Critical behavior of the flux-line tension in extreme type-II superconductors

Abstract

The entropic corrections to the flux-line energy of extreme type-II superconductors are computed using a schematic dual Villain model description of the flux quanta. We find that the temperature profile of the lower critical field vanishes polynomially at the transition with an exponent Х2/3 in the isotropic case, while it exhibits an inflection point for the case of weakly coupled layers in parallel magnetic field. It is argued that vestiges of these effects have already been observed in high-temperature superconductors. ͓S0163-1829͑96͒02425-3͔It is well known that type-II superconductors allow magnetic fields to penetrate in the form of flux lines quantized in units of the flux quantum, ⌽ 0 ϭhc/2e, for field values above the lower critical field given by H c1 (T)ϭ4(T)/⌽ 0 , where (T) denotes the flux-line free energy per unit length. 1 The London approximation yields 0 (T)ϭ(⌽ 0 /4 L ) 2 ln L / for this free energy, where both the London penetration length, L , and the coherence length, , vary as (T c0 ϪT) Ϫ1/2 near the critical temperature, T c0 , within the Ginzburg-Landau mean-field approximation. Spatial fluctuations of a flux line act to reduce this mean-field line tension, however, particularly near the transition temperature. Such entropic corrections to the lower critical field have been computed 2,3 and observed 4-6 in the case of hightemperature superconductors, where fluctuation effects are important due to the short coherence lengths and the large transition temperatures that characterize these materials.In this paper, we shall recompute the flux-line tension of extreme type-II superconductors ͑ L ӷ͒ in the absence of magnetic field utilizing the so-called ''frozen'' limit of lattice superconductors first studied by Peskin. 7-9 The principle idea here is to exploit the duality of quantized magnetic flux lines in superconductors to confining strings of electric flux in pure lattice gauge theories. Hence, only the entropic correction to the line tension due to flux-line wandering is accounted for. The energy functional for three-dimensional ͑3D͒ superconductors may be expressed as 7where (r) is the phase of the order parameter on a cubic lattice with spacing aЈϾ, and where the magnetic flux threading the plaquette at site r perpendicular to the ϭx,y,z direction reads ⌽ (r)ϭ(⌽ 0 /2) ͚ ,␥ ⑀ ␥ ⌬ A ␥ . Also, ⌬ (r)ϭ(rϩ )Ϫ(r) is the lattice difference operator. A ''frozen'' superconductor ͑FZS͒ is simply the thermodynamic state in the limit of infinite phase rigidity, s →ϱ. This means that the London penetration length is small in comparison to the effective lattice constant, aЈ, and hence that flux lines do not interact. In particular, upon making the gauge transformation A ϭA Ј ϩ⌬ , we immediately see that this limit implies A Ј ϭ2m , where m (r) is an integer field. Therefore, the energy functional ͑1͒ reduces to E FZ ϭ͚ r, (2e 0 ) Ϫ2 [2n (r)] 2 , where n (r) is the integer field describing unit line segments of quantized flux traversing the plaquette ͑r,͒ and satisfying the continuity equationi.e., n ϭ...

Critical behavior of layered superconducting films in a parallel magnetic field

“…Parallel fluctuations of the string give rise to an effective Josephson penetration length, λ J (T ) > Λ 0 , as well as to entropic pressure. 14,15,24 The latter is responsible for both the anomalous H −1 tail shown by the parallel magnetization (11) and for the parallel collective pinning effect discussed above. Although HTSC films that are equivalent to N Josephson-coupled layers (1) already exist, their physical properties have been examined only at temperatures far from the critical region.…”

“…This agrees with results based on the Coulomb gas analogy 14 for the LD model (1), as well as with those based on a model for "frozen" layered superconductors in the Meissner phase. 24 After making the replacements (B9) in the coherence factors above, which is valid for excitations near the Fermi surface, we recover the original form…”

“…This implies an inflection point in its temperature profile below T * . 14,24 The presence of a parallel magnetic field, however, will generally induce the phase θ ′ l (x) of the CDW order parameter, χ l = (−1) l χ 0 exp(iθ ′ l ), to wind. This can be seen explicitly from the Ginzburg-Landau (GL) equations that describe the present mean-field theory at "temperatures"…”

“…This means that ohmic flux flow sets in at temperatures above or equal to T dp by Eq. (24). Given the temperature dependence of the parallel field scale B * ∼ H 1 (see Fig.…”

Fermion analogy for layered superconducting films in parallel magnetic field

“…In particular, the ratio of the parallel lower-critical field to the bare one set by the LD functional is equal to γ * a/λ J ∼ α/aγ * , which is small if α ∼ a. This effect is a result of vortex wandering [9], [20].…”