外加磁場下磁偶極交互作用之自旋冰的相變化

面心最密堆積的晶格環境中,由正四面體作為基底組成的特殊結構,並且以自旋之間的交互作用為系統主要交互作用,此即為綠火晶自旋冰(Pyrochlore Spin Ice)。在挫折系統(Frustrated System)的研究中, 自旋冰系統的研究是研究者們感興趣的問題之一。 1956年起一直有著零星的相關研究,而在1980年代末期開始出現大量的研究成果,一方面是實驗有了越來越多的成果,一方面則是運算科技的突破。鈦酸鏑(Dy2T i2O7)以及鈦酸鈥(Ho2T i2O7)相繼在實驗中被發現,並被確認為自然界中的自旋冰材料。隨著數值方法的不斷演進,研究者們有辦法處理超過最近鄰(nearest neighbour)的交互作用,甚至能夠處理任兩自旋間的交互作用。由於特殊的幾何結構,在自旋冰系統中的自旋, 其自旋方向受到嚴格限制,僅能夠沿著易磁化方向軸(easy axis)的方向。而在較低溫時,自旋冰系統的一項重要限制-冰法則(ice rule)-開始掌控整個系統。在低溫環境下並外加磁場的自旋冰,其物理現象為何依然是有爭議的,例如使用最近鄰之交互作用不足以解釋許多實驗結果。在這篇論文中,我們所提出的研究成果主要為二,皆為外加一磁場並以電腦模擬其行為,其一為引入次近鄰之交互作用,其二為於外加一磁場引入任二自旋間之交互作用。我們於任二自旋之交互作用結果中,發現一種介於已知的兩種可能基態之有序結構,並提出一種隨磁場及溫度變化的可能相圖。本篇論文主要模擬之方法為古典蒙地卡羅(Classical Monte Carlo)。蒙地卡羅在模擬晶格之交互作用時,是一種相對簡單並且強大的工具, 然而在研究本項主題時,由於自旋冰的行為演化極度緩慢,尤其是在任二自旋之交互作用時,在低溫時是極度難以達成熱平衡的。因此我們運用許多演算法,如「平行回火」(parallel tempering)演算法以及「群集更新」(cluster update)演算法,如「迴路更新」(loop update)、「蟲演算法」(worm algorithm)等。為了達成本論文之研究目的,我們在符合蒙地卡羅法的假設下,改寫了慣常使用的「群集更新」演算法,藉此得到本論文的結果。在本論文中,第一章介紹挫折系統以及自旋冰。挫折系統的部份主要介紹「幾何挫折系統」(geometrical frustrated system)。第二章介紹所主要使用的模擬方法,包含其理論基礎,如馬可夫鏈過程(Markov chain process);實際應用所使用的方法,如「梅婥坡里斯單一自旋翻轉演算法」(Metropolis sing spin flip algorithm)、「迴路更新」、「蟲演算法」以及「平行回火」等。除了模擬的方法之外,第二章也會解釋如何以統計的辦法處理產生的數據。在本篇論文之中,我們使用的數值處理辦法為「靴子陷阱」(bootstrap)法,都會詳加解釋。第三章中,我們將介紹外加磁場下引入次近鄰之交互作用之模擬, 包含模擬時使用的模型、為了本模型而改寫的演算法以及模擬結果。我們所提出的結果將和僅考慮最近鄰交互作用之模擬結果比較; 第四章中則是介紹外加磁場下引入磁偶極交互作用之模擬,包含模擬時使用的模型、為了本模型而改寫的演算法以及模擬結果。在這個章節中,我們提出一個在溫度及磁場改變下的可能相圖,以及一種可能的新的有序結構。本論文的內容適合下列兩類型的讀者閱讀 • 對自旋冰系統有興趣,尤其對於[100]外加磁場下的自旋冰系統。 • 熟悉自旋冰系統之模擬,想處理引入更多鄰近交互作用之模擬, 而需要更有效率的可能演算法選擇。

關鍵字

自旋冰；磁偶極；鈦酸鏑；蒙地卡羅法；幾何挫折系統；單一自旋翻轉演算法；迴路更新；蟲演算法；平行回火

並列摘要

Geometrically frustrated system is one of the popular research issues for condensed matter researchers. There are many degenerate ground states for such a system. Pyrochlore spin ice is a typical geometrically frustrated system due to its special structure and single-ion anisotropy. By the constraint of ice rule, it is able to calculate the residual entropy of such a system, which is also confirmed by experiment. The residual entropy shows the high degeneracy of the ground state of the geometrically frustrated system. To study the system, many of the mathematical models are proposed. The simplest one is the nearest neighbor(NN) spin ice model. Althogh simulation results show that NN spin ice is qualitatively equivalant to the experiment re- sults, further neighbor interactions still need to be considered due to the high magnetic moment of Dy (≈10μB ). Therefore, dipolar spin ice becomes a bet- ter model to study pyrochlore spin ice. The purpose of this thesis is to study the behaviors of the dipolar spin ice in a [100] applied magnetic field. Before showing our research, both the NN spin ice in a [100] field and dipolar spin ice model will be introduced to enhance the background knowledge of the readers. We will present the phase transitions and the ground states of these two different models. In addition, to handle with the conditionally conver- gent dipolar term in dipolar spin ice model, we will introduce Ewald method to deal with the problem. Chapter (1) will cover the above discussions. In Chap. (2), Monte Carlo method, which is the main method we use to study in our researches, will be discussed. We will briefly introduce the back- ground knowledge of Monte Carlo method. Some of the algorithms based on the Monte Carlo method will be proposed, including single spin flip(SSF) Metropolis algorithm, loop update, worm algorithm and parallel tempering. By adding the effect of next nearest neighbor(NNN) interaction to NN spin ice in a [100] field, we observe that a ferromagnetic NNN interaction can enhance the effective magnetic field. The NNN interaction can have a relatively strong influence on the result of the NN spin ice model in a [100] field with a small value(∼ 10−4J1). The results and discussion will be present in Chap. (3). Our main results of dipolar spin ice in a [100] field will be shown in Chap. (5). Three plateaux of magnetization in different regime of temperature and applied magnetic field is the newly discovered and first time proposed in history. A new ground state, which is the mixture of two other ground states, is discovered. We find that dipolar spin ice experiences first-order phase tran- sition when the system enters one of the three ground states from spin ice state. The phase transition between three different ground states is also the first-order. A phase diagram of dipolar spin ice in different [100] fields and different temperatures is also proposed, which is another main contribution of our research. This thesis is intended for the reader who: • is interested in spin ice system, especially for those which is under a applied magnetic field. • is familiar with spin ice simulation and wants a better algorithm to over- come the complexity of further neighbour interaction or to have a more efficient algorithm to handle the slow equilibrium.

並列關鍵字

Metropolis sing spinle flip algorithm ； Classical Monte Carlo ； Dysprosium Titanate ； Pyrochlore ； Spin ice ； Worm algorithm ； parallel tempering ； loop update

參考文獻

[1] B. C. den Hertog and M. J. P. Gingras, “Dipolar interactions and origin of spin ice in ising pyrochlore magnets,” Phys. Rev. Lett., vol. 84, pp. 3430–3433, Apr 2000.

[2] L. D. C. Jaubert, J. T. Chalker, P. C. W. Holdsworth, and R. Moessner, “Three- dimensional kasteleyn transition: Spin ice in a [100] field,” Phys. Rev. Lett., vol. 100, p. 067207, 2008.

[3] R. G. Melko, B. C. den Hertog, and M. J. P. Gingras, “Long-range order at low temperatures in dipolar spin ice,” Phys. Rev. Lett., vol. 87, p. 067203, 2001.

[4] E. Ising, “Beitrag zur theorie des ferromagnetismus,” Zeitschrift fu ̈r Physik, vol. 31, pp. 253–258, 1925.

[5] W. H. Barnes, “The crystal structure of ice between 0 degrees c. and - 183 degrees c,” Proceedings of the Royal Society of London. Series A, vol. 125, no. 799, pp. 670– 693, 1929.

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外加磁場下磁偶極交互作用之自旋冰的相變化

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