low temperature scanning electron microscopy of ... · the quasiparticle and josephson current...

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 pm 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

652 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

3.1. Beam electron range and thermalization time 3.2. Electron beam induced non-equilibrium state

4.1. General response theory 4.2. Response at low temperatures 4.3. Response at higher temperatures

5.1. General aspects 5.2. Measuring techniques

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

7.1. Basic equations 7.2. Pair tunnelling current density 7.3. Quasiparticle tunnelling current density 7.4. Arrays of superconducting tunnel junctions

8. I . Basic equations 8.2. Pair current density 8.3. Quasiparficle current density 8.4. Disordered arrays of weak links

3. interaction of the beam electrons with the superconductor

4. Response of a superconductor to electron beam irradiation

5. The signal

6. LTSEM study of superconducting films

7. LTSEM study of superconducting tunnel junctions and circuits

8. LTSEM study of superconducting weak links

Acknowledgments References

653 655 655 656 659 659 661 663 664 666 669 672 672 673 678 679 684 691 691 691 693 694 699 708 716 723 724 726 733 734 735 735

Low remnperature scanning electron microscopy

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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 Joseph- son junctions, arrays of Josephson junctions, and superconducting microwave circuits

the spatial distribution of the critical current density in high temperature supercon-

the anisotropic phonon propagation in single crystals [50-55]; *defects in single crystals by phonon tomography [56-621;

[ 1 1-28];

ducting films and grain boundary junctions [20-491;


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

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 specific advantages 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

[47-491.

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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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 specific heat 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.

R Gross and D Koelle

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.


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)

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)

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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

I 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. I t 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



Figure 2. Opcned specimeii chiimher of thc ( 'amScan J ~ D V microscops sliowing thc small liquid helium tank with the sample holder installed. The liquid nitrogcn temperature radiation shield i s 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 b can be estimated as

(1) (m4/2)* rb =

31rNZe4vo'

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

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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 Eo and 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 pm 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 66 1

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 electron- stopping 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 non- equilibrium distribution of quasiparticles and phonons is obtained by solving the Boltz- mann 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


characterislic length ( pm ) bo-3 10-2 10-1 100 101 102

10

' Pb Beom Electron

, m COODipl POWS , w Phonaor

x 100 .--- E,, , c

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 clec- tmm 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 coefficients i ron [I%). Tht data for YBaiCu,O,.a are taken from [106-112].

Material rb(xIo-"s) rR (XIIPS) ra(xIO-vs) r , (X IO-qS) l + r , / r e

Pb 0.41 0.19 0.034 12.9 38 I In 0.58 0.79 0.17 18.4 110 Sn 0.59 2.3 0.11 2. I 21 Nb 0.49 0.15 0.0042 1.2 288 AI I .4 438 0.24 0.40 2.6 YECO 0.73 <0.1 <0.001 <0.01 -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

decay time and decay length of the electron beam induced response of the superwnduct- ing 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 non- equilibrium 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 ~ ( x , y , z), which has unit volume integral

dxdydz a(x,y,z)=I (4) s 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). I f the response F is proportional to the perturbation P, it can be calculated using a linear response function G as

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 ) s s 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

Low temper.ature scanning electron microscopy 665

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 non- equilibrium 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 re2xf<l / r - l MHzand v,<A/r=lOOcm sC’ . Whereas thescanningspe disusually

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, ~ ~ ( 4 . 2 K)-5 x IO-’W c K 2 K-’ [ 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 ~ ~ ( 7 7 K) - 5 x lO-’W K-‘ [ I 18-1251. That is, in most cases the thermal boundary

much smaller than 100 cms-’, the beam modulation frequency can be made iy‘ onsiderably

F

666

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~ise at 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

(01 e-BEAM (bl e-BEAM I QUASIPARTICLES

I -PHONOUS m COOPER PAIRS

I I I I DIFFUSION REGION

A I , / \ A L 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 / rB 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

where6 is the Diracdelta function and p ( / ) = ~ ( x - ~ , ( t ) ) ~ + ( y - y ~ ( r ) ) ~ i s thedistance from the beam focus in the film plane. For the calculation of the response F the 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 non- thermal treatment the response of the superconducting film is given by [ 5 , 61

Here, KO is the modified Bessel function of zero order.

ture rise 6T(p, [) and we have [S, 61 If a thermal treatment is appropriate, the response F(p , t ) corresponds to a tempera-

C, d a

T=-

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.

I f 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 Iph are 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

(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 DC and an AC component,

p((P, I ) Fdc(P? /) + /) (16) where the DC component 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 superconducting film/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 of the response represents a good approximation for high-T. films. For YBa2Cu307-s, we have ~ ~ ( 4 . 2 K)- 1 mW cm-* K-' and C ~ ( 4 . 2 K)13 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 ~ K ~ [126] 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 3 for 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.S are neglected solving the heat diffusion equation. Taking into account the temperature dependences of K and C results


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) or 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

(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

Po 1 p ( r , f ) = - ( I + e ' ~ ' ) ) - - ( r ( f ) )

2 r(t)

{ I +exp[ -kr+ i(wr-kr)]} (21) Po F(r, f) = __

Z ~ K F S ~

with

k = I / I , = & . (22 )

Here, DF.s = K ~ , ~ / C ~ . ~ is the 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 C . In this case the temperature field is given by equation (21) independent of the film thickness. For d - R and K ~ # K ~ the 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.J- at T=77 K, that is, for a typical beam modulation frequency of 10 kHz we have I, - 15 pm. 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

F ( r , t ) = F , 1 - 7 [l+e'"'] for r I 4 R / 2 123)

for I r l > R/2 (24)

( ,","I Fo R F(r, t ) =- [ 1 +e'"'] 3r

with

for beam modulation frequencies or<<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 I O 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-b films. 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


-a -4 0 4 8 r ( ”

Figure 8. Normalized response lunction F(r) in a Y B a D ~ , 0 7 - s 61m 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.ph is 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 .” S(x, y, f) = dx’ dy‘ d:’ dt’ H(x, y, 2 , x’, y’, z‘, f- f‘)I:(x’, y’, z‘, t’) (26) J J 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. Expres- sions for the LTSEM signal obtained in the study of superconducting thin films, Joseph- soil 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 (BE imaging). 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 non- linear, 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


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 C’B 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 of the sample voltage or current simultaneously. Equations (27) and (28) show that a finite signal is obtained only i f 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 W2 and 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 Rtun and 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

Ref. In 1 1 (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

~~

~~.~ .


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

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 mechan- isms controlling the critical current density in superconducting materials: Depairing, depinning and decoupling. In this section we will consider samples containing no so- called 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 n z * are, respectively, the density, the charge and the effective mass of the Cooper pairs, and to the 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 [ I5 l l . 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 of current flow is minimized. Sample regions with stronger pinning sites will carry a higher current density and vice versa.


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 , ( x , t ) = dt’- dx [P(x, t - t ’ , Tb+ST(x, I , ) , J B ) - P ( x , t - l ’ , T ~ , J B ) ] (31) s i d j 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 ' ' * ' ' ' ' ' ' ' ' ' I 87 aa a9 90 91 92 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 6V(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-b film (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 6V(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 (non- linear 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 ' ' I

-8 -4 0 4 a x--xi (!Jm)

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

the spatial resolution for the investigation of YBa2Cu,07-6-films by LTSEM has been shown recently [31, 341.

6.1.2. Measurefttent ofthc 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 JB[ap(x ) /

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

a~,(x)i [ a ~ , ( x ) i a ~ i .

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 ' I ' I ' l i Y I

-5 12 14 16 18 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 Jn>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-d films 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 ) Je=4.0xIO6Acm~~', (c) JB=4.5xIf16Acn~', ( 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 10pm 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

5 x IO6 A cm-2. Bright regions correspond to those regions yielding a high electron beam induced voltage signal. According to 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.

R Gross and D Koelfe

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 as 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 to 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

lowdy SJ(x, y) = 0.

6V(x ,y , l )U-- low dy [ dx JGpfx, y , t ) .

(38)

Here, the integration has to be done along the cross-sectional area Ai perpendicular to the current flow. The voltage change 6V( t ) is then obtained to

(39)

Similarly, for a voltage biased sample we have

JoL dx ~ E ( x , y) = o

and the electron beam induced change of the sample current is obtained to

6 r ( x , y , I ) U - 1 loL dx jo” dy 6p(x , y , I ) . L

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 fB through 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 IB is 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 two- dimensional.

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, 6p(y) i ( a p ( y ) / aT)6To is 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)- 8 f d X J . (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


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 cross- sectional 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 cross- sectional 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 of the critical current S l , ( x j , y ) according to equation (46). Assuming that aJ,/aT is about constant in the perturbed sample


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 cross- sectional area Ai together with the expected voltage signal SV(x , , y).

690

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 61c(x j , y )<<L(x t ) the resulting voltage signal for a current biased sample is given by

Ifd, A and GToare independent 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-sectional area 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 effective film 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, i t 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 cross- sectional area also may be determined by the current density in the adjacent cross- sectional 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 two- dimensional 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.


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 three- dimensional case. The integrations in section 6.2 have to be performed also in z- direction. 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 by 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 S- shaped 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 two- dimensional 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

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.

R Gross and D KoeIle

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, kB 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 AB 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. ,IL is 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 funnel junctions (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 ( 5 5 )

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 self- field 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

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


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. Supercon- ducting 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 non- linear, 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 non- linear 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

which for a one-dimensional junction with overlap geometry is given by [ 145, 146, 1841


bXx- $,,- sin 4 = 4,- P q L - 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 multi- mode 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 coefficients a 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 Joseph- son 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 (BZO) . 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 self- field 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~fieeld cffecls, 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

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 (SJJJ , cc 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 non- local 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) = STO exp(-r2/A2). The non-linear relation between S/:, and ETo is 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 be neglected. That is, the measured signal mainly contains information on on(x, J') and 6(&Y).

Low leniperafure scanning electron microscopy

1.4, . I I , I , I ,,I

701

".U

0.0 0.2 0.4 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 of 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 of 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

~

703

e - B E A M /

Figure 24. Imaging of the spatial distribution of the Josephson current density 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 ] ) .

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 Joseph- son 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


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 6I : (x , y ) allows the imaging of superconducting microshorts of the tunnelling barrier [13, 1951. In the presence of


& A -- Y 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 sine- Gordon 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

Sp(x , y, I ) = dr 27rr - &T(r, I ) aT r=rS

167)


where r is the distance to the position ( x , y ) of the beam focus. Replacing a by a + Sa and p by p + Sp 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 61(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 SPc<p) the change of the average junction voltage depends about linearly on the power loss, i.e.

~ V ( X , Y , t )=-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

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 cn that 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 ( I s > , = 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

(70) 0”W

e ddsr-[IK(i, Vb, Tb$.6T(?’,t))-K(r, Vb, &)I.

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 (aK( 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) = &Tu exp(-?/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 6To of 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) v

0.85 (iii) N 2 0.40 0.90 (ii) --. 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 quasiparticle tunnelling 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. - 0.25

nflC - 0.1 (i) 0.2 (ii) 0.3 (5) 0.4 (i)

N <

0.5 (v) N , 0.4 P

= 0.2

n n _._ 0.90 0.95 1.00 1.05 1.10

evb / A(o) Figure 29. Normalized electron beam indued 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 Tb= 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 6 a ST0 ( 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=1.8 K. 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 20

Frequency ( MHz )

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))4- versusf 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 of a superconducting tunnel junction and measuring SI.,, or, equivalently, 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.

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 Nb (b ) tunnel junction. Tbe signal is plotted i n 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 on in 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 un usually 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 RF 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 compw- able 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 RF 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 RF 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 two- dimensional images showing the spatial variation of the RF 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 pm wide and 400 pm 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 of 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 of superconducting tunnel junctions [ 19,281.

Figure 33. Voltage image of the standing microwave pattern inside a 400 pm 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

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)le will 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 (figure 34(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.

LOW temperature scanning electron microscopy

~

715

S W P L E 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


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 detector- junction, 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

Low temperature scanning electron microscopy

~

717

1 1 b l I 1

Figure 36. Two-dimensional voltage images of the injector tunnel junction showing the distribution of the tunnelling conductivity (a) and the detector junction 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 single junction 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 microelec- tronic 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. How- ever, 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 of temporal 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 RF 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

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 superconducting circuitry. 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 1 .o 1.5 Voltage ( 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 l c j -&Ic j>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 ld<lB, 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 V,cs2A2/e €or I, < IB . 6 Virr2A;/e for I, rr I , (75)

That is, only those junctions with critical currents IcirrIB yield 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- le . 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

6V.=- 6 L . ' arc, a v l 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 I e - I t i . 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 GBJS can 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 quasi- two-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 120 R Gross and D Koelle

Figure 38. Secondary electron image of 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 )

x 1 - 1 u X x , y ) av, aI var

(76) e a i 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 values of their tunnelling conductivity. The unknown bias voltage of the individual junctions also makes it difficult

Figure 38. Secondary electron image of 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 1 - 1 u X x , y ) av, e a i va,

(76)

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 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

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.

R Gross and D Koelfe

7.4.2. Two-diineiisional arrays. There are dimensional arrays. The most common

various junction configurations for two- 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 Joseph- son 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 two- dimensional 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 to that shown in figure 40(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 j th 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 to be reduced considerably in order to 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

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

density

. I ~ ( x , y ) = J , ( x , Y ) ~ ' [ 4 ( x , Y ) l (77)

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. Qualita- tively, 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<0.5TC and 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, i t 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

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.

R Gross and D KoeNe

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 bicrystal- GBJ [152, 2281 is shown in figure 42. A oaxis oriented YBazCu,O,-s film is grown

ELECTRON BEAM I 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-s thin 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. I t 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 profile 6T(y’) = 6To exp(-(y’-y)’/A’). In the calculation non-linear effects, which

0.0301 , I , , I , , ,

0.020 8

-3 .. - s 0.010 I

0.000 0

0.95 (Cj (iii)

[ii)

.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

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.


8.2.2. Finite voltage s tate4ynmnic phenomena. For I’fO 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

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 of 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

6V(y) cc - 6I$(y) cc -- aJc(y) F(@(y))2 dA 6To. (84) dT

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. Both signals are proportional to J&) and $ ( y ) and give information on these 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.


- H X 10pm

Figure 45. Voltage imagc ofa 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. The 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-a bicrystal (a) and step- edee (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 of 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


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 jm. The electron beam was scanned along the grain boundary extending in y-direction and the induced voltage signal is used to control the

132

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.

R Gross and D Koeile

I

0.0 0.2 0.4 0.6 0.8 1.0

0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0

\ 0.0

1 2 -0.5

v

\ 0.0

0.5 -0.5

0.0 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 OJI 0.2 0.4 0.6 0.8 1.0

;;;m .?a = 0

-0.5

-1.0 . . 0.0 0.2 0.4 0.6 0.8 1.0 (e) 1.0

f 7 :::mi bo = TI

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u / w 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 difference 4(c). and lhe normalized supercurrent 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 cover- ing 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


Figme 49. LTSFM voltage image or the supercurrent distribution in a 25 p m wide ramp- edge 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 to 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 current- voltage 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 to 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 to d o so, theoretical models for the description of the IVCS and the electron beam induced signal have to he worked out. Fortunately, most high-T, weak

134

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 dir- ordered 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. There- fore, 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 contrihu- tions 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 high- T, 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.

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