JOSEPHSON VORTEX LATTICE IN LAYERED SUPERCONDUCTORS
A. E. Kosheleva* M. J. W. Dodgsonh
" Materials Science Division, Argonne National Laboratory 60439, Argonne, Illinois
b Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge, CB3 OHE, UK
Institut de Physique, Université de Neuchâtel 2000, Neuchâtel, Switzerland,
Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
Received April 1, 2013
Dedicated to the memory of Professor Anatoly Larkin
Many superconducting materials are composed of weakly coupled conducting layers. Such a layered structure has a very strong influence on the properties of vortex matter in a magnetic field. This review focuses on the properties of the Josephson vortex lattice generated by the magnetic field applied in the direction of the layers. The theoretical description is based on the Lawrence-Doniach model in the London limit, which takes only the phase degree of freedom of the superconducting order parameter into account. In spite of its simplicity, this model leads to an amazingly rich set of phenomena. We review in detail the structure of an isolated vortex line and various properties of the vortex lattice, in both dilute and dense limits. In particular, we extensively discuss the influence of the layered structure and thermal fluctuations on the selection of lattice configurations at different magnetic fields.
DOI: 10.7868/S0044451013090071
1. INTRODUCTION
Layered superconductors arc materials made from a stack of alternating thin superconducting layers separated by nonsuperconducting regions. The superconducting layers are essentially two-dimensional (2D) as long as they are so thin that there is no variation in fields, or in the superconducting order parameter, across each layer. Such structures frequently occur naturally in anisotropic crystals. A layered superconductor can carry supercurrents along the layers, as well as between the layers. This is due to the Josephson tunneling of Cooper pairs fl] across the insulating regions that separate neighboring superconducting layers, i.e., each pair of neighboring layers forms one Josephson junction. In general, the ¿-axis (Josephson) supercurrents are weaker than the supercurrents along the lay-
* E-mail: koshelev'fflanl.gov
ers. A mere "layeredness" of atomic structure, however, does not automatically make a material a layered superconductor. When the interlayer electrical coupling is sufficiently strong, this discrete system of layers approximates to a continuous superconductor with uniaxial anisotropy. Hence, we are interested in the case where the approximation to a uniaxial continuous superconductor breaks down, which happens when the layer separation d is greater than the ¿-axis superconducting coherence length, d £c.
The most prominent example is the high-Tc cuprate superconductors, discovered in 198C [2 5], which led to a huge interest in physics of layered superconductors. The two most studied cuprate compounds, YBa-iCusOT (YBCO) and Bi2Sr2CaCu203. (BSCCO), have similar transition temperatures Tc « 90K and represent two important particular cases. YBCO is moderately anisotropic, with the anisotropy factor 7 «5 7, and its "layeredness" becomes essential at low temperatures when the e-axis coherence length drops below the
Fig. 1. Illustration of a dilute lattice of Josephson vortices generated in a layered superconductor by a magnetic field applied
along the layer direction
layer spacing d. On the other hand, BSCCO has a huge anisotropy factor, 7 « 400 — 1000, and behaves as a layered superconductor practically in the whole temperature range below Tc. Other naturally layered superconductors include the transition metal dichalco-genides [6, 7] and organic charge-transfer salts formed with the molecule BEDT-TTF [8,9]. An important new family of atomically layered superconducting materials, iron pnictides and chalcogenides, was discovered in 2008 [10] and is being extensively explored since then (see, e.g., reviews [11-13]). Anisotropy of most compounds is actually not very high and they typically behave as anisotropic three-dimensional materials. There are important exceptions, however. The most studied compound in which the layered structure is clearly essential is SmFeAsOi-^F^ [14] with Tc up to 55 K. For example, the Josephson nature of the in-plane vortices at low temperatures has been recently demonstrated in this compound [15]. Also, several iron pnictide compounds with extremely high anisotropy have been discovered [16-18]. Properties of these compounds remain mostly unexplored due to their rather complicated composition.
All layered superconductors share a very similar general behavior of the vortex matter generated by an external magnetic field, which is insensitive to the microscopic nature of superconductivity inside the layers. Several excellent review articles have been published in the past covering different aspects of the vortex matter in type-II superconductors [19-23]. Nevertheless, we feel that further progress in the understanding of the Josephson vortices in layered superconductors war-
rants a specialized review, providing more details and discussing important recent results.
This short review narrowly focuses on the vortex lattice that appears at magnetic fields applied along the layers. In this case, the flux line winds its phase around an area between two neighboring layers and is called a Josephson vortex in analogy with a vortex in a superconducting tunneling junction. The Josephson vortex contains out-of-plane currents that tunnel via the Josephson effect from layer to layer. The current distribution around a vortex is anisotropic. As a consequence, the vortex lattice is also anisotropic: it is a triangular lattice strongly stretched along the layers (see Fig. 1). In addition, the restriction to lie between the layers leads to commensurability effects and an energy barrier to tilting the field away from the layers. There are two very different regimes depending on the magnetic field strength Bx. The crossover field scale Bcr separating these two regimes is set by the anisotropy factor 7 and the layer periodicity d as Bcr = $o/(27T7d2), where $0 = hc/2e is the flux quantum. In the case of BSCCO, this field scale is around 0.5 tesla. In the dilute lattice regime, Bx < Bcr, the nonlinear cores of Josephson vortices are well separated and the distribution of currents and fields is very similar to that in continuous anisotropic superconductors [24]. The dense lattice regime is realized at high fields Bx > Bcr, where the cores of Josephson vortices overlap. In this regime, the Josephson vortices fill all layers homogeneously [25]. This state is characterized by rapid oscillations of the Josephson current and by very weak modulation of the in-plane current. In this re-
view, we characterize these two lattice regimes in more detail.
We do not consider the properties of vortices generated by a magnetic field applied perpendicular to the layers, along the c axis1). The structure of a e-axis vortex is very different from the structure of an in-plane vortex. In layered superconductors, a e-axis vortex can be viewed as a stack of weakly coupled pointlike pancake vortices. Properties of the pancake vortex lattice were also extensively explored, see, e.g., reviews [23] and [26] and the references therein.
Several experimental techniques have been employed to explore the Josephson vortex lattices. The dilute stretched lattice at small fields (< 100 G) has been directly observed in YBCO with Bitter decoration in [27], where the elliptical distribution of the flux around each Josephson vortex was also seen. At high fields (> 1 tesla), the commensurability between the e-axis parameter of the Josephson vortex lattice and the interlayer separation leads to magnetic field oscillations, which have been observed experimentally in un-derdoped YBCO in irreversible magnetization [28, 29] and nonlinear resistivity [30].
In much more anisotropic BSCCO, direct observation of Josephson vortices is not possible. However, when the magnetic field is tilted at small angles with respect to the layers, the e-axis field component generates the pancake-vortex stacks that preferably enter the superconductor along the Josephson vortices forming chains. Visualizing the flux of these chains, it is possible to find locations of vertical rows of the Joseph-son vortices and measure the in-plane lattice parameter uy. This was done using a variety of visualization techniques, such as Bitter decorations [31,32], scanning Hall probes [33], Lorentz microscopy [34,35] and magnet ooptical imaging [36,37]. These observations have been summarized in review [38].
Most extensively, properties of the Josephson vortex lattice were explored in BSCCO using e-axis transport in small-size mesas [39 43]. These studies revealed a very rich dynamical behavior of the lattice, which is beyond the scope of this review. The very important feature is that, due to low dissipation, the Joseph-son vortex lattice can be accelerated up to very high velocities. It is clear that understanding dynamics is not possible without good understanding of static lattice properties. The dynamic phenomenon closely related to static lattice configurations is magnetic-field oscillations of resistance for very slow lattice motion,
11 In the literature the layer plane and the axis perpendicular to the layers are frequently called "ab plane" and "c axis".
which have been discovered and explored in small-size BSCCO mesas [44 48]. The oscillation period can correspond to either the flux quantum or half the flux quantum per junction depending on the magnetic field and the lateral size of the mesa. An interplay between the bulk shearing interaction and the interaction with edges leads to very nontrivial evolution of lattice structures, which we consider in this review.
This review is organized as follows. We start in Sec. 2, w
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