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

科斯特利茨-索利斯相變(Kosterlitz–Thouless transition,以下簡稱KT相變)是發生於二維系統的特殊相變。當系統由高溫的無序態轉為低溫的有序態時，該相變並不會產生自發性的連續性對稱破缺，所以無法和通常一樣用連續性對稱性破缺來界定相變是否產生及其相變溫度。本篇論文採用兩種方法來探討KT相變:一是用數值模擬來研究一個有限大小的二維自旋晶格之KT相變。該晶格的交互作用可用鐵磁性XY模型來描述(自旋方向被限制在XY平面上)，並只考慮最近鄰的格點間之耦合；二是實際量測成長於砷化鎵基板上的奈米等級超導鋁薄膜因KT相變而產生之電性傳輸變化。這兩種系統(鐵磁與超導)均被預測有KT相變，且可用一樣的數學模型來分析。 採取數值模擬的目的在於確認二維的有限系統是否具有相變，以及該相變如果有的話)是否產生自發性的連續性對稱破缺。經由計算磁化強度的平均值以及其磁化強度大小(norm)的平均值，我們發現當溫度在0.9以下時(以最近鄰格點間的耦合強度J為單位)，晶格具有不為零的磁化強度，因此具有自發性的連續性對稱破缺。當溫度上升至0.9到1.0間，磁化強度迅速衰減至零左右，顯示系統由有序態經歷相變轉為無序態。在同一溫度區間，磁化率也對應到一極高峰值，亦佐證了相變的發生。藉由分析相關函數(correlation function)對距離r的函數形式，我們得到當溫度在0.95以下，相關函數以冪次函數(power law)的形式衰減；當溫度在1.025以上，相關函數則以指數函數(exponential function)形式衰減。此函數形式隨溫度的變化恰好符合KT相變的特徵。我們進一步藉由T—η(T)圖中η=0.25以及T—Jeff(T)圖中Jeff =2T/pi的溫度，決定了較精確的KT相變溫度，兩者分別是0.903以及0.896。這裡的η(T)是相關函數以冪次衰減時的指數，Jeff則是因屏蔽效應而被修正的有效耦合強度。 量測方面則是藉著觀察樣品是否展現由緊密束縛的渦流-反渦流對(vortex-anti-vortex pair)所對應的電性傳輸性質轉變至游離的渦流(或反渦流)所具有的傳輸性質，來間接觀察KT相變是否產生。當溫度在KT相變溫度以下時，I—V特性曲線呈現非歐姆定律的冪次形式，且指數項可表示成1+1/(2η)。經由尋找1+1/(2η)=3對應到的溫度，我們得到的相變溫度是2.17 K，但並未觀察到無限大樣品在經歷KT相變時，1+1/(2η)由3迅速下降至1(歐姆定律)的現象，顯示此樣品不適合被當作無限大的系統。當溫度在KT相變溫度以上時，電阻會反比於相關長度的二次方。透過 函數的曲線擬合，我們發現當通過樣品的電流Isd由10 μA, 100 μA 提升到140 μA時，KT相變溫度則由2.203 K, 2.174 K下降到2.161 K。推測相變溫度與電流的相依性是由於電流通過電阻造成樣品被些微加熱所致，才會當電流越高時，相變溫度會越低。我們進一步使用dynamical scaling來確認量測得到的I—V特性曲線是否符合KT相變遵守的scaling 函數。當Isd約在100-150 μA時，能使不同溫度的I—V特性曲線算得的scaling 函數曲線全部疊合在一起的參數顯示，KT相變溫度約等於2.17 K，與由I—V冪次曲線決定的相變溫度一致。此外運用scaling也確認了dynamical critical exponent z等於2，由此得知我們用來分析電性傳輸所以依據的渦流動力學是正確的。

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.