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目前对二维铁磁体的研究主要集中在范德瓦尔斯材料领域, 而无应力束缚的自支撑二维钙钛矿薄膜的成功制备为设计范德瓦耳斯材料之外的二维铁磁体提供了良好的契机. 钙钛矿氧化物SrRuO3作为典型的钙钛矿巡游铁磁体, 在诸多领域具有广阔应用前景. 本文结合第一性原理计算、对称性分析和蒙特卡罗模拟方法研究了其钙钛矿单层(化学式Sr2RuO4)的晶格动力学、基态结构、电子与磁性质以及电场调控效应, 并揭示了哈伯德参量U的影响. 证实单层基态结构为八面体旋转畸变产生的结构相(空间群$P4/mbm$), 具有铁磁半金属性质和面外易磁化轴. 铁磁性主要源于最近邻自旋之间的强铁磁交换作用. 利用自洽测定U值模拟出的居里温度为177 K, 与其块体相的值比较接近. 外加电场可以显著调制其电子和磁性质, 甚至诱导铁磁半金属相到铁磁金属相的转变. 本文为开发基于钙钛矿的二维铁磁体及利用电场调控二维磁性提供了借鉴.At present, the research on two-dimensional (2D) ferromagnets is mainly concentrated on van der Waals materials, while the successful preparation of strain-free freestanding 2D perovskite films provides a great opportunity for designing 2D ferromagnets beyond van der Waals materials. Perovskite oxide SrRuO3, a typical perovskite itinerant ferromagnet, has broad application prospects in many fields. In this work, the lattice dynamics, ground-state structure, electronic and magnetic properties of its perovskite monolayer with formula Sr2RuO4, as well as the effect of external electric field, are studied by combining first-principles calculation, symmetry analysis and Monte Carlo simulation. The influence of the Hubbard parameter U is also revealed. The results indicate that the ground-state structure under all U values presents the structural phase (space group P4/mbm) generated by octahedral rotation distortion. Similar to the SrRuO3 bulk, Sr2RuO4 has a monolayer ground-state phase that exhibits ferromagnetism, which is independent of the U value and thus robust. Density functional theory calculation using Hubbard parameter U predicts the ground-state phase of the monolayer to be a ferromagnetic half metal with an out-of-plane easy-magnetization axis, while excluding that the U parameter predicts the ground-state phase to be a ferromagnetic metallic state. The ferromagnetism mainly originates from the strong ferromagnetic exchange interaction between the nearest neighbor spin pairs. The simulated Curie temperature of the Sr2RuO4 monolayer is 177 K, which is close to the value (150 K) of its bulk phase. The out-of-plane electric field does not change the ground-state structure nor ferromagnetism of the Sr2RuO4 monolayer, but can significantly modulate its electronic property and magnetic property. When an external electric field exceeding 0.3 V/Å is applied, the system undergoes a transition from a ferromagnetic half-metal state to a ferromagnetic metallic state. This work indicates the potential application of Sr2RuO4 monolayer in low-dimensional spintrnic devices, and provides a reference for developing perovskite-based 2D ferromagnets and realizing the control of 2D magnetism by electric field.
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Keywords:
- two-dimensional ferromagnetism /
- perovskite /
- first principles
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图 1 Sr2RuO4单层原型相的(a)晶体结构和(b)声子谱
Fig. 1. (a) Schematic crystal structure and (b) phonon spectrum of the prototype phase of Sr2RuO4 monolayer.
图 2 Sr2RuO4单层每种晶格畸变模式的能量随模式幅度的变化, 涉及(a) 旋转(R)和倾斜(T), 以及(b) JT1和JT2扭曲模式. 插图为四种晶格扭曲模式的示意图, 其中弯曲的箭头代表八面体转动方向, 而直箭头表示阴离子位移方向
Fig. 2. Energy variation of each distortion mode with the mode amplitude for the Sr2RuO4 monolayer, involves (a) rotation (R) and tilt (T), and (b) JT1 and JT2 modes. The illustration shows four lattice distortion modes, where the curved arrows represent the direction of octahedral rotation and the straight arrows indicate the direction of anion displacement
图 3 Sr2RuO4单层基态相的(a) 晶体结构, (b) 声子谱, (c) 300 K 下的第一性分子动力学模拟
Fig. 3. Crystal structure (a), phonon spectrum (b), and first-principles molecular dynamics simulation (at 300 K) (c) of the ground-state phase of Sr2RuO4 monolayer.
图 4 Sr2RuO4单层基态相在不同U值下的上自旋和下自旋投影能带结构 (a) $ U_{{\mathrm{eff}}} = 1.2 $ eV; (b) $ U_{{\mathrm{eff}}} = 2 $ eV; (c) $ U_{{\mathrm{eff}}} = 0 $ eV
Fig. 4. Projected up and down spin band structures of the ground-state phase of Sr2RuO4 monolayer at different U values: (a) $ U_{{\mathrm{eff}}} = 1.2 $ eV; (b) $ U_{{\mathrm{eff}}} = 2 $ eV; (c) $ U_{{\mathrm{eff}}} = 0 $ eV.
图 5 (a) 不同反铁磁序的示意图和相对能量(相对于铁磁相), 以及$ U_{{\mathrm{eff}}} = 1.2 $ eV 时计算的(b)不同自旋面的磁各向异性能量曲线和 (c) 蒙特卡罗模拟的磁化和比热曲线
Fig. 5. (a) Schematic of different antiferromagnetic orders and their relative energy (to ferromagnetic order) for the ground-state phase. (b) Energy curves of magnetic anisotropy for different spin planes, and (c) magnetization and specific heat curves in Monte Carlo simulation calculated at $ U_{{\mathrm{eff}}} = 1.2 $ eV.
图 6 (a)上自旋带隙和磁矩及(b)磁交换相互作用、磁各向异性常数、居里温度随电场强度的变化
Fig. 6. (a) Variation of the up-spin band gap and magnetic moment, and (b) magnetic exchange interaction, magnetic anisotropy constant and Curie temperature with the electric-field intensity.
表 1 Sr2RuO4单层不同晶格畸变模式及其各种组合所产生的结构相的对称性和不同U值下的相对能量, 其中基态相的能量被设置为0. 符号“—”表示对应的结构相不稳定, 即经过结构优化后转变为其他结构相
Table 1. Symmetry and the relative energy at different U values of structural phases resulting from different distortion modes and their various combinations for the Sr2RuO4 monolayer. Energy of the ground-state phase is set to 0. The symbol “—” represents that the corresponding structural phase is unstable, that is, it is transformed into another structural phases after structural optimization.
Distortion modes Space group $ \Delta E $/(meV·f.u.–1) $ U_{{\mathrm{eff}}} = 0 $ $ U_{{\mathrm{eff}}} = 0.5 $ $ U_{{\mathrm{eff}}} = 1.2 $ $ U_{{\mathrm{eff}}} = 1.5 $ $ U_{{\mathrm{eff}}} = 2 $ $ {\mathrm{Para}} $ $ P4/mmm $ 154 172 225 227 214 ${\mathrm{ R}}(M_2^+) $ $ P4/mbm $ 0 0 0 0 0 $ {\mathrm{T}}(a, 0)(M_5^+) $ $ Pmna $ — — 228 225 214 $ {\mathrm{T}}(a, a)(M_5^+) $ $ Cmma $ — — 227 224 212 $ {\mathrm{JT}}_1(M_3^+) $ $ P4/mbm $ — — — — — $ {\mathrm{JT}}_2(M_4^+) $ $ P4/mmm $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{JT}}_1 $ $ Pbam $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{JT_2}} $ $ P4/m $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{T}}(a, 0) $ $ P2_1/c $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{T}}(a, a) $ $ C2/m $ — — — — — $ {\mathrm{T}}(a, 0) \oplus {\mathrm{JT}}_1 $ $ P2_1/c $ — — — — — $ {\mathrm{T}}(a, 0) \oplus {\mathrm{JT_2}} $ $ P2/m $ — — — — — $ {\mathrm{T}}(a, a) \oplus{\mathrm{ JT}}_1 $ $ C2/m $ — — — — — $ {\mathrm{T}}(a, a) \oplus {\mathrm{JT}}_2 $ $ C2/m $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{T}}(a, 0) \oplus {\mathrm{JT}}_1 $ $ P2_1/c $ — — — — — ${\mathrm{ R}} \oplus {\mathrm{T}}(a, 0) \oplus {\mathrm{JT}}_2 $ $ P\bar{1} $ — — — — — $ {\mathrm{R}} \oplus{\mathrm{ T}}(a, a) \oplus {\mathrm{JT}}_1 $ $ P\bar{1} $ — — — — — $ {\mathrm{R}} \oplus {\mathrm{T}}(a, a) \oplus {\mathrm{JT}}_2 $ $ P\bar{1} $ — — — — — 
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表 2 不同U值下计算的最近邻($ J_1 $)和次近邻($ J_2 $)磁交换作用参量, 磁各向异性常数(K), 磁矩(M)及居里温度($ T_{\mathrm{C }}$)
Table 2. Nearest neighbor ($ J_1 $) and next nearest neighbor ($ J_2 $) magnetic exchange interaction parameters, magnetic anisotropy constant (K), magnetic moment (M) and Curie temperature ($ T_{\mathrm{C}} $) calculated at different U values.
$ U_{{\mathrm{eff}}} $/eV $ J_1 $/meV $ J_2 $/meV K/meV M/$ \mu_{\mathrm{B}} $ $ T_{\mathrm{C}} $/K 0 11.48 –1.35 1.57 0.73 81 0.5 15.39 1.73 1.11 0.96 111 1.2 25.34 –2.41 1.83 1.43 177 1.5 31.18 –3.90 1.70 1.44 195 2 38.38 –8.53 1.81 1.47 202
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[1] Burch K S, Mandrus D, Park J G 2018 Nature 563 47
Google Scholar
[2] Gong C, Zhang X 2019 Science 363 706
Google Scholar
[3] 肖寒, 弭孟娟, 王以林 2021 70 127503
Google Scholar
Xiao H, Mi M J, Wang Y L 2021 Acta Phys. Sin. 70 127503
Google Scholar
[4] Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, Zhang X 2017 Nature 546 265
Google Scholar
[5] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P, Xu X 2017 Nature 546 270
Google Scholar
[6] Song T C, Cai X H, Tu M W Y, Zhang X, Huang B, Wilson N P, Seyler K L, Zhu L, Taniguchi T, Watanabe K, McGuire M A, Cobden D H, Xiao D, Yao W, Xu X D 2018 Science 360 1214
Google Scholar
[7] Klein D R, MacNeill D, Lado J L, Soriano D, Navarro-Moratalla E, Watanabe K, Taniguchi T, Manni S, Canfield P, Fernández-Rossier J, Jarillo-Herrero P 2018 Science 360 1218
Google Scholar
[8] Kim H H, Yang B, Patel T, Sfigakis F, Li C, Tian S, Lei H, Tsen A W 2018 Nano Lett. 18 4885
Google Scholar
[9] Wang Z, Zhang T Y, Ding M, et al. 2018 Nat. Nanotechnol. 13 554
Google Scholar
[10] Jiang X, Liu Q X, Xing J P, Liu N S, Guo Y, Liu Z F, Zhao J J 2021 Appl. Phys. Rev. 8 031305
Google Scholar
[11] Dagotto E, Hotta T, Moreo A 2001 Phys. Rep. 344 1
Google Scholar
[12] Dagotto E 2005 Science 309 257
Google Scholar
[13] Eerenstein W, Mathur N D, Scott J F 2006 Nature 442 759
Google Scholar
[14] Hong S S, Yu J H, Lu D, Marshall A F, Hikita Y, Cui Y, Hwang H Y 2017 Sci. Adv. 3 eaao5173
Google Scholar
[15] Ji D X, Cai S H, Paudel T R, Sun H Y, Zhang C C, Han L, Wei Y F, Zang Y P, Gu M, Zhang Y, Gao W P, Huyan H X, Guo W, Wu D, Gu Z B, Tsymbal E Y, Wang P, Nie Y F, Pan X Q 2019 Nature 570 87
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Google Scholar
[17] Zhang J T, Shen X F, Wang Y C, Ji C, Zhou Y, Wang J L, Huang F Z, Lu X M 2020 Phys. Rev. Lett. 125 017601
Google Scholar
[18] Zhang J T, Zhou Y, Wang F, Shen X F, Wang J L, Lu X M 2022 Phys. Rev. Lett. 129 117603
Google Scholar
[19] Zhou Y, Dong S, Shan C X, Ji K, Zhang J T 2022 Phys. Rev. B 105 075408
Google Scholar
[20] Shen X F, Wang F, Lu X M, Zhang J T 2023 Nano Lett. 23 735
Google Scholar
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Google Scholar
[22] Gertjan K, Lior K, Wolter S, Guus R, Steven D J, Chang-Beom E, A B D H, R B M 2012 Rev. Mod. Phys. 84 253
Google Scholar
[23] Anwar M S, Lee S R, Ishiguro R, Sugimoto Y, Tano Y, Kang S J, Shin Y J, Yonezawa S, Manske D, Takayanagi H, Noh T W, Maeno Y 2016 Nat. Commun. 7 13220
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[27] Kresse G, Hafner J 1993 Phys. Rev. B 47 558(R
Google Scholar
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Google Scholar
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Google Scholar
[30] Şaşıoğlu E, Friedrich C, Blügel S 2011 Phys. Rev. B 83 121101
Google Scholar
[31] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J, Sutton A P 1998 Phys. Rev. B 57 1505
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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