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The influence of electromagnetic field on material characteristics remains a pivotal concern in scientific researches. Nonetheless, in the realm of computational condensed matter physics, the extension of traditional density functional theory to scenarios inclusive of external electromagentic fields poses considerable challenges. These issues largely stem from the disruption of translational symmetry by external fields inherent in periodic systems, rendering Bloch's theorem inoperative. Consequently, the using the first-principles method to calculate material properties in the presence of external fields becomes an intricate task, especially in circumstances where the external field cannot be approximated as a minor perturbation. Over the past two decades, a significant number of scholars within the field of computational condensed matter physics have dedicated their efforts to the formulation and refinement of first-principles computational method adopted in handling periodic systems subjected to finite external fields. This work attempts to systematically summarize these theoretical methods and their applications in the broad spectrum, including but not limited to ferroelectric, piezoelectric, ferromagnetic, and multiferroic domains. In the first part of this paper, we provide a succinct exposition of modern theory of polarization and delineate the process of constructing two computation methods in finite electric fields predicated by this theory in conjunction with density functional theory. The succeeding segment focuses on the integration of external magnetic fields into density functional theory and examining the accompanying computational procedures alongside the challenges they present. In the third part, we firstly review the first-principles effective Hamiltonian method, which is widely used in the study of magnetic, ferroelectric and multiferroic systems, and its adaptability to the case involving external fields. Finally, we discuss the exciting developments of constructing effective Hamiltonian models by using machine learning neural network methods , and their extensions according to the external fields.
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Keywords:
- first-principles calculation /
- electromagnetic field /
- effective Hamiltonian /
- machine learning
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图 1 两种原子构成的系统的不同原胞选择[21], 两原子电荷分别为$ {Z}_{1}=+e $(空心圆)和$ {Z}_{2}=+3 e $(阴影圆) (a), (b)包含了完整的原子, 但相对位置不同; (c)原胞由一个完整的$ +e $电荷和4个$ +3 e/4 $电荷组成
Figure 1. Possible choices of unit cell for a system composed of two types of atoms having ionic charges $ {Z}_{1}=+e $ (open circles) and $ {Z}_{2}=+3 e $ (shaded circles) [21]: (a), (b) Unit cell is specified by two complete basis ions, but in different relative orientations; (c) unit cell is specified by “split basis” consisting of one complete $ +e $ charge and four charges $ +3 e/4 $ in a symmetric arrangement
图 2 固定电压法和固定电场法的差异[45]. 当材料发生应变$ \eta $时, 如果保持电压$ {{\Delta }}V $不变, 电场从$ E= {{{\Delta }}V}/{d} $变化为$ E= {{{\Delta }}V}/{[(1+\eta )d]} $; 如果保持电场$ E $不变, 电压从$ {{\Delta }}V $变为$ (1+ \eta ){{\Delta }}V $
Figure 2. Differences between fixed-E method and fixed-voltage method[45]. When a strain $ \eta $ is applied to the material, electric field will change from $ E= {{{\Delta }}V}/{d} $ to $ E= $$ {{{\Delta }}V}/{[(1+\eta )d]} $ if voltage is held fixed, or voltage will change from $ {{\Delta }}V $ to $ (1+\eta ){{\Delta }}V $ if electric field is held fixed
图 3 施加场的方向约束在[001], [110]或[111]方向时, PbTiO3中形式为$ \varepsilon \left(D\right) $($ \varepsilon $为外电场)(a)、$ D\left(P\right) $ (b)和$ P\left(\varepsilon \right) $ (c) 的电状态方程[51], 采取原子单位制
Figure 3. Electric equations of state of the form $ \varepsilon \left(D\right) $ (a), $ D\left(P\right) $ (b), and $ P\left(\varepsilon \right) $ (c) in PbTiO3, plotted for fields constrained to lie along the [001], [110], or [111] directions[51]. All units are a.u..
图 4 Cr2O3的横向响应贡献[70] . 固定离子响应$ {\alpha }^{\left({\mathrm{e}}{\mathrm{l}}\right)} $(空心方块)的贡献约为总响应的四分之一(实心方块); 响应的剩余部分(空心圆)来自于外场下的结构畸变, 利用波恩有效电荷计算得到
Figure 4. Contributions to the transverse response of Cr2O3 [70]. The clamped-ion response, $ {\alpha }^{\left({\mathrm{e}}{\mathrm{l}}\right)} $(open squares) contributes approximately one fourth of the total response (filled circles). The remainder of the response, computed using Born effective charges, is due to structural distortions in the applied field (open circles).
图 5 实空间波函数$ \psi \left(x, y\right) $通过两次傅里叶变换到倒空间函数$ c $[112] (a)当$ B=0 $时, $ f(x, {K}_{y}) $可被视为一系列一维周期函数或圆环; (b)当$ B=2{\mathrm{\pi }}/\left(ab\right) $时, MPBC使其变为一条长螺旋线. 由此介空间和倒空间内的波函数可等效为一维函数
Figure 5. The real-space wave function $ \psi (x, y) $ can be Fourier transformed into reciprocal space $ c $ in two steps[112]: (a) At $ B=0 $, $ f(x, {K}_{y}) $ can be regarded as a set of one-dimensional periodic functions, or rings; (b) at $ B=2{\mathrm{\pi }}/\left(ab\right) $, MPBC requires to be a long spiral. The resulting wavefunction in intermediate and reciprocal space is effectively one dimensional.
图 6 致密氘流体在零场和强场下的电子结构[112] (a)总电荷密度在B从0升到104 T时基本保持一致; (b) B = 0 (蓝色)和 B = 104 T (红色)时HOMO态在不同原子上的电荷密度分布
Figure 6. Electronic structure of dense deuterium fluid under zero and intense magnetic fields[112]: (a) Total charge density remains essentially the same as B goes from 0 to 104 T; (b) the charge densities of the HOMO state for B = 0 (blue) and B = 104 T (red) are distributed on different atoms.
图 7 CrGe(Se, Te)3 Janus单层的磁场-温度相图[173]. 相边界由热容、磁化率、局域自旋手性决定. 这8个相描述为破碎迷宫畴、斯格明子与嵌套斯格明子合相(I)、迷宫畴(II)、破碎迷宫畴与斯格明子混合相(III)、孤立斯格明子与嵌套斯格明子混合相(IV)、孤立斯格明子(V)、杂化斯格明子相(VI, 部分斯格明子合并, 部分斯格明子保持分离)、饱和铁磁态(VII)、顺磁态(VIII). 如图所示为相III ($ B=1.8{\mathrm{T}} $, $ T=4.14{\mathrm{K}} $)、相IV ($ B=1.8{\mathrm{T}} $, $ T=4.14{\mathrm{K}} $)、相V ($ B= $$ 2.4{\mathrm{T}} $, $ T=4.14{\mathrm{K}} $) 和相VI ($ B=2.4{\mathrm{T}} $, $ T=13.3{\mathrm{K}} $) 的代表性自旋结构
Figure 7. Magnetic field versus temperature phase diagram of the studied CrGe(Se, Te)3 Janus monolayer[173]. The phase boundaries are determined by heat capacity, magnetic susceptibility, local spin chirality, as well as snapshots. The eight phases depicted are as follows: fragmented labyrinth domain, skyrmion and skyrmionium mixed phase (I), labyrinth domain (II), fragmented labyrinth domain and skyrmion mixed phase (III), isolated skyrmion and skyrmionium mixed phase (IV), isolated skyrmion (V), hybrid skyrmion phase (VI, for which some skyrmions merge together but others remain isolated), saturated ferromagnetic state (VII), and paramagnetic state (VIII). Representative spin textures are shown for phase III ($ B=2.4{\mathrm{T}} $, $ T=4.14{\mathrm{K}}) $, phase IV ($ B=1.8{\mathrm{T}} $, $ T=4.14{\mathrm{K}} $), phase V ($ B=2.4{\mathrm{T}} $, $ T=4.14{\mathrm{K}} $), and phase VI ($ B=2.4{\mathrm{T}} $, $ T=13.3{\mathrm{K}} $).
图 8 铁电材料PbSc0.5Ta0.5O3 的电热效应[195] (a) 铁电材料PbSc0.5Ta0.5O3的极化强度$ P(\tilde{E}, T) $关于沿$ \langle 111\rangle $方向施加的电场$ \tilde{E} $和温度$ T $的函数; (b) 电热系数$ \alpha $在330 K时随电场E的函数关系. 在研究温度下时使$ \alpha $达到极大值的电场[$ \tilde{E}\left({\alpha }_{{\mathrm{m}}{\mathrm{a}}{\mathrm{x}}}\right) $]和固定温度时使得$ r\langle 11\bar{1}\rangle $达到最大值的电场[$ \tilde{E}\left(r\langle 11\bar{1}\rangle \right) $]也标记在图(a)中. $ {\chi }^{2}\tilde{E} $也在图(b)中标出以与$ \alpha $做对比. $ r\langle 11\bar{1}\rangle $是大致沿$ \left[11\bar{1}\right] $, $ \left[\bar{1}11\right] $或$ \left[\bar{1}11\right] $方向的局域偶极矩的比例
Figure 8. Electrocaloric effects of ferroelectric PbSc0.5Ta0.5O3 [195] : (a) Polarization $ P(\tilde{E}, T) $ of as a function of electric field $ \tilde{E} $ applied along $ \langle 111\rangle $ direction and temperature $ T $; (b) electrocaloric coefficient $ \alpha $ as a function of electric field at 330 K. The electric field for which α exhibits its maximum [$ \tilde{E}\left({\alpha }_{{\mathrm{m}}{\mathrm{a}}{\mathrm{x}}}\right) $] and the electric field at which $ r\langle 11\bar{1}\rangle $ exhibits its maximum [$ \tilde{E}\left(r\langle 11\bar{1}\rangle \right) $] for the investigated temperatures are shown in panel (a). $ {\chi }^{2}\tilde{E} $ is shown in panel (b) to compare it with α.$ r\langle 11\bar{1}\rangle $ is defined as the percentage of local dipoles lying near $ \left[11\bar{1}\right] $, $ \left[\bar{1}11\right] $, or $ \left[\bar{1}11\right] $ directions.
图 9 块体BiFeO3在各种$ {C}_{1} $和$ {C}_{2} $值下预测的磁性结构[211], 其中$ {C}_{1} $和$ {C}_{2} $分别是第一近邻和第二近邻的反自旋-电流相互作用系数 (a) 计算得到的相图与$ {C}_{1} $和$ {C}_{2} $的函数关系, 蓝色十字标志和黑色圆点分别代表来自前人选取的$ {C}_{1}{\mathrm{值}} $(此时$ {C}_{2}=0 $)和$ {C}_{2} $值(此时$ {C}_{1}=0 $), 蓝色三角表示通过密度泛函理论计算得到的结果, 黑线是磁场大小为18 T时$ \left[\bar{1}10\right] $螺旋相向反铁磁相转变的临界相; (b) 图示展示了5种类型的磁螺旋的传播方向, 对于每种类型, 红色、蓝色和黄色分别代表了不同传播方向的等效磁螺旋
Figure 9. Predicted magnetic structures at various $ {C}_{1} $ and $ {C}_{2} $ values for bulk BiFeO3 [211], $ {C}_{1} $ and $ {C}_{2} $ are coefficients of inverse spin-current interaction for 1st nearest neighbors and 2nd nearest neighbors, respectively: (a) Calculated phase diagram as functions of $ {C}_{1} $ and $ {C}_{2} $, the blue cross symbols and black circles are $ {C}_{1} $ (with $ {C}_{2}=0 $) or $ {C}_{2} $ (with $ {C}_{1}=0) $ values from previous studies, respectively, the blue triangle symbols are calculated by density functional theory, the black lines are determined by considering the critical magnetic field of 18 T changing the $ \left[\bar{1}10\right] $ cycloid to antiferromagnetism; (b) illustration of the propagation directions of the five types of cycloids, for each type, equivalent cycloids of different propagation directions are shown in red, blue, and yellow colors.
图 10 SpinGNN框架[229], SpinGNN包含海森伯边图神经网络(HEGNN)和自旋-距离边图神经网络(SEGNN) (a) HEGNN利用更新后的GNN边特征作为Heisenberg系数, 构建Heisenberg型的磁势; (b) SEGNN利用自旋-距离边晶体图, 以自旋矢量的点乘和键长初始化边, 可以构建一般的高阶磁势, $ \parallel $表示拼接
Figure 10. The SpinGNN framework [229], illustration of the SpinGNN including the Heisenberg Edge GNN (HEGNN) and Spin-Distance Edge GNN (SEGNN): (a) HEGNN utilizes the updated edge feature of GNN as the Heisenberg coefficients and builds a Heisenberg-based magnetic potential; (b) SEGNN utilizes the spin-distance edge crystal graph which initializes the edge with the dot product of the spin vector and bond length and builds a high-order general magnetic potential, $ \parallel $ denotes concatenation.
表 1 用固定电场方法计算的一些III-V半导体介电性质与实验的比较[42], 其中玻恩有效电荷张量在材料对称性下退化为标量, 且$ {d}_{123} $定义为$ {\chi }_{123}^{\left(2\right)}/2 $, LDA和PBE是计算时使用的交换关联泛函近似
Table 1. Computed dieletric properties of some III–V semiconductors by means of fixed-E method compared to experiment[42], Born effective charge tensor degenerates to a scalar due to the symmetry of the material and $ {d}_{123} $ is defined as $ {\chi }_{123}^{\left(2\right)}/2 $, LDA and PBE are different exchange-correlation functional approximations used in calculation.
Compound $ {Z}_{{\mathrm{A}}{\mathrm{l}}}^{*} $ $ {\varepsilon}_{{\mathrm{s}}{\mathrm{t}}{\mathrm{a}}{\mathrm{t}}{\mathrm{i}}{\mathrm{c}}} $ $ {\varepsilon}_{\infty } $ d123/(pm·V–1) AlP (LDA) 2.22 10.26 8.01 21.5 (PBE) 2.23 10.09 7.84 23.2 (Expt.) 2.28 9.8 7.5 AlAs (LDA) 2.18 11.05 8.75 32.7 (PBE) 2.17 10.89 8.80 38.8 (Expt.) 2.20 10.16 8.16 32 AlSb (LDA) 1.84 12.54 11.17 98.3 (PBE) 1.83 12.83 11.45 103 (Expt.) 1.93 11.68 9.88 98 
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