Polarization and magnetization properties of ferroelectric/ ferromagnetic layer films under external field

Zheng Wei; Du An

doi:10.7498/aps.68.20181879

Abstract

Authors and contacts

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1. 引　言
2. 理论模型
3. 数值计算结果与讨论
3.1 系统的热力学性质
3.2 外场下系统的极化磁化行为
4. 结　论

References

Cited By

Abstract

A ferroelectric/ferromagnetic bilayer film model is established. The electric moment of ferroelectric layer is described by continuous scalars, and the spins of ferromagnetic layer are described by classical vectors. The thermodynamic properties, polarization and magnetization behavior are simulated by using Monte Carlo method. The temperature dependence of internal energy, specific heat, polarization and magnetization of the system under zero field are given, and the polarization and magnetization behavior of the system under an external magnetic field and under an external electric field are studied respectively. Simulation results show that the values of internal energy, specific heat, polarization and magnetization of the bilayer films under no action of external field are obviously different from each other due to the fact that their interlayer coupling coefficients are different. When the interfacial coupling is weak (J_em = 0.01), the bilayer films exhibit their own thermodynamic properties. The interaction between ferroelectric layer and ferromagnetic layer increases with the increase of interlayer coupling coefficient. When the interfacial coupling increases to a certain extent (J_em = 0.5), the bilayer film is coupled into a whole and exhibits a uniform thermodynamic behavior. The phase transition temperature of the system increases significantly.In an external magnetic field, the ferromagnetic layer shows hysteresis behavior, and the ferroelectric layer also shows hysteresis behavior. At relatively low temperature (T = 0.08), the hysteresis loop of ferromagnetic layer and ferroelectric layer exhibit bias behavior, the area of hysteresis loop of ferroelectric layer is small when the interfacial coupling is weak. With the increase of interfacial coupling, the phenomenon of bias is more obvious, and the area of hysteresis loop of ferroelectric layer also increases significantly. When the interfacial coupling reaches J_em = 0.75, the polarization behavior of the ferroelectric layer fully responds to the magnetization behavior of the ferromagnetic layer, and neither the bias phenomenon of the ferromagnetic layer nor the bias phenomenon of the ferroelectric layer is still existent. As temperature increases (T = 0.4), the phenomenon of bias disappears even if the interlayer coupling is weak. In an external electric field, the hysteresis behavior of ferromagnetic layer and the hysteresis behavior of ferroelectric layer are similar to those in an external magnetic field. The difference is that the bias phenomenon of the bilayer film still exists for weak interfacial coupling at relatively high temperature (T = 0.4). The theoretical results are in good agreement with the experimental results reported in the literature.

Keywords:

polarization /
magnetization /
loop /
bias

PACS:

75.85.+t	(Magnetoelectric effects, multiferroics)
77.55.Nv	(Multiferroic/magnetoelectric films)
91.60.Pn	(Magnetic and electrical properties)
75.60.Ej	(Magnetization curves, hysteresis, Barkhausen and related effects)

Authors and contacts

Zheng Wei^1,2,
Du An¹

1.
College of Sciences, Northeastern University, Shenyang 110819, China
2.
College of Physics Science Technology, Shenyang Normal University, Shenyang 110034, China

Corresponding author: Du An, duan@mail.neu.edu.cn

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1. 引　言

近年来, 随着现代工业向着器件微型化、需求多样化发展, 对多功能材料的需求变得更为迫切. 多铁性材料就是一类典型的多功能材料, 能够同时具备铁电、铁磁、铁弹、压电、压磁等多种特性^[1], 而且由于多种有序态共存引起材料中新的耦合作用, 从而衍生出很多新的性能, 大大开拓了铁性材料的应用范围. 作为重要的先进功能材料, 多铁性材料已被广泛应用于换能器、传感器、存储器、微驱动器等电子器件^[2-6].

基于应用上的广泛需要, 对多铁性材料的研究已成为目前国际上新的研究热点. 其中多铁性复合薄膜材料由于其丰富的物理性质和在制备工艺上的优势是备受关注的热点之一. Zheng等^[7]首先报道了CoFe₂-BaTiO₃复合薄膜, CoFe₂铁磁纳米柱以垂直于薄膜平面的方式分散在BaTiO₃铁电薄膜基体中形成磁电复合薄膜, 分析了薄膜的磁电耦合效应; Ryu等^[8]制备了PZT-NiFe₂O₄复合薄膜, 以NiFe₂O₄铁磁纳米颗粒弥散分布在锆钛酸铅压电陶瓷(PZT)铁电薄膜基体中, 并研究了其磁电系数及与电场、磁场的关系; Deng等^[9]在SrTiO₃的单晶基片上外延生长了NiFe₂O₄-BaTiO₃双层结构薄膜, 并研究了其磁电性质. 国内外研究者在实验上采用物理和化学方法积极探索合成层状多铁性薄膜材料, 制备的各种薄膜材料展现了铁电性、铁磁性、介电性、磁电耦合效应、磁阻效应、磁电容效应等丰富的磁电性质^[10-18]. 在理论研究上, Liu等^[19]应用Landau-Ginsburg-Devonshire热力学理论计算了BaTiO₃-CoFe₂O₄复合薄膜的界面耦合; Duan等^[20]应用第一性原理计算了铁磁膜在外电场下的磁电效应; Nan等^[21]应用格林函数方法计算了磁场诱发极化; Sukhov等^[22]建立了一维链式模型计算了场驱动下的多铁结构的动力学. 本文在此基础上建立了铁电/铁磁双层薄膜模型, 利用蒙特卡罗模拟方法研究了其热力学性质, 并全面分析了其在外电场和外磁场下的极化磁化行为.

2. 理论模型

基于文献[7—9], 建立了由一个铁电层和一个铁磁层构成的双层膜体系(FE/FM双层膜), 如图1所示. 双层膜通过界面的铁电/铁磁耦合作用互相影响. 考虑到铁电/铁磁层晶体的极化特征, 假定铁电层极化为位移极化, 即电偶极子取向仅沿着 $z$ 方向; 铁磁层自旋磁矩取向为三维空间, 可投影到 $x$ , $y$ , $z$ 三个方向.

$图 1 FE/FM双层膜结构示意图\r\nFig. 1. Schematic of the ferroelectric/ferromagnetic double layer film.$

图 1 FE/FM双层膜结构示意图

Fig. 1. Schematic of the ferroelectric/ferromagnetic double layer film.

下载: 全尺寸图片幻灯片

对于该FE/FM双层膜系统, 总能量包含三部分:

$H = {H_{{\rm{FE}}}} + {H_{{\rm{FM}}}} + {H_{\rm{C}}},$

(1)

式中 ${H_{{\rm{FE}}}}$ 表示铁电层的能量, 表示为

$\begin{split}{H_{{\rm{FE}}}} = & \sum\limits_j \Bigg[\frac{1}{2}\alpha {{p}}_j^2 + \frac{1}{4}\beta {{p}}_j^4 \Bigg. \\ & \Bigg.+ \frac{1}{2}\kappa \sum\limits_{j'} {{{\left( {{{{p}}_j} - {{{p}}_{j'}}} \right)}^2}} - {{{p}}_j} \cdot {{E}} \Bigg] ,\end{split}$

(2)

其中前两项为Landau自由能, 第三项代表最近邻极矩之间的作用, 第四项为偶极子在电场中的势能; ${H_{{\rm{FM}}}}$ 表示铁磁层的能量, 表示为

${H_{{\rm{FM}}}} = - J\sum\limits_{\langle i,i'\rangle }^{} {{{{S}}_i} \cdot {{{S}}_{i'}}} - \sum\limits_i {D{{\left( {{S_{iz}}} \right)}^2}} - \sum\limits_i {{{{S}}_i} \cdot {{B}}} ,$

(3)

其中第一项为最近邻自旋之间的交换作用, 第二项为各向异性能, 第三项为自旋在磁场中的塞曼能;只考虑两层间最近邻电矩和磁矩的相互作用, 层间耦合能量 ${H_{\rm{C}}}$ 可以写为^[22]

${H_{\rm{C}}} = \sum\limits_{\langle j,i\rangle } {{J_{{\rm{em}}}}{{{P}}_j} \cdot {{{S}}_i}} ,$

(4)

其中 ${J_{{\rm{em}}}}$ 表示层间耦合系数.

按照统计系综理论, 可以得到体系各物理量的计算公式如下:

系统内能

$U = \frac{{\left\langle H \right\rangle }}{{{N_{{\rm{FM}}}} + {N_{{\rm{FE}}}}}},$

(5)

系统比热

$C \!=\! \frac{{\partial U}}{{\partial T}} = \frac{1}{{{{\left( {{k_{\rm{B}}}T} \right)}^2}\left( {{N_{{\rm{FM}}}} \!+\! {N_{{\rm{FE}}}}} \right)}}\left(\! {\left\langle {{H^2}} \right\rangle \!-\! {{\left\langle H \right\rangle }^2}} \right),$

(6)

铁磁层的磁化强度

${M_z} = \frac{{\left\langle {{S_z}} \right\rangle }}{{{N_{{\rm{FM}}}}}},$

(7)

磁化率

${\chi _{\rm{m}}} = \frac{1}{{{k_{\rm{B}}}T{N_{{\rm{FM}}}}}}\left( {\left\langle {{S_z}^2} \right\rangle - {{\left\langle {{S_z}} \right\rangle }^2}} \right),$

(8)

铁电层的极化强度

${P_z} = \frac{{\left\langle {{P_z}} \right\rangle }}{{{N_{{\rm{FE}}}}}},$

(9)

极化率

${\chi _{\rm{e}}} = \frac{1}{{{k_{\rm{B}}}T{N_{{\rm{FE}}}}}}\left( {\left\langle {{P_z}^2} \right\rangle - {{\left\langle {{P_z}} \right\rangle }^2}} \right),$

(10)

其中 ${S_z} = \sum\limits_{i = 1}^{{N_{{\rm{FM}}}}} {{S_{iz}}}$ , ${P_z} = \sum\limits_{j = 1}^{{N_{{\rm{FE}}}}} {{P_{jz}}}$ , ${N_{{\rm{FM}}}}$ 和 ${N_{{\rm{FE}}}}$ 分别代表铁磁层和铁电层内自旋和电矩的个数.

3. 数值计算结果与讨论

采用蒙特卡罗方法模拟体系的性质. 将各格点上的电矩 ${{{P}}_j}$ 按照从−1到1连续变量处理, 将各自旋 ${{{S}}_i}$ 按照幅值为1的经典矢量处理. 为方便起见, 对体系能量和各物理量进行约化处理, 并以铁磁层交换作用 $J$ 为约化单位, 取玻尔兹曼常数 ${k_{\rm{B}}}$ 为1. 其他参数取值如下: $\alpha = - {\rm{3}}.0$ , $\beta = 0.{\rm{4}}$ , $\kappa = 0.{\rm{4}}$ , $D = 0.{\rm{2}}$ . 模拟过程中取总步数为40000次, 其中前10000步数为达到平衡前的弛豫过程.

3.1 系统的热力学性质

首先研究体系的自发极化和磁化情况, 取外场为零, 计算体系的磁化强度、极化强度、能量、比热、磁化率和极化率等.

3.1.1 FE/FM薄膜的自发极化和磁化

无外场作用时, 不同界面耦合情况下, 体系的自发极化和磁化曲线如图2所示. 由于界面耦合系数为正, 铁电层的自发极化强度和铁磁层的自发磁化强度的方向总是相反的. 当温度趋于零温时, 体系的自发极化和磁化趋于饱和.

$图 2 自发极化和磁化随温度变化曲线\r\nFig. 2. Temperature dependencies of the spontaneous polarization and spontaneous magnetization of the system.$

图 2 自发极化和磁化随温度变化曲线

Fig. 2. Temperature dependencies of the spontaneous polarization and spontaneous magnetization of the system.

下载: 全尺寸图片幻灯片

当层间耦合作用非常弱( ${J_{{\rm{em}}}} = 0.01$ )时, 体系表现出独立的铁电层和铁磁层行为. 由于铁电层的转变温度(0.5)比铁磁层的转变温度(0.8)低, 所以铁电层的极化强度较早地开始下降. 随着层间相互作用的增加, 铁电层和铁磁层抗热扰动能力都在增强. 当层间耦合增强到0.5时, 体系的转变温度明显增大, 表现出一个整体的极化强度和磁化强度行为.

3.1.2 FE/FM薄膜的内能和比热

图3给出了系统的内能和比热随温度的变化. 数值模拟结果表明, 对应不同层间耦合系数 ${J_{{\rm{em}}}} = 0.01$ , $0.1$ , $0.5$ , 内能曲线明显不同. 由图3(a)可见, 随着层间耦合作用的增加, 相同温度下内能 $U$ 取值降低, 当 ${J_{{\rm{em}}}} = 0.5$ 时, 能量线明显下移.

$图 3 (a)内能和(b)比热随温度的变化\r\nFig. 3. Temperature dependencies of (a) energy and (b) specific heat.$

图 3 (a)内能和(b)比热随温度的变化

Fig. 3. Temperature dependencies of (a) energy and (b) specific heat.

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比热曲线表现了双层体系从相对独立到耦合成整体的过程, 如图3(b)所示. 当 ${J_{{\rm{em}}}} = 0.01$ 时, 曲线有两个峰值, 分别对应铁电层和铁磁层的相变温度. 当耦合作用增强时, 两峰都向右移动, 对应的相变温度增大, 且曲线左侧铁电层对应的峰逐渐降低并向右侧铁磁层对应的峰位靠近. 当 ${J_{{\rm{em}}}} = 0.5$ 时, 两个峰合成一个,峰值明显变高, 相变温度也明显变大. 表明在该参数值下该双层膜已耦合为一个整体, 这与图2 所显示的情况一致.

3.1.3 FE/FM薄膜的极化率和磁化率

该双层膜的极化率和磁化率随温度的变化如图4所示. 图4(a)清晰地反映了界面耦合作用对系统极化率的影响. 当 ${J_{{\rm{em}}}} = 0.01$ 时, 曲线有一峰值; 当界面耦合增强为 ${J_{{\rm{em}}}} = 0.1$ 时, 曲线形状发生明显变化, 原峰高度大大降低, 右侧产生一新的峰值, 即受到铁磁层的影响表现明显; 当界面耦合继续增强为 ${J_{{\rm{em}}}} = 0.5$ 时, 曲线峰值明显右移, 且极大值是原来的4倍多. 图4(b)为体系磁化率随温度的变化, 当界面耦合增强时, 曲线的峰值右移且峰值明显增大.

$图 4 双层膜极化率(a)和磁化率(b)随温度的变化\r\nFig. 4. Temperature dependencies of (a) electric susceptibility and (b) magnetic susceptibility.$

图 4 双层膜极化率(a)和磁化率(b)随温度的变化

Fig. 4. Temperature dependencies of (a) electric susceptibility and (b) magnetic susceptibility.

下载: 全尺寸图片幻灯片

对比图4(a)和图4(b)可发现, 当 ${J_{{\rm{em}}}} = 0.5$ 时, 两曲线峰值对应的温度值相同, 即界面耦合作用足够强时, 两层耦合为一个整体. 极化率和磁化率的极大值变大, 转变温度值增高.

3.2 外场下系统的极化磁化行为

研究在外场作用下的FE/FM双层膜结构的磁滞和电滞行为具有实际意义. 这里研究在低温情况下体系在外场中的极化和磁化行为.

3.2.1 磁场下系统的极化磁化行为

图5给出了比较低的温度( $T = 0.08$ )下, 双层膜在外磁场下的极化和磁化过程.

$图 5 双层膜在外磁场中的滞后回线($T = 0.08$) (a) ${J_{{\rm{em}}}} = 0.01$, $0.1$, $0.5$; (b) ${J_{{\rm{em}}}} = 1.0$\r\nFig. 5. Polarization and magnetization loops in external magnetic field at $T = 0.08$: (a) ${J_{{\rm{em}}}} = 0.01$, $0.1$, $0.5$; (b) ${J_{{\rm{em}}}} = 1.0$.$

图 5 双层膜在外磁场中的滞后回线(

$T = 0.08$

) (a)

${J_{{\rm{em}}}} = 0.01$

$0.1$

$0.5$

; (b)

${J_{{\rm{em}}}} = 1.0$

Fig. 5. Polarization and magnetization loops in external magnetic field at

$T = 0.08$

: (a)

${J_{{\rm{em}}}} = 0.01$

$0.1$

$0.5$

; (b)

${J_{{\rm{em}}}} = 1.0$

下载: 全尺寸图片幻灯片

由图5(a)可以看出, 在外磁场下, 铁磁层具有磁滞行为, 铁电层也表现出电滞行为. 当界面之间的耦合作用较小时, 电滞回线面积较小, 铁磁层的磁滞回线表现出偏置行为, 且随着层间耦合作用的增加, 交换偏置现象更加明显. 在比较低的温度下, 铁电层趋于饱和自发极化. 由于铁电层不直接受磁场作用, 只是通过界面耦合作用响应铁磁层的磁化行为, 所以在弱的界面耦合作用下, 电滞回线面积较小. 随着界面耦合作用的增加, 铁电层的响应增大, 电滞现象逐渐明显,对铁磁层而言, 铁电层相当于一个外场作用其上, 因此引发了铁磁层的交换偏置现象. 当界面耦合强度至 ${J_{{\rm{em}}}} = 1.0$ 时(图5(b)), 铁电层和铁磁层同步响应外磁场形成回线行为. 通过数值运算发现, 当界面耦合作用达到 ${J_{{\rm{em}}}} = 0.75$ 时, 铁电层的极化行为能够完全响应铁磁层的磁化行为, 交换偏置现象不复存在.

当温度增加到 $T = 0.4$ 时, 双层膜在外磁场下的滞后回线如图6所示. 发现即使界面耦合比较弱( ${J_{{\rm{em}}}} = 0.01$ ), 交换偏置现象也不存在, 铁电层能够完全响应铁磁层的磁化行为. 这正是由于随着温度升高, 体系受热扰动影响, 体系铁电层的自发极化减弱的缘故. 同样此时的矫顽力远远低于极低温时的矫顽力. 当界面耦合作用增加时, 对应的矫顽力也增大, 剩余极化强度也增加.

$图 6 双层膜在外磁场中的滞后回线($T = 0.4$, ${J_{{\rm{em}}}} = 0.01$, $0.1$, $0.3$, $0.5$)\r\nFig. 6. Polarization and magnetization loops in external magnetic field at $T = 0.4$, ${J_{{\rm{em}}}} = 0.01$, $0.1$, $0.3$, $0.5$.$

图 6 双层膜在外磁场中的滞后回线(

$T = 0.4$

${J_{{\rm{em}}}} = 0.01$

$0.1$

$0.3$

$0.5$

)

Fig. 6. Polarization and magnetization loops in external magnetic field at

$T = 0.4$

${J_{{\rm{em}}}} = 0.01$

$0.1$

$0.3$

$0.5$

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3.2.2 电场下系统的极化磁化行为

在外加电场时, 铁磁层和铁电层也分别表现出磁滞行为和电滞行为, 并显示交换偏置行为, 且其行为因层间耦合作用和温度的不同而发生变化, 变化规律与在外磁场中的情形类似. 不同的是, 即使温度也上升至 $T = 0.4$ , 交换偏置现象也未消失, 这是由于铁磁层各系数的选择使得其磁化状态较难改变, 这对实验制备多铁性薄膜是有实际意义的. 图7给出了电场作用下该双层膜的极化和磁化过程.

$图 7 双层膜在外电场中的滞后回线　(a) $T = 0.08$, ${J_{{\rm{em}}}} = 0.01$, $0.1$; (b) $T = 0.08$, ${J_{{\rm{em}}}} = 0.5$, $1.0$; (c) $T = 0.4$, ${J_{{\rm{em}}}} = 0.01$, $0.1$; (d) $T = 0.4$, ${J_{{\rm{em}}}} = 0.3$, $0.5$\r\nFig. 7. Magnetization and polarization loops in external electric field: (a) $T = 0.08$, ${J_{{\rm{em}}}} = 0.01$, $0.1$; (b) $T = 0.08$, ${J_{{\rm{em}}}} = 0.5$, $1.0$; (c) $T = 0.4$, ${J_{{\rm{em}}}} = 0.01$, $0.1$; (d) $T = 0.4$, ${J_{{\rm{em}}}} = 0.3$, $0.5$.$

图 7 双层膜在外电场中的滞后回线　(a)

$T = 0.08$

${J_{{\rm{em}}}} = 0.01$

$0.1$

; (b)

$T = 0.08$

${J_{{\rm{em}}}} = 0.5$

$1.0$

; (c)

$T = 0.4$

${J_{{\rm{em}}}} = 0.01$

$0.1$

; (d)

$T = 0.4$

${J_{{\rm{em}}}} = 0.3$

$0.5$

Fig. 7. Magnetization and polarization loops in external electric field: (a)

$T = 0.08$

${J_{{\rm{em}}}} = 0.01$

$0.1$

; (b)

$T = 0.08$

${J_{{\rm{em}}}} = 0.5$

$1.0$

; (c)

$T = 0.4$

${J_{{\rm{em}}}} = 0.01$

$0.1$

; (d)

$T = 0.4$

${J_{{\rm{em}}}} = 0.3$

$0.5$

下载: 全尺寸图片幻灯片

FE/FM双层膜结构的磁滞和电滞行为在诸多不同的铁电/铁磁复合薄膜实验中都有体现. 文献[16]的实验结果表明, Co/Co₃O₄/PZT复合薄膜在温度降至77 K时出现交换偏置现象, 且交换偏置场和矫顽场随温度的降低而增大, 当温度达到200 K时交换偏置现象不再出现. 在文献[11]中描述了BiFeO₃-CoFe₂O₄复合薄膜中电场诱发磁矩反转. 文献[16]中复合薄膜与纯PZT薄膜的电滞回线的比较, 以及文献[18]中CoFe₂O₄/(Ba_0.85Ca_0.15)(Ti_0.9Zr_0.1)O₃复合薄膜与纯BCTZO薄膜的电滞回线比较, 都表明铁磁层对电滞回线产生了明显影响. 文献[17]描述了BaTiO₃与缺氧的铁磁绝缘态 ${\rm{L}}{{\rm{a}}_{0.67}}{\rm{S}}{{\rm{r}}_{0.33}}{\rm{Mn}}{{\rm{O}}_{3 - \delta }}$ 复合薄膜在不同温度下磁场对电滞回线的影响, 在40 K时, 施加0.8 T的磁场对电滞回线的影响是显著的. 数值分析结果同这些实验结果相符, 并且给出了外场中极化磁化行为与层间耦合系数的关系, 这对实验制备是有实际意义的.

4. 结　论

利用蒙特卡罗方法研究了一种铁电/铁磁双层膜结构体系的热力学和极化、磁化行为. 模拟结果表明: 该双层膜结构的热力学性质与界面耦合作用的强弱有直接关系. 当界面耦合较弱时, 双层膜表现出各自的热力学性质, 比热和极化率、磁化率都在各自的相变温度处出现尖峰. 当层间耦合增强到一定程度时, 双层膜耦合为一个整体, 表现出统一的热力学性质, 体系有共同的相变温度. 当施加外场时, 体系的极化强度和磁化强度在外场中表现出滞后回线. 在外磁场中, 铁磁层回线表现出偏置行为, 而在外电场中, 铁电层的回线也表现出偏置行为. 界面耦合强度和温度影响滞后回线和偏置现象. 理论计算的结果与实验观测结果相符.

References

[1]	Schmid H 1994 Ferroelectrics 162 317 Google Scholar
[2]	Fiebig M 2005 J. Phys. D: Appl. Phys. 38 R123 Google Scholar
[3]	Eerenstein W, Mathur N D, Scott J F 2006 Nature 442 759 Google Scholar
[4]	Nan T, Hui Y, Rinaldi M, Sun N X 2013 Sci. Rep. 3 1985 Google Scholar
[5]	O'Handley R C, Huang J K, Bono D C, Simon J 2008 IEEE Sens. J. 8 57 Google Scholar
[6]	南策文 2015 中国科学: 技术科学 45 339 Google Scholar Nan C W 2015 Sci. Sin. Tech. 45 339 Google Scholar
[7]	Zheng H, Wang J, Lofland S E, Ma Z, Mohaddes-Ardabili L, Zhao T, Salamanca-Riba L, Shinde S R, Ogale S B, Bai F, Viehland D, Jia Y, Schlom D G, Wuttig M, Roytburd A, Ramesh R 2004 Science 303 661 Google Scholar
[8]	Ryu H, Murugavel P, Lee J H, Chae S C, Noh T W 2006 Appl. Phys. Lett. 89 102907 Google Scholar
[9]	Deng C Y, Zhang Y, Ma J, Lin Y H, Nan C W 2007 J. Appl. Phys. 102 074114 Google Scholar
[10]	Park J H, Jang H M, Kim H S, Park C G, Lee S G 2008 Appl. Phys. Lett. 92 062908 Google Scholar
[11]	Zavaliche F, Zheng H, Mohaddesardabili L, Yang S Y, Zhan Q, Shafer P, Reilly E, Chopdekar R, Jia Y, Wright P, Schlom D G, Suzuki Y, Ramesh R 2005 Nano Lett. 5 1793 Google Scholar
[12]	Greve H, Woltermann E, Quenzer H J, Wagner B, Quandt E 2010 Appl. Phys. Lett. 2010 96 182501 Google Scholar
[13]	He H C, Zhou J P, Wang J, Nan C W 2006 Appl. Phys. Lett. 89 052904 Google Scholar
[14]	Leufke P M, Kruk R, Brand R A, Hahn H 2013 Phys. Rev. B 87 094416 Google Scholar
[15]	Zurbuchen M A, Wu T, Saha S, Mitchell J 2005 Appl. Phys. Lett. 87 232908 Google Scholar
[16]	李永超, 周航, 潘丹峰, 张浩, 万建国 2015 物理学报 64 097701 Google Scholar Li Y C, Zhou H, Pan D F, Zhang H, Wan J G 2015 Acta Phys. Sin. 64 097701 Google Scholar
[17]	王建元, 白健英, 罗炳成, 王拴虎, 金克新, 陈长乐 2018 物理学报 67 017701 Google Scholar Wang J Y, Bai J Y, Luo B C, Wang S H, Jin K X, Chen C L 2018 Acta Phys. Sin. 67 017701 Google Scholar
[18]	Ramana E V, Zavasnik J, Graca M P F, Valente M A 2016 J. Appl. Phys. 120 074108 Google Scholar
[19]	Liu G, Nan C W, Xu Z K, Chen H 2005 J. Phys. D: Appl. Phys. 38 2321 Google Scholar
[20]	Duan C G, Velev J P, Sabirianov R F, Zhu Z, Jaswal S S, Tsymbal E Y 2008 Phys. Rev. Lett. 101 137201 Google Scholar
[21]	Nan C W, Liu G, Lin Y H, Chen H 2005 Phys. Rev. Lett. 94 197203 Google Scholar
[22]	Sukhov A, Jia C L, Horley P P, Berakdar J 2010 J. Phys.: Condens. Matter 22 352201 Google Scholar

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图 1 FE/FM双层膜结构示意图

Figure 1. Schematic of the ferroelectric/ferromagnetic double layer film.

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图 2 自发极化和磁化随温度变化曲线

Figure 2. Temperature dependencies of the spontaneous polarization and spontaneous magnetization of the system.

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图 3 (a)内能和(b)比热随温度的变化

Figure 3. Temperature dependencies of (a) energy and (b) specific heat.

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图 4 双层膜极化率(a)和磁化率(b)随温度的变化

Figure 4. Temperature dependencies of (a) electric susceptibility and (b) magnetic susceptibility.

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图 5 双层膜在外磁场中的滞后回线( $T = 0.08$ ) (a) ${J_{{\rm{em}}}} = 0.01$ , $0.1$ , $0.5$ ; (b) ${J_{{\rm{em}}}} = 1.0$

Figure 5. Polarization and magnetization loops in external magnetic field at $T = 0.08$ : (a) ${J_{{\rm{em}}}} = 0.01$ , $0.1$ , $0.5$ ; (b) ${J_{{\rm{em}}}} = 1.0$ .

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图 6 双层膜在外磁场中的滞后回线( $T = 0.4$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ , $0.3$ , $0.5$ )

Figure 6. Polarization and magnetization loops in external magnetic field at $T = 0.4$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ , $0.3$ , $0.5$ .

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图 7 双层膜在外电场中的滞后回线　(a) $T = 0.08$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ ; (b) $T = 0.08$ , ${J_{{\rm{em}}}} = 0.5$ , $1.0$ ; (c) $T = 0.4$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ ; (d) $T = 0.4$ , ${J_{{\rm{em}}}} = 0.3$ , $0.5$

Figure 7. Magnetization and polarization loops in external electric field: (a) $T = 0.08$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ ; (b) $T = 0.08$ , ${J_{{\rm{em}}}} = 0.5$ , $1.0$ ; (c) $T = 0.4$ , ${J_{{\rm{em}}}} = 0.01$ , $0.1$ ; (d) $T = 0.4$ , ${J_{{\rm{em}}}} = 0.3$ , $0.5$ .

DownLoad: Full-Size Img PowerPoint

[1]	Schmid H 1994 Ferroelectrics 162 317 Google Scholar
[2]	Fiebig M 2005 J. Phys. D: Appl. Phys. 38 R123 Google Scholar
[3]	Eerenstein W, Mathur N D, Scott J F 2006 Nature 442 759 Google Scholar
[4]	Nan T, Hui Y, Rinaldi M, Sun N X 2013 Sci. Rep. 3 1985 Google Scholar
[5]	O'Handley R C, Huang J K, Bono D C, Simon J 2008 IEEE Sens. J. 8 57 Google Scholar
[6]	南策文 2015 中国科学: 技术科学 45 339 Google Scholar Nan C W 2015 Sci. Sin. Tech. 45 339 Google Scholar
[7]	Zheng H, Wang J, Lofland S E, Ma Z, Mohaddes-Ardabili L, Zhao T, Salamanca-Riba L, Shinde S R, Ogale S B, Bai F, Viehland D, Jia Y, Schlom D G, Wuttig M, Roytburd A, Ramesh R 2004 Science 303 661 Google Scholar
[8]	Ryu H, Murugavel P, Lee J H, Chae S C, Noh T W 2006 Appl. Phys. Lett. 89 102907 Google Scholar
[9]	Deng C Y, Zhang Y, Ma J, Lin Y H, Nan C W 2007 J. Appl. Phys. 102 074114 Google Scholar
[10]	Park J H, Jang H M, Kim H S, Park C G, Lee S G 2008 Appl. Phys. Lett. 92 062908 Google Scholar
[11]	Zavaliche F, Zheng H, Mohaddesardabili L, Yang S Y, Zhan Q, Shafer P, Reilly E, Chopdekar R, Jia Y, Wright P, Schlom D G, Suzuki Y, Ramesh R 2005 Nano Lett. 5 1793 Google Scholar
[12]	Greve H, Woltermann E, Quenzer H J, Wagner B, Quandt E 2010 Appl. Phys. Lett. 2010 96 182501 Google Scholar
[13]	He H C, Zhou J P, Wang J, Nan C W 2006 Appl. Phys. Lett. 89 052904 Google Scholar
[14]	Leufke P M, Kruk R, Brand R A, Hahn H 2013 Phys. Rev. B 87 094416 Google Scholar
[15]	Zurbuchen M A, Wu T, Saha S, Mitchell J 2005 Appl. Phys. Lett. 87 232908 Google Scholar
[16]	李永超, 周航, 潘丹峰, 张浩, 万建国 2015 物理学报 64 097701 Google Scholar Li Y C, Zhou H, Pan D F, Zhang H, Wan J G 2015 Acta Phys. Sin. 64 097701 Google Scholar
[17]	王建元, 白健英, 罗炳成, 王拴虎, 金克新, 陈长乐 2018 物理学报 67 017701 Google Scholar Wang J Y, Bai J Y, Luo B C, Wang S H, Jin K X, Chen C L 2018 Acta Phys. Sin. 67 017701 Google Scholar
[18]	Ramana E V, Zavasnik J, Graca M P F, Valente M A 2016 J. Appl. Phys. 120 074108 Google Scholar
[19]	Liu G, Nan C W, Xu Z K, Chen H 2005 J. Phys. D: Appl. Phys. 38 2321 Google Scholar
[20]	Duan C G, Velev J P, Sabirianov R F, Zhu Z, Jaswal S S, Tsymbal E Y 2008 Phys. Rev. Lett. 101 137201 Google Scholar
[21]	Nan C W, Liu G, Lin Y H, Chen H 2005 Phys. Rev. Lett. 94 197203 Google Scholar
[22]	Sukhov A, Jia C L, Horley P P, Berakdar J 2010 J. Phys.: Condens. Matter 22 352201 Google Scholar

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