Peak-effect and surface crystal-glass transition for surface-pinned vortex array

The peak effect has been investigated in clean Nb crystals with artificially corrugated surfaces by measuring the linear surface impedance in the 1kHz1MHz frequency range. From a two-mode analysis of the complex spectra, we establish that vortex dynamics is governed by surface pinning and deduce the associated vortex slippage length. We demonstrate experimentally and theoretically that the peak effect is related to a transition from collective to individual surface pinning. A proper account of the peak-effect anomalies implies softening of the shear rigidity by disorder-induced lattice deformations. This leads to a vortex crystal-glass transition induced by surface defects. PACS numbers: 74.60.Ge Typeset using REVTEX 1 The peak effect (PE) is a well known anomaly of the mixed state due to vortex lattice (VL) softening in the presence of random defects [1,2]. It is observed at the upper critical field Bc2 of clean superconductors [3,4], well below Bc2 in disordered oxide superconductors (second peak), or at the vortex-melting line in very clean YBa2Cu3O7 crystals [5]. The physics of PE is the order-disorder transition, which is of interest for a broad range of elastic media. Recent developments concern vortices in conventional materials (Nb, NbSe2), where PE is investigated in great detail [6–8]. Originally believed to be a crossover from collective to individual vortex pinning according to the Larkin-Ovchinnikov (LO) scenario [2], PE is now considered as a genuine transition [6,8], with a quasi-long-range order (Bragg-Glass) in the low-field phase [9,10]. However, strong limitations for a quantitative analysis of the phenomenon arise from the poor knowledge about the random defects responsible for PE. In this work we consider a new situation where this difficulty can be overcome : the peak effect due to the surface pinning of vortices. The pinning landscape is provided by the surface corrugation ζ(r) of a slab in perpendicular or tilted magnetic field; a controlled roughness ζ∗ ∼ 1-10nm is produced by ion beam etching (IBE) and measured by atomicforce microscopy (AFM). Far from the upper critical field Bc2 the intervortex interaction is strong and suppresses the surface pinning. This means that the surface pinning is collective similar to the collective pinning in the bulk [2]. But close to Bc2 the VL distortion by surface pinning become essential, and near the surface the ordered VL transforms to the disordered vortex array, which we call surface vortex glass. Formation of the surface vortex glass is accompanied by the crossover from collective to individual pinning. This results in the growth of the critical current, i.e. in PE. Experiments are performed on a series of millimeter-thick slabs of very pure Nb. Starting from electro-chemically polished samples, a controlled corrugation is introduced by ionbeam etching (IBE) the surfaces with a 1.5mA/cm flux of 500eV-Ar ions for 10–100 min. Given the sputter yield, 0.6 Nb atom per Ar, and the atomic spacing c = 0.26 nm, the average etching rate is of the order of 100 nm/min. From this stochastic process, one expects a roughness amplitude ζ∗ ≃ (c∆z) and a white spectrum Sζ(k) = ∫ dr e−ikr〈ζ(r+ 2 R)ζ(R)〉R = c∆z/π in the range 0− c of wave number k. In practice, the wave number cut-off kc < 1/c due crystal rearrangement associated with local heating; whence a reduction of ζ∗ by a factor kcc∼10. This picture is confirmed by AFM-inspection, which shows a flat spectrum below 1μm and a power law dependence Sζ(k)∼ 1/k–1/k above. The etched surfaces have a broad-band corrugation, in the range k = 1–100μm covering the vortex reciprocal-lattice unit Q = a 0 ≃ (50nm), which is very effective for VL-pinning. Here a0 = √ φ0/πB is the radius of the vortex Wigner-Seitz cell (on the order of the intervortex spacing) and φ0 = h/2e is the flux quantum. The pinning strength is deduced form the AC response. The AC surface impedance Z(ω) = −iμ0ωλAC is determined by the effective penetration depth given by 1 λAC = 1 LS + ( 1 λ2C + iωμ0σf )1/2 ; LS = lSB μ0ε . (1) Here σf is the flux-flow resistivity, εφ0 is the vortex-line tension (energy per unit length), and λC the Campbell penetration depth for bulk pinning. The surface pinning is taken into account by the length LS ∼ 0.1–100μm, which can simulate the effect of bulk pinning characterized by the Campbell depth λC . This expression was derived within the frame of the two-mode electrodynamics [11,12], which incorporates the surface pinning by introducing a phenomenological boundary condition,