1.School of Microelectronics, Faculty of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China 2.Key Laboratory of New Processing Technology for Nonferrous Metals and Materials, Ministry of Education, Guangxi Key Laboratory of Optical and Electronic Materials and Devices, College of Materials Science and Engineering, Guilin University of Technology, Guilin 541004, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11974268, 11574246) and the Natural Science Foundation of Guangxi, China (Grant Nos. AA138162, AA294014, GA245006)
Received Date:30 November 2020
Accepted Date:27 December 2020
Available Online:31 May 2021
Published Online:05 June 2021
Abstract: Relaxor ferroelectric is a very special type of ferroelectric material, which has important applications in communication, sensor, ultrasound, energy conversion, and aerospace industry. Unlike normal ferroelectric, a relaxor undergoes a diffuse phase transition in the cooling process, and its macroscopic polarization does not occur suddenly, but polar nano region appears while the macro-symmetry does not change significantly. As the transition from the paraelecric to the ferroelectric phase is a gradual process with a broad dielectric peak, relaxor ferroelectric has no definite Curie temperature (TC), and the temperature corresponding to the maximum dielectric constant (Tm) and the Burns temperature (TB) are often used as their characteristic temperatures. Here, in order to understand the diffuse phase transition and its internal mechanism, we build a modified Ising model by introducing an energy potential well that affects the spin variable (which is regarded as electric dipole in this research) and simulate the phase transition process using this model, which results in significantly smoothed phase transition with respect to temperature, exhibiting relaxor characteristics with diffuse phase transitions. More precisely, it is found that by applying the energy potential well to the dipoles in the system, the ferroelectric phase transition can be significantly broadened, that is, a diffused phase transition appears, showing strong relaxation characteristics that, as the temperature gradually increases, the average electric dipole moment does not change abruptly while the peak value of its permittivity decreases with the energy potential well. Moreover, at a temperature much higher than the transition temperature of the usual Ising model, the system can still maintain a certain polarization, which is in line with relaxor characteristics. By comparing to a previously proposed statistical model, it is found that the relaxation phenomenon is due to the fact that dipoles in the system are constrained by the given potential well, therefore difficult to flip, making the overall polarizability deviate from that of conventional ferroelectrics. Our results therefore show that the existence of dipole energy potential well is an important factor in the relaxation phenomenon of ferroelectric. This modified Ising model, which accounts for the constrained dipoles statistically, is then used to investigate the thermal hysteresis effect of relaxor ferroelectrics in order to understand its origin. By comparing to experimental results, we are able to clarify the physics of the thermal hysteresis of relaxor ferroelectric, deepening our understanding from the theoretical and simulation perspective. Keywords:relaxor ferroelectrics/ Ising model/ diffuse phase transition/ Monte-Carlo simulations
其中$ {\chi }_{1} $和$ {\chi }_{2} $分别表示两种电偶极子的极化率, 可以进行一定的选取. 例如, 可以使用(9)式描述弛豫铁电体的极化率与温度的关系[9], 结果将在后面进行讨论(图3): 图 3 升温过程与降温过程电极化率随温度变化的拟合, 实线为(9)式的拟合结果 (a) 升温过程; (b) 降温过程 Figure3. The fitting of the electrical polarization with the temperature during the heating process and the cooling process, the solid line is the fitting result of Eq. (9): (a) Heating process; (b) cooling process
3.模拟结果及讨论基于上述改进的伊辛模型, 设置20 × 20 × 20的晶格进行模拟, 通过产生随机数$ \alpha $并以$\alpha > $$ {\mathrm{exp}}\bigg({-\dfrac{{E}_{\mathrm{B}}}{{k}_{\mathrm{B}}T}} \bigg)$的标准随机选择该温度下的晶格作为陷入能量势阱的特殊格点. 这些被随机选择的晶格偶极子在翻转时被能量势阱所束缚无法自由翻转, 从而固定在其位置上, 而其他晶格上的偶极子是自由偶极子, 在翻转时仍然按照普通伊辛模型的方法进行翻转. 在模拟中采用$ J=1 $, $ {k}_{\mathrm{B}}=1 $的单位, 并使用约化温度, 以$ J/{k}_{\mathrm{B}} $为单位, 这样能够更好地显示出$ {E}_{\mathrm{B}} $的作用. 为了保证系统在模拟中达到平衡态, 先让系统的每个晶格上的偶极子翻转n次(本文n = 40000), 并且忽略掉前m次(本文m = 10000), 而选取剩余的次数中的系统构型获得平均的电偶极矩、电极化率等. 这样能够保证系统与恒温热库充分接触, 并忽略不平衡过程的数据, 统计出最后结果的平均值, 获得其物理性能. 为了研究弛豫铁电体的特性, 模拟过程可以选择升温过程或者降温过程; 并且在升温过程中可以考虑使用不同的初始状态, 例如低温下所有偶极子处于+1的状态(长程有序铁电态)或者所有偶极子处于随机取值的状态(顺电态)等作为初始态. 图1(a)给出从初始态为所有偶极子处于+1状态的升温结果, 图1(b)给出了偶极子从随机初始化态的降温结果. 从图1(a)可以发现, 随着设置的能量势阱$ {E}_{\mathrm{B}} $绝对值的增加(0, 1, 2, 5), 平均电偶极矩随着温度的曲线发生了剧烈的变化, 和无外电场无能量势阱的情况相比, 能量势阱越大, 相变完成时的温度越高, 并且即便在很高的温度时极化也没有完全变为0, 这与弛豫体特征比较接近, 即在居里温度以上很宽区域内仍然存在自发极化. 图 1 平均电偶极矩随温度的变化 (a) 初始态为极化状态的升温过程; (b) 初始态随机状态的降温过程 Figure1. Temperature dependence of the average electric dipole moment: (a) Heating process from the initial state with all dipoles being +1; (b) cooling process from the initial state with random dipoles.
总的看来, 设置能量势阱以后, 系统的相变呈现弥散现象, 相变不再发生在一个狭窄的温度范围内, 而是在一个较为宽泛的温度区间; 同时系统相变点变得不明显, 平均电偶极矩没有在某个温度出现急剧下降. 也就是说, 通过给伊辛模型引入能量势阱, 确实能够再现弛豫体的一些特征现象. 图1(a)中曲线的形成是由于系统部分格点上的偶极子在较低温度下被能量势阱束缚, 无法自由翻转, 只能随着温度的升高缓慢地从能量势阱的束缚中释放出来, 因而增加了相变的弥散程度. 此外, 如图1(b)所示, 在降温过程中, 当温度从$ T=20 $(以J/kB为单位, 下同)开始逐渐降低时, 在能量势阱存在的情况下, 平均电偶极矩逐渐增大. 但是与常规的伊辛模型不同, 即使在最低的温度下, 整个系统仍然不可以达到完全极化状态(即偶极子按同方向整齐排列, 平均电偶极矩为1). 这是因为能量势阱随着温度的降低, 束缚住更多的自由偶极子, 这些被束缚住的偶极子可能处于不同的状态(+1, –1), 而这些偶极子被能量势阱束缚, 所以宏观上低温状态下平均电偶极矩依然不能为1. 从电极化率的角度看, 引入能量势阱之后系统的相变确实发生在一个很宽泛的温度范围内, 如图2(b)和图2(c)所示. 可以确定的是, 随着能量势阱绝对值的变大, 系统极化率最高点对应的温度提高了, 并且能量势阱越大, 其居里温度越不明确, 发生的相变越弥散. 这和无能量势阱下的相变曲线(图2(a))完全不同. 图 2 不同能量势阱下电极化率随温度的变化 (a) 无能量势阱, 初始态为极化态的升温过程; (b) 存在能量势阱, 初始态为极化态的升温过程; (c) 存在能量势阱, 初始态为随机态的降温过程 Figure2. Polarizability versus temperature with different $ {E}_{\mathrm{B}}: $ (a) Heating process from an initial state with all dipoles being +1 for EB = 0; (b) heating process from an initial state with all dipoles being +1 with nonzero EB; (c) cooling process from an initial state with random dipoles with nonzero EB
图 3 升温过程与降温过程电极化率随温度变化的拟合, 实线为(9)式的拟合结果 (a) 升温过程; (b) 降温过程
图 1 平均电偶极矩随温度的变化 (a) 初始态为极化状态的升温过程; (b) 初始态随机状态的降温过程
图 2 不同能量势阱下电极化率随温度的变化 (a) 无能量势阱, 初始态为极化态的升温过程; (b) 存在能量势阱, 初始态为极化态的升温过程; (c) 存在能量势阱, 初始态为随机态的降温过程
图 4 升温和降温过程极化率随温度变化的对比
图 5 
