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Tamir et al This content was downloaded from IP address 165.230.225.223 on 02/12/2017 at 00:18 Physics ± Uspekhi 53 (1) 1 ± 49 (2010) # 2010 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences REVIEWS OF TOPICAL PROBLEMS PACS numbers: 74.62. ± c, 74.78. ± w, 74.81. ± g Superconductor±insulator quantum phase transition V F Gantmakher, V T Dolgopolov DOI: 10.3367/UFNe.0180.201001a.0003 Contents 1. Introduction 1 1.1 Superconducting state, electron pairing; 1.2 Superconductor±insulator transition as a quantum phase transition; 1.3 Role of disorder. Granular superconductors; 1.4 Fermionic and bosonic scenarios for the transition; 1.5 Berezinskii±Kosterlitz±Thouless transition 2. Microscopic approaches to the problem of the superconductor±insulator transition 7 2.1 Fermionic mechanism for the superconductivity suppression; 2.2 Model of a granular superconductor; 2.3 Bose± Einstein condensation of a bosonic gas; 2.4 Bosons at lattice sites; 2.5 Superconducting fluctuations in a strong magnetic field in the framework of the Bardeen±Cooper±Schrieffer model; 2.6 Fermions at lattice sites. Numerical models 3. Scaling hypothesis 15 3.1 General theory of quantum phase transitions as applied to superconducting transitions; 3.2 Scaling for two- dimensional systems and the role of a magnetic field; 3.3 Two-parametric scaling 4. Experimental 22 4.1 Ultrathin films on cold substrates; 4.2 Variable-composition materials; 4.3 High-temperature superconductors; 4.4 Crossover from superconductor±metal to superconductor±insulator transitions; 4.5 Current±voltage character- istics and nonlinear phenomena 5. Related systems 39 5.1 Regular arrays of Josephson junctions; 5.2 Superconductor±insulator type transitions in an atomic trap 6. Concluding discussion 43 6.1 Scenarios of the transition; 6.2 Role of macroscopic inhomogeneities; 6.3 Localized pairs; 6.4 Pseudogap; 6.5 Scaling References 48 Abstract. The current understanding of the superconductor ± fluctuations in a strong magnetic field) or on the theory of the insulator transition is discussed level by level in a cyclic spiral- Bose±Einstein condensation. A special discussion is given to like manner. At the first level, physical phenomena and pro- phenomenological scaling theories. Experimental investiga- cesses are discussed which, while of no formal relevance to the tions, primarily transport measurements, make the contents of topic of transitions, are important for their implementation and the third level and are for convenience classified by the type of observation; these include superconductivity in low electron material used (ultrathin films, variable composition materials, density materials, transport and magnetoresistance in super- high-temperature superconductors, superconductor±poor metal conducting island films and in highly resistive granular materi- transitions). As a separate topic, data on nonlinear phenomena als with superconducting grains, and the Berezinskii± near the superconductor±insulator transition are presented. At Kosterlitz±Thouless transition. The second level discusses and the final, summarizing, level the basic aspects of the problem are summarizes results from various microscopic approaches to the enumerated again to identify where further research is needed problem, whether based on the Bardeen±Cooper±Schrieffer and how this research can be carried out. Some relatively new theory (the disorder-induced reduction in the superconducting results, potentially of key importance in resolving the remaining transition temperature; the key role of Coulomb blockade in problems, are also discussed. high-resistance granular superconductors; superconducting 1. Introduction V F Gantmakher, V T Dolgopolov Institute of Solid State Physics, Russian Academy of Sciences, As temperature decreases, many metals pass from the ul. Institutskaya 2, 142432 Chernogolovka, Moscow region, normal to the superconducting state which is phenomen- Russian Federation ologically characterized by the possibility of a dissipationless Tel. (7-496) 522 54 25, (7-496) 522 29 46. Fax (7-496) 524 97 01 electric current and by the Meissner effect. As a result of a E-mail: [email protected], [email protected] change in some external parameter (for example, magnetic field strength), the superconductivity can be destroyed. In Received 30 June 2009 Uspekhi Fizicheskikh Nauk 180 (1) 3 ± 53 (2010) the overwhelming majority of cases, this leads to the return DOI: 10.3367/UFNr.0180.201001a.0003 of the superconducting material to the metallic state. Translated by S N Gorin; edited by A Radzig However, it has been revealed in the last three decades that there are electron systems in which the breakdown of 2 V F Gantmakher, V T Dolgopolov Physics ± Uspekhi 53 (1) superconductivity leads to the transition to an insulator include high-temperature superconductors in which the rather than to a normal metal. At first, such a transition superconductivity is caused by charge carriers moving in seemed surprising, and numerous efforts were undertaken in CuO2 crystallographic planes. As in any two-dimensional order to experimentally check its reality and to theoretically (2D) system, the density of states g0 in the CuO2 planes in the explain its mechanism. It was revealed that the insulator can normal state is independent of the charge carrier concentra- 4 1 prove to be quite extraordinary; moreover, upon breakdown tion and, according to measurements, is g0 2:5 Â 10 K of superconductivity with the formation of a normal metal, per structural element in each CuO2 crystal plane [to the metal can also be unusual. This review is devoted to a approximately one and the same magnitude in all families of discussion of the state of the art in experiment and theory in the cuprate superconductors (see, e.g., Ref. [2])]. Assuming, this field. for the sake of estimation, that D is on the order of the superconducting transition temperature Tc, we obtain the 1.1 Superconducting state, electron pairing average distance between the pairs in CuO2 planes: 1=2 By the term `superconducting state', we understand the state s g0Tc 25 A at Tc 100 K. This value is compar- of metal which, at a sufficiently low temperature, has an able with the typical coherence length z 20 A in high- electrical resistance exactly equal to zero at the zero temperature superconductors. frequency, thus indicating the existence of a macroscopic The existence of exotic superconductors, for which coherence of electron wave functions. This state is brought inequality (2) is violated, forced researchers to turn to about as a result of superconducting interactions between another model of superconductivity Ð the Bose±Einstein charge carriers. Such an interaction is something more condensation (BEC) of the gas of electron pairs considered general than superconductivity itself, since it can either lead as bosons with a charge 2e [3] Ð and to investigate the to or not lead to superconductivity. crossover from the BCS to the BEC model (see, e.g., the According to the Bardeen±Cooper±Schrieffer (BCS) review [4]).One of the essential differences between these theory, the transition to the superconducting state is accom- models consists in the assumption of the state of the electron panied by and is caused by a rearrangement of the electronic gas at temperatures exceeding the transition temperature. The spectrum with the appearance of a gap with a width of 2D at BEC model implies the presence of bosons on both sides of the Fermi level. The superconducting state is characterized by the transition. An argument in favor of the existence of a complex order parameter superconductors with the transition occurring in the BEC Á scenario is the presence of a pseudogap in some exotic F r D exp ij r ; 1 superconductors. It is assumed that the pseudogap is the binding energy of electron pairs above the transition in which the value of the gap D in the spectrum is used as the temperature (for more detail, see the end of Section 4.3 modulus. If the phase j r of the order parameter has a devoted to high-temperature superconductors). gradient, j r 6 const, then a particle flow exists in the In the BCS model, the Cooper pairs for T > Tc appear system. Since the particles are charged, the occurrence of a only as a result of superconducting fluctuations; the equili- gradient indicates the presence of a current in the ground brium concentration of pairs exists only for T < Tc. The state. crossover from the BCS to the BEC model consists in The rearrangement of the spectrum can be represented as decreasing gradually the relative size of Cooper pairs and a result of a binding of electrons from the vicinity of the Fermi appearing the pairs on both sides of the transition, which are level (with momenta p and p and oppositely directed spins) correlated in phase in the superconducting state and uncorre- into Cooper pairs with a binding energy 2D. The binding lated in the normal state. The conception that in super- occurs as a result of the effective mutual attraction of conducting materials with a comparatively low electron electrons located in the crystal lattice, which competes with density the equilibrium electron pairs can exist for T > Tc the Coulomb repulsion.