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Probing the symmetry of the pairing state of unconventional superconductors by SQUID Interferometer Measurements
PHYSICA ELSEVIER
PhysicaC 282-287 (1997) 128-131
Probing the Symmetry of the Pairing State of Unconventional Superconductors by SQUID Interferometer Measurements* D. J. Van Harlingen Department of Physics and Materials Research Laboratory, 104 S. Goodwin Avenue, Urbana, IL 61801
Phase-sensitive SQUID interferometer experiments have emerged as the most powerful technique for determining the symmetry of the pairing state of unconventional superconductors. This approach uses Josephson tunneling to probe specific directions in k-space and long-range phase coherence to measure the phase anisotropy of the order parameter, allowing a direct and definitive determination of the symmetry. Developed for testing the symmetry of the cuprates, corner SQUID and junction experiments indicate a sign change in the order parameter in orthogonal in-plane directions in YBCO, strong evidence for dx2-y2 pairing. Now, this technique is being applied to other classes of superconductors suspected to be unconventional.
INTRODUCTION One of the primary success stories of high temperature superconductivity has been the determination that the cuprates exhibit unconventional pairing with a dx2-y2 symmetry. At the first three M2S meetings, there was little indication that order parameter symmetry would be an important issue. The possibility of d-wave symmetry arising from magnetic interactions had been raised [1], but had initially been thought to be too weak to account for the high transition temperatures of the cuprates. By M2S-IV in Grenoble in 1994, however, the situation had changed dramatically. Detailed calculations suggested that magnetic spin fluctuation pairing in a dx2-y2 channel could account for many of the normal and superconducting state properties of the cuprates [2], motivating intense experimental activity directed toward testing the symmetry. Key measurements of NMR relaxation rates [3], the low temperature penetration depth [4], and angleresolved photoemission spectroscopy [5] gave strong evidence for substantial anisotropy in the order parameter magnitude, consistent with dx2-y2 symmetry. More than any other experiments, however, it was the development of the SQUID interferometer technique [6,7], with its unique
ability to probe the phase anisotropy of the order parameter, that unambiguously established dx2-y2 as the dominant pairing symmetry in the cuprates. Thus, as we meet in Beijing for M2S-V, there is now general agreement on the symmetry, and the focus now turns to issues of the microscopic mechanism and implications of the unconventional symmetry on the properties and applications of the high temperature superconductors. In this paper, I will focus on illuminating some of the physical concepts underlying the phasesensitive interferometer experiments, summarize the evidence that demonstrates the dx2-y2 symmetry, and discuss briefly the application of this technique to other materials that are candidates for unconventional superconductivity. 1.
PAIRING SYMMETRY CANDIDATES
The most direct approach to distinguishing different pairing symmetries is to measure the anisotropy of the phase of the order parameter. The value of this is illustrated in Figure 1, which shows the magnitude and phase of the order parameter vs. k-space angle for candidate states, Whereas the magnitude anisotropy is rather similar for all but the isotropic s-wave state, distinguishable only by
* In collaboration with D. M. Ginsberg, A. J. Leggett, D. A. Wollman, and B. D. Yanoff. Research supported by the Science and Technology Center for Superconductivity under grant NSF-DMR-91-20000. 0921-4534/97/$17.00 © ElsevierScienceB.V. All rightsreserved. PII S0921-4534(97)00238-4
D.J. Van Harlingen/Physica C 282-287 (1997) 128-131
PAIRING STATE
MAGNITUDE
RELATIVE
PHASE
IAI isotropic s
/
'I!
0
i .L
i
ilxqb i
,~,
i.,,
z:,,
~.,
dx2~2
H anisotropic
s
o
extended
s
129
junctions probe the order parameter in orthogonal directions, and the response of the SQUID to applied magnetic fields is sensitive to a phase shift inside the crystal arising from the symmetry. In this regard, the corner SQUID is simply an interferometer in which the internal order parameter phase drop shifts the diffraction pattern, as in a two-slit optical interference experiment. Figure 2(b) shows the shift of the critical current modulation for the d-wave state compared to a conventional s-wave SQUID, demonstrating the sensitivity to the order parameter phase anisotropy. Figure 2(c) shows measured results for 21 corner SQUIDs and 11 edge SQUIDs (control samples with both junctions on the same face). The phase shift of order 7z in all corner devices is strong evidence for pairing in a d-wave channel; the spread is due to vortices trapped near the SQUID loop that couple a flux up to 0.200 into the loop.
'0L[ % ' , Q
(a)
's
'
i
0.5
0.0 dx2.y2
+ i
-2
-1
"
~'
1
';"
2
Magnetic flux (~0)
dxy
8
7
0 Figure I. Magnitude and phase anisotropy of the order parameter for some candidate pairing symmetries.
8 the behavior near the nodes, the phase anisotropy is distinctly different. The objective of the SQUID interferometer technique is to measure the relative phase in orthogonal directions. This single measurement is sufficient to distinguish s-wave from d-wave, and can also identify a complex mixture which breaks time-reversal symmetry.
2. The Corner SQUID Experiment The design of the corner SQUID experiment is shown in Figure 2(a). The circuit is a bimetallic dc SQUID, in which Josephson junctions on the a and b faces of the cuprate crystal are connected by a loop of a conventional superconductor. The
1 0
(c)
f .....
-10~
-05~
OOn
comer SQUIDs
05~
10z
I 5~
t 20~
Phase shift
Figure 2. Corner SQUID experiment: (a) plan, (b) critical current modulation vs. symmetry, (c) results. Several key principles are important in this experiment, many of which have not been fully understood or appreciated: (a) Domain structure. The corner experiment measures the global phase shift between orthogonal k-space directions but between spatially-separated regions of the crystal adjacent to the junctions in real space. As a result, the range of the order parameter
130
D.J. Van Harlingen/Physica C 282-287 (1997) 128-131
domains and, in particular, how the order parameter behaves across grain and twin boundaries are important. The only way to determine this is to measure the phase response in both twinned and untwinned single crystals. Our measurements in crystals of these types [6] have established that the order parameter has gyroscopic symmetry, that is, the order parameter maintains the same sign orientation across grain and twin boundaries, forming a single domain throughout the crystal. This result is crucial for the SQUID interferometer technique to make a definitive determination of the microscopic symmetry, especially in heavilytwinned thin film samples in which subsequent versions of the interferometer experiment have been carried out. (b) Directionality. The measurement of the phase anisotropy depends on probing the order parameter in specific k-space directions. This relies on the directionality of the Josephson tunneling orthogonal to the junction barrier. Although this is known to occur in insulator-barrier junctions for which the charge transport is solely by tunneling, it also applies to any planar junction (such as the SNS junctions used in our experiments) since Josephson tunneling involves phase-coherent electron-hole pair conduction. The grain boundary junctions used in thin film interferometers certainly do not strictly maintain directionality because of the wandering of the boundary, but their ability to resolve the orthogonal phase shift indicates that directionality is maintained on average. (c) Origin of phase shift. The phase shift observed in the corner SQUID experiment and its variations occurs inside the crystal between orthogonal directions. A common misconception is that the phase shift can be described in terms of n-junctions, Josephson junctions in which the minimum coupling energy is at junction phase n, corresponding to a negative critical current. This concept was introduced [8] to describe Josephson junctions in which the tunneling occurs via magnetic spin-flip scattering, adding a n-phase shift inside the junction. To date, such junctions have not been realized, and have nothing to do with the phase shifts and the spontaneous circulating currents observed in interferometer experiments. The junctions in these experiments are all ordinary Josephson junctions, with their minimum energy at zero phase. In
particular, it is clear in the SQUID experiments that 7t-junctions cannot account for the observed phenomena since there are two identical junctions. 3. VARIATIONS Following our initial corner SQUID tests, several subsequent experiments based on the same principle were carried out, contributing further evidence for the d-wave symmetry.
3.1 Singlecornerjunction Perhaps the most qualitatively clear indication of the d-wave symmetry comes from measurements of the critical current modulation pattern of a single Josephson junction that straddles the corner of a cuprate crystal [9], as in Figure 3(a). This is the single junction equivalent of the corner SQUID experiment, essentially a SQUID without the loop. For s-wave symmetry, such a junction exhibits the ordinary Fraunhofer diffraction pattern familiar from single slit optical interference. However, in YBCO-Pb junctions, we observe the dramatically modified pattern in Figure 3(b), characteristic of a phase shift of n between the orthogonal segments. The most telling feature is the precipitous drop in the critical current at zero field, demonstrating the cancellation of the critical currents through the two sections. Recent observation of complex critical current modulation patterns of 45 ° grain boundaries can be understood in terms of multiple corner junctions formed by the wandering boundary [10].
~
(a)
~~= 20 40 s°I~ ~ 60 0 -150 -100 -50
0
50
100
150
Applied magnetic field (raG)
Figure 3. Design (a) and measured diffraction pattern (b) for a single YBCO-Pb comer Josephson junction.
3.2 Spontaneouscirculatingcurrents An alternate approach to measuring the modified critical current modulation of the SQUID (or single junction) in a d-wave superconductor is to detect the spontaneous (zero applied field) circulating current
D.J Van Harlingen/Physica C 282-287 (1997) 128-131
required to maintain phase coherence around the SQUID loop. This has been achieved using SQUID magnetometry in the corner SQUID geometry by the Maryland group [11] and first in a tricrystal ring geometry by the IBM group [12]. The ring sample consists of a cuprate thin film grown on a tricrystal substrate, forming three superconducting regions separated by grain boundary Josephson junctions. This system is exactly equivalent to the corner SQUID, except that the magnetic field response depends on whether one, two, or all three of the cuprate sections exhibit a n-phase shift (i.e., straddle a node). In both configurations, the spontaneous circulating currents only flow if the loop inductance is sufficiently large to make it energeticallyfavorable to generate a current (as opposed to flipping the phase on one junction to its higherenergy zero-current state). The generated magnetic flux approaches ½~0 for large loop inductance. 3.3 c-axis supercurrents A somewhat different set of experiments involves Josephson tunneling into the c-axis face of YBCO samples. The observed finite Josephson supercurrents [13] are inconsistent with a state of pure d-wave symmetry since such currents are expected to average to zero, implying the existence of an s-wave component in the order parameter. Recent measurements of single c-axis junctions straddling a twin boundary exhibit the same critical current diffraction patterns as we observed in corner junctions (Figure 3), indicating that this s-wave component is largely the result of the orthorhombic structure of YBCO [ 14].
4. FUTURE DIRECTIONS The results of the corner SQUID, corner junction, spontaneous current, grain boundary diffraction, and c-axis tunneling experiments now all point conclusively to a state with at least predominately dx2-y2 symmetry. With the unconventional nature of the pairing symmetry in the cuprates established, attention is now being focused in three directions: (a) Determining the microscopic mechanism. Surprisingly and somewhat disappointingly, knowing the symmetry has not significantly accelerated the search for the microscopic
131
mechanism. Although the dx2-y2 symmetry was first predicted from magnetic pairing mechanisms, it is now recognized that many interactions can result in pairing in a d-wave channel. (b) Implications o f unconventional symmetry. The nodes and sign change associated with the dwave pairing affect many thermodynamic, transport, optical, and interface properties of the cuprates. Work is only beginning to explore the exciting science and potential technological impact resulting from the d-wave symmetry. (c) Application to other superconductors. The interferometer experiments we pioneered can also be used to test the symmetry of other candidates for unconventional superconductivity. We are currently carrying out experiments designed to determine the symmetry of the heavy fermion superconductor UPt 3, suspected to be p-wave, and the organic superconductor •-(ET)2Cu[N(CN)2]Br, which may be also be d-wave. REFERENCES
I. N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett. 62, 961 (1989). 2. P. A. Monthoux, A. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992). 3. J. Martindale et al., Phys. Rev. B 47, 9155 (1993). 4. W . N . Hardy et al., Phys. Rev. Lett. 70, 3999 (1993). 5. T. P. Devereaux et al., Phys. Rev. Lett. 72, 396 (1994). 6. D. A. Wollman et al., Phys. Rev. Lett. 71, 2134 (1993). 7. D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995). 8. L . N . Bulaevski, V. V. Kuzii, and A. A. Sobyanin, JETP Lett. 25, 290 (1977). 9. D. A. Wollman et al., Phys. Rev. Lett. 74, 797 (1995). 10. J. Mannhartetal., Z. Phys. B 101,175 (1996). 11. C. C Tsuei et al., Phys. Rev. Lett. 73, 593 (1994). 12. A. Mathai et al., Phys. Rev. Lett. 74, 4523 (1995). 13. A. G. Sun et al., Phys. Rev. Lett. 72, 2267 (1994). 14. J. Clarke et al., preprint.
PhysicaC 282-287 (1997) 128-131
Probing the Symmetry of the Pairing State of Unconventional Superconductors by SQUID Interferometer Measurements* D. J. Van Harlingen Department of Physics and Materials Research Laboratory, 104 S. Goodwin Avenue, Urbana, IL 61801
Phase-sensitive SQUID interferometer experiments have emerged as the most powerful technique for determining the symmetry of the pairing state of unconventional superconductors. This approach uses Josephson tunneling to probe specific directions in k-space and long-range phase coherence to measure the phase anisotropy of the order parameter, allowing a direct and definitive determination of the symmetry. Developed for testing the symmetry of the cuprates, corner SQUID and junction experiments indicate a sign change in the order parameter in orthogonal in-plane directions in YBCO, strong evidence for dx2-y2 pairing. Now, this technique is being applied to other classes of superconductors suspected to be unconventional.
INTRODUCTION One of the primary success stories of high temperature superconductivity has been the determination that the cuprates exhibit unconventional pairing with a dx2-y2 symmetry. At the first three M2S meetings, there was little indication that order parameter symmetry would be an important issue. The possibility of d-wave symmetry arising from magnetic interactions had been raised [1], but had initially been thought to be too weak to account for the high transition temperatures of the cuprates. By M2S-IV in Grenoble in 1994, however, the situation had changed dramatically. Detailed calculations suggested that magnetic spin fluctuation pairing in a dx2-y2 channel could account for many of the normal and superconducting state properties of the cuprates [2], motivating intense experimental activity directed toward testing the symmetry. Key measurements of NMR relaxation rates [3], the low temperature penetration depth [4], and angleresolved photoemission spectroscopy [5] gave strong evidence for substantial anisotropy in the order parameter magnitude, consistent with dx2-y2 symmetry. More than any other experiments, however, it was the development of the SQUID interferometer technique [6,7], with its unique
ability to probe the phase anisotropy of the order parameter, that unambiguously established dx2-y2 as the dominant pairing symmetry in the cuprates. Thus, as we meet in Beijing for M2S-V, there is now general agreement on the symmetry, and the focus now turns to issues of the microscopic mechanism and implications of the unconventional symmetry on the properties and applications of the high temperature superconductors. In this paper, I will focus on illuminating some of the physical concepts underlying the phasesensitive interferometer experiments, summarize the evidence that demonstrates the dx2-y2 symmetry, and discuss briefly the application of this technique to other materials that are candidates for unconventional superconductivity. 1.
PAIRING SYMMETRY CANDIDATES
The most direct approach to distinguishing different pairing symmetries is to measure the anisotropy of the phase of the order parameter. The value of this is illustrated in Figure 1, which shows the magnitude and phase of the order parameter vs. k-space angle for candidate states, Whereas the magnitude anisotropy is rather similar for all but the isotropic s-wave state, distinguishable only by
* In collaboration with D. M. Ginsberg, A. J. Leggett, D. A. Wollman, and B. D. Yanoff. Research supported by the Science and Technology Center for Superconductivity under grant NSF-DMR-91-20000. 0921-4534/97/$17.00 © ElsevierScienceB.V. All rightsreserved. PII S0921-4534(97)00238-4
D.J. Van Harlingen/Physica C 282-287 (1997) 128-131
PAIRING STATE
MAGNITUDE
RELATIVE
PHASE
IAI isotropic s
/
'I!
0
i .L
i
ilxqb i
,~,
i.,,
z:,,
~.,
dx2~2
H anisotropic
s
o
extended
s
129
junctions probe the order parameter in orthogonal directions, and the response of the SQUID to applied magnetic fields is sensitive to a phase shift inside the crystal arising from the symmetry. In this regard, the corner SQUID is simply an interferometer in which the internal order parameter phase drop shifts the diffraction pattern, as in a two-slit optical interference experiment. Figure 2(b) shows the shift of the critical current modulation for the d-wave state compared to a conventional s-wave SQUID, demonstrating the sensitivity to the order parameter phase anisotropy. Figure 2(c) shows measured results for 21 corner SQUIDs and 11 edge SQUIDs (control samples with both junctions on the same face). The phase shift of order 7z in all corner devices is strong evidence for pairing in a d-wave channel; the spread is due to vortices trapped near the SQUID loop that couple a flux up to 0.200 into the loop.
'0L[ % ' , Q
(a)
's
'
i
0.5
0.0 dx2.y2
+ i
-2
-1
"
~'
1
';"
2
Magnetic flux (~0)
dxy
8
7
0 Figure I. Magnitude and phase anisotropy of the order parameter for some candidate pairing symmetries.
8 the behavior near the nodes, the phase anisotropy is distinctly different. The objective of the SQUID interferometer technique is to measure the relative phase in orthogonal directions. This single measurement is sufficient to distinguish s-wave from d-wave, and can also identify a complex mixture which breaks time-reversal symmetry.
2. The Corner SQUID Experiment The design of the corner SQUID experiment is shown in Figure 2(a). The circuit is a bimetallic dc SQUID, in which Josephson junctions on the a and b faces of the cuprate crystal are connected by a loop of a conventional superconductor. The
1 0
(c)
f .....
-10~
-05~
OOn
comer SQUIDs
05~
10z
I 5~
t 20~
Phase shift
Figure 2. Corner SQUID experiment: (a) plan, (b) critical current modulation vs. symmetry, (c) results. Several key principles are important in this experiment, many of which have not been fully understood or appreciated: (a) Domain structure. The corner experiment measures the global phase shift between orthogonal k-space directions but between spatially-separated regions of the crystal adjacent to the junctions in real space. As a result, the range of the order parameter
130
D.J. Van Harlingen/Physica C 282-287 (1997) 128-131
domains and, in particular, how the order parameter behaves across grain and twin boundaries are important. The only way to determine this is to measure the phase response in both twinned and untwinned single crystals. Our measurements in crystals of these types [6] have established that the order parameter has gyroscopic symmetry, that is, the order parameter maintains the same sign orientation across grain and twin boundaries, forming a single domain throughout the crystal. This result is crucial for the SQUID interferometer technique to make a definitive determination of the microscopic symmetry, especially in heavilytwinned thin film samples in which subsequent versions of the interferometer experiment have been carried out. (b) Directionality. The measurement of the phase anisotropy depends on probing the order parameter in specific k-space directions. This relies on the directionality of the Josephson tunneling orthogonal to the junction barrier. Although this is known to occur in insulator-barrier junctions for which the charge transport is solely by tunneling, it also applies to any planar junction (such as the SNS junctions used in our experiments) since Josephson tunneling involves phase-coherent electron-hole pair conduction. The grain boundary junctions used in thin film interferometers certainly do not strictly maintain directionality because of the wandering of the boundary, but their ability to resolve the orthogonal phase shift indicates that directionality is maintained on average. (c) Origin of phase shift. The phase shift observed in the corner SQUID experiment and its variations occurs inside the crystal between orthogonal directions. A common misconception is that the phase shift can be described in terms of n-junctions, Josephson junctions in which the minimum coupling energy is at junction phase n, corresponding to a negative critical current. This concept was introduced [8] to describe Josephson junctions in which the tunneling occurs via magnetic spin-flip scattering, adding a n-phase shift inside the junction. To date, such junctions have not been realized, and have nothing to do with the phase shifts and the spontaneous circulating currents observed in interferometer experiments. The junctions in these experiments are all ordinary Josephson junctions, with their minimum energy at zero phase. In
particular, it is clear in the SQUID experiments that 7t-junctions cannot account for the observed phenomena since there are two identical junctions. 3. VARIATIONS Following our initial corner SQUID tests, several subsequent experiments based on the same principle were carried out, contributing further evidence for the d-wave symmetry.
3.1 Singlecornerjunction Perhaps the most qualitatively clear indication of the d-wave symmetry comes from measurements of the critical current modulation pattern of a single Josephson junction that straddles the corner of a cuprate crystal [9], as in Figure 3(a). This is the single junction equivalent of the corner SQUID experiment, essentially a SQUID without the loop. For s-wave symmetry, such a junction exhibits the ordinary Fraunhofer diffraction pattern familiar from single slit optical interference. However, in YBCO-Pb junctions, we observe the dramatically modified pattern in Figure 3(b), characteristic of a phase shift of n between the orthogonal segments. The most telling feature is the precipitous drop in the critical current at zero field, demonstrating the cancellation of the critical currents through the two sections. Recent observation of complex critical current modulation patterns of 45 ° grain boundaries can be understood in terms of multiple corner junctions formed by the wandering boundary [10].
~
(a)
~~= 20 40 s°I~ ~ 60 0 -150 -100 -50
0
50
100
150
Applied magnetic field (raG)
Figure 3. Design (a) and measured diffraction pattern (b) for a single YBCO-Pb comer Josephson junction.
3.2 Spontaneouscirculatingcurrents An alternate approach to measuring the modified critical current modulation of the SQUID (or single junction) in a d-wave superconductor is to detect the spontaneous (zero applied field) circulating current
D.J Van Harlingen/Physica C 282-287 (1997) 128-131
required to maintain phase coherence around the SQUID loop. This has been achieved using SQUID magnetometry in the corner SQUID geometry by the Maryland group [11] and first in a tricrystal ring geometry by the IBM group [12]. The ring sample consists of a cuprate thin film grown on a tricrystal substrate, forming three superconducting regions separated by grain boundary Josephson junctions. This system is exactly equivalent to the corner SQUID, except that the magnetic field response depends on whether one, two, or all three of the cuprate sections exhibit a n-phase shift (i.e., straddle a node). In both configurations, the spontaneous circulating currents only flow if the loop inductance is sufficiently large to make it energeticallyfavorable to generate a current (as opposed to flipping the phase on one junction to its higherenergy zero-current state). The generated magnetic flux approaches ½~0 for large loop inductance. 3.3 c-axis supercurrents A somewhat different set of experiments involves Josephson tunneling into the c-axis face of YBCO samples. The observed finite Josephson supercurrents [13] are inconsistent with a state of pure d-wave symmetry since such currents are expected to average to zero, implying the existence of an s-wave component in the order parameter. Recent measurements of single c-axis junctions straddling a twin boundary exhibit the same critical current diffraction patterns as we observed in corner junctions (Figure 3), indicating that this s-wave component is largely the result of the orthorhombic structure of YBCO [ 14].
4. FUTURE DIRECTIONS The results of the corner SQUID, corner junction, spontaneous current, grain boundary diffraction, and c-axis tunneling experiments now all point conclusively to a state with at least predominately dx2-y2 symmetry. With the unconventional nature of the pairing symmetry in the cuprates established, attention is now being focused in three directions: (a) Determining the microscopic mechanism. Surprisingly and somewhat disappointingly, knowing the symmetry has not significantly accelerated the search for the microscopic
131
mechanism. Although the dx2-y2 symmetry was first predicted from magnetic pairing mechanisms, it is now recognized that many interactions can result in pairing in a d-wave channel. (b) Implications o f unconventional symmetry. The nodes and sign change associated with the dwave pairing affect many thermodynamic, transport, optical, and interface properties of the cuprates. Work is only beginning to explore the exciting science and potential technological impact resulting from the d-wave symmetry. (c) Application to other superconductors. The interferometer experiments we pioneered can also be used to test the symmetry of other candidates for unconventional superconductivity. We are currently carrying out experiments designed to determine the symmetry of the heavy fermion superconductor UPt 3, suspected to be p-wave, and the organic superconductor •-(ET)2Cu[N(CN)2]Br, which may be also be d-wave. REFERENCES
I. N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett. 62, 961 (1989). 2. P. A. Monthoux, A. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992). 3. J. Martindale et al., Phys. Rev. B 47, 9155 (1993). 4. W . N . Hardy et al., Phys. Rev. Lett. 70, 3999 (1993). 5. T. P. Devereaux et al., Phys. Rev. Lett. 72, 396 (1994). 6. D. A. Wollman et al., Phys. Rev. Lett. 71, 2134 (1993). 7. D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995). 8. L . N . Bulaevski, V. V. Kuzii, and A. A. Sobyanin, JETP Lett. 25, 290 (1977). 9. D. A. Wollman et al., Phys. Rev. Lett. 74, 797 (1995). 10. J. Mannhartetal., Z. Phys. B 101,175 (1996). 11. C. C Tsuei et al., Phys. Rev. Lett. 73, 593 (1994). 12. A. Mathai et al., Phys. Rev. Lett. 74, 4523 (1995). 13. A. G. Sun et al., Phys. Rev. Lett. 72, 2267 (1994). 14. J. Clarke et al., preprint.