Critical properties of Bose-glass superconductors

– We study vortex lines in high-temperature superconductors with columnar defects produced by heavy ion irradiation. We reconsider scaling theory for the Bose glass transition with tilted magnetic fields, and propose, e.g., a new scaling form for the shape of the Bose glass phase boundary, which is relevant for experiments. We also consider Monte Carlo simulations for a vortex model with a screened interaction. Critical exponents are determined from scaling analysis of Monte Carlo data for current-voltage characteristics and other quantities. The dynamic critical exponent is found to be z = 4.6± 0.3. A rich variety of new vortex phase transitions has been established in high-temperature superconductors [1]. Bose glass physics was originally suggested, by Nelson and Vinokur [2], to apply for vortex phase transitions in systems with artificially introduced columnar defects, produced as permanent damage tracks from heavy ion irradiation of the sample. Columnar defects act as optimal pinning centers, leading to a considerable increase of critical currents and fields, as compared to the unirradiated sample [3, 2, 4]. Bose glass theory makes a set of distinct predictions, e.g., for the dynamic scaling of vortex transport properties, and the response to tilted magnetic fields [2, 5, 6]. Such behavior has been observed in various experiments [7, 8, 9, 10, 11, 12]. Vortex lines in 3D superconductors with columnar defects can be mapped to imaginary time world lines of bosons in 2+1 dimensions on a disordered substrate [2]. The superconducting glass phase for the vortex lines corresponds to the insulating Bose glass (BG) phase for the bosons. The dissipative vortex line liquid phase corresponds to the superconducting phase for bosons. This mapping gives useful information about the equilibrium properties in the columnar defect problem [13]. However, dynamical properties do not follow from the mapping. A tilted magnetic fieldH⊥ with respect to the columns enters like an imaginary vector potential for the bosons [2], leading to a localization problem in non-Hermitian quantum mechanics, which has received considerable attention recently [14, 15]. The vortices want to stay localized on the columns, which results in a transverse Meissner effect with a divergent tilt modulus c44. The BG phase boundary in the (T,H⊥)-plane has a sharp cusp at H⊥ = 0, which distinguishes Typeset using EURO-TEX 2 EUROPHYSICS LETTERS the Bose glass from the isotropic point-defect vortex glass which has a smooth phase boundary [2]. Such a feature in the transition line is seen in experiments [8, 9, 12], and provides support for the Bose glass theory. Other types of correlated disorder, like splayed columnar defects [16], and planar defects [2], also strongly influence the properties of the glass phase. In this paper we reconsider scaling theory for the Bose glass transition for the case of tilted magnetic fields, which leads to certain important modifications of predictions in earlier work [2], e.g., for the form of the BG phase boundary. We further present Monte Carlo (MC) simulations for various thermodynamic and dynamic quantities, which verifies the scaling predictions derived below, and give an estimate of the dynamical critical exponent z. We first discuss scaling theory. A detailed scaling theory has been developed for the superconducting phase transition by Fisher et al. [17], and generalized to anisotropic Bose glass scaling in Refs. [2, 5, 6]. In the present problem it is crucial to correctly distinguish between the scaling of the B and H fields. These relations have sometimes been mixed up in the literature. On approaching the transition the correlation lengths in the directions perpendicular and parallel to the columnar defects are assumed to diverge as ξ ≡ ξ⊥ ∼ |T −TBG| −ν and ξ‖ ∼ ξ ζ , respectively, where ζ is an anisotropy exponent. Due to screening of the interaction and the correlated disorder, the correlation volume is anisotropic with ζ = 2 [13]. In case of unscreened long-range interactions we expect instead ζ ≈ 1. The correlation time diverges as τ ∼ ξ where z is the dynamical critical exponent. The vector potential enters in the combination ∇− (2πi/Φ0)A, and therefore scales as A⊥ ∼ ξ , A‖ ∼ ξ −1 ‖ . From hyperscaling the free energy density scales as f ∼ ξ , where d is the dimension, and therefore the current density scales as J⊥ = ∂f ∂A⊥ ∼ ξ, J‖ = ∂f ∂A‖ ∼ ξ, and the electric field as E⊥ = − 1 c ∂A⊥ ∂t ∼ ξ , E‖ = − 1 c ∂A‖ ∂t ∼ ξ . The scaling of the magnetic field is obtained from B = −4π ∂f ∂H (or from ∇×H = 4π c J), which gives H⊥ ∼ ξ , H‖ ∼ ξ 3−d−ζ , and the flux density scales as B⊥ = (∇ × A)⊥ ∼ ξ −1−ζ , B‖ = (∇ × A)‖ ∼ ξ . The appropriate scaling combinations involving the magnetic field are therefore H⊥ξ d−2 and H‖ξ , which differs from those found, e.g., in Ref. [2]. This has several experimentally relevant consequences as we will discuss below. The linear resistivity scales as ρ⊥ = ξ ρ̃±(H⊥ξ ) (1) ρ‖ = ξ ρ̃ ‖ ±(H⊥ξ ), (2) and the I-V characteristic as E⊥ = ξ Ẽ ±(J⊥ξ , H⊥ξ ) (3) E‖ = ξ Ẽ ‖ ±(J‖ξ , H⊥ξ ), (4) where ρ̃ and Ẽ are scaling functions. For the magnetic permeability we obtain μ⊥ = ξ d−3−ζ μ̃⊥(H⊥ξ ) (5) μ‖ = ξ d−5+ζ μ̃‖(H⊥ξ ). (6) As an independent check of the correctness of this scaling law we can again use the mapping to dirty bosons in (2+1)D. The permeability μ⊥ corresponds to the superfluid density ρs for the bosons, whose scaling was derived in Ref. [13] with the result ρs ∼ ξ 3−d−ζ . This scaling law coincides with Eq. (5) for d = 3 [18]. Equation (6) corresponds to the compressibility of the bosons κ ∼ ξ, again with agreement for d = 3. To obtain the scaling of the BG phase boundary for small tilt, Eq. (5) gives μ⊥ = t μ̃⊥(H⊥t ), where t = |T − TBG|, and following Ref. [2] we assume that the Bose glass phase persists up to a finite tilt, so μ̃⊥(x) J. LIDMAR and M. WALLIN: CRITICAL PROPERTIES OF BOSE-GLASS SUPERCONDUCTORS 3 has a singularity at a finite value x = xc. At this point the critical tilt field vanishes when T → TBG as TBG − T ∼ |H⊥| 1/ν , (7) for d = 3. Equations (5-7) differ from the corresponding ones in Ref. [2], and in particular our Eq. (7) does not include a factor 3 in the exponent. The shape of the phase boundary still has a cusp at H⊥ = 0, but not as sharp as the earlier predictions suggested. Another consequence of experimental interest is the scaling relation ρ⊥ ∼ H z−ζ ⊥ for the resistivity at T = TBG. We now turn to our Monte Carlo simulations. We concentrate on low magnetic fields, B . Φ0/λ , where the vortex interaction is effectively short ranged, and also restrict our study to fields below the matching field, where the density of columnar defects is greater than the vortex density. Vortex dynamics has been studied previously in simulations of the nonlinear current-voltage (I-V) characteristics [5] for finite applied supercurrents. In this paper we present extensive simulation results for the linear resistance, and compare with dynamical scaling of the I-V characteristics. We choose to study the dirty Boson action with an onsite repulsion [13], which gives a coarse-grained representation of the physical system,