The influence of the Hall force on the vortex dynamics in type II superconductors

The effect of the Hall force on the pinning of vortices in type II superconductors is considered. A field theoretic formulation of the pinning problem allows a non-perturbative treatment of the influence of quenched disorder. A selfconsistent theory is constructed using the diagrammatic functional method for the effective action, and an expression for the pinning force for independent vortices as well as vortex lattices is obtained. We find that the pinning force for a single vortex is suppressed by the Hall force at low temperatures while it is increased at high temperatures. The effect of the Hall force is more pronounced on a single vortex than on a vortex lattice. The results of the self-consistent theory are shown to be in good agreement with numerical simulations. PACS numbers: 74.60.Ge, 05.40+j, 03.65.Db Typeset using REVTEX 1 The advent of high temperature superconductors has led to a renewed interest in vortex dynamics. We shall consider the influence of quenched disorder on the vortex dynamics in type II superconductors in the presence of a Hall force. The description of the vortex dynamics will be based on the phenomenological Langevin equation müRt + ηu̇Rt + ∑ R ΦRR′uR′t = αu̇Rt × n̂−∇V (R+ uRt) + FRt + ξRt, (1) where uRt is the displacement at time t of the vortex which initially has equilibrium position R, η is the friction coefficient, and m is a possible mass (per unit length) of the vortex. The dynamic matrix, ΦRR′, of the hexagonal Abrikosov vortex lattice describes the interaction between the vortices in the harmonic approximation. Having a thin superconducting film in mind the system is two-dimensional (normal to n̂) and the dynamic matrix is specified within the continuum theory of elastic media by the compression modulus, c11, and the shear modulus, c66, Φq = φ0 B ( c11q 2 x + c66q 2 y (c11 − c66)qxqy (c11 − c66)qxqy c66q x + c11q y ) , (2) where φ0/B is equal to the area of the unit cell of the vortex lattice, and φ0 = h/2e is the flux quantum. The force (per unit length) on the right hand side of eq. (1) consists of the Hall force characterized by the parameter α, and FRt = φ0 j(R, t)× n̂ is the Lorentz force due to the transport current density j, and the thermal white noise stochastic force, ξRt, is specified according to the fluctuation-dissipation theorem 〈ξα Rtξβ R′t′〉 = 2ηkBTδ(t− t′)δαβδRR′, and V is the pinning potential due to quenched disorder. The pinning is described by a Gaussian distributed stochastic potential with zero mean, and thus characterized by its correlation function (where now the brackets denote averaging with respect to the quenched disorder) 〈V (x)V (x′)〉 = ν(x−x′) = ν0/(4πa) exp(−|x−x′|2/(4a2)), taken to be a Gaussian function with range a and strength ν0. Upon averaging with respect to the quenched disorder the average restoring force, FR = −∑R′ ΦRR′〈〈uR′t〉〉, of the lattice vanishes. On the average, corresponding to the lattice reaching a steady state velocity v = 〈〈u̇〉〉, there will be a balance, F+ Ff + FH + Fp = 0, between the Lorentz force, F, the friction force, Ff = −ηv, the Hall force, FH = αv × n̂, and the pinning force, Fp = −〈〈∇V 〉〉. The pinning force is due to time-reversal symmetry invariant under reversal of the direction of the magnetic field, and is therefore antiparallel to the velocity. Thus, the pinning yields a renormalization of the friction coefficient in terms of a velocity dependent effective friction coefficient, Ff + Fp ≡ −ηeff(v)v, which reduces in the absence of disorder to the bare friction coefficient η, and has previously only been determined to lowest order in the disorder. The relationship between the average vortex velocity and the induced electric field, E = v×B, leads to the expressions for the resistivity tensor and Hall angle ρ = φ0B η eff + α 2 (