Abstract
Phase transitions in two-dimensional (2D) systems have received considerable attention recently. In several experiments the 2D crystal under consideration is exposed to the force field created by a periodic substrate. Among other situations this is the case of a 2D lattice of superconducting vortices interacting with a periodic pinning potential. As pointed out by Martinoli and coworkers1-3 some years ago, thin superconducting films, whose thickness is periodically modulated in one dimension, provide such a system. In this lecture we discuss the static and dynamic behaviour of this model system in which the 2D vortex lattice can be driven through a variety of phases4, 5 simply by changing the conditions of flux-line density and/or temperature. In particular, we show how measurements of the critical currents and of the complex rf impedance of thickness-modulated layers can be used to probe the transition of the 2D vortex lattice from a “locked” commensurate (C) phase in registry with the substrate periodicity to a “floating” incommensurate (I) solid phase exhibiting 2D topological order6, 7 or to a fluid-like phase.
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References
O. Daldini, P. Martinoli, J. L. Olsen, and G. Berner, Vortex-line pinning by thickness modulation of superconducting films, Phys. Rev. Lett. 32:218 (1974).
P. Martinoli, O. Daldini, C. Leemann, and B. Van den Brandt, Josephson oscillation of a moving vortex lattice, Phys. Rev. Lett. 36:382 (1976).
P. Martinoli, Static and dynamic interaction of superconducting vortices with a periodic pinning potential, Phys. Rev. B 17:1175 (1978).
P. Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Rep. Prog. Phys. 45:587 (1982).
J. Villain, Theories of commensurate-incommensurate transitions on surfaces, in: “Ordering in two dimensions”, S. K. Sinha, ed., North Holland, Inc., New York (1980).
J. M. Kosterlitz, and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6:1181 (1973).
D. R. Nelson, and B. I. Halperin, Dislocation-mediated melting in two dimensions, Phys. Rev. B 19:2457 (1979).
V. L. Pokrovsky, and A. L. Talapov, The theory of two-dimensional incommensurate crystals, Zh. Eksp. Teor. Fiz. 78:269 (1980) [Sov. Phys. JETP 51:134 (1980)].
P. Martinoli, H. Beck, M. Nsabimana, and G.-A. Racine, Locking-unlocking transition of a two-dimensional lattice of superconducting vortices, Physica 107B:455 (1981).
P. Martinoli, M. Nsabimana, G.-A. Racine, H. Beck, and J. R. Clem, Locked and unlocked phases of a two-dimensional lattice of superconducting vortices, Helv. Phys. Acta 55:655 (1982).
A. T. Fiory, Measurements of the shear modulus of the superconducting mixed state of thin films, Phys. Rev. B 8:5039 (1973).
F. C. Frank, and J. H. van der Merwe, One-dimensional dislocations, Proc. Roy. Soc. (London) A198:205 (1949).
J. Bardeen, and M. J. Stephen, Theory of the motion of vortices in superconductors, Phys. Rev. 140A:1197 (1965).
M. Tinkham, in: “Introduction to superconductivity”, McGraw-Hill, Inc., New York (1975).
M. W. Puga, E. Simanek, and H. Beck, Renormalization-group approach to commensurate-incommensurate transitions in two dimensions, Phys. Rev. B 26:2673 (1982).
M. W. Puga, E. Simanek, and H. Beck, Commensurate-incommensurate transitions, to appear in Helv. Phys. Acta (1983).
B. A. Huberman, and S. Doniach, Melting of two-dimensional vortex lattices, Phys. Rev. Lett. 43:950 (1979).
D. S. Fisher, Flux-lattice melting in thin-film superconductors, Phys. Rev. B 22:1190 (1980).
J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model, Phys. Rev. B 16:1217 (1977) [Erratum, Phys. Rev. B 17:1477 (1978)].
S. Ostlund, Relation between lattice and continuum theories of two-dimensional solids, Phys. Rev. B 23:2235 (1981).
S. Ostlund, Incommensurate and commensurate phases in asymmetric clock models, Phys. Rev. B 24:398 (1981).
J. Villain, and P. Bak, Two-dimensional Ising model with competing interactions: floating phase, walls and dislocations, J. Physique 42:657 (1981).
S. N. Coppersmith, D. S. Fisher, B. I. Halperin, P. A. Lee, and W. F. Brinkman, Dislocations and the commensurate-incommensurate transition in two dimensions, Phys. Rev. Lett. 46:549 (1981).
T. Bohr, V. L. Pokrovsky, and A. L. Talapov, Commensurate-incommensurate phase transition in continuous media containing dislocations, Pis’ma Zh. Eksp. Teor. Fiz. 35:165 (1982) [Sov. Phys. JETP Lett. 35:203 (1982)].
T. Bohr, Dislocations in the commensurate-incommensurate transition, Phys. Rev. B 25:6981 (1982).
F. D. M. Haidane, P. Bak, and T. Bohr, Phase diagramms of surface structures from Bethe-Ansatz solutions of the quantum sine-Gordon model, preprint (1982).
S. E. Burkov, and V. L. Pokrovsky, Critical currents and electric fields of two-dimensional systems, J. Low Temp. Phys. 44:423 (1981).
P. Martinoli, J. L. Olsen, and J. R. Clem, Superconducting vortices in periodic pinning structures, J. Less-Common Metals 62:315 (1978).
P. Martinoli, and J. R. Clem, Pinning in periodic superconducting structures, in: “Inhomogeneous superconductors-1979”, D. U. Gubser, T. L. Francavilla, S. A. Wolf, and J. R. Leibowitz, ed., American Institute of Physics, New York (1980).
A. Schmid, and W. Hauger, On the theory of vortex motion in an inhomogeneous superconducting film, J. Low Temp. Phys. 11:667 (1973).
J. R. Clem, Simple model for the vortex core in a type II superconductor, J. Low Temp. Phys. 18:427 (1975).
P. Martinoli, O. Daldini, C. Leemann, and E. Stocker, A. C. quantum interference in superconducting films with periodically modulated thickness, Solid State Comm. 17:205 (1975).
V. L. Pokrovsky, private communication.
A. T. Fiory, and A. F. Hebard, Radio-frequency complex-impedance measurements on thin film two-dimensional superconductors, in: “Inhomogeneous superconductors-1979”, D. V. Gubser, T. L. Francavilla, S. A. Wolf, and J. R. Leibowitz, ed., American Institute of Physics, New York (1980).
A. F. Hebard, and A. T. Fiory, Recent experimental results on vortex processes in thin-film superconductors, in: “Ordering in two dimensions”, S. K. Sinha, ed., North Holland, Inc., New York (1980).
A. F. Hebard, and A. T. Fiory, Vortex dynamics in two-dimensional superconductors, Physica 109 & 110 B:1637 (1982).
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Martinoli, P., Beck, H., Nsabimana, M., Racine, G.A. (1984). Static and Dynamic Properties of Commensurate and Incommensurate Phases of a Two-Dimensional Lattice of Superconducting Vortices. In: Goldman, A.M., Wolf, S.A. (eds) Percolation, Localization, and Superconductivity. NATO Science Series, vol 109. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-9394-2_14
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DOI: https://doi.org/10.1007/978-1-4615-9394-2_14
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