Peak Effect in Superconductors: Melting of Larkin Domains

Motivated by the recent observations of the peak effect in high-Tc YBCO superconductors, we reexamine the origin of this unusual phenomenon. We show that the mechanism based on the k-dependence (nonlocality) of the vortex-lattice tilt modulus C44(k) cannot account for the essential feature of the peak effect. We propose a scenario in which the peak effect is related to the melting of Larkin domains. In our model, the rise of critical current with increasing temperature is a result of a crossover from the Larkin pinning length to the length scale set by thermally excited free dislocations. 74.60. Ge, 64.70.Dv Typeset using REVTEX 1 About 35 years ago Le Blanc and Little [1] discovered a striking phenomenon, later known as the “peak effect”, in a conventional superconductor Nb that the sample can carry more supercurrent at a higher temperature (or field) slightly below Bc2(T ) where it becomes normal. Over the years, the peak effect was found to be ubiquitous in conventional superconductors [2–4] and it has been observed recently in the high-Tc superconductor YBCO crystals [5,6]. Pippard [7] and Larkin and Ovchinnikov [8] proposed that the peak effect is a result of an anomalous softening of the vortex lattice. The basic physics is that a soft lattice can be pinned more strongly than a more rigid lattice. In fact, an infinitely rigid lattice cannot be pinned at all by random pinning. The unresolved problem, however, is the mechanism which leads to the abrupt loss of the vortex-lattice rigidity. In this paper, we first reexamine the standard interpretations of the peak effect and show that the mechanism of an anomalous softening of the wavevector dependent tilt modulus C44(k) does not account for the essential features of the peak effect. We then propose a scenario in which a melting of the “Larkin domains” leads to the peak effect. The rise of the critical current with increasing temperature is a result of a crossover of two length scales: from the Larkin pinning length to the average separation between thermally excited free dislocations [9]. Let us first recall briefly the general features of the peak effect phenomenon in both conventional superconductors and high-Tc superconducting YBCO single crystals. In samples with high values of the critical current density jc at a fixed magnetic field B, one usually finds that jc decreases to zero monotonically with increasing temperature and the peak effect is absent [2,4–6]. But in samples with low jc (weak pinning), the temperature dependence of jc can be quite different. Fig. 1 is a plot of the critical current density as a function of temperature extracted from Ref. [5] for a YBCO crystal. With increasing temperature, jc initially decreases monotonically, then suddenly rises, reaches a peak before finally dropping to zero. Experimentally, the peak effect is identified as a dip in resistance [2–4,6], a dip in the in-phase part of ac susceptibility [5], or a peak in critical current density jc (obtained with the standard voltage criterion) [2–4,6] as a function of T or B. In Nb and other low-Tc 2 superconductors, the onset (the rise of jc) temperature Tp of the peak effect is very close to Tc(B) and Tc(B) − Tp ∼ 0.5 K. In YBCO, Tc(B) − Tp ∼ 5 K is about 10 times larger. However, (Tc(B) − Tp)/Tc(B) ∼ 0.05–0.1 is about the same for both low-Tc and high-Tc superconductors. The problem under consideration here is why the critical current rises with increasing temperature. Pippard [7] proposed that the rise of the critical current has to result from a rapidly decreasing rigidity of the vortex lattice. The rigidity of the vortex lattice prevents the vortex lines from taking advantage of the valleys of random pinning potential. Thus a rapidly decreasing rigidity would allow the lattice to conform better to the pinning potential and enhance critical current. This idea on pinning was subsequently put forward more rigorously by Larkin and Ovchinnikov (LO) in their theory of collective pinning [8]. It was shown by Larkin [10] that in the presence of random pinning the vortex lattice loses its long-range translational order and breaks up into domains of correlated regions in which the vortex lines interact elastically. LO argued [8] that the critical current density is determined by the fluctuations of random potential in a domain and the pinning force density jcB = (nf /Vc) , where n and f are the density and strength of the pins, Vc the volume of the domain. The size of the Larkin domains can be estimated by a simple energy consideration. The vortex lattice deforms to take advantage of the random pinning potential at the cost of the elastic energy. The total unit volume energy change is [8] δF = C66( rp R ) + C44( rp L ) − frp( n V ), (1) where C66 is the shear modulus of the lattice, rp the range of the pinning potential, R and L are the transverse (to the field) and longitudinal (along the field) dimensions of the domain, and V = RL. The minimization of Eq. (1) gives the pinning lengths Rc and Lc: Rc ∼ C 3/2 66 C 1/2 44 r 2 p/nf , Lc = (C44/C66) Rc. In very thin samples with a perpendicular field, if the pinning is so weak that Lc is greater than the sample thickness, the problem becomes two-dimensional (2D) and only Rc ∼ C66rp/n f is relevant. In the LO theory, since jcB = (nf /Vc) , the peak effect can be accounted for if the volume of Larkin domain 3 Vc drops faster than nf 2 in some field or temperature range. The central question here is what mechanism does that. It was found by Brandt [11] that, near upper critical field Bc2 the vortex fields overlap strongly and the tilt modulus C44 becomes nonlocal: it softens substantially for short wavelength tilt deformation. Most experiments of the peak effect on low-Tc materials are carried out as a function of field while keeping the temperature constant, the peak effect manifests itself as a peak in jc (or a dip in resistance) near Bc2(T ). It is thus natural to relate the C44(k) softening mechanism to the peak effect. LO found [8] that this mechanism leads to an exponential form for Vc when Rc becomes smaller than λ ′ = λ/(1 − b), where λ is the penetration depth and b = B/Bc2, and jc ∼ exp(−BC 3/2 66 khr 2 p/W ), with W = nf 2 and kh = 1/λ . It was customary to assume [8] a scaling function for the field dependence of W , W ∼ b(1− b), and with C66 ∼ b(1− b) 2 [12] and k h ∼ (1− b), one indeed finds that jc rises exponentially with B for b ∼ 1. The above interpretation of the peak effect has two major difficulties. The first is that if this mechanism is the relevant one, it should also account for the temperature dependence: jc rises with increasing T in the peak regime. Giving the most liberal estimate for the temperature dependence of the parameters, however, the above mechanism fails to explain why jc rises with T . The elementary pinning force f is a function of the local gradient of the amplitude of the gap function, or f ∼ a1|∆(r)| . For Tc smearing pins, a1 ∼ (1 − t), |∆(r)| ∼ (1 − t), t = T/Tc, f ∼ (1 − t) , and for pins that do not smear Tc, f ∼ (1 − t). Thus W ∼ (1− t) or (1− t). Without melting, C66 ∼ (1− t) and kh ∼ (1− t) . Thus jc either does not change with T or decreases exponentially with increasing temperature and jc never increases with increasing T , according to this mechanism. The second difficulty of this mechanism is that the peak effect has been observed in thin films [3] and in very thin NbSe2 crystals with pinning weak enough such that Lc exceeds the sample thickness [4,13], in which C44 does not seem to play any role. Therefore we believe that this mechanism cannot be responsible for the peak effect. Another modulus of the vortex lattice which enters the pinning problem is the shear