurrent and the electromagnetic waves to separate
units (fluxons) and the dynamics usually is discussed in krms of fluxon motion.
In order to model the dynamic behaviour of real Josephson junctions dissipative losses
due to the quasiparticle tunnelling current and due to the finite surface resistance o f the
superconducting electrodes have to be taken into account. Due to the loss terms real
Josephson junctions have to be described by the perturbed sine-Gordon equation (SCE),
698
R Gross and D Koelle
which for a one-dimensional junction with overlap geometry is given by [ 145, 146, 1841
bXx-$,,- sin 4 = 4,PqL-
r.
Here, the spatial coordinate is normalised to k, and the time to the plasma frequency
wp of the Josephson junction. The indices denote the partial derivatives with respect to
the indicated variables. The loss terms a4, and p&v, represent the quasiparticle tunnelling and the surface losses and r is the bias current normalized to the critical current
of the junction. In a steady state situation the losses are compensated by the energy
input of the bias current supplied by the external circuit.
There is no analytical solution of the PSGE. There are mainly two different approaches
to obtain approximate solutions, namely the multi-mode theory [ 1851 developed by
Enpuku et a1 and the perturbation theory [ 1861 of McLaughlin and Scott. In the multimode theory, which represents a natural extension ofthesinglemodeapproach ofTanaka
and Kulik [187, 1881, the solutions of the PSGE are approximated by a series of spatial
Fourier modes with unknown time-dependent amplitudes. This approach works well for
strong boundary conditions. In the perturbation theory the known solutions of the unperturbed SGE are modified treating the damping coefficientsa and /3 as small perturbations.
This approach works well for weak boundary conditions. For more general cases, the
PSGE has to be solved numerically using appropriate boundary conditions.
Qualitatively, in the case of weak boundary conditions the dynamics of large Josephson junctions can be well described in terms of Ruxon motion. Fluxons travel along
the Josephson junction transmission line accelerated by the bias current and are reflected
as anti-fluxons at the edges of the junction. If the average energy loss is compensated
by the energy input of the bias current, a steady state motion is obtained. We note that
the resonant motion of fluxons in a Josephson transmission line results in resonant
structures that manifest themselves as current steps in their current-voltage characteristics, which are denoted as zero-field steps [145, 1891 (B=O) and Fiske steps [145, 1901
( B Z O ) . In the case of strong boundary conditions there are no longer pulse-shaped
fluxons. Here, it is more appropriate to describe the dissipative modes in terms of a
non-linear interaction of the oscillating Josephson current with the cavity modes of the
Josephson transmission line. For sufficiently strong boundary conditions it is sufficient
to take into account only a single cavity mode. A single mode theory for the description
of resonant states in Josephson junctions (Fiske steps [190], zero field steps [188]) has
been developed. The multi-mode theory extends the single-mode approach to the case
of weaker boundary conditions.
Similar to the static case the exact boundary conditions are only known for simple
one-dimensional junction geometries (e.g. in-line or overlap geometry). For more complex junction geometries the unknown boundary conditions represent a severe problem
for solving the SGE or PSGE. That is, the detailed dynamics of large Josephson junctions
is difficult to predict theoretically. Therefore, the direct imaging of dynamic phenomena
in large Josephson junctions by LTSEM is highly interesting. In particular, the LTSEM
imaging provides a critical test for the theoretical predictions of model calculations.
7.1.3. Origin of spatial StrucIures in superconducting rimncl junctions. Spatial structures
in superconducting tunnel junctions can originate from spatial variations of
the tunnelling resistance, R.(x, y ) ,
the energy gap of the superconducting electrodes, A(x,y),
and the phase difference function, b ( x , y ) .
Law teinperature scanning electron microscopy
699
For the various types of weak links spatial structures can arise by similar reasons.
Equation (59) shows that for tunnel junctions spatial variations of the quasiparticle
tunnelling current density can be caused only by inhomogeneities of R , and
In
most cases spatial variations of R,, are caused by inhomogeneities of the thin insulating
tunnelling barrier. Inhomogeneities of the energy gaps can arise from material imperfections, temperaturegradients, or due to an external perturbation such as laser irradiation
or quasiparticle injection [IO, 104, 191, 1921. Spatial variations of the quasiparticle tunnelling current density also can be caused by a standing microwave patterns generated
by microwave irradiation of the tunnel junctions due to photon assisted tunnelling
[IS, 19,201.
The pair tunnelling current density is influenced by spatial inhomogeneities of R,
and Al.2 in the same way as the quasiparticle tunnelling current density. However, even
for tunnel junctions with perfectly homogeneous R, and A,.* strong spatial variations
of the pair tunnelling current density including a change of sign are caused by spatial
variation of the phase difference. A spatial inhomogeneous phase difference function
d(x, y ) is caused by an applied magnetic field (small and large junctions), by the selffield of the tunnelling current (large junctions), or by trapped magnetic flux quanta
[12-15]. Beyond the static structures that are observable only in the zero voltage state,
resonant structures can arise in the finite voltage state due to the nonlinear interaction
of the Josephson oscillation with the cavity modes of the Josephson junction acting as
a transmission line resonator or due to the resonant motion offluxons in the Josephson
transmission line. These resonant structures also can be imaged by LTSEM [ 19-28].
7.2. Pair tunnelling current density
As a first example of a useful application of LTSEM for studying Josephson junctions
and weak links we discuss the imaging of the spatial distribution of the pair tunnelling
current density. Since the local pair tunnelling current density is determined both by
the local phase di/jkence and tunnelling conductivity, LTSEM images contain spatially
resolved information on bolh of these quantities. For low-T, superconductors high
quality Josephson lunnel junctions with a sufficiently homogeneous tunnelling conductivity (cn(x,
y)=const) can be fabricated. For such junctions the LTSEM images unambiguously yield information on the phase difference function $ ( x , y ) . For high-T,
superconductors the fabrication of planar type tunnel junctions is difficult and has not
yet been reported. The principle arrangement for the measuring of the spatial variation
of the pair tunnelling current density was shown i n figure I I . In the following we assume
that the focused electron beam irradiation simply results in an increase of the local
sample temperature within an area of radius A. This is a good approximation for most
experimental situations. For the case of a non-thermal effect of the electron beam
perturbation most results presented below are qualitatively the same. A detailed discussion of the effect of the electron beam irradiation was given in section 4 and in [6] and
[104]. Furthermore, we assume that both electrodes are perturbed about equally and
that the scanning speed of the electron beam is slow so that transient effects due to the
beam scanning can be neglected.
7.2.I . Slatic siruclures-vortex states, se~fieeldcffecls, trapped vorlices. I n our discussion we first consider the zero voltage state. Here, according to equation (57) the phase
difference function does not vary with time resulting in static structures. The local
electron beam perturbation of the Josephson junction results in a change 8Js of the
700
R Gross and D Koelle
local pair current density both due to a change of the maximum pair tunnelling current
density J, and the pahse difference 6.For small perturbation ( S J J J , c c l ) the electron
beam induced change of the pair tunnelling current can be expressed as
61,(x, 7 )= SC(x, y ) + 61f(x, y )
(62)
where ( x , y ) is the position of the focused electron beam on the sample surface.
S/:(x,y) and @x, J') are, respectively, the change of the pair tunnelling current due
to the local change of the maximum pair tunnelling current density and the local phase
difference and are given by
and
6/$(x, y , t ) =
1
A,vn
dr2mJC(r)cos @(r)J@(u,I ) .
(64)
Here, r is the distance to the position (x, y) of the electron beam. Equations (63) and
(64) represent the local and the non-local effect of the electron beam perturbation. For
the local effect the static linear response functions of equation (26) is ( a J , / W ) sin 4.
The local effect, S/:, results from a local reduction of J , by increasing the local sample
temperature. For the local effect there is no significant contribution to SI, for I' 7 A. In
contrast, the non-local contribution, J/$, is caused by a global change of the phase
ditference function by the local electron beam perturbation. This contribution results
from the local increase of the penetration depths AL and A,. Due to the macroscopic
phase coherence the beam induced global change of the phase difference function results
in a signal both from the perturbed ( r < A ) and unperturbed ( ? > A ) parts of thejunction
[193-195). Non-local effects have to be taken into account only for large junctions or
for large bcam perturbation. The interpretation of the nonlocal signal contribution can
be complicated and will not be discussed here. An explanation of the non-local signal
contributions from simple physical arguments is given in [I51 and [ 1951.
In our LTSEM experiments the electron beam perturbation is kept as small as possible
in order to be able to treat the electron beam as a passive probe. I n this case the nonlocal effect can be neglected and the local response is linear. We note that for large
beam perturbation the local response becomes non-linear. In figure 22 we have plotted
SI:, versus the electron beam induced temperature rise for various reduced temperatures
T/T,taking into account the non-linear response [ 1041. The curves were calculated
using a Gaussian temperature profile ST(r) = STOexp(-r2/A2). The non-linear relation
between S/:, and ETois clearly visible in figure 22.
In most experiments the electron beam induced change of the maximum Josephson
current is measured by the MCCD technique (see section 5 . 2 ) . Neglecting non-local
signal contributions we have S/s=JIf. Assuming 2Jc/i3T=constant within the perturbed junction region, according to equation (50) the signal S is given by
S ( . ~ , y ) = - S I : ( . ~ , y ) a - J ~ ( x , r ) s i n @ ( x , ~ . ) c c - s i n @ ( x , ~ ) A ( ~ , ~ ' ) / R . (65)
(~,~.).
Here we have used the proportionality aJ,(T),'dTcrJ, where Jn=Jc(T=O). Spatial
variations of the energy gap of the electrode material usually are very small and can
J') and
be neglected. That is, the measured signal mainly contains information on on(x,
6(&Y).
Low leniperafure scanning electron microscopy
1.4,
.
,
I
I
0.2
0.4
I
,
I
701
I,,
".U
0.0
0.6
0.8
1.0
6TdTc
Figure 22. Calculated 61: vwsus 6 6 dependence for an electron beam induced temperature
field T(r.)=6Toexp(-r'/Az).
For small tunnel junctions in the absence of an external magnetic field we have
@(x,y)=const and the measured signal is directly proportional to the local critical
current density Jc(x, y ) or the local tunnelling conductivity un(x,y ) = l/R.(x, y). That
is, scanning the sample and measuring -6/,(x, y ) synchronously, a two-dimensional
image of un(x,y ) is obtained. As will be shown in section 7.3, un(x,y ) can more easily
be imaged for both small and large tunnel junctions by measuring the electron beam
induced change of the quasiparticle tunnelling current SI,&, y ) . In particular, since
the quasiparticle tunnelling current is not influenced by the phase difference function,
the measurement of SI,,(x, y ) yields information only on U&, y).
For tunnel junctions with spatially homogeneous tunnellingconductivity (U,& y) =
const) the measurement of S l : ( x , y ) allows the imaging of @(x,y). In the following,
we consider Josephson junctions with a spatially homogeneous tunnelling conductivity.
Of course, the homogeneity of un(x,y ) has to be examined experimentally in order to
be sure that the measured structures result only from spatial variations of the phase
difference. As discussed above, spatial variations of the phase difference function and,
hence, the pair tunnelling current density can be caused by an external magnetic field
parallel to the tunnelling barrier resulting in a gradient of the phase difference according
to equation (51). Furthermore, a spatially varying phase difference function is obtained
by the self-field of the tunnelling current in large junctions or by trapped magnetic flux
quanta. With on(x,y)=const, according to equation (65) the electron beam induced
signal only depends on +(x, y ) and is directly proportional to sin @(J,y).
We will first show the effect of an external magnetic field applied parallel to the
tunnelling barrier. A typical experimental result is shown in figure 23. The LTSEM images
are obtained for a 19pm wide and 97pm long Josephson junction (L/AJ=1.6). A
homogeneous magnetic field parallel to the tunnelling barrier results in a constant
gradient of the phase difference along the junction. This, in turn, causes a sinusoidal
modulation of the Josephson current density resulting in so-called vortex states of the
junction. The evolution of the different vortex states with increasing magnetic field are
clearly shown in figure 23. The different vortex states can be imaged by LTSEM and the
experimental results agree well with theoretical expectations.
For large one-dimensional junctions (L/AJ>>1) the amplitude of the measured sinusoidal signal, +ZS, usually increases or decreases along the junction depending on the
702
R GPOSS
and D Koelle
Figure W . Imaging of the spafial distribution of the Josephson current dcnsity in a Pbln/
oxide;Pbln tunnel junction ai different magnetic Gelds applied parallel to the tunnelling
barrier. The magnetic field increases from ( 0 ) lo ( d ) .The measured signal, -6f,(x. j),is
plotted in vertical direction during the horizontal Scans (Y-modulation). In ( c ) Ihe sample
geometry and the scanning direclion is shown schematically (from 11951).
applied magnetic field [ 131. This behaviour is shown in figure 24 where we have imaged
the 4-5 vortex state of a long PbIn/oxide/Pbln tunnel junction (L/L,= 14.4). The
different images are recorded at slightly different magnetic field values. Image ( 6 ) was
recorded at a field value corresponding to a local maximum o f the magnetic interference
pattern I,(B) where dl,/dB=O. The images (a) and (c) were recorded at a slightly
smaller and larger field where dI,/dE> 0 and dl,/dB CO, respectively. The observed
behaviour results from the non-local response of the tunnel junction in agreement with
the predictions by Chang et a1 [193, 1941. In addition to the imaging of the different
vortex states, the restriction of the Josephson current to the edges of the junction at
zero magnetic field (Meissner effect) could be confirmed directly by LTSEM imaging
[13, 1951. From such measurements the value of the Josephson penetration depth can
be inferred directly.
Next we will show the effect of the self-field of the Josephson current. As discussed
above the self-field can be neglected for small Josephson junctions and obtains a growing
influence on the distribution of the Josephson current density with increasing junction
size. A typical example for the effect of the self-field is shown in figure 25. The LTSEM
images show the spatial distribution o f the Josephson current density of a 37 p m wide
and 85 p m long crossline Josephson junction at zero applied magnetic field for diferent
kinds of the current feed. The Josephson penetration depth of this junction is about
13 pm, i.e. it represents a large junction. According to figure 25, the tunnelling area is
~
Low temperature scanning electron microscopy
e -BEAM
703
/
in a long
Pbln/oxide/Pbln tunnel junction (L/L,= 14.4) at different magnetic fields close to a local
maximum of the I@) dependence (Ta=4.2 K). The magnetic field which i s applied parallel
to the tunnelling barrier increases from ((I) to ( e ) .The measured signal, -6f,(x, j),is plotted
in vertical direction during the horizontal scans (Y-modulation). The sample geometry and
the scan direction is indicated at lhe bottom (from [ I I]).
Figure 24. Imaging of the spatial distribution of the Josephson current density
subdivided into two independent regions carrying a substantial Josephson current
density. For a completely symmetrical current feed the Josephson current is clearly
restricted to two zones at the edges of the junctions with a width ofabout 2& (Meissner
effect). These two regions are separated by a Meissner-like zone in the middle of the
junction. For an asymmetric current feed the current flow is restricted to the upper or
lower part of the junction depending on the current feed. The measured distribution is
in good agreement with model calculations based on an iteration method [15, 1961.
Now we will discuss the effect of transverse magnetic flux quanta trapped in Josephson junctions on the distribution of the Josephson current density. Figure 26 shows the
spatial variation OF the Josephson current density (Y-modulation presentation) of a
PbIn/oxide/Pbln tunnel junction measured by LTSEM. The sample contains two transverse magnetic flux quanta trapped at the upper right hand and lower left hand side of
the tunnel junction area. The vortices enter the tunnel junction via the barrier and leave
it piercing through the top or bottom electrode [ 151. Assuming that such kind of vortex
contains Nflux quanta the phase difference 6 between the junction electrodes will vary
by 2nN along a closed path in the barrier plane around the vortex. Since the Josephson
currenl density is proportional to sin 0 it will change its sign N times on a path around
the vortex. Accordingly, it is evident that the vortices trapped in the tunnel junction of
figure 26 contain only a single flux quantum. Comparing the experimental results to
model calculations shown in figure 27, the location, the vorticity, and the flux content
704
R Gross and D Koelle
Figure 25. Measured Josephson current density distribution of a Pbln/oxide/Pbln tunnel
junction for diKerent kinds of the current feed as indicated by the arrows. The measurcd
signal is plotted in vertical direction during the horizontal scans (Y-modulation). The tunnel
junction has a crossline geometry, the triangles mark the edges of the tunnelling area (from
Ref. [15]).
of the trapped vortices can be obtained unambiguously by the LTSEM imaging technique
[IS]. Furthermore, LrsEhf can be used to measure the pinning force by which trapped
vortices are kept at their position. This is done by applying a Lorentz force via a current
through one of the junction electrodes and by imaging at which applied Lorentz force
the vortices are depinned. In this way the pinning force of individual vortices can be
determined [IS, 1961.
We note that by spatially averaging measuring techniques (e.g. current-voltage
characteristics, magnetic field dependence of the critical current) it is almost impossible
to obtain information on the location and the flux content of trapped vortices in
Josephson junctions or on the effect of the current feed on the distribution of the
Josephson Current density in large area junctions. Therefore, the spatially resolved
imaging by LTSEM is highly interesting, in particular for the evaluation of the validity
of model calculations for the spatial distribution of the Josephson current density.
7.2.2. Inraging of microshorts. The measurement of 6 I : ( x , y ) allows the imaging of
superconducting microshorts of the tunnelling barrier [13, 1951. In the presence of
Low temperature scanning electron microscopy
705
-&
Y
A
SO pm
Figure 26. LTSEM voltage image showing the Josephson current density distribution in a
Pbln,'oxide/Pbln tunnel junction containing two trapped magnetic flux quanta as indicated
at the top. The electron beam signal is plotted in vertical direction during the horizontal
Scans (Y-modulation), the triangles mark the edges of the tunnelling area (from [Is]).
superconducting microshorts, which are shortening the tunnelling barrier, most of the
supercurrent is carried by the microshorts. The microshorts can be viewed as junction
regions with a critical current density JE'(x, y) which is by several orders of magnitude
higher than the maximum Josephson current density. Therefore, according to equation
(65) the electron beam induced change of the pair current, Sl,(x,y), is much larger at
the positions of tbe microshorts. In a two-dimensional 61& y ) image the position of
the microshort is marked by a sharp peak of the electron beam induced signal. For
example, the tunnelling barrier of Pbln/oxide/Pbln tunnel junctions is known to get
superconducting shorts on ihermal cycling between liquid helium and room temperature
due to the formation of hillocks and whiskers. These microshorts could be imaged
directly by LTSEM [ 1951. Furthermore, it is possible to image superconducting shorts
parallel to the junction barrier that are located outside the tunnelling area. These shorts
are caused by pinholes in the insulating layer separating the base electrode and the
wiring of the top electrode.
7.2.3. Resonant struclures--cavify resonances, soliton oscillafions. In the finite voltage
state the phase difference function varies with time resulting in dynamic phenomena
determined by the PSGE together with the boundary conditions. It would be highly
interesting to directly image the temporal and spatial evolution of the dynamic processes
directly by LTSEM. However, the temporal resolution of the LTSEM imaging technique
is determined by the characteristic decay time of the electron beam induced non-equilibrium state (see section 4), which typically ranges between IO-'and
s. The dynamic
706
R Gross and D KoeNe
Figure 27. Calculated Josephson current density for a superconducting tunnel junction
containing two transverse flux quanta of diKercnt vorticity (S and Q).The position ofthe
flux quanta is the same as in figure 26. The Josephson current density is plotted in vertical
direction over the junction area (from [Is]).
phenomena in the Josephson junction, in contrast, occurs on a much smaller time scale.
Here, the typical time scale ranges between about IO-'and IO-'* s for a junction voltage
between I pm and 1 mV. Therefore, the temporal evolution of dynamic phenomena in
Josephson junctions cannot be imaged directly by LTsEbi. However, it is possible to
image the dissipative resonant modes in tunnel junctions resulting in static time-averaged structures such as zero-field or Fiske modes (see section 7.1.2).
In the following we will give a qualitative discussion of the electron beam induced
voltage signal for a current biased Josephson junction and its relation to the local sineGordon dynamics. A more detailed discussion is given in [22] and [23]. We only will
consider small electron beam perturbations allowing us to apply linear response theory.
Supposing that the effect of the focused electron beam irradiation is simply an increase
of the local sample temperature the electron beam induced change of the local damping
coefficients a and p in the PSGE (equation (61)) can be expressed as
(66)
and
S p ( x , y, I ) =
dr 27rr &T(r,I )
aT r=rS
167)
Low temperature scanning electron microscopy
707
where r is the distance to the position ( x , y ) of the beam focus. Replacing a by a + Sa
and p by p + S p in equation (61) the electron beam induced loss term can be taken
into account in the PSGE. For a current biased junction, i.e. if the power supply of the
external circuit is kept constant, the additional damping due to the local electron beam
irradiation cannot be compensated and manifests itself as a reduction SV(x,y) of the
average DC voltage across the junction. On the other hand, if the junction voltage is
kept constant, the additional loss due to the electron beam irradiation is compensated
by an increase 6 1 ( x , y ) of the junction current. The electron beam induced change of
the junction voltage or current can be measured as a function of the beam position in
order to obtain a two-dimensional LTSEM voltage or current image.
The typical time scale of the dynamic phenomena in Josephson is much shorter
than the time scale for variations of the electron beam induced perturbation and hence
of 6a and Sa.Therefore, one has to use time averaged equations. Extending the analysis
of McLaughlin and Scott the time average of the electron beam induced additional
power loss can be expressed as [22,23].
( W X ,
Y , t)>r=-
Y , O(+?(x, Y , t)>,--GP(x,.Y, f)(+:,(x, Y , Oh.
(68)
Here, ( ),denotes quantities that are time averaged over many periods of the Josephson
oscillation. Usually the change 6 V of the average junction voltage is a complicated
function of the power loss. However, for small perturbation (6acca and S P c < p ) the
change of the average junction voltage depends about linearly on the power loss, i.e.
~ V ( X , Yt)=-Sa(x,y,
,
t)<h?x,y, t ) > , - s p ( x , y , t)(+:dx,y, t ) > , .
(69)
Equation (69) shows that the electron beam induced voltage signal is proportional to
the time average of the square of the local electric field +? and the square of the time
derivative of the magnetic field +.,: Hence, two-dimensional voltage images yield information on these quantities. Which quantity is dominating in a particular experiment
mainly depends on the prefactors Sa and 6p.Experiments performed with Nb/AIO,/
Nb tunnel junctions showed that for temperatures well below T, the quasiparticle losses
can be neglected as compared to the surface losses 122, 231, That is, the L T s m voltage
images show the spatial variation of (&).
Recently, single- and multi-mode cavity resonances in Josephson transmission lines
as well as soliton oscillations and flux flow behaviour have been imaged using LTSEM
121, 22, 231. Based on the above analysis the LTSEM images can be compared directly
to the local dynamics expected from the sine-Gordon equation.
7.2.4. Moveinent ofmagneticflux quunza by /he electron beam An interesting application
of LTSEM is the movement of magnetic flux quanta within a superconducting thin-film
structure by the focused electron beam. In particular, the controlled movement and
positioning of magnetic flux quanta allows the study of the influence of trapped magnetic
flux quanta on the distribution of the Josephson current density in superconducting
tunnel junctions. The nucleation and movement of flux quanta by the focused electron
beam can be understood as follows. Initially the electron beam is positioned at the edge
of a superconducting thin film and is increasing the film temperature locally above T,.
Applying a small magnetic field and moving the electron beam slowly to a position
inside of the film, an Abrikosov vortex is nucleated and trapped in the thin film. Moving
the beam the vortex moves along with the normal domain generated by the focused
electron beam irradiation. In this way the vortex can be moved by the electron beam
708
R Gross and D Koelle
in a controlled way. Moreover, by choosing the electron beam parameters in a proper
way it is possible to generate transverse trapped vortices in superconducting tunnel
junctions [15, 195, 1961. Such vortices pierce only one junction electrode and leave the
sandwich structure via the barrier (see figure 26). Recently, LTSE.M has been successfully
used to introduce individual flux quanta into annular Josephson junctions allowing the
controlled study of soliton dynamics in these junctions [24, 251.
7.3. Quasiparficle funtielling current densify
For V>O there is a finite current of normal excitations or quasiparticles between the
weakly coupled superconductors. According to equation (59) for superconducting tunnel junctions the quasiparticle tunnelling current density Jqp(x.y ) depends both on the
local tunnelling conductivity CJ&, y ) and the energy gaps A1.2(x,
y ) of the superconducting electrodes. For the various types of weak links Jqp(x,y ) in general depends on the
local coupling strength of the weakly coupled superconductors and their energy gaps
[ 176- 1801. The imaging of the electron beam induced change of the quasiparticle current
density allows us to obtain information on these quantities. For superconducting tunnel
junctions an important application of LTSEM is the imaging of spatial variations of the
tunnelling conductivity cnthat are caused by inhomogeneities of the tunnelling barrier
17, 9-16], A further important application of LTSEM is the imaging of spatial variations
of the energy gap of the superconducting electrodes caused by external perturbations
such as quasiparticle injection [ 10, 191, 1921. Variations of the energy gap due to imperfections of the superconducting material usually are small.
For a voltage biased tunnel junction the pair current density oscillates sinusoidally
at a frequency 2nf=ZeV/h(f=483,6 MHz at V = 1 pV) resulting in a vanishing time
average of the supercurrent ( ( I s ) , = O ) . In this case the total time averaged tunnelling
current at D O is solely given by the quasiparticle tunnelling current. Furthermore,
since the characteristic decay time z of the electron beam induced perturbation is much
larger than the period of the Josephson oscillation one has to consider time averaged
expressions in calculating the electron beam induced change of the tunnelling current.
Since the time average of the pair current vanishes, the electron beam induced change
of the tunnelling current is equal to the change of the quasiparticle tunnelling current.
For the current biased case the situation is more complex. Were we have (V(r)),=
const but V ( t ) fconst. This results in a non-sinusoidal oscillation of the supercurrent
and in an eventually non-vanishing time average. Hence, the electron beam induced
change of the total tunnelling current no longer is equal to the change of the quasiparticle tunnelling current alone. In general, for an underdamped curtent biased tunnel
junction one has ( I , ) , = O only for small junctions ( W ,Le&) at B=O. A detailed
discussion of this problem is given in [104, 145, 181, 1821. Our following analysis is
restricted to those cases where (Is>,=O. Furthermore, we will assume that the effect of
the electron beam irradiation simply results in a local temperature rise.
For a voltage biased tunnel junction the electron beam induced change of the quasiparticle tunnelling current can be expressed as
0”W
ddsr-[IK(i,
e
Vb,
Tb$.6T(?’,t))-K(r, Vb, &)I.
(70)
Low terriperature scanning electron niicroscopy
709
For small perturbation (8i“o/Tb<< I ) equation (70) can be simplified to
where r is the distance to the position (x,y) of the electron beam and K(r, Vb, T )
represents the integral on the right hand side of equation (59). Hence, the static linear
response function of equation (26) is given by ( a K ( i’b)/aT)c./e. For a current biased
junction the electron beam induced change of the junction voltage is given by
as long as the area perturbed by the electron beam is small compared to the total area
of the tunnel junction (Apr<<Atun). Here, a V / d I is the differential resistance of the
unperturbed junction at the bias point.
We note that for large beam perturbation the response function becomes non-linear,
since the quasi-particle tunnelling current is a highly non-linear function of temperature.
In figure 28 we have plotted SI,, calculated according to equation (70) versus 6To/T,
for various values Vb<2A(0)/e of the bias voltage taking into account the non-linear
response [ 1041. The cunres were calculated using a Gaussian temperature profile ST(r)=
&Tuexp(-?/A2). The non-linear relation between SI,, and ST, is evident. Similar
dependencies are obtained also for Vb> 2A(Oj/e [ 1041.
Figure 29 shows the electron beam induced change of the quasiparticle lunnelling
current versus the bias voltage for a symmetrical tunnel junction (AI =AI) at various
values 6Toof the electron beam induced temperature rise. The dependencies are calculated assuming ST(r)=ST, for r < h and ST(r)=O for r > A . The strong variation of
SI,, with varying bias voltage at constant ST, originates from the sharp increase of
Jqp(I’b) at Vb=?h(T)/e. For small perturbation, i.e. for small ST,, ST,,(x, y ) shows
a sharp peak at evb=2A(x, y ) / e , which broadens more and more with increasing ST,.
Figure 29 shows that the measurement of SI,, as a function o f (Vb) for fixed beam
position ( x , y ) allows the determination of the local energy gap [104].
1,
/ T,
= 0.25
N
0
0.80 (i)
2 0.40
0.85 (iii)
0.90(ii)
v
N
--.
g 0.20
0.00
0.0
0.2
0.4
0.6
0.8
1.0
6TO / Tc
Figure 28. Normalized electron beam induced change of the quasiparticletunnelling current
calculated according to equation (70) versus normalized temperature rise for different values
of the bias voltage at Tb= 1.8 K. The dependencies were calculated using a Gaussian temperature profile 6 q r ) =SToexp(-?/A2).
I10
R Gross and D Koelle
1, / T.
N
-
-
0.25
nflC
<
0.1 (i)
0.2 (ii)
0.3 (5)
0.4 (i)
0.5 (v)
,
0.4
= 0.2
N
P
n_._
n
0.90
0.95
1.00
evb
1.05
1.10
/ A(o)
Figure 29. Normalized electron beam i n d u e d change of the quasiparticle tunnelling current
versus the bias vollage for different values ofJhe temperature rise a t T,= 1.8 K. The dependencies were olculated according to equation (70) using ST(r)=STo for lrl SA and
6T(r)= O for I rl >A.
Figure 30 shows the experimental 61,,(6T0) dependencies measured for a PbIn/
oxide/PbIn tunnel junction at T b = 1.8 K . The thickness of the base and top electrode
was 200 and 400 nm, respectively. Fitting the experimental data to the theoretical dependencies yields the decay length A and the heat transfer coefficient a between the superconducting electrodes and the substrate [104]. For the PbIn tunnel junction of figure
30 we obtained A=6.6pm and a = 1.9 W cm-’ K-’ at Tb=1.8 K. Due to the large
thickness of the junction electrodes and the good thermal conductivity K of the PbIn
films, the thermal decay length A = (Kd/a)”* is large for this sample resulting in a bad
spatial resolution. However, since the thermal relaxation time I= Cd/a also is quite
large, the spatial resolution can be improved by modulating the electron beam at high
frequencies 2nf2 I / r as discussed in section 4.
0
2
4
ST0
6
a
(K)
Figure 30. Normalized electron beam induced change of the quasiparticle tunnelling current
versus normalized temperature rise for diRerent values of the bias voltage at Tb=1.8K.
The symbols represent the experimental data obtained with a PbIn/oxide/Pbln tunnel
junction. The lines are calculated dependencies using a Gaussian tempcrature profile
ST(r) = 6T,exp(-r*/A’).
Low temperature scanning electron microscopy
71 I
0
0
4
8
12
16
Frequency ( MHz )
20
Figure 31. Dependence of the electron beam induced change of the quasiparticle tunnelling
current of a PblnjoxidejPbln tunnel junction on the modulation frequency of the electron
beam for Tb= 1.8 and 4.2 K. The lines represent linear fits to the data.
In figure 31 we have plotted the frequency dependence of SI,. According to
the theoretical analysis of section 4, S I q , ( f ) should be proportional to [ 1 + (27”~)’]-’’~,
that is, plotting [(61.,p(0)/SIqp(f))4versusf should yield a straight line [ 1041.
As shown in figure 31 this is indeed observed. The slope of the straight line directly
gives the thermal decay time T. Fitting the data yields the thermal relaxation time. For
the investigated PbIn/oxide/Pbln tunnel junction we obtained z = 125 ns and 115 ns
for Tb=1.8 and 4.2 K, respectively. Modulating the electron beam at 2 n f ~l/z results
in an improvement of the spatial resolution but also in a decrease of the induced signal.
For the sample of figure 30 a modulation Frequency of 20 MHz should result in an
increase of the spatial resolution by a factor of about 4 (see equation (IS)). This agrees
well with the experimental observation 11041.
7.3.I . Imaging of the quasiparticle tunnelling conductivity. As shown above, the electron
beam induced change of the quasiparticle tunnelling current depends both on the local
values of CT” and A,,’. However, for a bias point well below or well above the sumgap
voltage (A, +A2)/e the value of the integral K( V,, AI,*) depends only very weakly on
the value of the energy gap. Hence, if the electron beam induced temperature rise is
constant for all beam positions, K(r) can be assumed constant in equations (70) and
(71) and we obtain
that is, SI,&,
y ) oc crn(x,y). Therefore, scanning the electron beam across the surface
SV,, as a
function of the beam coordinates (x.J ) yields a two-dimensional image showing the
spatial variation of the tunnelling conductivity. Usually a bias point in the subgap
regime ( Vb<(Al +A2)/e) is used, since here SI,,(x,y) is much larger than for a bias
point well above the sumgap voltage [ 1041. During such measurements the Josephson
current is suppressed by an applied magnetic field.
of a superconducting tunnel junction and measuring SI.,, or, equivalently,
712
R Gross and D Koclle
Figure 32. Two-dimensional voltage images S V ( s , y ) showing the spatial variation of the
tunnelling conductivity a.(x,y) = I/Rn(x,y) of a Pbln/oxide/Pbln ( 0 ) and a Nb1AIO.J
N b ( b ) tunnel junction. Tbe signal is plotted in vertical direction during the horizontal
SCMS(Y-modulation). T6=4.2 K, beam vollage=26 kV, beam current= IOpA.
Typical examples of two-dimensional LTSEM images of U&, y ) are shown in figure
32. The images represent voltage images 8V&, y ) recorded for current biased samples.
The signal, -6V&,y), is plotted along the ydirection (y-modulation representation)
for a series of horizontal scans along a Pbln/oxide/PbIn (a) and a Nb/AIO,/Nb (b)
tunnel junction. The variation of the signal directly indicates the spatial variation of
the tunnelling conductivity. Variations of onin most cases originate from an improper
fabrication process or structural and chemical imperfections of the thin oxide layer
forming the tunnelling barrier. For Pbln/oxide/Pbln tunnel junctions the variation of
unusually is large (>30%). Here, the inhomogeneity of the tunnelling barrier is likely
to be caused by strain effects due to the thermal cycling of the samples [ 104, 1951. Lead
junctions are notoriously known to short when, after being immersed into liquid helium,
their temperature reaches about 250 K due to the growth of whiskers and hillocks. For
Nb/AIO,/Nb tunnel junctions the spatial homogeneity of the tunnelling barrier and,
hence, of un(x,y) is much better as indicated in figure 32(b). For high quality Nb/
AI0,JNb tunnel junctions variation of un smaller than 1% were observed.
We note that the imaging technique discussed in this subsection has been successfully
applied to series arrays of superconducting tunnel junctions [195,201). The imaging of
un(x,y)by LTSEM allows a direct spatially resolved quality test for individual tunnel
junctions and complete circuits. We note that for particular applications of these devices
(e.g. x-ray spectrometers [67-77]) a highly homogeneous tunnelling conductivity is
required making a direct control of this junction property desirable.
t o w temperature scunning electron microscopy
713
7.3.2. Imaging of the inicrowave field distribution. A further imaging technique based
on LTSEM is the direct probing of the local RF amplitude inside of microwave irradiated
Josephson tunnel junctions. When a Josephson junction is irradiated with an external
R F field, its current-voltage curve is modified in a characteristic way. For example,
equidistant current steps at voltages V.=(U.fnhv)/e, n= 1,2, 3 , . . . , were discovered
in 1962 by Dayem and Martin [ 1971 and explained by Tien and Gordon [ 1981 as photon
assisted tunnelling of quasiparticles. Due to the impedence mismatch between the
Josephson junction transmission line and the surrounding structure the microwave is
multiply reflected at the edges of the junction. For junctions with dimensions compwable or larger than the wavelength of the microwave this results in a standing microwave
pattern in the junction [ 199,2001. This presence of a spatially inhomogeneous R F field
V,&, y ) in the junction causes a spatially inhomogeneous quasiparticle tunnelling current density Jqp(x,y).For Vb<2A/e, Jqp(i,y)is increased by the local R F field, i.e.,
Jq,(x,y ) is large in those regions where V&, y ) is large. For V b > 2A/e this behaviour
is reversed. It is evident, that in those regions where Jqp(x,y ) is already large the electron
beam induced change Sl,,(x, y ) of the quasiparticle tunnelling current is small. That
is, for Vb<2A/e, SJqp(x,y)is small in those regions where V&y) is large [IS, 261.
For Vb>2A/e the behaviour is reversed. Here, SIqp(x,y)is large where V,,(x,y) is
large. For any value of the bias voltage, SIq,(x,y) is a direct function of the local
RF field V,,(x, y ) . Hence, scanning the junction and measuring SIqp(x,y ) yields twodimensional images showing the spatial variation of the R F field.
Figure 33 shows a typical example for the study of standing-wave patterns in superconducting tunnel junctions. The voltage image SV,,(x, y ) shows the standing microwave pattern inside of a 20 p m wide and 400 p m long Nb/AIO,/Nb tunnel junction.
The voltage signal is plotted vertically during the horizontal scans (Y-modulation presentation). The tunnel junction was irradiated directly by a 70 GHz microwave [4]. The
microwave enters at the right hand side of the junction and is reflected at its left hand
side. From the characteristic properties o f the RF field propagation inside of the tunnel
junction the Swihart velocity and the damping constant can be derived. For the sample
of figure 33 a normalized Swihart velocity E/c=0.029 and a damping constant of
(350 pm)-’ are obtained at Tb=4.2 K. These values are close to the expected ones. We
note that this imaging technique c a n be applied also to series arrays o f superconducting
tunnel junctions [ 19,281.
Figure 33. Voltage image of the standing microwave pattern inside a 400 p m long and
20 pm wide Nb/AlOJNb tunnel junction at Tb=4.2 K . The microwave (70 GHz) enters
from the right hand side (from 1171).
114
R Gross and D Koelle
7.3.3. Imaging of the energy gap disrribution. For a bias point close to the sumgap
voltage ( Vb=(A, +A2)/e) the local quasiparticle tunnelling current density Jqp(x,y )
strongly depends on the value of the local energy gaps of ihe electrodes. J,,(x,y)
increases sharply at v b = (AI@, y ) t A&, y ) ) / e . In the same way, the electron beam
induced change of the quasiparticle tunnelling current density Slqp(x,p) strongly
depends on whether the bias voltage is smaller or larger than the local sumgap voltage
A , + 2 ( ~ ,y ) / e . This behaviour is shown in figure 29, where we have plotted 61q,,(v b ) for
different values of the electron beam induced temperature rise 6 T , . For small perturbation, 61q,(x, y ) approximately can be expressed as
For small beam perturbation 6K( v b , x,y) has a sharp peak at L'b=&+2(x, y ) / e , that
is, 6lqP(v b ) shows a sharp peak for a bias voltage equal to the local sumgap voltage
(see figure 29). Therefore, scanning a superconducting tunnel junction with a spatially
inhomogeneous sumgap only those regions with V b = A I +&x,y)v)lewill yield a large
signal. The spatial distribution of the sumgap inside of the junction can be obtained
by recording a series of LTSEM images at different values of the bias voltage. The local
value of the sumgap can be determined quantitatively by keeping the electron beam
position fixed and varying the bias voltage continuously. The peak in the resulting
61q,,(Vb) dependence yields the local value of the sumgap.
With the highly developed fabrication techniques for superconducting tunnel
junctions the spatial variations of the energy gap in the superconducting electrodes due
to material imperfections usually are very small. Therefore, the imaging of the spatial
distribution of the energy gap in superconducting tunnel junctions mainly becomes
interesting for studying the spatial structures developed in thin-film superconductors
driven far from thermal equilibrium by quasiparticle injection via tunnelling
[104, 191, 1921. In this non-equilibrium state theory predicts a gap instability resulting
in the formation of a spatially inhomogeneous state [202-2061. In this state domains
with different values of the energy gap exist simultaneously in the superconductor. I n
general, the study of dynamic equilibrium states in dissipative systems is of strong
interest [207]. LTsmi provides the possibility for the spatially resolved observation of
such states in thin-film superconductors with high spatial resolution.
Experimentally, the behaviour of superconducting thin films under strong quasiparticle injection can be studied using a double tunnel junction configuration as shown in
the inset of figure 34. The bottom and the middle film form the injector junction used
to inject quasiparticles into the middle film. The middle and the top layer form the
detector junction. The quasiparticle tunnelling characteristics of the injector and detector junction are shown in figure 34. The tunnelling conductivity of the detector junction
is much smaller than that of the injector junction. In this way the detector junction can
be considered as a passive probe of the superconducting properties of the middle film.
The quasiparticle tunnelling characteristics of the detector junction (figure34(b))clearly
show how the energy gap of the middle film is reduced with increasing quasiparticle
injection. However, the integral tunnelling characteristics do not give information on
the spatial structure of a possible multiple gap state.
~
715
LOW temperature scanning electron microscopy
SWPLE
98-A
OETECTOR
VOLTAGE l m V l
Figure 41. Quasiparticle tunnelling characteristics of the injector ( U ) and the detector junction ( b ) for different injection currents. The inret shows a cross-sectional view of the double
tunnel junction configuration.
For an injector voltage at the sumgap voltage (bias point (b) to (g) in figure 34)
theory predicts that during the injection of excess quasiparticles two stable values of
the energy gap can coexist: a reduced gap AI and an unperturbed gap A3. For a certain
fixed injector voltage Vu, a first order phase transition takes place and the relative size
of the domains with energy gaps AI and A3 is determined by the total injection current.
For a fixed injector current, a second order phase transition occurs at V= Vu and the
given current fixes the relative size of the phase volumes. The phase boundaries are
stationary and stable. A current change is accompanied by a motion of the phase
boundary. For the injector voltage we have A I + A3 < e Vu< 2Aj. A more detailed discussion and further theoretical references can be found in [104, 191, 1921.
Figure 35 shows two-dimensional LTSEM images indicating the spatial distribution
of the energy gap in the injector junction. During the recording of these images no
current was applied to the detector junction. The bias points used for the recording of
the different images are indicated in the inset as black dots. Image (a) was recorded in
the subgap regime and shows the spatial distribution of the tunnelling conductivity of
the injector junction. Obviously there is an inhomogeneity of the tunnelling barrier
about in the middle of the junction. The images (b)-(g) were obtained with the bias
points in the gap regime. Since we have A I+A3<eVb<2A3 for the bias points in the
vertically rising part of the IVC and since the electron beam induced signal is maximum
where Vb=2A(x, y ) , the transition region between the domains of small gap AI and
large gap A3 generates the maximum signal. Image (b) shows that the small gap domain
nucleates just at the point of increased tunnelling conductivity. That is, the inhomogeneities of the tunnelling barrier act as nucleation centres for the small-gap domain. With
increasing injector current the small-gap domain grows in size until it occupies the total
junction area. Of course, an accurate determination of the two energy gaps A I and A3
is obtained from the ivcs of the detector junction at fixed injector current and their
derivatives [191, 1921.
For injector bias points well above the gap regime, i.e. at even higher injection
currents, the energy gap distribution in the middle film can no longer be determined
716
R Gross and D Koelle
Figure 35. Two-dimensional voltage images of the injector tunnel junction showing the
distribution of the tunnelling conductivity ((I) and the energy gap ( h ) - ( g )at T,= 1.8 K .
The bias p i n t s used for the recording of the difTerent images are indicated in the inset.
The elcctron beam induced voltage signal is plotted in vertical direction during the horizontal scans. The arrows mark the junclion area within the field of view. Beam voltage =26 kV,
beam currenl=6SpA. modulation frequency=ZOkHz (from [191]).
from LTSEM images of the injector junction. In this case LTSEM images of the detectorjunction, which is biased in the gap regime, can be used to obtain information on the
gap structure in the middle film. A typical example is shown in figure 36. Here, image
(a) shows the tunnelling conductivity of the injector junction, which is slightly higher
on the left hand side of the tunnelling area. Image (b) is an LTSEM voltage image of
the detector junction at large injection current. In this image the maximum signal is
obtained where the sum of the energy gaps of the middle and the top film is equal to
the bias voltage ( Vb= 1.9 mV). Figure 36 clearly shows that we have a small gap domain
at the left hand side, where the injection current density is high, and a large gap domain
at the right hand side. That is, the LTSEM images indicate that the formation of the
small and large gap region for injection voltages well above the sumgap voltage is
correlated to spatial inhomogeneities of the injector current density [ 191, 1921.
7.4. Arrays of superconducting tunnel junctions
Large one- and two-dimensional arrays of Josephson junctions become increasingly
interesting both because of their fundamental properties [lo-2121 and their
~
717
Low temperature scanning electron microscopy
1
1
bl
I
1
Figure 36. Two-dimensional voltage images of the injector tunnel junction showing the
distribution of the tunnelling conductivity ( a )and the detectorjunction showing distribution
of the energy gap ( b ) at Tb= 1.8 K. During the recording of (6) the injector current was
45 mA and the detector bias vollage was I9 mV. The electron beam induced voltage signal
is plotted in vertical direction during the horizontal scans. The arrows mark the junction
area within the field of view. Beam voltage=26 kV, beam current=15oA, modulation
frequency=20 kHz.
cryoelectronic applications [208-2121. Josephson junctions are natural voltage-controlled oscillators providing tunable sources operating up to the superconducting gap
frequency (a few THz for low- and tens of THz for high-T, materials). This makes
them attractive as high frequency devices. However, the problems of the very low power
and source impedence as well as the broad linewidth of singlejunction oscillators restrict
tbeir application [211]. These problems can be overcome with Josephson junction
arrays. For example, for a series array of N junctions the available power and the
impedence increases by a factor of N , whereas the linewidth of the radiation decreases
by a factor of l / N , if the junctions are phase-locked. Using two-dimensional arrays it
is possible to adjust the array parameters, such as the impedance and available microwave power, arbitrarily over a wide range according to the requirements of the particular
application. A further interesting application of series arrays of Josephson junctions is
the frequency based voltage standard [213-2151. In general, superconducting microelectronic circuits with varying degree of complexity find increasing application. Today
these circuits are fabricated by means of thin film technology and LTSEM is ideally
suited for their spatially resolved analysis. In this section we discuss the investigation
of Josephson junction arrays by means of LTSEM.
Two-dimensional arrays are highly interesting for tunable microwave sources. However, high output power and narrow radiation linewidth is obtained only if the junctions
of the array are phase-locked. In general, arrays of Josephson junctions also can be
considered as dissipative open systems. The formation oftemporal and spatial structures
(e.g. clusters of phase-locked junctions) are expected if the arrays are driven far from
thermal equilibrium. The mechanism of phase-locking and pattern formation in arrays
ofdifferent degree of complexity is not understood in detail. For the further clarification
of this subject the additional spatially resolved information provided by LTSEM is highly
interesting. In order to obtain spatially resolved information on the R F properties of
Josephson junction arrays one can use the change of the microwave power emitted by
the array as a response signal. The emitted power can be measured using a detector
junction coupled to the array by means of a DC blocking capacitor [217, 2181. The
718
R Gross and D Koelle
quasiparticle tunnelling current of the detector junction depends on the incident microwave power as discussed in section 7.3.2. Scanning the array and measuring the change
of the emitted microwave power bPrr(x,y ) simultaneously, two-dimensional 'microwave power' images are obtained. Such images show the sensitivity of the phase-locking
among the individual junctions on the local perturbation of the array by the focused
electron beam. For example, if the phase-locking in parf of the array is destroyed by
the electron beam irradiation, the emitted microwave power is expected to decrease
considerably.
7.4.I. One-dintensional arrays. One-dimensional arrays are formed by N tunnel
junctions connected in series or in parallel. Most of the imaging techniques discussed
in the last sections for single junctions also work for series arrays. In the following we
will restrict our discussion mainly to the imaging of variations of the integral parameters
of the individual junctions within an array such as their critical current values. This
allows the spatially resolved investigation of the overall superconductingcircuitry. Since
the area of the individual junctions within arrays often is quite small and in the range
of the spatial resolution of LTSEM, this overall analysis is suficient. Of course, it is
possible to study the properties of individual junctions of the array such as their spatial
distribution of the tunnelling conductivity with the spatial resolution of the LTSEM
technique 1195, 2011.
As a first example we discuss the imaging of the distribution of the critical current
values o f the individual junctions in a series array of N under-damped symmetrical
Josephson junctions (A, =Az). A typical IVC of such an array with critical currents
0.0
0.5
Voltage
1.o
1.5
( N ZA / e )
Figure 37. Current-voltage characteristic of a series array of 100 superconducting tunnel
junctions.
ranging between I,"'" and I,""" is shown in figure 37. For LTSEM imaging the array is
current biased and the change of the voltage drop across the total array is measured.
Irradiating the ith junction its critical current I,! is reduced by 81, due to the local
heating effect of the electron beam. As long as lcj-&Icj>IB no change of the array
voltage can be detected, since the junction stays in the zero voltage state upon irradiation. For IC,- &Iti< I B , the junction is switched into the voltage state and the change
of the array voltage is equal to 2AJe. For l d < l B ,the junction is in the voltage state
already without electron beam irradiation and the beam induced change of the junction
Low teinperature scanning electron microscopy
I19
voltage is small. Hence, for small beam perturbation (61ci/Iti<<l)
we have
svi=o
for Iti> I%
6 Virr2A;/e
6 V,cs2A2/e
for I, rr I ,
€or I, <IB.
(75)
That is, only those junctions with critical currents IcirrIByield a large change of the
array voltage. Scanning the array at fixed bias current and recording the induced change
of the array voltage as a function of the beam coordinates results in a voltage image
showing those junctions with ICi- l e . The distribution of the critical current values
within the array is obtained by recording a series of LTSBM images at different bias
current values.
For overdamped Josephson junctions, which have non-hysteretic current-voltage
characteristics, the situation is different. Here, the transition from the zero voltage to
the voltage slate is continuous and the voltage signal can be expressed as
avl
6L.
6V.=-
'
arc,
I=le
As will be shown below (see section 8.2.2) aV/aI,; and, hence, the electron beam
induced voltage signal has a sharp maximum at Ie-Iti. Therefore, the distribution of
the critical current values within an array is obtained by scanning the array and
recording the electron beam induced voltage signal as a function of the bias current.
Figure 38 shows a typical experimental result obtained with a series array of 30
YBa2Cu307-s bicrystal grain boundary Josephson junctions (GBJs). The array is
obtained by patterning a 5 pm wide, meandershaped YBCO line along the single grain
boundary in the YBCO film, which is obtained by the epitaxial deposition of the film
on a SrTi03 bicrystal substrate. The GBJS are located along the single grain boundary
where the YBCO line is crossing it. Therefore, all GBJScan be scanned by a single linescan.
In figure 38 the electron beam induced voltage signals of a large number of linescans
recorded for increasing values of the bias current are displayed. Since on the video
screen the linescans recorded for different bias current are displayed vertically, a quasitwo-dimensional voltage image showing the 6 Vj(IB)dependences of all GBJS is obtained.
The peaks in the SV,(IB)dependences (dark spots) mark the critical currents Iti of the
different GBJS. That is, the LTSEM voltage image directly shows the distribution of the
critical current values within the array. Vatying the applied magnetic field by this
technique the magnetic field dependence of each GBJ of the array can be determined
[491.
A particularly interesting application of LTSEM is the imaging of the distribution of
the critical current values in a series array during the irradiation with microwaves. As
shown in section 2.2 it is possible to irradiate the sample mounted on the low temperature stage of the SEM directly by microwaves 141. Since the critical current values
depend on the local amplitude of the microwave field, LTSEM allows the imaging of the
microwave distribution in the series array [ 191. This application of LTSEM is particularly
interesting for the optimization of the frequency based Josephson voltage standard
where a large series array of several thousand junctions has to be coupled homogeneously to an external microwave field.
Next we discuss the imaging of the homogeneity of the tunnelling conductivity of
the different junctions of an array. Applying a magnetic field parallel to the tunnelling
barrier of the individual junctions the Josephson current of the array can be suppressed.
120
R Gross and D Koelle
Figure 38. Secondary electron image o f a scrim array of 30 YBa2Cu10, ~6 bicrystal GnJs
( a ) and LTSEM voltage image ( h ) showing thc bias current dependence of the electron beam
induced voltage signal for the different G R J ~of the array at T=35 K . The electron beam
always is scanned along the grain boundary. The scans recorded for diKerent bias currents
are displaced vertically. The dark spots (large voltage signal) mark the critical current values
of the different O B J ~ .
Current biasing the array in the subgap regime ( Vb< 2NA/e) and recording the electron
beam change of the array voltage yields the signal
6 v b , ( x , y ) = - ~ ~S I i , ( x , y )
aI var
x u1
X x ,-y ) 1av,
e
ai
(76)
va,
in analogy to the single junction case. Here, aVJJI is the differential resistance of the
ith junction at its bias point. By scanning an individual junction of the array and
measuring 6 Vb,(x, y) the distribution of the tunnelling conductivity of this junction is
obtained. Unfortunately, for a series array the values of the bias voltage and the differential resistance of the individual junctions are not known exactly and may be different
from junction to junction. Therefore, different signal levels obtained for different
junctions cannot unambiguously he attributed to different
different values of their tunnelling
conductivity. The unknown bias voltage of the individual junctions also makes it difficult
Low temperature scanning eIeclron microscopy
721
Figure 39. Secondary electron image ( a ) and LTSEM voltage image ( h ) of a small part of
a series array of window type Pbln/oxide/Pbln tunnel junctions. The LTSFM image clearly
shows the malfunctioning of some of the junctions. The secondary electron image suggests
that the malfunctioning is caused by the formation of hillocks inside the tunnelling area
resulting in superconducting microshorts (from [ZOI]).
to measure the energy gap distribution of the junctions for a bias point in the gap
regime as described in section 7.3.3.
Figure 39 shows a secondary electron image and a LTSEM voltage image of a small
part of a PbIn/oxide/PbIn series array. The array was current biased in the subgap
regime during the recording, the Josephson current was suppressed by an applied magnetic field. In the voltage image the tunnelling areas of the individual junctions are
clearly visible. The bright and dark regions correspond to regions yielding a large and
vanishingly small voltage signal, respectively. The completely dark tunnelling areas
represent shorted junctions. For these junctions the supercurrent is not suppressed
completely. Therefore, they stay in the zero voltage state and yield no electron beam
induced voltage signal. Hence, they can easily he identified by the LTSEM imaging
method. For the investigated array the malfunctioning junctions could he correlated to
anomalies of the microstructure of the electrode material resulting from the fabrication
process. Of course, a magnified view allows the imaging of the distribution of the
tunnelling conductivity of a single junction.
For a parallel array the measurement of the variation of the parameters of the
individuat junctions (e.g. critical current, tunnelling conductivity) by LTSEM is difficult.
Applying a transport current one only knows the integral array current and voltage.
That is, in contrast to a series array, the current across each individual junction is no
longer known. Of course, one still can measure the change of the array critical current
(zero voltage state) or the change of the array voltage (finite voltage state) dependent
on the position (x, y ) of the electron beam focus. However, the resulting 61.(x, y ) or
6 V(x,y ) images are more difficult to interpret. The problem is to find an unambiguous
relation between the electron beam induced change of the global quantity and the local
array properties. This problem has not been solved yet. In particular, a major obstacle
722
R
Gross and D Koelfe
related to this problem may be non-local signal contributions due to the macroscopic
quantum coherence in the array.
If non-local signal contributions can be neglected the distribution of the critical
current values i: of the individual junctions can be obtained by measuring the electron
beam induced change, -6I:, of the critical current of the parallel array. Approximately
Si:(x,) is proportional to the critical current I: of the ith junction. Hence, scanning the
array and measuring 61, as a function of the beam position, variations of the critical
current values of the array junctions can be imaged.
7.4.2. Two-diineiisional arrays. There are various junction configurations for twodimensional arrays. The most common configurations are shown in figure -10.
Figure 40. Typical con6gurations of two-dimensional arrays of Josephson junctions.
The behaviour of two-dimensional arrays may be quite complex. The presence of
superconducting loops enables the low-frequency interaction of the individual Josephson junctions due to the flux quantization. The most evident manifestation of this
interaction is a complicated dependence of the critical current of the array on a
magnetic field applied perpendicular to the plane of the array [219-2211. In the
finite voltage state of the array the microwave interaction also may be important.
The microwave interaction depends on a characteristic interaction radius, which is
determined by the inductance of the elementary cells of the array and the critical
current of the junctions [ 1451. Coherent behaviour of the array should be observable,
if the interaction radius is of the same order as the spatial dimension of the array.
However. the details of the interaction of the individual junctions within twodimensional arrays are not yet known.
It is expected that the spatially resolved analysis by LTSEM can provide new information on the nature of phase locking in two-dimensional Josephson arrays. However, at
present an unambiguous interpretation of the electron beam induced response signal
measured during the scanning of two-dimensional arrays is not possible due to the
complexity of these systems. Recently, two-dimensional Josephson junctions arrays,
which were coupled to detector junctions via blocking capacitors, have been studied by
LTSEM [28]. By measuring the change of the emitted microwave power as a function
of the beam position, a standing microwave pattern in the blocking capacitor due to
an impedence mismatch along the coupling circuit could be imaged. Using the same
imaging technique clusters of coherently oscillating Josephson junctions within a large
two-dimensional array could be imaged by LTSEM [216]. LTSEM experiments clarifying
the nature of phase-locking in two-dimensional arrays are currently in progress.
Low ternperature scanning electron rnicroscopy
123
Figure 41. LTSEM voltage images of a two-dimensional array (configuration IC)
of figure
40) o f YBalCu,07 ~6 step-edge junctions. The LTSEM images show the "arialion of the
critical current of the junctions in the different vertical lines. One line is open so that no
signal can be detected. T = l l K ; Tb=0.9 (a), 1.4 ( b ) , 2.6 (e) and 9.0 mA.
As an example for the measurement of the variation of the critical current values
within two-dimensional arrays, figure 41 shows LTSEM images of a two-dimensional
array of YBa2Cu307-s step-edge junctions. The array configuration is equivalent t o
that shown in figure 4 0 ( c ) . The array is obtained by fabricating I O about 5 p m wide
and 0.3 p m deep grooves into a SrTiOl substrate. By the deposition of I O parallel
YBa2Cu3O7_6 lines ( 5 p m wide, 200 nm thick) running perpendicular across the grooves
a two-dimensional array of I O x 20 step-edge junctions is obtained. Each YBa2Cu10,-s
line contains step-edge junctions where it runs across the edge of the grooves. In the
LTSEM experiment a constant current is fed across the array and the electron beam
induced change of the array voltage is measured. Note that the bias current l a flowing
along the jth parallel line is not exactly known. However, due to the large number of
junctions (20) along each line we can suppose that the average resistance of the IO
parallel lines is quite similar. That is, we can assume Ik- IB/lO. In the images of figure
41 only those junctions with I : < & yield a voltage signal (bright regions). The LTSEM
images are recorded at different bias current values and directly show the spread of the
critical current values within the step-edge junction array. In particular, figure 41 shows
that a single YBa2Cu307-a line is open (Ilg=O) and hence does not yield any voltage
signal. Furthermore, it shows that the spread of the critical current values is high and
has t o be reduced considerably in order t o obtain phase-locking among the junctions.
8.
LTSEM
study of superconducting wesk links
In this section we discuss the spatially resolved investigation of weak links by LTSEM.
The term weak link means the conducting junction between bulk superconducting
electrodes, which has a critical current density much smaller than that of the electrode
material [176].The term weak link is introduced to distinguish between those weakly
coupled superconducting structures with direct, non-tunnel type conductivity and the
724
R Gross and D Koelle
well known tunnel junctions discussed in the last section. In weak links the electrical
contact between the electrodes is provided by some link fabricated of a normal metal
(N) or a superconductor (5). It is well known that the Josephson cffect is clearly exhibited
in weak links, for which the spacing between the electrodes is smaller than the coherence
length 6. of the weak link material [ 1761. For such weak links it is unimportant whether
the link material is super- or normal conducting. In SNS type weak links the Josephson
current is due to the proximity effect. That is, due lo the ability of the superconducting
condensate to conserve amplitude and phase in a normal metal at distances of the order
of the coherence length 6.. Small size weak links are promising for various applications
of the Josephson effect. Here, a particular advantage of weak links over tunnel junctions
is their low capacitance, which usually results in non-hysteretic I V G (overdamped junction). A major disadvantage of most weak links is their small J,R. product.
For most of the low-T. superconducting materials the fabricaion of tunnel junctions
of high quality is possible. Therefore, lov+T, cryoelectric circuits requiring Josephson
elements usually are realized using tunnel junctions (for a recent review see [222]). The
most common junction geometry is the planar sandwich geometry. In contrast, to our
knowledge no well defined Josephson tunnel junctions have been fabricated using the
high temperature superconductors with critical temperatures above 77 K. Tunnel junction characteristics have been obtained only for the isotropic 30 K superconductor
BaKBiO [223-2251. The major obstacles preventing the fabrication of high-T, tunnel
junctions are the extremely short coherence length and the strong electronic anisotropy
of the high-T. materials. Any viable technology of high-): superconducting electronics
must accommodate the consequences of these materials issues and also the high temperatures required for film growth.
Due to the difficulty of making high-T, tunnel junctions a large variety of different
weak link structures have been fabricated until now [ 152,2261. These structures include
bicrystal [ 152,226-2321, bi-epitaxial [223-2351, and step-edge [236-2381 grain boundary
junctions, junctions with artificial (normal and semiconducting) barriers 1239-2431, and
microbridges with a weakened/normalized superconductor link [244, 2451. Almost all
of these weak link type structures show non-hysteretic current-voltage characteristics
as expected for overdamped Josephson junctions (pc< 1) [145]. At present, the nature
of the electric transport is not known in detail for most of the different high-T, weak
links [152]. However, despite the low J,R. products of these overdamped Josephson
elements ( 4 0 0 pV at 77 K), they are useful for several applications (e.& SQUIDS [226]).
The application in more complex cryoelectionic circuits requires a controllable and
reproducible fabrication and a better understanding of the electric transport characteristics. As will be shown below the spatially resolved analysis of the high-T, Josephson
elements by LTSEM is highly interesting both with respect to the understanding of their
electric transport characteristics and to the further improvement of their fabrication
processes.
8.I . Basic equations
The basic equations describing superconducting weak links are similar to those given
in section 7.1 for Josephson tunnel junctions. For Y=O the phase difference 4 between
the electrodes of the weak link is constant in time. In this case a constant supercurrent
Low temperafure scanning electron microscopy
725
.I~(x,y)=J,(x,Y)~'[4(x,Y)l
(77)
density
flows through the weak link. Here, 56 is a 27z periodic function which is single valued
only for sufficiently short weak links [14S]. Sufficiently short means that the effective
spacing between the electrodes is less than -35.. The basic principles of how the
Josephson effect takes place in weak links with strong current concentration has been
given by Aslamazov and Larkin [246]. A comprehensive overview for the various types
of weak links including expressions for the magnitude of the maximum Josephson
current density J, and its temperature dependence is given in [ 1761. Expressions for the
spatial variation of the phase difference and the magnetic field dependence of the critical
current of large weak links can be derived in the same way as for tunnel junctions
taking into account the modified current-phase relation of the weak links. We note
that F(4) often is quite close to sin 4. In this case the expressions given in section
7.1.1 for the tunnel junctions directly apply also for the weak links.
For V#O the phase difference between the electrodes of a weak link will vary with
time according to equation (57) resulting in time dependent supercurrent
Up to now no single theory for the description of the weak link behaviour for arbitrary
F(#)relationship has been developed. The existing microscopic theory 1247, 248) is
very complex and is usually not used for analysing the dynamics of weak links. Qualitatively, theory predicts that the behaviour of the supercurrent, in particular its frequency
dependence, is close to that of tunnel junctions [249]. In contrast, the properties of the
normal current density Jn are quite different. The expressions for J, deviate strongly
from the expression obtained for tunnel junctions (equation (59)). For tunnel junctions
one always has Jqp<VIR, and, furthermore, Jqp(V ) approaches the ohmic line V / R .
for I VI >>2A/e. In contrast, for weak links one has J.> VIR. at least at I VI >2A/e and
the J.1 V ) dependence approaches
+
J, = V / R . J, sign V
(79)
at I VI > > a l e . Here, J, is the so-called excess current density, which often is close to
J, for T<0.5TCand R. is the norma1 resistance times area. The difference in the normal
current behaviour between tunnel junctions and weak links is caused by the transfer of
quasiparticles with energies less than A, which exist only in the weak link structures
[247].
At constant voltage, the supercurrent through a weak link oscillates and its time
average should disappear allowing the direct measurement of Jn(Y ) . However, at high
frequencies the intrinsic impedance of most weak links is much smaller than the impedance of the external circuit. Therefore, the electric charge carried by the oscillating
supercurrent cannot pass through the external circuit and has to be compensated either
by a normal or displacement current across the weak link. This situation corresponds
to the current biased case. That is, it is very difficult to establish a voltage bias for weak
links at high frequencies. For a current biased weak link, however, it is almost impossible
to measure Jn( V ) due to the small capacitance of the weak links. Note that for tunnel
junctions the capacitance is usually quite high (&>> I) resulting in a large displacement
current. Due to the large capacitance for tunnel junctions the oscillating supercurrent
126
R Gross and D KoeNe
is fully compensated by the displacement current or, equivalently, the AC component
of the voltage is totally shunted by the capacitance of the structure. This results in an
almost constant junction voltage V ( t ) E ( V ) , and, hence, (I,>,=O. This allows the
measurement of Jqp(V ) even in the current biased case as pointed out in section 7.1.2.
In contrast, for the majority of weak links the opposite is true (&<c I). That is, the
displacement current is vanishingly small and the oscillating supercurrent has to be
compensated by a normal current resulting in a strong temporal variation of the weak
link voltage. This, in turn results in a complex oscillation and non-zero time average
of the pair current for V > 0 making the determination of J.( V ) difficult.
8.2. Pair current densify
In this section we discuss the application of LTEM to the imaging of the spatial distribution of the pair tunnelling current density in superconducting weak links. Here we will
restrict our discussion mainly to the investigation of high-T, grain boundary junctions
(GFA) [I 52, 1531. The typical sample geometry of a a - b tilt YBa2Cu,0,-s bicrystalGBJ [152, 2281 is shown in figure 42. A oaxis oriented YBazCu,O,-s film is grown
I
ELECTRON BEAM
i
,-GRAIN
BOUNDARY
Figure 42. Sketch of the sample configuration for the investigation of thin-film bicrystal
grain boundary junctions by LTSEM.
epitaxially on a SrTi03 bicrystal with a misorientation angle 8. In this way a
YBa2Cu30.r-sthin film bicrystal with the same misorientation angle is obtained. The
GBJ is obtained by patterning a narrow line of width W into the bicrystalline film
straddling the grain boundary. The area of the GBJ is given by the width W of the line
times the thickness d of the film. The film thickness typically ranges between 20 and
400 nm and is small compared to the Josephson penetration depth. That is, GBJS usually
can be considered as one-dimensional junctions. Furthermore, the penetration depth
of the electron beam is comparable to or even larger than the film thickness. Therefore,
we can assume that the GBJ is perturbed about homogeneously across the whole film
thickness (2-direction).
The local pair tunnelling current density of weak links is determined by the local
coupling strength and phase difference between the superconducting electrodes. It is
evident that the electron beam induced signal will contain information on these quantities. Hence, LTSEM allows the imaging of spatial variations of these quantities, which
are caused by similar reasons as discussed in section 7.1.3. In the same way as for
tunnel junctions, LTSEM images are obtained by scanning the electron beam across the
Low temperature scanning electron niicroscopy
127
weak link and measuring the induced response sigual as a function of the beam position.
The signal usually is the change of the weak link current or voltage for the voltage and
current biased situation, respectively.
8.2.1. Zero uoltuge slate-static structures. In analogy to equations (63) and (64)the
electron beam induced change of the maximum pair current bl,(x,y) of the weak link
is composed of a local, 61:, and a non-local contribution, &If,
and can be expressed
in the same way as for tunnel junctions (equations (62)-(64)). For the one-dimensional
CBI geometry shown in figure 42 we obtain
and
for small electron beam perturbation (6J,/J,<< 1). In most LTSEM experiments the electron beam perturbation is kept small, i.e., the beam can be treated as a passive probe.
In this case non-local signal contributions usually can be neglected and the electron
beam induced change of the pair current is solely given by SI:. With the assumption
that aJ,/aTis about constant within the perturbed area and approximating the integral
over the temperature profile by 2A6T0 the signal S is given by
S( y ) = - 61:( y )
- aJ,(y)/aTS(q5 (~‘))2dA6
To.
(82)
For most weak links we have aJ,(T)/aTaJ, where J,=J,(T=O).
That is, the
measured signal is proportional to J d , F(q5), TO, and A. In the experiment 6To and
A are about constant, i.e., the signal yields information on J&) and $(U). In figure
43 we have plotted -61: versns the electron beam induced temperature rise for different
reduced temperatures Tb/Tc.The curves were calculated using a Gaussian temperature
In the calculation non-linear effects, which
profile 6T(y’)= 6To exp(-(y’-y)’/A’).
0.0301
8
,
,
I
0.020
0.95
-.s.
,
I
(Cj
,
,
,
(iii)
-3
[ii)
0.010
I
0.000
0.OO
0.02
0.04
0.06
0.08
0.10
W T C
Figure 43. Calculated normalized electron beam induced change of the maximum pair
current o f a bicrystal grain boundary junction versus normalized temperature rise at ditTerent
reduced temperatures. For the calculation a Gaussian temperature prohle Sr(y‘) =
6Toexp(-(y’-y)’/A2) and J . ( T ) = &(I-T/T$ was used.
128
R Gross and D Koelle
become important for large 6To,have been taken into account. The non-linear sample
response results in a deviation from the linear 61,(6To)dependence for large 6To.The
calculated sample response agrees well with the measured SI,(GTo) dependencies.
For small weak links (WGA,), b(y)-constant in the absence of an external magnetic field. In this case the measured signal is proportional to Jco(y).That is, scanning
the weak link and measuring -61: as a function of the beam coordinate yields a LTSEM
image showing the spatial variation of the maximum pair current density. For large
weak links (W$A,) or in the presence of a magnetic field applied parallel to the weak
link area, spatial variations of the LTSEM signal originate both from spatial variations
of Jd and @ making an unambiguous interpretation of the LTSEM signal more difficult.
The electron beam induced change of the maximum pair current is measured by the
MCCD technique described in section 5.2. Note that weak links usually have non-hysteretic IVCS whereas those of tunnel junctions are hysteretic. With respect to the MCCD
technique this means that for weak links one has to use a small threshold voltage for
a precise measurement of the maximum pair current (typically less than 1OpV). In
contrast, for superconducting tunnel junctions the threshold voltage can be close to the
sumgap voltage (typically a few mV). Therefore, the precise measurement of the maximum pair current by the MCCD technique is more difficult for weak links than for
tunnel junctions. In particular, this is the case if the IVC of the weak link is rounded
by thermal noise.
the phase difference @ is
varying in time resulting in a time dependent pair current. The characteristic time scale
of the dynamic phenomena occurring in the finite voltage state of weak links is given
by the Josephson oscillation (typically ps to ns) is by several orders of magnitude
shorter than the temporal resolution of the LTSEM technique, which is given by the
relaxation time r of the electron beam induced non-equilibrium state (typically tens of
ns to ps). Therefore, as for tunnel junctions LTSEM does not allow the time resolved
measurement of the dynamics of weak links. Furthermore, for the calculation of the
electron beam induced response signal time averaged expressions have to be used. This
is completely analogous to the case of tunnel junctions.
As discussed above it is difficult to establish a voltage bias for the low impedance
weak links. Therefore we will discuss only the current biased case in the following. Due
to the vanishingly small displacement current of weak links the temporal oscillation of
the pair current density has to be compensated by an oscillating normal current in
order to keep the net current constant. This is possible only if the weak link voltage
and the phase difference oscillate with the Josephson frequency and its harmonics. Due
to the phase oscillations the calculation of the IVC becomes complex. Usually, the IVCS
of w a k links cannot be expressed analytically for the arbitrary case. An exception is
the resistively shunted junction (RSJ)model [145, 1461 (pc --t 0). Fortunately, most
high-T, weak links can be modelled well by the RSJ model [152]. Note, however, that
the RSJ model can be applied only to small weak links ( W’< A,), which can be considered
as so-called lumped elements and can be characterized by their integral critical
current I,.
We first discuss the response signal for small weak links. In order to be able to
apply the RSJ model we assume that the current-phase relation of the weak link is
about sinusoidal. According to the RsJ-model the time average of the weak link voltage
8.2.2. Finite voltage state4ynmnic phenomena. For I’fO
Low temperature scanning eiecrron microscopy
129
Figure 44. Calculated normalized electron beam induced voltage signal versus normalized
bias current for four different values of the electron beam induced change of the maximum
pair current. The voltage signal is calculated according to the RSJ model.
where V.=J,R, is the characteristic weak link voltage. Equation (83) shows that the
resistive branch of the IVC has a parabolic shape. Using the RSJ expression for the time
averaged voltage we can calculate the electron beam induced change 6V o f the weak
link voltage. In figure 44 we have plotted the normalized voltage change versus the
normalized bias current. The curves have been calculated according to equation (83)
for different values of 6 / : / f c .The voltage signal is maximum at I n = I c . Therefore, for
measurements in the resistive state usually a bias current value close to I, is used. Figure
44 shows that the electron beam induced voltage signal is about proportional to the
&Ii/Ic, that is, we have
6 V ( y )cc - 6 I $ ( y )cc --aJc(y) F(@(y))2dA 6To.
dT
(84)
For small electron beam perturbation SV(y) approximately is given by
According to equations (82), (84) and (85) the measurement of SI;'in the zero voltage
state and the measurement of 6 V in the resistive state yield about the same information.
and $ ( y ) and give information on these
Both signals are proportional to J&)
quantities.
For large weak links the RSJ model can no longer be applied. The time average of
the weak link voltagecan be determined by solving the perturbed sine-Gordon equation.
To our knowledge this has not yet been done in the limit of high damping applying to
most weak links. Therefore, any quantitative calculation of the electron beam induced
voltage signal is dificult a! present.
730
R
Gross and D Koelle
- H
X
10pm
Figure 45. Voltage imagc o f a bicrystal GEJ (Tb=14 K ) showing the maximum pair currenl
density along the grain boundary. The grain boundary extends in j-direction as indicated
at the top. The electron beam is scanned in x-direction. T h e voltage signal is plotted
vertically during the horizontal Scans (from [?SI).
Figure 45 shows a LTSEM voltage image of a Y Ba2Cu,0,-s bicrystal GBJ. The CBI
was current biased at I s > I, during the experiment. Since the investigated OBI is small
and there was no applied magnetic field the measured voltage signal is about proportional to the local pair current density. That is, the voltage image shows the spatial
distribution of the maximum pair current density along the grain boundary. Spatial
variations of the maximum pair current density of about 20-50% as shown in figure
45 are typical for bicrystal GBJS. The origin of the observed inhomogeneities is not yet
clear. However it is expected that they are related to variations of the microstructure
and oxygen content of the superconducting film along the grain boundary.
In figure 46 the LTSEM voltage signal along a YBafu300r-abicrystal (a) and stepedee (b) GBJ is shown. The measured voltage signal is about proportional to the I O C ~
maximum pair current density. As shown by the single line scans, the spatial variation
of J , along the step-edge GBJ is much larger than along the bicrystal CBI. Our LTSSEM
analysis o f both bicrystal and step-edge GBJS showed that the spatial variation of the
maximum pair current density in general is much larger for the step-edge GBJS 12501.
Although the origin of this behaviour is not clear in detail, it is likely that the strong
inhomogeneities observed for the step-edge GBJS are related to the inhomogeneous
properties of the substrate steps prepared by ion beam milling [236,237]. The spatially
resolved analysis of high-T, GBJ by LTSEM shows that the spatial homogeneity of these
weak links is quite poor at present and should be improved considerably. Thereby,
LTSEM is expected to provide important information on possible origins of the observed
in homogeneities.
A magnetic field applied parallel to the grain boundary causes a linear increase of
the phase difference along the grain boundary resulting in a sinusoidal oscillation of
Low temperature scanning electron microscopy
731
Figure 46. Electron beam induced voltage signal showing the maximum pair currcnt density
along a YBa,Cu,0,-8 bicrystal ( 0 ) and step-edge GBJ ( h ) .
Figure47. (a) LTSFM voltage image ofthe 4-5 vortex state in a 23 pm wide YBa2Cu107-s
grain boundary Josephson junction at T=83 K. The edges of the GRJ are indicated by the
arrows. The position of the grain boundary is marked by the broken line. Rright and
dark regions correspond to sample regions yielding a positive and negative voltage signal,
respectively. (b) Single linescan along the grain boundary with thc voltage signal plotted
vertically during the horizontal scan (from [461).
the pair current density (see equations (49) and (51)). This oscillation (static vortex
state) can be imaged by LTSEM. In figure 47 the spatial variation of the electron beam
induced voltage signal along a YBa2Cu30,-a hicrystal CBI is shown in the presence of
an applied magnetic field. The voltage signal was measured for a current biased grain
boundary having a width of 23 j m . The electron beam was scanned along the grain
boundary extending in y-direction and the induced voltage signal is used to control the
132
R Gross and D Koeile
brightness of the video screen (a). In (b) the voltage signal for a single linescan along
the grain boundary is plotted. The interpretation of the LTSEM image is equivalent to
that of figure 23. The spatial modulation of the pair current density by the applied field
(static vortex state) is clearly visible, Figure47shows the4-5 vortex state ofthe bicrystal
GBJ. Varying the magnetic field the different vortex states of the CBI with up to more
than 10 Josephson vortices could be imaged [46]. Until now, static vortex states could
be observed only for YBa2Cu,07-6 bicrystal GBJS. For the other types of high-T, weak
links the strong spatial inhomogeneity of the maximum pair current density (see e.g.
figure 46(b)) and flux trapping makes the observation of the flux states difficult.
;;;m
:::m
0.0
\
v
1 2
I
0.0
.?a = 0
-0.5
0.2
0.4
0.6
0.8
1.0
-1.0
..
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
(e) 1.0
f
7
\
-0.5
00..55
0.0
0.0
0.0 bo = TI
0.2
0.4
0.6
u/w
0.8
1.0
-1.0
OJI
0.2
u/w
Figure 48. Sketch of the geometry of the investigated ramp-edge Josephson junctions ( a ) ,
variation of the magnetic field H (61, the phase difference4(c). and lhe normalized supercur-
rent density J,/J. ((d) and [e))along the ramp-edge junction (y-direction) in the presence
of two vortices trapped at y/W-0.3 and 0.8 as skelched in (a).For the Calculation of the
supercurrent distribution a starling phase & = O ( d ) and 40=n/6 (e) was used.
Figure 48 shows the effect of the trapping of magnetic flux quanta in YBa2Cu307_6
ramp-edge Josephson junctions. In the ramp geometry shown in figure 48(a) the weak
coupling between the two YBCO electrodes is obtained through a thin (-10-30nm)
PBCO barrier layer 1251). Cooling down the sample in a non-vanishing magnetic field
(e.g. earth magnetic field), Abrikosov vortices are often trapped in the YBCO film covering the ramp. Trapping in the ramp region is favoured, since the quality of this part
of the film usually is degraded due to the etching process of the ramp. The high quality
bottom film, in contrast, expels the flux in order to stay in the Meissner state. In this
Low temperature scanning electron microscopy
733
Figme 49. LTSFM voltage image or the supercurrent distribution in a 25 p m wide rampedge Joscphson junction a1 T = 35 K . Bright and dark regions correspond to sample regions
yielding a positive and negative voltage signal, respectively. On top of the voltage image
we have plotted the LTSEM voltage signal for a single line scan along the )-direction (from
~451).
way the magnetic flux leaves the sandwich structure parallel t o the barrier. The magnetic
flux parallel to the barrier has strong influence on the pair current density distribution.
Figure 48(b-e) schematically shows the spatially inhomogeneous magnetic field parallel to the barrier due to two trapped Ahrikosov vortices i n the ramp region as well
as the resulting phaseshift 4 ( y ) and pair current density distribution JJy). Since the
field distribution is spatially inhomogeneous, the increase of the phase difference Q
along the junction is no longer linear (equation (51)) and, hence, the oscillation of the
pair current density not sinusoidal (equation (49)). Figure 49 shows a typical LTSEM
voltage image of a YBCO ramp-edge junction. The LTSEM image, which was recorded
at zero applied magnetic field, directly shows the spatial oscillation of the pair current
density due to trapped Ahrikosov vortices. After warming up the sample above T, and
cooling down again, usually a different number of flux quanta with different trapping
positions is frozen in [4S]. The trapped vortices result in a strong reduction of the
integral pair current and an unusual magnetic field dependence. The LTSEM study of
ramp-edge Josephsonjunctions showed that ambient magnetic fields have to be shielded
effectively to avoid flux trapping in the ramp-edge geometry.
8.3. Quasiparticle current density
In contrast to superconducting tunnel junctions, in the resistive state (I VI > O ) the total
current density across weak links is composed both of a pair and a quasiparticle current
density. As discussed above this makes the determination of the quasipartide currentvoltage characteristic difficult for the arbitrary case. Similarly, the electron beam
induced change of the weak link current at I VI 20 will have a, pair current component
61,(x, y) and a quasiparticle component 61,,(x, y ) . Due t o the lack of a suitable theoretical description of the IVC of most weak links the calculation of these components is
difficult. Therefore, from the measurement of the electron beam induced change of the
total weak link current we cannot obtain information on the local quasiparticle current
at present. In order t o d o so,theoretical models for the description of the IVCS and the
electron beam induced signal have t o he worked out. Fortunately, most high-T, weak
134
R Gross and D Koelle
links can he modelled reasonably well by the RSI model. Within this model the quasiparticle current density increases linearly with voltage and the normal conductivity
U" is proportional to the maximum pair current density. That is, within this model
the quasiparticle current density has the same spatial distribution as the pair current
density.
8.4. Disordered arrays of weak links
In the high temperature superconductors grain boundaries are known to act as weak
links [ 152, 1531. Therefore, polycrystalline samples of these materials represent
disordered networks of weak links. LTSEM can he used for the investigation of the
current transport in such samples. Historically, the first high-T, thin-film samples investigated by LTSEM have been polycrystalline YBCO films. The important result provided
by the LTSEM analysis was the observation that the critical current density in these films
was limited by the grain boundaries whereas the critical current density within the
individual grain could he significantly higher [29]. Ofcourse, due to the limited penetration depth of the electron beam the application of LTSEM is restricted to thin-film
ELECTRON BEAM
6
e
Figure 50. Sketch of a polycrystalline high-T, superconducting film representing a dirordered network of weak links.
samples. Figure 50 shows a sketch of a polycrystalline high-T, film. The grain boundaries
within the microbridge structure are indicated by the lines. It is evident that such sample
represents a two-dimensional disordered array of weak links.
The current transport along a narrow superconducting line containing a network
of grain boundaries as shown in figure 50 is determined by the maximum pair current
density across the grain boundaries and the phase difference between the individual
grains. Due to the variations of the parameters of the different grain boundaries it is
difficult to make any prediction on the current flow. The situation becomes even more
complicated in the presence of a magnetic field applied perpendicular to the film. Therefore, the application of LTSEM to the investigation of the current transport in general
is interesting. However, the clear interpretation of the electron beam induced signal
(e.g. the change of the voltage drop for the current biased film) usually is complicated.
The reasons for that are similar to those discussed above with respect to the investigation
of large weak links. In particular, for a network of weak links non-local signal contrihutions are-expected to play a more important role than for a single weak link.
Low tenperarure scanning electron microscopy
735
For the LTSEM analysis of a grain boundary network as shown in figure 50 it is
advantageous that the size of the individual grains is larger than the spatial resolution
limit of the LTSEM imaging method (-1 pm). We have fabricated polycrystalline highT, films with an average grain size of several microns by growing the high-l; material
epitaxially on a SrTi03 polycrystal. In this case the grain size of the film is determined
by the grain size of the polycrystalline substrate. The current transport along narrow
lines patterned into such films has been studied recently by LTSEM [157]. Although the
measured global sample response to the local electron beam irradiation is complex and
not understood in detail the following important results were obtained.
the maximum pair current density across the grain boundaries is by several orders
of magnitude smaller than of the grain material;
the spatial distribution of the supercurrenl in the grain boundary network depends
sensitively on the applied magnetic field;
the different grains have different critical temperatures most likely due to their
different crystallographic orientation.
The further interpretation o f the LTSEM response signal requires an intensive theoretical analysis of the current transport in disordered Josephson arrays.
Acknowledgments
The experimental and theoretical work summarized in this article was stimulated and
supported by the many former and present co-workers of the authors. The authors
would like to tbank L Alff, A Beck, J Bosch, C C Chi, J R Clem, T Doderer, R Eichele,
G Fischer, M Flik, L Freytag, 0 M Froehlich, R Gerdemann, M Hartmann, F Hebrank,
E Held, K Hipler, R P Huebener, K-D Husemann, W Klein, F Kober, M Koyanagi,
Ch Kriille, S Lemke, J Mannhart, A Mam, K M Mayer, B Mayer, J Neimeyer, T
Nissel, H Pavlicek, D Quenter, F Schmidl, H Seifert, C C Tsuei and A V Ustinov for
their contributions and J Fischer, K-H Freudemann, M Kleinmann, H Muller, Th
Nissel and H-G Wener for their technical assistance.
Financial support of this work by grants of the Deutsche Forschungsgemeinschaft,
the Stiftung Volkswagenwerk, and the Bundesminister fur Forschung und Technologic
(project No 13N5802-4) is gratefully acknowledged.
References
[I] Reimer L 1985 Scanning Electron Miemreopy (Heidelberg: Springer)
Goldstein 1 I 1975 Procticol Scanning Eieefron Microscopy (New York: Plenum)
Ehrenbcrg W and Gibbons D J 198 I Electron Bombardment induced Conductiuity and ifs Applicafions
(London: Academic)
[Z] Seifert H 1982 Cryogenics 22 675
131 Gross R, Bosch H-G. Fischer 1 and Huebener R P 1989 Cryogenics 29 716
141 Doderer Th,Wencr H-C. Moeck R. Becker C and Huebener R P 1990 Cryogenics 30 65
151 Clcm J R and Huebener R P 1980 J. Appl. Phys. 51 2764
[ 6 ] Gross R and Koyanagi M 1985 J . Low Temp. Phj's. 60 277
[7] Huebener R P 1984 Rep. Pro& Phys. 41 175
IS] Huebener R P 1987 Advances in Electronics aiid Electron Plry.sics Vol 70, P W Hawkes (New York:
Academic); 1989 Supepereondtretiuc Quontrrin Elecrronics ed V Kose (Berlin: Springer) pp 205-25
[9] Epperlein P W, Seifert H and Huebener R P 1982 Phys. Left. 92A 146; 1983 Plrys. Len. 95A 326
736
R Gross and D Koelle
[IO] Gross R, Koyanagi M, Seifert H a n d Huebener R P 1985 P1v:s. Lptt. 109.4 298; 1986 J. Low Tenip.
Phys, 62 245
[ I I ] Bosch J, Gross R, Koyanagi M and Huebener R P 1985 Phys. Reu. Lett. 54 1448
1121 Mannhart 1, Bosch J, Gross R and Huebener R P 1987 P l y Rev. B 35 5267
[I31 Bosch I, Gross R, Koyanagi M and Hueben'er R P 1987 J. Lon' Temp. Phys. 68 215
[I41 Mannhart J. Bosch J. Gross R and Huebener R P 1987 Phys. Lett. lZZA 439
[ I S ] Mannhart J, B o s h 1, Gross R and Huebener R P 1987 J . Low Temp. Pliys. 70 459
116) Gross R, Doderer T, Huebener R P, Kober F, Koelle D, Kruelle C, Mannhart J, Mayer 6, Quenter
D Ibtinov A 1991 Plysira B 169 415
[E71 DorderL- T. Quenter B, Mayer B, Kruelle C, Ustinov A V , Huebener R P, Niemayer I, Fromknecht
R, Pocpel R. Klein U, Dmmschneider P Hinken J H 1991 Nonlinear Superconducriue Electrurrics
and Josepltson Deuices ed N F Pedersen. M RUSSO,A Davidson, G Costabile and S Pagano (New
York: Plenum) p 353
I181 Krulle C A, Doderer T. Quenter D and Huebener R P 1991 Appl. Pltjs. Ldr. 59 3042
[IS] Doderer T, Huebener R P, Kruelle C A . Mayer B. Niemayer J. Poepel R Quenter D 1992 Supercorrductiue Devices and rlieir Applications ed H Koch and H Luebbig (Berlin: Springer) p 419
(201 Doderer Th 1991 Thesis University of Titbingen (unpublished)
1211 Mayer 6, Doderer T, Huebener R P and Ustinov A V 1991 Phys. Rev. B 44 12463
[ZZ] Lachenmann S G 1992 Diploriia nresis University of Tfibingen (unpublished)
(231 Lachenmann S G, Doderer T, Huebener R P. Quenter D, Niemeyer 1and Pepel R 1993 Phys. Reo,
B 48 3295
[24] Ustinov A V, Doderer T, Huebener R P, Mygind J, Obonzov V A and Pedersen N F 1993 IEEE
Trans. Appl. Supercond. 3 2287; 1993 IEEE Trons. Appl. Supercond, 3 2508
1251 Ustinov A V, Doderer T, Huebener R P, Pedcrsen N F, Mayer Band Obomov V A 1992 P l y . Rev.
Lett. 69 1815; 1993 Phys. Rev. B 4 7 944
[26] Quenter D, Stehle S, Doderer T, Kriille C A, Huebener R P, MGller F, Niemeyer J, PBpel R, Weimann
T and Barfknecht A T 1993 Appl. P h p . L E I I . 63 2135
I271 Kriielle C A, Doderer T, Quenter D, Huebener R P, Piiper R and Neimeyer J 1993 Phys. Rev. E
submitted for publication
[28] Doderer T. HoRmann D, Huebener R P, Kirchmann N, Krudle C A, Lachenmann S, Qventer D,
Schmidt 1, Slehle S, Niemeyer J and Pdpel R 1993 IEEE Trans. Appl. Supereond. 3 2724
1291 Gross R. Bosch 1. Huebener R P. Mannhart I. Tsuei C C. Scheuennann M.. Oorvsko
. , M M and Chi
C C 1988 Nu& 332 818
1301 Huebener R P. Gross R and Bosch J 1988 Z. f l w s . B 70 425
i3lj Gross R, Harimann M, Hipler K, Hucbener R P, Kober F, Koelle D 1989 IEEE Trans. M a p 25
2250
[32] Gross R, Mannhart I. Hipler K, Huebener R P, Chaudhari P, Dimos D, Tsuei C C. Sclluberr J and
PODDe U 1989 And. P/WS. L p f l . 55 2\32
[33] Koelie'D, Kober F;Hartmann M, Gross R Huebener R P,Roas B, Schultz L a n d Saemann-lschenko
G 1990 Plrysica C IGl79
[34] Hartmann M, Hipler K, Koelle D, Kober F, Bemhardt K. Sermet T and Gross R 1989 Z.P h y ~B
15 423
[35] Mannliart J. Gross R, Hipler R, Huebener R P, Tsuei C C, Dimos D and Chaudhari P 1989 Science
245 839
(361 Mannhafi J, Gross R, Huebener R P, Chaudhari P, Dimos D and Tsuei C C 1990 Cvyqenics 30 397
1371 Mannhart J, Huebener R P, Kober F, Koelle D, Chaudhari P, Dimos D, Gross R, Gupta A and Tsuei
C C I990 Pltpico A 168 345
[38] Monroe D, Brocklesby W S, Farrow R C, Hong M and Liou S H 1988 Appl. Plrys, f i l l . 53 I210
(391 Frenkel A, Clausen E, Chang C C, Venkatesan T, Lin P S D, Wu X D, Inam A and Lalevic B 1989
App/, Phys. Lert. 55 91 1
[SO] Frenkel A, Chang C C, Clausen E, Venkatesan T, Lin P S D , Wu X D, Inam A and Lalevic B 1990
J. Marer. RCS 5 691
[41] Gurtovoi V L, Chernykh A V and Yermolayev A B 1990 Suprrcond. 3 43
1421 Feng Y J, Cheng Q H. Wu P H and Liu H M 1992 J . Appl, P1ty.r. 72 5350; 1992 J. Low, Temp Plys.
14 356
[43] Hollin C A , Abcll IS, Goodyear Y W and Chew N G I993 Applie~Sdsr,percortd~,cli~iry
ed H C Freyhardt
(Oberwsel: DGM) pp 493-6
(441 Husemann K-D. Gerdemann R. Marx A, Mayer B. Nissel T, Gross R, Huebener R P 1993 Applied
Superconducliuify ed H C Freyhardt (Oberwsel: DGM) pp 1195-8
..
Low
t e n t p e r a l u r e scanning electron
microscopy
131
1451 Marx A. Husemann K-D, Mayer B, Nissel Th., Gross R, Verhoeven M A J and Gerritsma G J 1993
Appl, Pltys. Lett. accepted for publication
I461 Fischer G M,Mayer B, Gross R. Nissel TIL.,Husemann K-D, Huebener R P, Freltoft T, Shen Y and
Vase P 1993 Science submitted for publication
[47] Husemann K-D, Gross R, Huebener R P and Roas B 1993 Appl. Pltys. Lett. 62 2971
[48] AIR L, Mayer B, Schuster S, Froehlich 0 M, Gerdemann R, Beck A and Gross R 1993 J , Appl. Pltys.
75 1842
[49] Alff L, Gerdemann R, Husemann K-D, Beck A, Tiduble T, Mayer B and Gross R 1993 Applied
Superconductivity ed H C Freyhardt (Obenvsel: DGM) pp 1199-202
[SO] Eichele R, Seifert Hand Huebener R P Appl. Plrys. Leu. 38 383
(511 Eichele R, Seifert H and Huebener R P 1982 Z. Phys. B 48 89
[ S Z ] Eichele R 1983 Thesis University of Tiibingen
[53] Huebener R P and Metzger W 1985 Scanning Elemon Microscopy 11 617
1541 Metzger W and Huebener R P 1988 Z. P h y s B 73 33
[55] Kittel H, Held E, Klein W and Huebener R P 1989 Z. Phys. B 71 79
1561 Mehger W, Huebener R P, Haug R and Habermeier H U 1985 Appl. Phys. Lett. 47 1051
[57] Metzger W 1987 Thesis University of Tiibingen (unpublished)
[58] Klein W, Held E and Huebener R P 1987 2.Plys, B 69 69
1591 Klein W 1989 Thesis University olTiibingen (unpublished)
1601 Held E. Klein W and Huebener R P 1989 Z. Pliys. B 75 17: 1989 2. Phys. B 15 223 (1989)
[61] Held E 1989 Thesis University of Tiibingen (unpublished)
1621 Huebener R P, Held E and Klein W 1990 Mat. Science Eng. E 5 157
1631 Mayer K M, Gross R. Parisi J, Peinke J and Huebener R P 1987 Solid Sture Coiiunun. 63 55
1641 Mayer K M 1988 Thesis University of Tiibingen (unpublished)
[65] Mayer K M, Parisi J, Peinke J and Huebener R P 1988 Physicn D 22 306
1661 Mayer K M, Parisi J and Huebener R P 1988 Z. Phys. B I1 171
[67] Gross R, Klass U and Huebener R P 1987 Superconductive Particle Detectorsed A Barone (Singapore:
World Scientific)
1681 Gross R, Klass U and Bender H 1987 Pvoc. ISEC’87 (Tokyo: Japan Society of Applied Physics)
p 345
[69] Gross R, Klass U and Koyanagi M 1988 Low Temperature Detcctors/or Neutrinos ond Dork Mutter
II ed L Gonralez-Mestres and D Perret-Gallix (Paris: Editions Frontieres) p 255
170) Hebrank F, Lemke S,Huebener R P and Gross R 1990 Nucl. Instrunt. Me2hods A 288 541
I711 Lemke S, Hebrank F, Huebener R P and Gross R 1991 Low Temperatwe Detectors for Neutrinos and
Dark Matter Illed L Brogiato, D V Camin and E Fiorini (Paris: Editions Frontieres)
[72] Hebrank F, Lemke S, Deeg M, Cross R and Huebener R P 1992 Low Temperature Detectors for
Neutrinos and Dark hfatte? W e d N Booth and G L Salmon (Paris: Editions Frontieres) p 297
[73] Kebrank F, Lemke S, Breunig M, Gross R, Doderer T and Huebener R P 1993 IEE Trans. Appl.
Supercond. 3 2084
174) Lemke S, Hebrank F, Fominaya F, Gross R, Huebener R P, Rando N, ,‘an Dordrecht A, Huebner
P. Videler P and Peacock A 1993 Proc. LT-20 (Eugene) Physlcu B to appear
I751 Hebrank F, Lemke S, Fominaya F, Gross R and Huebener R P 1993 J. Loa. Temp. Phj’s. 93 647
[76] Lemke S. Hebrank F, Fominaya F. Gross R, Huebener R P, Wiemann Th.. Niemeyer J, Schnakenberg
U and Benecke W 1993 J. Low Temp. Plrys. 93 617
[77] Lemke S, Hebrank F,Gross R, Huebener R P, Wiemann Th., Poepel R, Niemeyer J, Schnakenberg
U and Eenecke W 1993 J. Appl. Ph.w 13 2659
I781 Beiser L 1974 Las@rScarwing Systems Lose,.Applieotims “01 2 (New York: Academic) pp 53-159
1791 Wilson T 1984 Theory and Practice o/Scunaing Optical Microscope (London: Academic) pp 213
[SO] Wilson T 1985 Sconning I 79
[81] Wilke V 1985 Seaming 7 88
[U] Maher E F 1985 Sconning 7 61
I831 Chi C C, Loy M T and Cronemeyer D C 1982 Appl. P/zy,s. Leu. 40 437
[84] Scheuemnn M, Lhota J R, Kuo P K and Chen J T 1983 Phys, Re”. Lett. 50 74
IS51 Lhota J R, Scheuemann M, Kuo P K and Chen J T 1983 IEEE Trans Magn. 19 1204
[86] Lhota 1 R, Scheuermann M, Kuo P K and Chen J T 1984 Appl. Phys. Lett. 44 255
[87] Chi C C, Loy M T and Cronemeyer D C 1984 Phys. Rev. B 29 4908
1881 Scheuermann M and Chi C C 1985 Phys. Reo. B 31 4676
[89] Konovodchenko V A, Sivakov A G and Zhuravel A P 1986 Cryogenics 26 531
[go] Dimitrenko I M, Zhurawl A P, Sivakov A G 1992 Sou. J . Low Temp. Plrys. 18 676
138
R Gross and D Koelle
1911 Sivakov A G, Zhuravel A P, Dimitrenko I M and Volotskaya V G 1992 Sou. J. Supwcond.: Cl>wn.
T e c h 5 1746
I921 Belz A and Grill W 1993 Appl. Phys. A 56 119
[93] Binnig 0 and Rolirer H 1984 Physics B 127 37
(941 for a recent reYiew of scanning tunnelling microscopy see Scanning Tunneling Microscopy 1993 ed J
Stroscio (New York: Academic)
[95] Smith D P E and Binnig C 1986 Reu. Sci. Insfmm.57 2630
[96] Elrod SA, deLozanne A L and Quate C F 1984 App1. Phys. Lerf.45 1240
[971 Hess H F 1989 P h j ~Reu.
.
Lett. 62 214: 1990 J . Vac. Science Teclmol. A 8 450; 1990 P h j ~Rev.
.
Left.
64 271 I; 1991 P/i)'sica B 169 422
[98] Burger J, Meepagala S C and Wolf E L 1989 Rev. Sei. Insrrunz. 60 135
e . 831
[99] Bcrthe R. Hartmann U and Heiden C 1990 Appl. Phys. Lett. 57 2351; 1988 J. M i c ~ ~ s152
IlOO] Kashiwaya S. Koyanagi M and Shoji A 1992 AppL Pliys. h l f . 61 1847; 1991 IEEE Trans. Mngn.
M A G 2 7 837
[ l o l l Oxford Instruments Limited 1989 C F 302 liquid helium cold stage
[IO21 BirkhoA R D 1958 Handhuch der Physik vol 34 ed S Riigp (Berlin: Springer)
[ 1031 Nonequilibriton Supcrcondwriuity. Plmnoiis and Kapilra Boundories 1980 ed K E Gray (New York:
Plenum)
[IO41 Gross R 1987 Thesis University ofTiibingen (unpublished)
[IO51 Kaplan S B, Chi C C, Langenbrrg D N, Chang 1 1, Jafarey S and Scalapino D J Ph)ts. Reu. B 14
4854
[I061 Han S G. Vardeny Z V. Wong K S, Symoko 0 G and Koren G 1990 Phys. Rev. Lerf. 65 2708; 1991
IEEE Tram. Mag". M A G 2 7 1548
[IO71 Chwalek J M, Uher C. Witaker J F, Mourou G A, Agostinelli J and Lelental M Appr. Phys, h i t . 57
I696 ( 1990)
[IO81 Bronon S D er 01 1990 Solid Stare Comiinm. 74 1305;
Face D W er a1 1991 IEEE Trans Mngn. MAG-27 1556
[IO91 Kaplan S B. C C Chi, Chaudhari P, Dimos D, Gross R, Gupta A and Koren G P1iys. Rev. B 43 8627
(1991)
[IIO] Albrecht W, Kruse Th. and Kurr H Phys. Rev. Lett. 69 1451 (1992)
[ I I I ] Carr G L er nl Appl. Phys. Left. 57 2725 (1990)
II
. I21. Semenov A D. Gol'tsman G N. Gopidze I G . Sereeeev
- A V. Gershenzon E M.. Lann- P T and Renk
K F 1992 Appl. f l i p . Lett. 60 903
III31 Swartz E T and Pohl R 0 1989 Rev. Mod. Phyx 61 605
[I 141 Nahum M, Verghese S, Richards P L a n d Char K 1991 AppL P/~ys.Lett. 69 2034
[I151 Levey C G, Etemad S and Inam A 1992 Appl. Phys. Lett. 60 126
[ 1161 Marshall C D, Fishman I M, Dorfman R C. Eom C G and Faycr M D 1992 Phys. REU.B 45 10009
[I171 Terzijska B M. Wawryk R. Dimitrov D A, Marucha C and Rafalowin J 1992 Cryogenb 32 53
[IIS] Uher C 1991 Mechanism of Heat Conduction in High Temperature Superconductors 3rd Ann. ConJ
oil Superconductivity and Applicafiorr (Bufalo. NY. 1989) (Buffalo, NY: New York State Institute
of Superconductivity)
[I 191 Cohn J L, Skelton E F, Wolf S A, Liu J 2 and Shelton R N 1992 Phys. Rev. B 45 13144
{IZO] Yu R C, Salomon M B, Lu J P and Lee W C 1992 Plrys. Rev. Leu. 69 1431
[ I Z I ] Morelli D T, lieremans J and Swets D E 1987 Phys. Reo. B 36 3917
[ 1221 Uher C and Kaiser A B 1987 Phys, Rev. B 36 5680
11231 Izbizky M A, Esquinari P and Fainstein C 1988 Phys. Rev. B 38 9220
1124) Braun E, Jackel G, Roden B, Sereni J G and Wohlleben D 1988 2.Phys. B 71 169
[I251 Gottwick U, Held R, Sparn G, Steglich F. Reitschel H.Ewert D, Renker B, Bauhofer W. von Molnar
S, Wilhelm M and Hoenig H E 1987 Europhys. Lett, 4 1183
[I261 Hagen S I, Wang 2 Z and Ong N P 1989 Phys. Rev. B 40 9389
[ 1271 Little W A 1959 Con. J. Phys. 37 334
Cheeke J D N, Ettinger H and FIebral B 1976 Con. J. Phys. 54 1749
Anderson A C 1981 The Kapitza Thermal Boundary Resistance between Two Solids Non-Equilib
rium Superconductivity, Phonons, and Kapitza Boundaries ad K E G a y (New York: Plenum)
[ 1281 Kaplan S B 1979 J . Low Temp. Phy.?. 37 343
[I291 Seifert H, Huebener R P and Epperlein P W 1983 Phys. Lett. 97A 421
[I301 Carslaw H S a n d Jaeger J C 1959 Conduction of Ifeat in Solids (London: Oxford University Press)
[I311 Flik M I and Tien C L 1990 J. AppL Pltys. 67 362
[I321 Flik M I and Tien C L 1991 J. Ifcal Transfir. 112 872; 1991 J. Heal Transfer 112 I O
Low teniperature scanning electron microscopy
739
[ I 331 Flik M I 1991 Appl. Mech. Reo. 44 93
1134) Flik M I and Goodson K E 1991 Proc. d S M E / J S M E Them” EngineeFing Jolnt CO@ (Reno. N V )
[I351 Feuerbaum H P 1982 Scanning 5 14
[I 361 Everhart T E. Wells 0 C and Oatley C W 1964 J. Elecrroclzeni. Soc. I 1 I 929
[I371 Malta R K 1968 Scanwing Elcerron. Microsc. IlTRl 131
[I381 Crostliwait D Land Ivy F W 1969 Rev. Sci. Instr. 40 1158
[ 1391 Gopinath A, Gopinathan K G and Thomas P R 1978 Scmmitw Electron. Microsc. IITRI 375
[I401 Balk L J, Feuerbaum H P, Kubalek E and Mensel E 1976 Seaming Electron. Microsc. IlTRI 615
[I411 Menzel E and Kubalek E 1981 Scomzinp Electron. Microsc. IlTRl 305
[I421 Reiners W, Herman K D and Kubalek E 1988 Scanning Microscopy 2 161
[I431 Jenkins K A, lmmedialo M J and Heidel D F 1990 Scnnning I2 34
[ISS] Jenkins K A and Oh B 1991 J . Appl. Ph.vs. 70 7179
[I451 Likharcv K K 1986 Dynamics o/JosepIisort Junctions and Circuits (New York: Gordon and Breach)
[I461 Barone A and Paterno G 1982 Pliysics and Application o/rhe Josephson EJlect (New York: Wiley)
[I471 Orlando T P and Delin K A 1991 Foundations of Applied Superconductiuify (New York: AddisonWesley)
[ 148) Manaharl J 1991 Cri/irnl Currenls in High-T, Su#mconducIors (Kyoto: Toshiba International School
o f Superconductivity)
Mannhart J and Tsuei C C 1989 Z.Phys. B 77 53
[ 1491 Tinkham’ M 1975 Introdricrio,t Io Superconducrivity (New York: McGraw-Hill)
[I501 Tsuei C C, Mannhart J and Dimos D 1989 Proc. of Topical Cor$ on High T, Superconducting Thin
Fjfins (AIP Conference Proceedings 182) ed G Margariton, R l o p 1 and M Omellion (New York:
AIP)
[ I51 ] Huebener R P 1979 Mognerie Flux Struclwes in Supercondueiors (Springer Series in Solid State Sciences
6 ) (Berlin: Springer)
[ 1.521 Gross R 1993 Grain Boundary Josephson Junctions in the High Temperature Superconductors Infer/aces in Supm.condurting S y ~ t o n sed S Shinde and D Rudman (New York: Springer)
[I531 Chaudhari P 1991 Physicn C 185-189 292
[IS41 Gross R, Chaudhari P, Dimos D, Gupta A and Koren G 1990 Pliys. Rm. Letf. 64 228
[I551 For a recent review ofthe field see Proc. MIS-HTSC I l l CO@ (Kanazawa) 1991 Physica C 185189
[I 561 Kes P 1992 Flux Pinning and the Critical Current Concise Encyclopedia o/Mngnetic nnd Superconductirig Moteriols ed J E Evetts (Oxford: Pergatnon)
[ I 571 Koelle D 1992 Thesis University of Tuebingen (unpublished)
[I581 Beck A, Koelle D,AltT L, Husemann K-D and Gross R 1993 IEEE Trans. Appl. Supercond. 3 1620
[ 1591 Landauer R 1978 Phys. Today 31 23
11601 Biittiker M and Landauer R 1982 Nonlineor Phenomeito at Phme Tramifions and Insrabiliries ed T
Riste (New York: Plenum)
[I611 Skocpol W J, Beasley M R and Tinkham M 1974 J. Appl. Phys. 45 4054; 1974 J. Low Temp. Pliys.
16 145
11621 Dharmadurai G 1980 Phys. Star. Sol. 62 I 1
[ 1631 Skocpol W J 1982 Nonequilibr.ium Superconductiuily, Phonons, nnd Kopitia Boundaries ed K Gray
(New York: Plenum) pp 559-92
(1641 Eichele R,Freytag L, Seifcrt H and Huebener R P 1983 3. Low Temp. Pliys. 52 449
[I651 Freytag Land Huebener R P 1985 J . Low Terrtp. Phys, 60 377
[I661 Freytag L. Huebener R P and Seiferl H 1985 J. L o w Temp. Phys. 60 365
[I671 Josephson B D 1962 Phys. Len. I 251; 1965 Adu. Pltys. 14 419
(1681 Ambegaokar V and BaratoBA 1963 Phys, Reo. Lcll. 10 486 (erratum II IM)
11691 Owen C S and Scalapino D 11967 Phys. Rev. I 6 4 538
[I701 Yamashila T and Onodera Y 1967 J . dppl. Phys. 38 3523
[I711 Yamashita T. Kunita M and Onodera Y 1968 J. Appl. P h s . 39 5396; 1968 Jap. J . AapL Phys. 7 288
[I721 Vaglio R 1976 J. Low Temp. P h s . 25 299
[I731 Barone A, Esposito F, Likharev K K, Semenov V K.Todorov B N and Vaglio R 1982 J. Appi. Phys.
53 5802 (1982)
[I741 Vabishchevich P N, Vasenko S A, Likharev K K and Semenov V K 1984 Sou. Phys.-JETP 59 662
[I751 Giaever T 1960 Phys. Reo. Leu. 5 147; 1960 Pliys. Rev. L e n 5 464
[I761 Likharev K K 1979 Reu. Mod. Phys. SI 116
[I771 Kupriyanov M ?U and Likharev K K 1990 Sou. Phys. Usp. 160 49
[I781 Kupriyanov M Yu and Likharev K K 1990 IEEE Trans. Mogn. MAG27 2460
[I791 Golubov A A and Kupriyanov M Yu 1991 Plrys. Left. 154A 181
R Gross and D Koelle
740
I180) Golubov A A and Kupriyanov M Yu 1989 Sou. Phys.-JETP 69 805
[ISl] Hahlbohm H D, Liibbig H and Luther H 1975 Proc. LT-I4 p 140
[I821 McDonald D G, Johnson E G and Harris R E 1976 Phys. Reu. B 13 1028
Harris R E 1976 Phys. Reu. B 13 3818
[I831 Krumhansl J A 1991 Phys. Toduy 3 33
[I841 Eilbeck J C, Lomdahl P S,Olsen 0 H and Samuelsen M R 1985 J. Appl. Phys. 57 861
[I851 Enpuku K, Yoshida K and Jrie F 1981 J. Appppl. Phys. 52 344
[I861 McLaughlin D Wand Scott A C 1978 Plfys. Rev. A 18 1652; 1977 Appl. Phys. Len. 30 545
11871 Kulik I O 1965 JKTP Latf. 2 84
[ I881 Tanaka K 1979 Solid Stufe Commun. 29 443
[ 1891 Fulton T A and Dynes R C 1973 Solid Stute Conrmun. 12 57
1190) Fiske M D I964 Reu. Mod. Pliys. 36 221
[I911 Gross R, Schmid D B and Huebener R P 1985 J. Lon, Temp. Phys. 62 314
[I921 Gross R and Schmid D B 1987 Jopun J. Appl. P h p 26-3 1601
[I931 Chang J J, Ho C H and Scalapino D J 1985 Phys. &U. B 29 2843; 1985 Phys. Rev. B 31 5826
[I941 Chang J J and Ho C H 1984 AppL Plrys. Lett. 45 182
[I951 Bosch J 1986 Thesis University of TGbingcn
[I961 Mannhart J 1987 'l'Ize.~is
University of Tiibingen
11971 Dayem A H and Martin R J 1962 Phys. Rev. Left. 8 246
[I981 Tien P K and Gordon J P 1963 Phj85. Reu. 129 647
11991 Werthamer N R 1966 Pltys. Reo. 147 255
[200] Hamilton C A and Shapiro S 1970 Phjs. Reu. B 2 4494
[20l] Bosch J, Gross R, Huebencr R P and Niemeyer J 1985 Appl. P h p Leu. 47 1004
12021 Elesin V F and Kopaev Yu V 1981 Sou. Phys..-Usp. ?A 116
12031 Elesin V F 1979 Sou. fhys.-JETP 49 1121
(2041 Eckero U, Schmid A, Schmutz M and Schon G 1979 J . Los Temp. Phys. 36 643
[ZOS] SchBn G and Trembley A M 1979 Phys. Rev. Lett. 42 1086
[206] Trembley A M 1982 Nonequilibrium Superconductivity, Phonons, und Kupilzu Boundaries ed K Gray
(New York: Plcnum) pp 309-40
12071 Haken H 1983 Advanced Syfergefics (Berlin: Springer); 1973 Synergetics (Vienna: Teubner)
12081 Lindelof P E and Hamen J B 1984 Reo. Mod. Phys. 56 431
[209] Lobb C J 1984 Physics B 126 319
(2101 Hansen J B, Finnegan T F and Lindelof P E 1981 IEEE Trans. Mugn. MAG-I1 95
[211] Lukens J 1990 Superconductins Deuices ed S T Ruggiero and Rudman D A (New York: Academic)
12121 Jain A K, Likharov K K. Lukens J E and Sauvageau J E 1984 Phys. Rep. 109 309
12131 Niemcyer J. Hinken J Hand Kautz R L 1985 Appl. Phys. Lett. 47 1051
[214] Niemeyer 1. Hinken J Hand hieicr W 1984 IEEE Trans. Insfrum, M e a . IM-33 311
I2151 Niemeyer J 1989 Superconducting Qtmntum Elecfronics ed V Kose (Berlin: Springer) pp 228-54
(2161 HoHman D 1993 Dip/o,nu Tltcsir University of Tiibingen
[217] Bcnz S P and Burroughs C J 1991 Appl, Pliys. Leu. 58 2162
12181 Benr S P and Burroughs C J 1991 Supercond. Sei.Technol.4 561
[219] Tinkham M, Abraham D Wand b b b C J 1983 fhyw Rev. B 28 6578
12201 Hasley T C 1985 Phys. Rev. B 31 5728
i221j Pannetier 8,Chaussy J, Rammal J and Villegier J C 1985 Phys. Rev. Letf. 53 1845
12221 Ruggiero S T and Rudman D A (eds) 1990 Superconduering Deuices (New York: Academic)
[223] Hiang Q, Zasadzinski J F, Grey K E, Richards D R and Hinks D C 1990 App/. Phys. Left, 51 2356
12241 Pargellis A N, Sharifi F, Dynes R C, Hellman E S,Rosamilia J M and Hartford E H 1991 Appl. Phy5.
Letf. 58 95
[225] Marlin S,Hellman E S, Kussmaul A and Hartford E H 1993 Phys. Rev. B 47 14510
12261 Gross R 1992 SQUIDS Employing High-T, Supereonducforsin Superconducting Quunfum lnterferenre
Devices ed A Barone (Singapore: World Scientific)
[E71 Chaudliari P. Mannhart J, Dimos D, Tsuei C C, Chi C C, Oprysko M M and Scheuermann M 1988
Phys. Rev. Lett 6Q 1653; 1988 Phys Rev. Lett. 61 219: 1988 P h p Rev. Left, 61 2476
12281 Dimos D, Chaudhari P and Mannhart J 1990 Phys. Reu. B 41 4038
12293 Gross R, Chaudbari P, Dim- D, Gupta A and Koren G I990 Phys. Reu. Left. 64 228
(2301 Gross R, Chaudhari P, Kawasaki M and Gupta A 1990 Phys. Reo. B 42 10735
[231] Gross R, Chaudhari P, Kawasaki M, Ketchen M Nand Gupta A 1991 IEEE Truns. M a p . MAG27
2565
Low temperature scanning electron microscopy
I4 I
[232] lvanov Z G, Nilsson P A , Winkler D, Alarco J A, Claeson T, Stepantsov E A, Tzalenchuk A Ya 1991
Appl. Plrys. Lett. 59 3030
[233] Char K, Coclough M S, Garrison S M, Newman N and Zaharchuk G 1991 Appi. Plryr. Lett. 59 133
I2341 Char K, Coclough M S, Lee L P and Zaharchuk G 1991 Appl. Pl1)~.Lett. 59 2177; 1991 Proc.
Superconducting Deuices ond their Applicaliorts ed H Koch and H Lcibbig (Berlin: Springer)
pp 172-9
[235] Lee L P. Char K, Coclough M S and Zaharchuk G 1991 Appl. Phys. Letr. 59 3051
[236] Simon R, Bulman J B, Butch J F, Coons S B, Daly K P, Dozier W D, Hu R, Lee A E, Luine J A,
Matt C E and Zani M J 1991 lEEE Trans. Magn. MAG-27 3209; 1991 Appl. Phys, Len. 58 543
[237] Cui G. Zhang Y, Hcrmann K , Buchal Ch., Schubert 1, Zander W, Braginski A I and Heiden Ch. 1991
Nonlinear Eiectronics and Joseplison Deuices ed N F Pedersen, M Russo, A Davidson, G Costabile
and S Pagano (New York: Plenum); 1991 Supercond. Sci. Tech. 4 583; 1991 Physic0 C 175 545
I2381 Vildic M, Fried1 G, Uhl D, Dahlmanr G, Kohler H, Meyer H, Bommel F and Saemann-Ischenko G
1993 IEEE Tram. Appi. Supercond. 3 2351
[239] Koren G, Aharoni E, Polturak E and Cohen D 1991 Appl. Phys. Lett. 58 634
[240] G a o J, Aamink W A M, Gerritsma G J and Rogalla H 1990 Physica C 171 126
[241] Schwarz D B, Mankiewich P M, Howard R E, Jackel L D, Straughn B L, Burhart E G and Dayem
A H 1989 IEEE Trans. Magn. MAG25 1298
[242] Akoh H, Camerlingo C and Takada S 1990 Appl. P l ? p Left. 56 1487
Lee M, Lew D, Eom C B, Geballe T H and Beasley M R 1990 Appl. P l z s . Lett. 51 1150
[243] Inam A, Ramesh R, Rogers C T, Remschnig K, Farrow L, Hart D, Venkatesan T and Wilkens B
1990 Appl, Phys. Len. 57 2484
Inam A, Ramesh R, Rogers C T , Wilkens B, Remschnig K, Hart D and Barmer J 1991 IEEE Trans.
Mogn. 27 1603
12441 Simon R W et a1 1990 Proc. Con! on the Science and Teclmology of Thin Films (Denver, CO)
12451 White A E. Short K T. Dynes R C, Leir A F J. Anzlowar M. Baldwin K W. Polakos P A. Fulton
T A and'Dunkelberger i N 198S.Appr. Phys. i d t , 53 1010
[246] Aslamazov L G and Larkin A I 1968 Sou. Phys.JETP 55 323; 1969 JETP Lett. 9 150
I2471 Artemenko S N, Volkov A F and Zajtsev A V 1979 Sou. P h p - J E T P 76 1816
12481 Zaitsev A V 1983 Sou. Pli:s.-JETP 78 221
i249j Kruglov A H 1981 sou. P/zys.-soridstate 23 3404
12501 Schmidl F, AIR L, Gross R, Husemann K-D, Schneidewind H and Seidel P 1993 IEEE Trans. Appl.
Supercond. 3 2349
[25l] Gao J, Aarnink W A M, Gerritsma G J and Rogalla H 1990 J. Appr. Plrys. 72 515; 1990 Pl,ysica C
171 126; 1991 IEEE Trans M a p . 21 3062
. .