科斯特利茨-索利斯相變之研究:二維XY模型之數值模擬及四奈米鋁薄膜之二維超導

Abstract

Kosterlitz–Thouless transition (KT transition), which is unique for not involving any spontaneous continuous symmetry breaking (commonly used as an indication for phase transition) when a two-dimensional (2D) system transits from disordered to ordered state, is the main subject of this thesis. We study it by two different approaches: one is a numerical simulation of the XY model on a 2D spin lattice with a ferromagnetic Hamiltonian, considering only nearest neighbors coupling; the other is a series of measurements of electrical transport properties conducted on a superconducting aluminum nano-film, fabricated on a GaAs substrate. Both systems are predicted to show KT transition, which is related to the excitation of topological charges (vortices and anti-vortices). The simulation is aimed at verifying whether there is a phase transition on a finite 2D spin lattice, and if there is, does it spontaneously break the continuous symmetry. We discover that the average of magnetization and the average of its norm can be nonzero when temperature is below 0.9, in unit of the bared coupling strength J, which indicates the O(2) continuous symmetry is broken by spin alignment. The temperature range where the ordering vanishes is 0.9—1.0. The susceptibility shows a strong peak inside this range. The phase transition is identified as KT transition by checking the functional form of the correlation function at low/high temperature (below 0.95/above 1.025), which shows power law (hence the correlation length is infinite)/exponential decay as a function of two points separation r. The transition temperature TKT=0.903 and TKT=0.896 is further extracted from T—η(T) and T—Jeff(T) plot respectively, where η(T) is the exponent of the power law decay correlation function and Jeff(T) is the modified coupling strength due to the mutual screening effect between vortex-anti-vortex pairs. The measurements performed on the aluminum nano-film is for detection of the electrical transport properties induced by bounded vortex-anti-vortex pair and free vortex (anti-vortex) excitation. Thus they can serve as an indirectly evidence of the existence of KT phase transition in our device, if a shifting of transport properties from vortex-anti-vortex pair to free topological charge excitation is observed. For temperature below TKT, I—V curve follows a power law and the exponent equals 1+1/(2η), where η(TKT)=0.25. By extracting the slope from log-log scale I—V curves, we determined that TKT =2.17 K, although a universal jump at TKT such that 1/(2η) drops from 2 to 0, which characterized the KT transition on an infinite sample, is not seen. The reason is attributed to the finite sample size. For temperature above TKT, the resistance is inversely proportional to the square of correlation length. By fitting the T—R curves with source drain currents Isd=10 μA, 100 μA, and 140 μA, the transition temperature TKT is found to decreases from 2.203 K, 2.174 K to 2.161 K. The Joule heating is inferred to cause this current dependence. A universal scaling function is utilized for further confirmation. With Isd=100—150 μA, when all the scaling function curves calculating from I—V curves at different temperatures collapse into a single one, the corresponding TKT is about 2.17 K. In addition to the TKT thus determined, the dynamical critical exponent z is also confirmed to be 2, suggesting the vortex dynamic on which our transport properties prediction based is correct.