低维磁性模型中钉扎—退钉扎相变的Monte Carlo模拟研究
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摘要
畴壁运动与许多物理现象相关,在过去的十几年里,一直是研究的热点,其中铁性材料(如铁磁材料、铁电材料等)中的畴壁运动受到了广泛的关注。在不同的驱动力类型和强度作用下,畴壁呈现出各种不同的运动形态和相变现象。另外,不同的系统背景,如温度、掺杂类型等也影响畴壁的形态和相变现象,以及普适类。理论上,畴壁运动是典型的非平衡态动力学过程；应用上,畴壁运动与新型的存储设备及逻辑运算元件的开发相关。因此,不管在理论上还是应用上,畴壁运动都是非常重要的研究课题。过去一些年,描述畴壁运动的典型方法是无序淬火的QEW方程。该方程未考虑微观相互作用,理论上也存在一定的不自洽性,并且理论与实验结果存在差别。在本论文中,我们引入含无序相互作用的微观格点模型,如XY模型、伊辛模型、多态时钟(Clock)模型等,研究低维磁性薄膜的畴壁运动并与实验进行比较。具体包括KT相变温度附近,XY模型涡旋态的弛豫过程；零温下,恒定外场驱动下随机场伊辛模型的钉扎-退钉扎相变普适类；零温下,多态时钟模型钉扎-退钉扎相变的短时动力学特性等。主要结果呈现在第二、三、四章。
第一章,我们主要对畴壁运动的研究进行综述,包括畴壁的定义、畴壁运动的理论模型和实验研究、畴壁运动的形态和临界现象等。对有序-无序相变、KT相变、钉扎-退钉扎相变及普适类进行了介绍。我们概述远离平衡态弛豫动力学的短时动力学方法,蒙特卡罗模拟方法的Metropolis算法和Heatbath算法。我们主要讨论的模型是二维微观格点模型,包括标量模型、连续矢量模型和离散矢量模型。最后,介绍本论文研究的动机和研究的主要内容。
第二章,我们研究KT相变温度附近二维XY模型的涡旋态弛豫动力学过程。我们引入局域赝磁化强度的定义,从而有效地描述动力学系统的对称结构。系统地分析赝磁化强度和Binder累积量的动力学标度行为,并测量动力学演化的临界指数。标度行为中包含非常强的对数修正,并且需要引入新的临界指数。为了验证强对数修正是由涡旋引起的这一观点,我们测量涡旋数密度的动力学演化,发现涡旋态中心区域涡旋数密度很大,甚至超过完全无序态的情形。为了验证拓扑缺陷的动力学效应,我们也模拟了二维XY模型的自旋波态弛豫动力学过程,类似地定义了赝磁化强度并测定了各临界指数。结果表明,赝磁化强度的定义是有效且可靠的。在涡旋态中心区域引入有限量的淬火无序,系统的普适类将会发生改变。进一步地,基于线性长波近似的方法,我们进行了理论解析计算,得到与数值模拟相一致的结果。
第三章,我们研究二维随机场伊辛模型钉扎-退钉扎相变的普适类。系统地测量相变的临界外场、动力学和静态临界指数、各粗糙指数。临界指数随着随机场的分布形式和强度变化,相变的强普适性缺失。根据粗糙指数ζ, ζloc和ζs三者的关系,二维随机场伊辛模型与QEW方程处于不同的普适类,属于新的反常普适类,该普适类特征为ζ≠ζloc≠ζs且ζloc≠1.对随机场均匀分布的情形,随着随机场强度的变化,系统从二级相变过渡到一级相变。但对高斯分布的情形,只有二级相变存在。与唯象的QEW方程比较,格点模型更好地描述实验结果。
第四章,我们将伊辛模型推广到矢量模型,研究二维随机场p态时钟模型钉扎-退钉扎相变的弛豫动力学特性,特别强调磁畴取向的动力学效应。磁畴不同的初始方向角θR会改变临界外场和各临界指数,临界外场与θR存在相当异常的依赖关系,对最小的θR,随机场的动力学效应最显著。相同的态数目p,不同的随机场强度和分布形式对应不同的临界外场。相同的强度和分布形式,不同的p也对应不同的临界外场。临界指数随着磁畴方向角、随机场强度和分布形式、态数目p等明显变化,相变的强普适性缺失。均匀分布时会出现一级相变到二级相变的过渡现象,而高斯分布时只存在二级相变。除一级相变外的所有情形,粗糙指数ζ,ζloc和ζs之间的关系与随机场伊辛模型类似,表明p态时钟模型的钉扎-退钉扎相变也属于新的反常普适类。
第五章,我们对论文的主要结果进行总结。
Domain-wall dynamics is related to many important physical phenomena. The mo-tion of domain-wall has attracted much interest recently. The domain-wall motion occurs in ferroic materials, e.g., ferromagnetic materials and ferroelectric materials. The dynamics and the morphology of interface are depend on the strength and type of the driving force. And they vary with the temperature and the form of quenched randomness. The study of domain-wall is very important both in the theoretical and experimental fields. Theoreti-cally, the motion of domain-wall is a typical nonequilibrium dynamics. Experimentally, it is related to the new classes of storage instrument and the logic operation components. For the domain-wall motion in magnetic films, the Edwards-Wilkinson equation with quenched disorder (QEW) is a typical theoretical approach. The QEW equation is a phe-nomenological model, without detailed microscopic structures and interactions of materi-als. Furthermore, the theoretical self-inconsistence is puzzling in the QEW equation. In this dissertation, we focus on the dynamics of domain-wall in the low-dimensional magnetic film. Using Monte Carlo methods and short-time dynamics, we study the domain-wall dynamics of different lattice models, e.g., two-dimensional XY model, Ising model with quenched disorder and p-state clock model with quenched disorder. The two-dimensional XY model is discussed at Kosterlitz-Thouless (KT) transition temperature. Ising model and p-state clock model are discussed at zero temperature. Chapter2,3and4are the main results of our work.
In Chapter1, we give a brief introduction to the dynamics of domain-wall. It includes the definition of domain wall, theoretical models, the morphology of magnetic domain wall and critical phenomena. The characteristic of ordered-disordered phase transition, KT phase transition and the universality class of the depinning transition are summarized. The main approaches are Monte Carlo method and the short-time dynamic approach. The mod- els include scalar model, continuous vector model and discrete vector model. Finally, we show the research motivation and the main research content.
In Chapter2, we investigate the dynamic relaxation of a vortex state at the KT phase transition temperature of the two-dimensional XY model. A local pseudo-magnetization is introduced to describe the central symmetric structure of the dynamic systems. Based on the short-time dynamic approach, the dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. A strong logarithmic correction to scaling is detected in the core of the vortex state. The vor-tex density in the core even exceeds that of the disordered initial state. Therefore, it is not surprising that a strong logarithmic correction to scaling emerges. To illustrate the dynamic effect of the topological defect and the reliability of the defined pseudo-magnetization, sim-ilar analysis for the dynamic relaxation from a spin-wave state is also performed. We verify that a limited amount of quenched disorder in the core of the vortex state may alter the dy-namic universality class. Finally, theoretical calculations based on the linearized long-wave approximation are presented.
In Chapter3, we systematically study the universality class of the depinning phase transition in the two-dimensional random-field Ising model with constant external field. We accurately locate the depinning transition field and determine both dynamic and static crit-ical exponents. The critical exponents vary significantly with the strength and form of the random fields. The strong universality of the depinning phase transition is violated. From the roughness exponents ζ,ζloc and ζs of domain-wall, one may judge that the depinning transition of the Ising model with quenched disorder belongs to a new dynamic universality class with ζ≠ζloc≠ζs and ζloc≠1. The crossover from the second-order transition to the first-order one is observed for the uniform distribution of the random fields which is bounded, but it is not present for the Gaussian distribution which is unbounded. The critical exponents exhibit independence from the updating schemes of the Monte Carlo algorithm. We verify that the random-single-spin-flip dynamics is robust. The DRFIM model does not suffer from the theoretical self-inconsistence as in the QEW equation, and its results are closer to experiments.
In Chapter4, we investigate the short-time dynamic behavior of the depinning transi-tion in the two-dimensional p-state clock model for different random fields, with different constant driving fields respectively. The depinning transition field exhibits rather unusual dependence on the initial orientation. Unexpectedly, the dynamic effect of the random fields is the most significant for the domain interface with the smallest θR. We accurately determine the depinning transition field and both dynamic and static critical exponents. The results show that the critical exponents vary significantly with the initial orientation and both of the form and strength of the random fields. Similarly, the crossover from the first-order phase transition to the second-order one is obtained for the case of uniform dis-tribution of the random fields, but it is not present for the case of Gaussian distribution. For all cases, the roughness exponents ζ, ζloc and ζs are carefully obtained, and the results indicate that the depinning transition of the random-field p-state clock model belongs to the new dynamic universality class, which is the same as the random-field Ising model.
In Chapter5, the main conclusions of this dissertation are summarized.
第一章,我们主要对畴壁运动的研究进行综述,包括畴壁的定义、畴壁运动的理论模型和实验研究、畴壁运动的形态和临界现象等。对有序-无序相变、KT相变、钉扎-退钉扎相变及普适类进行了介绍。我们概述远离平衡态弛豫动力学的短时动力学方法,蒙特卡罗模拟方法的Metropolis算法和Heatbath算法。我们主要讨论的模型是二维微观格点模型,包括标量模型、连续矢量模型和离散矢量模型。最后,介绍本论文研究的动机和研究的主要内容。
第二章,我们研究KT相变温度附近二维XY模型的涡旋态弛豫动力学过程。我们引入局域赝磁化强度的定义,从而有效地描述动力学系统的对称结构。系统地分析赝磁化强度和Binder累积量的动力学标度行为,并测量动力学演化的临界指数。标度行为中包含非常强的对数修正,并且需要引入新的临界指数。为了验证强对数修正是由涡旋引起的这一观点,我们测量涡旋数密度的动力学演化,发现涡旋态中心区域涡旋数密度很大,甚至超过完全无序态的情形。为了验证拓扑缺陷的动力学效应,我们也模拟了二维XY模型的自旋波态弛豫动力学过程,类似地定义了赝磁化强度并测定了各临界指数。结果表明,赝磁化强度的定义是有效且可靠的。在涡旋态中心区域引入有限量的淬火无序,系统的普适类将会发生改变。进一步地,基于线性长波近似的方法,我们进行了理论解析计算,得到与数值模拟相一致的结果。
第三章,我们研究二维随机场伊辛模型钉扎-退钉扎相变的普适类。系统地测量相变的临界外场、动力学和静态临界指数、各粗糙指数。临界指数随着随机场的分布形式和强度变化,相变的强普适性缺失。根据粗糙指数ζ, ζloc和ζs三者的关系,二维随机场伊辛模型与QEW方程处于不同的普适类,属于新的反常普适类,该普适类特征为ζ≠ζloc≠ζs且ζloc≠1.对随机场均匀分布的情形,随着随机场强度的变化,系统从二级相变过渡到一级相变。但对高斯分布的情形,只有二级相变存在。与唯象的QEW方程比较,格点模型更好地描述实验结果。
第四章,我们将伊辛模型推广到矢量模型,研究二维随机场p态时钟模型钉扎-退钉扎相变的弛豫动力学特性,特别强调磁畴取向的动力学效应。磁畴不同的初始方向角θR会改变临界外场和各临界指数,临界外场与θR存在相当异常的依赖关系,对最小的θR,随机场的动力学效应最显著。相同的态数目p,不同的随机场强度和分布形式对应不同的临界外场。相同的强度和分布形式,不同的p也对应不同的临界外场。临界指数随着磁畴方向角、随机场强度和分布形式、态数目p等明显变化,相变的强普适性缺失。均匀分布时会出现一级相变到二级相变的过渡现象,而高斯分布时只存在二级相变。除一级相变外的所有情形,粗糙指数ζ,ζloc和ζs之间的关系与随机场伊辛模型类似,表明p态时钟模型的钉扎-退钉扎相变也属于新的反常普适类。
第五章,我们对论文的主要结果进行总结。
Domain-wall dynamics is related to many important physical phenomena. The mo-tion of domain-wall has attracted much interest recently. The domain-wall motion occurs in ferroic materials, e.g., ferromagnetic materials and ferroelectric materials. The dynamics and the morphology of interface are depend on the strength and type of the driving force. And they vary with the temperature and the form of quenched randomness. The study of domain-wall is very important both in the theoretical and experimental fields. Theoreti-cally, the motion of domain-wall is a typical nonequilibrium dynamics. Experimentally, it is related to the new classes of storage instrument and the logic operation components. For the domain-wall motion in magnetic films, the Edwards-Wilkinson equation with quenched disorder (QEW) is a typical theoretical approach. The QEW equation is a phe-nomenological model, without detailed microscopic structures and interactions of materi-als. Furthermore, the theoretical self-inconsistence is puzzling in the QEW equation. In this dissertation, we focus on the dynamics of domain-wall in the low-dimensional magnetic film. Using Monte Carlo methods and short-time dynamics, we study the domain-wall dynamics of different lattice models, e.g., two-dimensional XY model, Ising model with quenched disorder and p-state clock model with quenched disorder. The two-dimensional XY model is discussed at Kosterlitz-Thouless (KT) transition temperature. Ising model and p-state clock model are discussed at zero temperature. Chapter2,3and4are the main results of our work.
In Chapter1, we give a brief introduction to the dynamics of domain-wall. It includes the definition of domain wall, theoretical models, the morphology of magnetic domain wall and critical phenomena. The characteristic of ordered-disordered phase transition, KT phase transition and the universality class of the depinning transition are summarized. The main approaches are Monte Carlo method and the short-time dynamic approach. The mod- els include scalar model, continuous vector model and discrete vector model. Finally, we show the research motivation and the main research content.
In Chapter2, we investigate the dynamic relaxation of a vortex state at the KT phase transition temperature of the two-dimensional XY model. A local pseudo-magnetization is introduced to describe the central symmetric structure of the dynamic systems. Based on the short-time dynamic approach, the dynamic scaling behavior of the pseudo-magnetization and Binder cumulant is carefully analyzed, and the critical exponents are determined. A strong logarithmic correction to scaling is detected in the core of the vortex state. The vor-tex density in the core even exceeds that of the disordered initial state. Therefore, it is not surprising that a strong logarithmic correction to scaling emerges. To illustrate the dynamic effect of the topological defect and the reliability of the defined pseudo-magnetization, sim-ilar analysis for the dynamic relaxation from a spin-wave state is also performed. We verify that a limited amount of quenched disorder in the core of the vortex state may alter the dy-namic universality class. Finally, theoretical calculations based on the linearized long-wave approximation are presented.
In Chapter3, we systematically study the universality class of the depinning phase transition in the two-dimensional random-field Ising model with constant external field. We accurately locate the depinning transition field and determine both dynamic and static crit-ical exponents. The critical exponents vary significantly with the strength and form of the random fields. The strong universality of the depinning phase transition is violated. From the roughness exponents ζ,ζloc and ζs of domain-wall, one may judge that the depinning transition of the Ising model with quenched disorder belongs to a new dynamic universality class with ζ≠ζloc≠ζs and ζloc≠1. The crossover from the second-order transition to the first-order one is observed for the uniform distribution of the random fields which is bounded, but it is not present for the Gaussian distribution which is unbounded. The critical exponents exhibit independence from the updating schemes of the Monte Carlo algorithm. We verify that the random-single-spin-flip dynamics is robust. The DRFIM model does not suffer from the theoretical self-inconsistence as in the QEW equation, and its results are closer to experiments.
In Chapter4, we investigate the short-time dynamic behavior of the depinning transi-tion in the two-dimensional p-state clock model for different random fields, with different constant driving fields respectively. The depinning transition field exhibits rather unusual dependence on the initial orientation. Unexpectedly, the dynamic effect of the random fields is the most significant for the domain interface with the smallest θR. We accurately determine the depinning transition field and both dynamic and static critical exponents. The results show that the critical exponents vary significantly with the initial orientation and both of the form and strength of the random fields. Similarly, the crossover from the first-order phase transition to the second-order one is obtained for the case of uniform dis-tribution of the random fields, but it is not present for the case of Gaussian distribution. For all cases, the roughness exponents ζ, ζloc and ζs are carefully obtained, and the results indicate that the depinning transition of the random-field p-state clock model belongs to the new dynamic universality class, which is the same as the random-field Ising model.
In Chapter5, the main conclusions of this dissertation are summarized.
引文
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