Constraints on Topological Defect Formation in First-order Superconducting Phase Transitions J. P´aramos Instituto Superior T´ecnico, Departamento de F´ısica, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail address: x [email protected] In this work we address the impact of a cubic term addition to the Ginzburg-Landau mean- field potential, and study the consequences on the description of first order phase transitions in superconductors. Constraints are obtained from experiment and used to assess consequences on topological defect creation. No fundamental changes in either the Kibble-Zurek or Hindmarsh- Rajantie predictions are found. I. INTRODUCTION which are known as flux tubes or vortices, lines of non- null magnetic field trapped inside the superconductor. Experiments targeted at observing defect densities in The following work is based on the research reported in high-T materials [10] were not in accordance with the Ref. [1]. It pursues the objective of empathizing the anal- c density predictions of the Kibble-Zurek (K-Z) mecha- ogy between accessible condensed matter systems and the nism [2]. This, however, is to be expected, since the currently accepted framework for the evolution of the K-Z prediction should be accurate only for global gauge Universe, notably the symmetry-breaking phase transi- symmetry breaking, when the geodesic rule for phase an- tions [2, 3] it has undergone after the Big-Bang. A key as- gle summation is valid. A new defect generation mech- pect to this comparison is the creation of topological de- anism, based on a local gauge treatment by Hindmarsh fects, frustrations of the unbroken phase within the bro- and Rajantie (H-R) [11], leads to an (additive) predic- ken one, arising from the continuity of the order param- tion. This, although well below the first Carmi-Polturak eter values. These are generally categorized according experimental sensitivity, is in reasonable agreement with to the homotopy group of the quocient of the unbroken the second. symmetry groups to the broken one and which enables for comparison of different physical phenomena. These As a starting point for this research, we note that the objects can appear as magnetic monopoles or point-like above experiments were both conducted in type-II ma- defects, cosmic strings, vortices or flux tubes, magnetic terials, which exhibit a higher critical temperature and domain walls or textures. Besides its mere aesthetical are therefore easier to manipulate, leading the current value, this analogy can provide a powerful probe into the trend in experimental superconductivity. These materi- early stages of the evolution of the Universe, since direct, als display a second order phase transition, with no re- hands-on experimental tests are unattainable: the exis- lease of latent heat. On the other hand, Type-I materials tence of more accessible systems that exhibit a formally are metastable, showing different responses to a magnetic similar behavior could provide crucial clues to many cos- field when in normal-superconductor or superconductor- mologically relevant issues. normal phase transitions. In this work, we try to account These “cosmology in the laboratory” experiments can for this more elaborate behavior and to estimate to which arXiv:hep-ph/0403011v2 27 Apr 2004 be found in various systems, ranging from vortices in extent it affects the defect density predictions for type-I superfluid phase transitions of 4He and 3He (see e.g. superconductors. Ref. [4, 5]), which exhibit common features with cos- Type-I and type-II superconductors are commonly dis- mic strings [6], to liquid crystals undergoing an isotropic- tinguished according to their Ginzburg-Landau (G-L) pa- nematic phase transition [7, 8]. Polymer chains were rameter κ = λ/ξ, the ratio between the magnetic field shown to also possess analogous thermodynamic and penetration length λ and the coherence length ξ of the transitional behavior [9]. However, most of these sys- order parameter. In the presence of a gauge field A~, these tems lack the existence of a quantity analogous to the characteristic length scales are obtained from the free en- magnetic field, which could be a key player in the early ergy density evolution of the Universe and formation of structure. For that reason, superconductors are a case of special in- terest. These comprise phase transitions involving a lo- 1 e 2 1 cal gauge symmetry-breaking process, during which the F (Φ) = i¯h~ Φ A~Φ + V (Φ) + ~µ (~ A~) , photon acquires a “mass” and, therefore, a penetration 2me ∇ − c 2 · ∇× length, giving rise to the Meissner effect: the expulsion of (1) the magnetic field from a superconducting material, with where ~µ is the sample’s magnetic moment, me is the elec- formation of shielding “supercurrents” on its surface. tron mass and Φ is the order parameter. The G-L po- This symmetry breaking originates topological defects tential is usually written as [6] 667 In the normal-to-superconductor phase transition, a term similar to γ Φ 3 also arises if one takes into ac- count gauge field− fluctuations| | [13, 14], producing e γ =8µ √πµ T . (6) 0 ¯hc 0 c This result enables a first order phase transition for all values of κ. Thermal fluctuations [14] and non local BCS effects [15] describe crossover behavior between first and second order transitions. FIG. 1: Characteristic Potential curves. In the following, we adopt a potential of the form of Eq. (5) and constrain γ(T ) based on experimental data. The results are then compared with both TFT 1-loop radia- tive corrections and the the results of Ref. [13] (valid only β at temperature close to T , that is, t 1). The intro- V (Φ) = αΦ2 + Φ4 , (2) c → 2 duction of γ(T ) in the potential (2) can produce changes in both the K-Z and H-R defect generating mechanisms. where α is assumed to be linear with temperature, α = Also, a possible nucleation suppression due to the po- α′(t 1), t T/T , α′ and β are constants, and T is the − ≡ c c tential barrier can significantly reduce the number of ob- critical temperature. One obtains served defects. The obtained constraints on γ(T ) are used to access the impact on these claims. m c2 β λ = e , (3) 4πe2 α II. TEMPERATURE SENSITIVITY BOUND s | | and A superconductor undergoing a first-order phase tran- sition crosses different supercritical fields, displaying a metastable behavior, as shown in the phase diagram ξ =¯h/ 2me α . (4) | | of Figure 1. The superheating curve is given by the 2 p dV d V The coherence length at zero temperature is ξ0 = condition = 2 = 0, for Φ = 0, equivalent to ′ dΦ dΦ 6 ¯h/√2meα , with κ √β. The transition is second order α =9γ2/16β. The supercooling curve is given by the con- ∼ 2 if κ > 1/√2, and ξ0 is typically less than 0.04 µm; d V (Φ) ∼ dition dΦ2 = 0 for Φ = 0, correspondingto α = 0. The for κ < 1/√2, the transition is first order and ξ typi- 0 (unobservable) critical curve is given by V (0) = V (Φc) cally greater than 0.08 µm. It is commonly accepted dV (Φc) and = 0, where Φ is the non-vanishing minimum that, if there is no∼ applied magnetic field, one always dΦ c of the potential. This corresponds to α = γ2/2β. has a second-order phase transition, for all values of κ. ′ First order transitions arise from the external field term Assuming α = α (t 1) and γ(t) = δ t, we obtain for the superheating curve− in Eq. (1) if the sample has a characteristic dimension l > λ. This degeneracy of the phase transition at H =0 2 is, however, only verifiable to the current experimental ′ 9 δ α (t 1) = t2 , (7) sensitivity, and it can be argued that there is some yet − 16 β undetected intrinsic metastability, regardless of the ap- plied magnetic field. and Bearing in mind the analogy between condensed mat- ter and cosmology, we now briefly look at phase tran- 2 sitions in high energy physics. In thermal field theory 2 9 δ tsh = 1+ ′ . (8) 9 δ2 ∼ 16 α β (TFT) a first order phase transition arises due to 1-loop 1+ 1 ′ radiative corrections to a potential similar to that of Eq. − 4 α β q (2); a barrier between minima of the potential is created, Due to the presence of the cubic term in the potential of as these corrections give rise to a cubic scalar field term Eq. (5), the superheating curve shows a zero-field shift 2 ′ in temperature from Tc by (9δ /16α β)Tc. This shift, β if detected, would indicate an intrinsic metastability, in V (Φ) = αΦ2 γ Φ 3 + Φ4 , (5) − | | 2 the sense that it does not depend on the existence of an applied field. Since such temperature shift has not where γ(T ) = (√2/4π)e3T [12]. As before, β is assumed yet been signaled, the current experimental temperature ′ −3 to be constant and α = α (t 1) to depend linearly with sensitivity being ∆texp 10 [16], a bound on the slope temperature. − of γ is ∼ 668 TABLE I: Critical properties of Sn and Al TABLE II: Derived quantities and bounds for δ Material Tc (K) Hc(0) (G) ξ0 (µm) λ (nm) Material Sn Al ′ −25 −27 Sn 3.7 309 0.23 34 α (J) 1.15 10 2.38 10 3 × −54 × −56 Al 1.2 105 1.6 16 β (J.m ) 4.72 10 2.16 10 ′ × − × − α (eV 2) 3.61 10 1 7.45 10 3 × − × − β 9.45 10 4 4.32 10 6 × × 9 δ2 bound tsh shift δ(eV ) tsh shift δ(eV ) ′ < ∆texp .