KIRTLEY J. R.

Tafuri, Francesco

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Physics 2, 92 (2009)ViewpointCan superconducting rings provide clues to the early development of theuniverse?John R. KirtleyCenter for Probing the Nanoscale,Stanford University, McCullough Building, 476 Lomita Mall, Stanford, CA 94305-4045, USAFrancesco TafuriDipartimento di Ingegneria dell’Informazione, Seconda Università di Napoli, and CNR-INFM Coherentia/NEST, Via Roma 29, 81031Aversa (CE), ItalyPublished November 2, 2009Defects—in the form of vortices in superconductors or “strings” in the fabric of the universe—can reveal thestate of a system at the time it was cooled.Subject Areas: Mesoscopics, CosmologyA Viewpoint on:Spontaneous fluxoid formation in superconducting loopsR. Monaco, J. Mygind, R. J. Rivers and V. P. KosheletsPhys. Rev. B 80, 180501 (2009) – Published November 2, 2009Watching a metal transform into a superconductor, itmay not be obvious that this transition provides accessto some of the same physics that governed the cool-ing of the universe following the Big Bang. Yet at theroot of both of these phenomena—albeit at astronom-ically different energy scales—is the question of howdefects form in a continuous phase transition. In a pa-per appearing in Physical Review B[1], Roberto Monacoat the Università di Salerno in Italy, Jesper Mygind atthe Technical University of Denmark, Ray Rivers at Im-perial College London, UK, and Valery Koshelets atthe Russian Academy of Science in Moscow have takenideas about the early development of the universe andapplied them to the dynamics of the superconductingphase transition in metal rings. In doing so, they intro-duce an elegant way to measure the tiny flux quanta thatare trapped at the centers of these rings, and develop anew understanding of how the cooling process works inmesoscopic systems.Just as there is a lattice mismatch between regions ofa liquid that crystallize with different orientations, de-fects formed as a result of the universe going through asymmetry breaking phase transition. Symmetry princi-ples and how they are broken are fundamental conceptsin physics, including our picture of what happened af-ter the Big Bang. The unification of the electromag-netic, weak, and strong forces came from realizing thatat high energies, a fundamental symmetry relates theparticles that mediate these interactions. However, overtime as the universe cooled to our “low-energy” world,this symmetry was broken, which is why the W and Zbosons, which mediate the weak interaction, have mass,while the photon, which mediates the electromagneticforce, does not.Theorists postulated that the defects associated withthe rapid cooling of the universe—imagined as strings,monopoles, and domain walls—could serve as imprintsof the universe at the point at which they were formed.Within this context, Tom Kibble proposed an intuitivepicture of the “freezing out” of the early universe andthe typical length scale between defects [2, 3]. Under-lying his picture was the idea that causality—that a fi-nite time delay is needed for information to be trans-ferred between different regions of a system—governedthe number of defects.Wojciech Zurek [4] noted that although Kibble’s ideasinvolved processes with energies far too high to be ac-cessible in the laboratory, they could be tested in therealm of condensed matter by studying the defects,called vortices, that form in superfluids and supercon-ductors. Figure 1 provides a helpful visualization ofKibble and Zurek’s basic idea for a superconductor. Thetransition from the normal (metallic) to the supercon-ducting state is a symmetry breaking transition and theorder parameter that describes the onset of the super-conducting phase has an amplitude ψ and a phase ϕ.Assume for simplicity that in the rings in Fig. 1, ψ isspatially independent and the ring is sufficiently nar-row that ϕ depends only on the azimuthal angle θ. Asthe ring cools through the superconducting transitiontemperature Tc, ψ will become larger, but different sec-tions of the ring will randomly assume different valuesof ϕ. This happens because locally, the superconductingphase is forming faster than it can communicate withDOI: 10.1103/Physics.2.92URL: http://link.aps.org/doi/10.1103/Physics.2.92c© 2009 American Physical SocietyPhysics 2, 92 (2009)FIG. 1: As a metal ring is cooled below its superconductingtransition temperature Tc, the phase of different regions of thering will be out of sync. The winding lines indicate the rotationin the phase of the superconducting wave function around thering. (Illustration: Alan Stonebraker)other regions in the ring. This will produce “kinks”or “defects,” regions where ϕ would eventually (at lowtemperature) change by a factor of 2pi. These kinks canbe removed by annihilation with anti-kinks if the coher-ence length ξ—the region over which the Cooper pairsin the superconductor share the same phase—is longenough and if the cooling rate is sufficiently slow. How-ever, as the system cools this annihilation process pro-ceeds less and less rapidly, until at some point the kinks“freeze in” because parts of the system cannot commu-nicate with other parts quickly enough.In the case of a superconducting ring, the net changein phase around the ring is limited to 2pin, where n is aninteger. Associated with this quantized phase are quan-tized values of supercurrents circulating around the ringand magnetic flux threading through it. For this rea-son, this process of forming defects is also called spon-taneous fluxoid generation.Kibble and Zurek showed that causality governs thefinal probability of finding a spontaneous fluxoid in thering after it is cooled. Specifically, causality implies thatthe probability of finding a single fluxoid in the ring(when cooled in zero external field) depends on the cool-ing (quench) rate τ−1Q as ∝ (τQ/τ0)−σ, where τ0 is a char-acteristic time and the exponent σ depends on the de-tails of how the correlation length ξ and the relaxationtime τ vary with temperature close to the transition. Fora superconducting ring and assuming a mean-field ap-proximation, σ = 1/4.Kibble and Zurek’s prediction has been tested in anumber of experimental systems by looking for vor-tices in superfluid 4He (see Refs. [5, 6]) and 3He (seeRefs. [7, 8]), in superconducting films [9], and super-conducting rings interrupted by thin regions of insu-lator (called Josephson weak links) [10–13]. However,experimental verification of the Kibble-Zurek proposalhas been lacking in what is arguably the simplest sys-tem—superconducting rings without Josephson weaklinks—because it was experimentally difficult to deter-mine the presence or absence of single flux quanta insuperconducting rings. Recent advances in magneticmicroscopy techniques have now made such measure-ments routine.In superconductors, an electromagnetic field adds tothe scalar order parameter field because of the electri-cally charged Cooper pairs, causing novel effects, in-cluding fractional vortices in systems involving intrin-sic or extrinsic shifts in the quantum mechanical phase[14, 15]. In collaboration with C. C. Tsuei, we (Kirtleyand Tafuri) used a scanning SQUID microscope to imagea number of rings after repeated cooling in various mag-netic fields and cooling rates to determine the probabil-ity of spontaneous fluxoid generation [16]. We showedthat a second mechanism prevailed, in which the fi-nal density of fluxoids depended on a balance betweenthermal generation and the relaxation rates of fluxoids[17–19]. We argued that the thermal fluctuation mecha-nism is complementary to the “causal” mechanism andshould be considered in attempts to understand phasetransitions, both in the laboratory and in the early uni-verse.These experiments were done on rings with parame-ters that favored the thermal activation mechanism overthe causal (Kibble-Zurek) mechanism. In their work,Monaco et al. study rings that are made of a differentmaterial and are narrow enough to suppress the thermalmechanism. They chose an experimental geometry inwhich the final fluxoid state of a single test ring could bedetermined by detecting the suppression in the criticalcurrent of a Josephson junction placed next to the ring.This, along with a carefully chosen thermal conduc-tivity between the ring and the thermal bath, allowedrapid multiple heating and quenching cycles of the ringwith varying quenching rates. Monaco et al. foundthat the probability of spontaneous fluxoid generationin the ring followed the expected power law in quench-ing rate, but the exponent σ was close to 1/2, a factorof 2 larger than the standard Kibble-Zurek prediction.Monaco et al. attribute their finding of a much weakerdependence on cooling rate than theory to the fact thatthey are studying a small ring: since the ring’s circum-ference is much smaller than the coherence length at thetemperature at which the fluctuations are frozen in, thekinks are more effectively annihilated. Rings with a cir-cumference that is large relative to the coherence lengthwould, in this view, have the Kibble-Zurek exponent of1/4. This explains not only their results on rings withoutJosephson weak links, but may also explain previous re-sults on annular junctions, which also showed the sameexponent. It remains to be seen whether these new in-sights about phase transitions in superconducting ringsare relevant to cosmological phase transitions.The spontaneous formation of topological defects re-mains a stimulating problem common to diverse fieldsDOI: 10.1103/Physics.2.92URL: http://link.aps.org/doi/10.1103/Physics.2.92c© 2009 American Physical SocietyPhysics 2, 92 (2009)of physics and various types of phase transitions. Whatis most fascinating about the class of condensed matterexperiments that were inspired by cosmology is the pos-sibility of reading their memories—the information thatwas frozen into the topological defects at the time theywere formed.References[1] R. Monaco, J. Mygind, R. J. Rivers, and V. P. Koshelets, Phys. Rev.B 80, 180501 (2009).[2] T. W. B. Kibble, J. Phys. A 9, 1387 (1976).[3] T. W. B. Kibble, Phys. Today 60, No.9, 47 (2007).[4] W. H. Zurek, Nature 317, 505 (1985).[5] P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock, ,and C. D. H. Williams, Nature 368, 315 (1994).[6] M. E. Dodd, P. C. Hendry, N. S. Lawson, P. V. E. McClintock, andC. D. H. Williams, Phys. Rev. Lett. 81, 3703 (1998).[7] C. Bäuerle, Yu. M. Bunkov, S. N. Fisher, H. Godfrin, and G. R.Pickett, Nature 382, 332 (1996).[8] V. M. H. Ruutu, V. B. Eltsov, A. J. Gill, T. W. B. Kibble, M. Krusius,Yu. G. Makhlin, B. Plaçais, G. E. Volovik, and Wen Xu, Nature 382,334 (1996).[9] R. Carmi and E. Polturak, Phys. Rev. B 60, 7595 (1999).[10] R. Carmi, E. Polturak, and G. Koren, Phys. Rev. Lett. 84, 4966(2000).[11] E. Kavoussanaki, R. Monaco, and R. J. Rivers, Phys. Rev. Lett. 85,3452 (2000).[12] R. Monaco, J. Mygind, and R. J. Rivers, Phys. Rev. Lett. 89, 080603(2002).[13] R. Monaco, J. Mygind, and R. J. Rivers, Phys. Rev. B 67, 104506(2003).[14] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).[15] F. Tafuri and J. R. Kirtley, Rep. Prog. Phys. 68, 2573 (2005).[16] J. R. Kirtley, C. C. Tsuei, and F. Tafuri, Phys. Rev. Lett. 90, 257001(2003).[17] M. Hindmarsh and A. Rajantie, Phys. Rev. Lett. 85, 4660 (2000).[18] M. Ghinovker, B. Ya. Shapiro, and I. Shapiro, Europhys. Lett. 53,240 (2001).[19] M. Donaire, T. W. B. Kibble, and A. Rajante, New J. Phys. 9, 148(2007).DOI: 10.1103/Physics.2.92URL: http://link.aps.org/doi/10.1103/Physics.2.92c© 2009 American Physical SocietyPhysics 2, 92 (2009)About the AuthorsJohn R. KirtleyJohn R. Kirtley received his B.A. in 1971 and his Ph.D. in 1976, both in physics, from theUniversity of California, Santa Barbara. After a post-doctoral position at The Universityof Pennsylvania, he joined IBM’s Thomas J. Watson Research Center in 1978, where he re-mained until becoming a Research Staff Member Emeritus in 2006. He is now a consultingprofessor in the Applied Physics Department at Stanford University. In 1998 he shared theOliver E. Buckley prize in condensed matter physics with C. C. Tsuei, D. J. van Harlingen,and D. M. Ginsberg. He is a fellow of the American Physical Society and of the AmericanAssociation for the Advancement of Science, won the Humboldt Forschungspreis in 2007,was a Jubileum Professor at Chalmers University in Sweden the same year, and will holda Chair d’Excellence from the CNR in Grenoble, France, 2010–2013.Francesco TafuriFrancesco Tafuri received his Ph.D. in physics at the University of Napoli “Federico II” in1993, carrying out part of his research activity at Stony Brook University. Since 1994 hehas been at the Second University of Napoli, first as an Assistant Professor and, since 2002,as Associate Professor. In 2001, he was a visiting scientist at the IBM’s Thomas J. WatsonResearch Center. Most of his experimental work has focused on the study of the Joseph-son effect and macroscopic quantum phenomena in superconductors with a high criticaltemperature and vortex matter in unconventional systems and oxide nanostructures.DOI: 10.1103/Physics.2.92URL: http://link.aps.org/doi/10.1103/Physics.2.92c© 2009 American Physical Society

Can superconducting rings provide clues to the early development of the universe?

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