Vortex dynamics equation in type-II superconductors in a temperature gradient

443 Brazilian Journal of Physics, vol. 40, no. 4, December, 2010 Vortex Dynamics Equation in Type-II Superconductors in a Temperature Gradient R. Vega Monroy∗ and J. Sarmiento Castillo Facultad de Ciencias Básicas. Universidad del Atlántico Km. 7, Via a Pto. Colombia, Barranquilla, Colombia D. Puerta Torres Facultad de Ciencias Exactas. Universidad de Cartagena Plaza de la Artillerı́a N. 30-84, Cartagena, Colombia (Received on 27 August, 2010) In this work we determined a vortex dynamics equation in a temperature gradient in the frame of the time dependent Ginzburg-Landau equation. In this sense, we derived a local solvability condition, which governs the vortex dynamics. Also, we calculated the explicit form for the force coefficients, which are the keys for the understanding of the balance equation due to vortex interactions with the environment. Keywords: Type-II Superconductors; Vortex Equation; Vortex Balance Equation. 1. INTRODUCTION In type-II superconductors, the vortices are the ones in charge for the magnetic properties of these systems since every vortex carries a magnetic flux quantum. In the last years the interest in the vortex motion is associated to many non peculiar properties in HTSC not found in conventional typeII superconductors. In particular, one of the most important effects encountered in HTSC is the Hall anomaly [1,2]. In this sense, it is known that, under the action of the Lorentzs force vortices acquire a movement and therefore losses appear in the superconducting state. The equation of motion, which governs the vortex dynamics in type-II superconductors, has been subject to a great amount of works, which have helped us to understand this phenomenon in these systems. In addition, the interest in the vortex motion is emphasized by the responsibility of this dynamics in a great variety of transport phenomena in type II superconductors. Generally, the vortex dynamics has been considered on the hydrodynamical two fluid model [3,4], where the relative motion between the superfluid and the vortex generates the Magnus force. Next, the normal component reacts to this motion, producing the longitudinal viscous drag force and the transversal Iordanski’s force, which are the two components of the medium force. An attempt to describe the vortex dynamics was done by Dorsey [5] in the frame of the time dependent Ginzburg-Landau equation following previous developments of Gorkov and Kopnin [6]. In this work, Dorsey formulates a solubility condition through with a vortex equation can be obtained. Today, there are several attempts to construct a unified theory about the vortex motion. Some approximations use the sophisticated many body formalism [7,8,9]. Other authors apply a simpler theory based on the kinetic Boltzmann equation to study the dynamic behavior of the vortex structures [10], but so far the vortex dynamics is an open question for the solid state physics community. The purpose of the present work is to contribute to a better understanding of the fascinating phenomenon of the vortex motion. In this connection, the goal of this paper is to determine a vortex dynamics equation in a temperature gradient in the frame of the time dependent Ginzburg-Landau equation. Such kind ∗ Electronic address: of equation has been introduced in a heuristic way in many works [11] to satisfy experimental data [12,13,14]. The paper is organized as follows: In Section 2, we have obtained the basics equations of the work. In Section 3, some dynamic coefficients are calculated and finally in Section 4 we summarized the main results. 2. BASIC EQUATIONS The present analysis follows the works done by Dorsey [5] in order to obtain the equation, which describes the vortex dynamics. Let us write the dimensionless time dependent Ginzburg-Landau equation for the complex order parameter in the form: " #2   ~∇ ∂ γ + iφ ψ = − i~A ψ + ψ − |ψ|2 ψ. (1) ∂t κ In the above equation γ is the dimensionless relaxation time, κ is the Ginzburg-Landau parameter, Φ and ~A are the electric potential and the magnetic vector respectively. The order parameter ψ in terms of the amplitude f (~r,t) and the fase χ(~r,t) can be represented as follows ψ(~r,t) = f (~r,t) exp[iχ(~r,t)]. The relaxation time has a complex character and can be written γ = γ1 + iγ2 . The appearance of the imaginary part in the previous expression is a necessary condition for the gauge invariance conservation of equation (1) [10]. Relaxation processes that entail to dispersion in the vortex dynamics are of two types: the first is associated to the Bardeen- Stephens mechanism of dissipation and the second process is an intrinsic relaxation mechanism that governs the approach of the order parameter to its equilibrium state due to variations in the chemical potential. This mechanism is associated to the change of the order parameter in time because the electrons are forced to pair and de-pair with respect to different potentials due to the vortex motion. In consequence, this processe is the responsible that the relaxation time acquires a complex character as was shown by Kopnin [10] and it is framed in the parameter γ2 γ2 = − ~ ∂ν , 2λ ∂µ 444 R. Vega Monroy et al. where ν and µ are the density of states and the chemical potential respectively. Introducing ψ and γ in the expression (1) and in addition, introducing the invariant forms for the magnetic vector and the scalar potential ~ = ~A − 1 ~∇χ, P = Φ + ∂χ , Q κ ∂t (2) one can obtain two equations, the first for the real part and the other one for the imaginary part as follows: γ1 ∂f 1 − γ2 f P = 2 ~∇2 f + Q2 f − f − f 3 , ∂t κ γ1 f P − γ2 ∂f 2~ ~ f ~ + Q.∇ f + ~∇.Q = 0. ∂t κ κ (3) (4) To form a closed system of equations, we need to derive an equation for the magnetic vector. In this connection, the dimensionless equations for the superconducting and normal current are: 1 ~ (ψ ∗ ~∇ψ − ψ~∇ψ∗) − |ψ|2 ~A = − f 2 Q, J~s = 2κi ~ 1 ∂Q 1 J~n = σ (− ~∇P − ) + b(n)~∇T, κ ∂t κ (n) where f0 and f1 are the amplitudes of the order parameter associated to the equilibrium and non equilibrium states respectively. In addition, in the vortex motion it is possible to consider that vortices move independently in the first approximation of the limit B  Hc2, so that one can find an equation of motion for every individual vortex in presence of a temperature gradient. In this limit, assuming that the ~ and P are funcvortices move uniformly, the quantities f , Q tions solely of~r −~vLt, where~r is the electron position vector and ~vL is the vortex velocity. Thus, the temporary derivative can be written in terms of spatial derivative by means of ∂ vL · ~∇, so if we introduce this relation in (3), (7) and ∂t = −~ (9), we obtain the following set of equations for the equilibrium state: (5) (6) where σ(n) and b(n) are the electric and thermoelectric conductivities in the normal state respectively. In this sense, the dimensionless equation for the magnetic vector is: ~ ~∇ × (...truncated)