H
Tutorial on Frontiers in Vortex Physics

This appendix introduces the study of vortex matter and describes some of the leading avenues of research. The experimental exploration of vortex matter, which is one of the many topics uncovered by HTS research, has become an important part of modern condensed-matter physics and depends on high magnetic fields.

The study of vortex matter in high fields over the full phase diagram is one of the most appealing research areas in condensed-matter physics. It is interesting not only because vortex system properties govern the charge transport and magnetic properties of HTS materials and, more generally, all Type II superconductors, but also because the behavior of these complex objects is generic for almost all disordered condensed-matter systems, so that cutting-edge contemporary concepts, involved theoretical methods, and the most advanced experimental techniques will be required to understand them. The study of vortices may lead to the physical realization of systems in which the entire statistical and quantum mechanics of strongly correlated systems with tunable parameters can be studied experimentally. A complex interplay between quantum and thermal fluctuations and disorder determines both the thermodynamic and the dynamic properties of vortex matter. Studies of vortex matter melting and the role of disorder have advanced the knowledge of solid and liquid thermodynamics. Last, but not least, several new phenomena, such as dynamic melting, have been discovered in vortex matter. These discoveries have led to the development of a novel and hitherto almost unexplored branch of physics, the physics of dynamic phase transitions. Another



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Opportunities in High Magnetic Field Science H Tutorial on Frontiers in Vortex Physics This appendix introduces the study of vortex matter and describes some of the leading avenues of research. The experimental exploration of vortex matter, which is one of the many topics uncovered by HTS research, has become an important part of modern condensed-matter physics and depends on high magnetic fields. The study of vortex matter in high fields over the full phase diagram is one of the most appealing research areas in condensed-matter physics. It is interesting not only because vortex system properties govern the charge transport and magnetic properties of HTS materials and, more generally, all Type II superconductors, but also because the behavior of these complex objects is generic for almost all disordered condensed-matter systems, so that cutting-edge contemporary concepts, involved theoretical methods, and the most advanced experimental techniques will be required to understand them. The study of vortices may lead to the physical realization of systems in which the entire statistical and quantum mechanics of strongly correlated systems with tunable parameters can be studied experimentally. A complex interplay between quantum and thermal fluctuations and disorder determines both the thermodynamic and the dynamic properties of vortex matter. Studies of vortex matter melting and the role of disorder have advanced the knowledge of solid and liquid thermodynamics. Last, but not least, several new phenomena, such as dynamic melting, have been discovered in vortex matter. These discoveries have led to the development of a novel and hitherto almost unexplored branch of physics, the physics of dynamic phase transitions. Another

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Opportunities in High Magnetic Field Science example of such a phenomenon is the Josephson plasma resonance effect, which holds great technological promise, as evidenced by the high level of Japanese investment in this area. A vortex system is a tunable, correlated system. The density of vortex lines in a superconductor depends on the applied magnetic field, so that by varying the external field, one can control the density of the “particles” in the system and hence also their interactions. Vortex separation is ~80 nm at 1 T, 14 nm at 10 T, 6 nm at 50 T, and 4 nm at 100 T. The radius of strong interaction between vortices is determined by λ, generally 200-400 nm for HTS materials. At distances exceeding this scale, interactions are screened and become exponentially weaker. Most experiments—as well as practical applications—are conducted in the range where the vortex spacing is much less than the London depth, which ensures retention of the tunable property of these strongly correlated systems. Additionally, the degree of vortex disorder can often be changed by altering its density using magnetic field strength as the controlling variable. Thus, by adjusting both the temperature and the magnetic field one can scan the entire range of possible interactions. In general, the quantum mechanics of particles in N dimensions is equivalent to the statistical mechanics of elastic lines in an N + 1 dimensional space. Thus the physics of vortex systems in a bulk sample can be mapped onto the quantum mechanics of a two-dimensional system of repulsively interacting bosons. In particular, a vortex lattice can correspond to a two-dimensional Wigner crystal, while a vortex liquid maps onto a two-dimensional superfluid. As tunable systems, vortex states in high magnetic fields offer a unique possibility for study of the fundamental localization problem in strongly correlated systems. One of the most important questions being investigated today is how the mixed superconducting state depends on the collective properties of the vortices when a transport current is applied. Hydrodynamic interactions of the applied current with the circular currents of the vortices cause the vortices to migrate (via the so-called Magnus force), and energy is dissipated. Thus, in the presence of vortices, a superconductor cannot carry a current without energy loss unless the vortices are pinned in place. Since the first field penetration at Hc1 occurs at ~0.01 T, even weak currents generate vortices in superconducting devices. A closely related topic is the behavior of very small superconductor samples in relatively high magnetic fields (a few tesla). For instance, transport properties of superconducting point contacts, which will be an issue in most of the next generation of superconducting devices, is an increasingly important research topic. At the tip of such a contact, superconductivity disappears, and charge transport occurs exclusively via Andreev states. Extensive theoretical research is being carried out on this kind of nanoscale superconductivity, and corresponding experimental investigations are needed. Another important topic concerns superconductor

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Opportunities in High Magnetic Field Science samples where one of the physical dimensions is smaller than the superconducting coherence length, which is a situation often seen in thin superconducting films, wires, and granules. An applied magnetic field induces a quantum phase transition between the superconducting and normal states at moderate (of the order of 1 T) fields in such samples. Measurement of the transport and thermodynamic properties of nanometer-size superconductors in the vicinity of their critical points should result in dramatic advances in the understanding of critical quantum behavior. Experimental studies of vortex lattice materials can provide insights into phenomena that cannot be as easily accessed with the usual atomic solids. By introducing disorder in vortex lattice materials in a controlled way, using irradiation with high-energy electrons, protons, or heavy ions or chemical means, systems can be obtained in which the thermodynamic properties of glasses can be studied with a precision that is not achievable with other systems, such as spin glasses in random magnets. The study of vortex physics thus offers unique opportunities for exploring the fundamental statistical mechanics of condensed-matter systems. The superconducting state is characterized by the macroscopic quantum coherence of the whole sample, and it has been experimentally observed that melting of vortex lattices results in a loss of quantum coherence within superconductors. However, it is not known if the lattice melting in question is associated with a complete loss of coherence. There may be a range of magnetic fields and temperatures for which the vortex liquid loses only its transverse coherence, and correlations along the vortex lines persist. Experiments using the Josephson plasma resonance effect in high magnetic fields may provide decisive answers. There are three main directions in the study of vortex matter today: What factors control the pinning of vortices? A defining scientific and technological property of superconductors is their ability to carry current without loss. Vortices are present in superconductors in all practical situations, and when a current is applied to a superconductor, the vortices present will start to move unless immobilized by inhomogeneities in the material, which can be either natural or artificially engineered. If the vortices move, there will be energy loss. The effectiveness of this immobilization, or “pinning,” effect in any given superconducting material is quantified by its so-called critical current, the maximum current that it can carry without dissipation. Finite critical currents exist only in the vortex solid state and become zero as the vortex lattice melts. The quest to increase the critical currents of existing superconducting materials has thus motivated studies of vortex pinning and research into the thermodynamics of the vortex lattice. How do vortices behave dynamically? Even at low temperatures, and with small applied currents, vortices slowly move owing to thermal or quantum

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Opportunities in High Magnetic Field Science fluctuations. This effect is known as vortex creep. Vortex dynamics is the study of pinning-dominated vortex creep. What (other) novel phenomena do Type II superconductors exhibit? The search for qualitatively new effects that have no analogue in nonvortex matter or that were previously overlooked has become the third important line of investigation. Dynamic phase transitions and the so-called Josephson plasma resonance are good examples of phenomena of this kind. THERMODYNAMIC PROPERTIES OF VORTEX MATTER Phase transitions in the presence of disorder are a major concern in condensed-matter physics, and the melting of vortex lattices is a first-order phase transition that has received a lot of attention. In this case, the disorder in question is provided by impurities, crystalline defects, and the like, in the material in which the vortices exist, or it can be artificially introduced into the material via various irradiation methods. Though a good deal is known about the general behavior of this kind of melting, very little is known about the local fluctuations and the factors that determine the local properties of this phase transition. Owing to high anisotropy and very short coherence length (on the order of tens of angstroms), thermal fluctuations play a primary role in determining the physical properties of HTS materials. In conventional superconductors, the vortex-melting line was not detected since it lies very close to the upper critical field Hc2. In HTS materials, however, the melted phase—the vortex liquid—occupies a significant part of the vortex phase diagram. Experimentally, vortex melting was unambiguously established by a combination of transport measurements that revealed the onset of finite resistivity marking the disappearance of vortex pinning; neutron scattering measurements that demonstrated a loss of the long-range order; and thermodynamic measurements that showed a peak in heat capacity, which is the unambiguous signature of a thermodynamic phase transition. The thermodynamics of vortex systems are governed by the interplay between disorder and thermal fluctuations. Disorder transforms the vortex lattice into a vortex glass, a pinning-dominated phase with a highly nonlinear response to small external currents. Artificially manufactured defects can be introduced into a superconductor by irradiating it with heavy ions such as Pb, Au, or U. These impurities enhance pinning because of a geometrical match between the pinning site (the cylindrical amorphous track of the heavy ion through the material) and the linear vortex. Such impurities can extend the vortex glassy phase to higher fields and temperatures, forming a so-called Bose glass. Interestingly, a high density of defects can destroy a topologically defect-free vortex lattice—disorder-induced melting—and lead to the formation of an amorphous vortex glass phase at higher magnetic fields.

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Opportunities in High Magnetic Field Science The dynamics of nonlinear creep motion is governed by the fact that interactions between disorder and elastic degrees of freedom give rise to effective motion barriers that grow as a power of the applied force. One of the fundamental questions of the physics of glasses is how these growing barriers relate to the proposed hierarchical structure of the glass itself. Studies of noise associated with vortex motion will also advance understanding of the statistics of low-lying states in glasses. DYNAMIC PROPERTIES OF VORTEX SYSTEMS Even before the discovery of high-temperature superconductors, it was predicted that the thermally activated dynamics of disorder-dominated systems with internal degrees of freedom would exhibit highly nonlinear responses. For instance, in response to an infinitesimal applied driving force, the system velocity v appeared to show not only a nonlinear but also a nonanalytic dependence on the force F: v ~ exp(–const/Fµ), where the exponent, µ, depends on the dimensionality and the characteristic scale of the moving elastic medium and on the dimensionality of the space this medium moves in. For a magnetic domain wall moving in a magnetic film, µ = 1/4; for the vortex lattice in a bulk superconductor at very small currents, µ = 1/2. This theoretical result is highly counterintuitive: Textbook wisdom suggests that all physical systems demonstrate a linear response—that is, v is proportional to F (e.g., Ohm’s law) when forces are small. Initially, this result was met with skepticism and disbelief. Experiments on magnetic relaxation in HTS materials in 1990 confirmed that vortex motion at low temperatures obeys the prediction just discussed. Since then, extensive studies of the peculiarities of vortex dynamics in low-temperature vortex phases—vortex and Bose glasses—has become the focus of both experimental and theoretical research on superconductors. Investigation of vortex creep dynamics has been extended to the quantum case (quantum creep), and vortex creep now appears to be generic in all disordered systems. Recently, creep motion was even found for the motion of domain walls and dislocations. It is thus a fundamental characteristic of glassy phases, and the recognition of its importance has motivated intense interest in the dynamics of low-temperature vortex creep, one of the most exciting and fundamental discoveries in the physics of superconductivity of the last decade. The dissipation processes associated with the vortex motion are related to the scattering of the normal carriers confined within the normal vortex core—the so-called quasi-particle Andreev states. Experimental studies of the AC response of vortex lattice materials in high magnetic fields should provide access to these

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Opportunities in High Magnetic Field Science mechanisms and will help elucidate the microscopic nature of the electronic states in HTS materials. Another fundamental phenomenon discovered in the context of vortex lattice motion—in this case in the high-driving-force regime—is dynamic melting. When driven in the random environment, the periodic structure (vortex lattice) experiences random forces due to encounters with randomly distributed pinning sites. These collisions cause random fluctuations in the positions of vortices. The effect of this positional disorder therefore resembles the effect of temperature, and one can discuss it in terms of the temperature, the so-called shaking temperature, that would have the same disordering effect. It is proportional to the strength of the disorder present, and inversely proportional to the driving force. Thus, if the disorder in a material is sufficiently strong, the effective shaking temperature may become large enough to melt the vortex lattice. This dynamic melting, the transition from the coherent motion of the vortex lattice to the plastic dynamics of amorphous vortex structures, has been observed experimentally in superconducting films. The effect is also very general: In particular, the effect of abrupt switching in the dynamics of charge density waves is one manifestation of the transition between the plastic and coherent motions of the vortex lattice. NONSTANDARD EFFECTS Although vortex physics governs many aspects of superconductivity, it has some manifestation in HTS materials (particularly those that are ceramics), which, because they have a layered structure, have no analogues in conventional superconductors. In the layered superconductors, the vortices induced by the penetrating magnetic field actually appear as sets of aligned vortex discs (“pancake” vortices). Discs in adjacent layers couple via the Josephson interaction; discs in more remote layers interact magnetically. At high magnetic fields, the vortex system becomes so dense that interactions between pancakes within the same layers are stronger than interlayer interactions, and a decoupling transition occurs. The vortex system becomes a “gas” of pancake vortices. (A similar effect can be caused by introducing disorder into the system. Hypothetically, melting the vortex lattice would give rise to a loss of coherence along the magnetic field lines.) The resulting gas of pancakes behaves like a gas of charged particles, with Josephson interactions between pancakes in adjacent layers taking the place of Coulomb interactions between particles. This state of liquid vortex matter is called a Josephson plasma. Electromagnetic excitations, called plasmons, have been found in Josephson plasmas just as in normal plasmas, as have the usual plasma resonant absorptions, which are called Josephson plasma resonances. Josephson plasma resonance is one of the main diagnostics of the vortex liquid state because the frequency of these

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Opportunities in High Magnetic Field Science resonances directly measures the degree of correlations between the layers. The study of Josephson plasmas has become a major focus of superconductivity research in Japan, motivated in part by its potential technological application. Interactions between a moving Josephson plasma and the underlying periodic potential of the lattice of the material in which it exists may make it possible to generate sharp signals at terahertz frequencies. Devices exploiting these effects might include high-resolution filters, high-sensitivity detectors, and AC field generators in the terahertz frequency range (for terahertz lasers).