CRITICAL CURRENTS IN QUASIPERIODIC PINNING ARRAYS: ONE-DIMENSIONAL CHAINS AND PENROSE LATTICES
Foundations of Quantum Mechanics in the Light of New Technology
We have studied the critical depinning current J c versus the applied magnetic flux Φ, for quasiperiodic (QP) one-dimensional (1D) chains and 2D arrays of pinning centers placed on the nodes of a five-fold Penrose lattice. In 1D QP chains, the peaks in J c (Φ) are determined by a sequence of harmonics of the long and short segments of the chain. The critical current J c (Φ) has a remarkable selfsimilarity. In 2D QP pinning arrays, we predict analytically and numerically the main features of J c
... ain features of J c (Φ), and demonstrate that the Penrose lattice of pinning sites (which has many built-in periods) provides an enormous enhancement of J c (Φ), even compared to triangular and random pinning site arrays. This huge increase in J c (Φ) could be useful for applications. Model We perform simulated annealing simulations of • The force due to the vortex-vortex interaction is • The pinning force is N p is the number of pinning sites, f p (expressed in f 0 ) is the maximum pinning force of each short-range parabolic potential well located at r k (p) , r p is the range of the pinning potential, Θ is the Heaviside step function. • All the lengths (fields) are expressed in units of λ (Φ 0 /λ 2 ). • In the equation of motion, f i T is the thermal stochastic force, and f i d is the driving force; η is the viscosity. 1D Quasicrystal 2D Quasicrystal: Penrose lattice Five-fold Penrose lattice Stability of the maxima of J c Strong enhancement of the critical current J c in a Penrose-lattice array of pinning sites Critical current J c in a Penrose-lattice array of pinning sites J c for increasing sample size Self-similarity of J c in real space Self-similarity of J c in k-space Critical current J c for 1D QP chains Quasiperiodic (Penrose) lattice provides an unusually broad critical current J c (Φ), that could be useful for practical applications demanding high J c 's over a wide range of fields A Penrose lattice is a 2D quasiperiodic (QP) structure, or 2D quasicrystal. Construction of a 1D QP chain (1D quasicrystal): Iteratively apply the Fibonacci rule L LS, S L LSLLSLSLLSLLSLSLLSLS ... For an infinite QP sequence, the ratio of the numbers of L to S elements is the golden mean τ = (1 + √5)/2). The nth point where a new segment (e.g., for S = 1 and L = τ) begins is: where [x] denotes the integer part of x. This sequence exhibits self-similarity and has a hierarchy of built-in periods. • Has a local rotational (five-or ten-fold) symmetry, but does not have translational longrange order. • Is constructed using certain simple shapes combined according to specific local rules, and can extend to infinity without any defects. • Self-similar diffraction patterns of a Penrose lattice exhibit a dense set of "Bragg" peaks. The dimensionless difference of the pinning and elastic energies is Here E pin = U pin β n pin ; E el = C 11 [(a eq -b) / a eq ] 2 ; n pin is the density of pinning centers, β (H ≤ H 1 ) = H/H 1 = B/(Φ 0 n pin ), and β (H > H 1 ) = 1 is the fraction of occupied pinning sites, a eq = [2/(3 1/2 ) β n pin ] 1/2 and b are the inter-vortex distances in triangular and distorted lattices (b = a/τ for H = H 1 and b = a for H = H v/t ), and C 11 = B 2 / [4π(1 + λ 2 k 2 )] is the compressibility modulus for local deformations. Near matching fields, J c has a peak only if f diff > 0. References : V.R. Misko, S. Savel'ev, and F. Nori, Phys. Rev. Lett. 95, 177007 (2005); Phys. Rev. B 74, 024522 (2006); cond-mat/0502480.