Magnetic exchange interaction in two-dimensional lattice under generalized Bloch condition

Zhao Hong-Yan; Jiang Ling-Zi; Zhu Yan; Pan Yan-Fei; Fan Ji-Yu; Ma Chun-Lan

doi:10.7498/aps.71.20211317

Abstract

Two-dimensional magnetic material which has been rapidly developed in recent years, has potential applications in developing spintronic devices. In order to understand the magnetic properties of two-dimensional magnetic materials, it is necessary to comprehend the magnetic interaction which is estimated by the exchange parameters between the magnetic atoms. The calculation of the magnetic exchange parameters is based on the first-principle. The commonly used method of determining the values of exchange parameters is energy-mapping. However, this method has some disadvantages. In this paper, the spin-spiral dispersion relationship is derived under the Heisenberg interaction and the Dzyaloshinskii-Moriya (DM) interaction through the generalized Bloch condition of three common two-dimensional magnetic structures: a tetragonal structure, a hexagonal structure in which the cell contains one magnetic atom, a hexagonal structure in which the cell contains two magnetic atoms. The magnetic exchange parameters of some materials are calculated through the first principle. These materials are MnB, VSe ₂ MnSTe and Cr ₂I ₃Cl ₃. Among them, the MnSTe and Cr ₂I ₃Cl ₃ are two-dimensional Janus materials, which means that they have space-reversal symmetry broken, that is why there is DM interaction in the system.

Keywords:

Authors and contacts

1.
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210006, China
2.
Key Laboratory of Aerospace Information Materials and Physics, Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210006, China
3.
Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, China

Corresponding author: Zhu Yan, yzhu@nuaa.edu.cn ; Ma Chun-Lan, wlxmcl@mail.usts.edu.cn

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References

[1]	Bi C, Liu Y H, Newhouse-Illige T, Xu M, Rosales M, Freeland J W, Mryasov O, Zhang S F, Velthuis S G E, Wang W G 2014 Phys. Rev. Lett. 113 267202
[2]	Novoselov K S, Mishchenko A, Carvalho A, Castro Neto A H 2016 Science 353 9439 Google Scholar
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[7]	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 D 2017 Nature 546 270 Google Scholar
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[11]	Dzyaloshinsky I 1958 J. Phys. Chem. Solids 4 241 Google Scholar
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[14]	Zhu Y, Ma C L, Shi D N, Zhang K C 2014 Phys. Lett. A 378 2234 Google Scholar
[15]	Yang H X, Thiaville A, Rohart S, Fert A, Chshiev M 2015 Phys. Rev. Lett. 115 267210 Google Scholar
[16]	Pan Y, Zhu Y, Shi D N, Wei X Y, Ma C L, Zhang K C 2015 J. Alloys Compd. 644 341 Google Scholar
[17]	Xiang H J, Lee C, Koo H J, Gong X G 2013 Dalton Trans. 42 823 Google Scholar
[18]	Marsman M, Hafner J 2002 Phys. Rev. B 66 224409 Google Scholar
[19]	Sandratskii L M 1991 J. Phys. Condens. Matter 3 8565 Google Scholar
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[21]	Hector A L, Jura M, Levason W, Reid S D, Reid G 2009 New J. Chem. 33 641 Google Scholar
[22]	Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[23]	Blöchl P E 1994 Phys. Rev. B 50 17953 Google Scholar
[24]	Farooq M U, Hashmi A, Khan I, Hong J S 2017 Sci. Rep. 7 17101 Google Scholar
[25]	Liu C, Fu B T, Yin H B, Zhang G B, Dong C 2020 Appl. Phys. Lett. 117 103101 Google Scholar
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[29]	Fert A, Cros V, Sampaio J 2013 Nat. Nanotechnol. 8 152 Google Scholar

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图 1 奈尔型斯格明子的磁矩结构示意图

Figure 1. Spin configuration of Néel skyrmions.

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图 2 VSe ₂原子结构示意图　(a) 俯视图; (b) 侧视图

Figure 2. The view of the lattice structure for VSe ₂: (a) Top view; (b) side view.

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图 3 (a)六角结构的原胞基矢和倒格基矢的示意图; (b)六角结构的磁性原子及各近邻原子的分布; (c) 磁性原子的磁矩变化坐标系(蓝色坐标轴)以及二维晶格坐标系(黑色坐标轴)示意图

Figure 3. (a) The labeled a ₁ and a ₂ are basis vectors and b ₁ and b ₂ are reciprocal lattice vectors. (b) distribution of neighboring atoms; (c) blue axis and black axis represent the coordinate system of magnetic moment and two dimensional lattice , respectively.

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图 4 离散点是VSe ₂体系通过程序计算得到的自旋螺旋能量色散关系 E( q ), 其中 q 是自旋螺旋的波矢; 实线是拟合曲线

Figure 4. Scatter symbols are energy dispersion E( q ) as a function of the spiral wave vector q calculated by program, lines are fitted ones.

DownLoad: Full-Size Img PowerPoint

图 5 MnB原子结构示意图　(a) 俯视图; (b) 侧视图

Figure 5. Structure of MnB: (a) Vertical view; (b) side view.

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图 6 离散点是MnB体系通过程序计算得到的自旋螺旋能量色散关系 E( q ), 其中 q 是自旋螺旋的波矢; 实线是拟合曲线

Figure 6. Scatter symbols are energy dispersion E( q ) of MnB as a function of the spiral wave vector q calculated by program, lines are fitted ones.

DownLoad: Full-Size Img PowerPoint

图 7 离散点是MnB体系通过程序计算得到的自旋螺旋能量色散关系 E( q ), 其中 q 是自旋螺旋的波矢; 实线是拟合曲线

Figure 7. Scatter symbols are energy dispersion E( q ) of MnB as a function of the spiral wave vector q calculated by program, lines are fitted ones.

DownLoad: Full-Size Img PowerPoint

图 8 MnSTe原子结构示意图　(a) 俯视图; (b) 侧视图

Figure 8. Atomic structure of MnSTe: (a) Vertical view; (b) side view.

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图 9 (a)离散点是MnSTe体系通过程序计算得到的自旋螺旋能量色散关系 E( q )和 E(– q ), 其中 q 是自旋螺旋的波矢, 实线是拟合曲线; (b)离散点是MnSTe体系通过程序计算的 E( q )与 E(– q )之间的能量差 E _DMI( q ), 实线是拟合曲线

Figure 9. (a) Scatter symbols are energy dispersion E( q ) and E(– q ) of MnSTe as a function of the spiral wave vector q calculated by program, lines are fitted ones; (b) scatter symbols are E _DMI( q ) which means the difference between E( q ) and E(– q ), lines are fitted ones.

DownLoad: Full-Size Img PowerPoint

图 10 Cr ₂I ₃Cl ₃原子结构示意图　(a) 俯视图; (b) 侧视图

Figure 10. Atomic structure of Cr ₂I ₃Cl ₃: (a) Vertical view; (b) side view.

DownLoad: Full-Size Img PowerPoint

图 11 (a)离散点是Cr ₂I ₃Cl ₃体系通过程序计算得到的自旋螺旋能量色散关系 E( q )和 E(– q ), 其中 q 是自旋螺旋的波矢, 实线是拟合曲线; (b)离散点是Cr ₂I ₃Cl ₃体系通过程序计算的 E( q )与 E(– q )之间的能量差 E _DMI( q ), 实线是拟合曲线

Figure 11. (a) Scatter symbols are energy dispersion E( q ) and E(– q ) of Cr ₂I ₃Cl ₃ as a function of the spiral wave vector q calculated by program, lines are fitted ones; (b) scatter symbols are E _DMI( q ) which means the difference between E( q ) and E(– q ), lines are fitted ones.

DownLoad: Full-Size Img PowerPoint

表 1 VSe ₂结构中磁性原子各近邻的海森伯交换参数大小(单位: meV)

Table 1. Calculated parameters of Heisenberg exchange J of VSe ₂, J is considered to the eighth neighbor. (The unit of J is meV).

J ₁	J ₂	J ₃	J ₄	J ₅	J ₆		J ₇	J ₈
27.24	12.81	–2.8	0.59	–1	–1.5		–0.5	0.38

DownLoad: CSV

表 2 MnB结构( U= 3 eV)中磁性原子各近邻的海森伯交换参数大小(单位: meV)

Table 2. Calculated parameters of Heisenberg exchange J of MnB, J is considered to the eighth neighbor. (The unit of J is meV).

J ₁	J ₂	J ₃	J ₄	J ₅	J ₆	J ₇	J ₈
21.22	44.5	–2.52	–4.53	0.952	1.08	0.569	1.27

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表 3 MnSTe结构中磁性原子各近邻的海森伯交换参数大小(单位: meV)

Table 3. Calculated parameters of Heisenberg exchange J of MnSTe, J is considered to the eighth neighbor. (The unit of J is meV).

J ₁	J ₂	J ₃	J ₄	J ₅	J ₆	J ₇	J ₈
5.49	4.64	6	–1.09	–1.32	0.765	–0.046	0.76

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表 4 MnSTe结构中磁性原子各近邻的DM交换参数大小(单位: meV)

Table 4. Calculated parameters of DM exchange d of MnSTe, d is considered to the forth neighbor. (The unit of d is meV).

d ₁	d ₂	d ₃	d ₄
5.46	1.77	2.78	–0.248

DownLoad: CSV

表 5 Cr ₂I ₃Cl ₃结构中磁性原子各近邻的海森伯交换参数大小(单位: meV)

Table 5. Calculated parameters of Heisenberg exchange J of Cr ₂I ₃Cl ₃, J is considered to the eighth neighbor. (The unit of J is meV).

J ₁	J ₂	J ₃	J ₄	J ₅	J ₆	J ₇	J ₈
12.92	–0.35	2.11	3.55	–1.46	–0.67	–0.11	0.841

DownLoad: CSV

表 6 Cr ₂I ₃Cl ₃结构中磁性原子各近邻的DM交换参数大小(单位: meV)

Table 6. Calculated parameters of DM exchange d of Cr ₂I ₃Cl ₃, d is considered to the forth neighbor. (The unit of d is meV).

d ₁	d ₂	d ₃	d ₄
–0.88	0.378	1.04	–0.0493

DownLoad: CSV

[1]	Bi C, Liu Y H, Newhouse-Illige T, Xu M, Rosales M, Freeland J W, Mryasov O, Zhang S F, Velthuis S G E, Wang W G 2014 Phys. Rev. Lett. 113 267202
[2]	Novoselov K S, Mishchenko A, Carvalho A, Castro Neto A H 2016 Science 353 9439 Google Scholar
[3]	Wang Z, Gutiérrez-Lezama I, Ubrig N, Kroner M, Gibertini M, Taniguchi T, Watanabe K, Imamoğlu A, Giannini E, Morpurgo A F 2018 Nat. Commun. 9 2516 Google Scholar
[4]	Hu C, Zhang D, Yan F G, Li Y C, Lv Q S, Zhu W K, Wei Z M, Chang K, Wang K Y 2020 Sci. Bull. 65 1072 Google Scholar
[5]	Mermin N D, Wagner H 1966 Phys. Rev. Lett. 17 1133 Google Scholar
[6]	Gong C, Li L, Li Z L, Ji H W, Stern A, Xia Y, Cao T, Bao W, Wang C Z, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, Zhang X 2017 Nature 546 265 Google Scholar
[7]	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 D 2017 Nature 546 270 Google Scholar
[8]	Deng Y, Yu Y, Song Y, Zhang J Z, Wang N Z, Sun Z Y, Yi Y F, Wu Y Z, Wu S W, Zhu J Y, Wang J, Chen X H, Zhang Y B 2018 Nature 563 94 Google Scholar
[9]	Bonilla M, Kolekar S, Ma Y J, Diaz H C, Kalappattil V, Das R, Eggers T, Gutierrez H R, Phan M H, Batzill M 2018 Nat. Nanotechnol. 13 289 Google Scholar
[10]	O'Hara D J, Zhu T, Trout A H, Ahmed A S, Luo Y Q, Lee C H, Brenner M R, Rajan S, Gupta J, McComb D W, Kawakami R K 2018 Nano. Lett. 18 3125 Google Scholar
[11]	Dzyaloshinsky I 1958 J. Phys. Chem. Solids 4 241 Google Scholar
[12]	Moriya T 1960 Phys. Rev. 120 91 Google Scholar
[13]	Muehlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubauer A, Georgii R, Böni P 2009 Science 323 915 Google Scholar
[14]	Zhu Y, Ma C L, Shi D N, Zhang K C 2014 Phys. Lett. A 378 2234 Google Scholar
[15]	Yang H X, Thiaville A, Rohart S, Fert A, Chshiev M 2015 Phys. Rev. Lett. 115 267210 Google Scholar
[16]	Pan Y, Zhu Y, Shi D N, Wei X Y, Ma C L, Zhang K C 2015 J. Alloys Compd. 644 341 Google Scholar
[17]	Xiang H J, Lee C, Koo H J, Gong X G 2013 Dalton Trans. 42 823 Google Scholar
[18]	Marsman M, Hafner J 2002 Phys. Rev. B 66 224409 Google Scholar
[19]	Sandratskii L M 1991 J. Phys. Condens. Matter 3 8565 Google Scholar
[20]	Knöpfle K, Sandratskii L M, Kübler J 2000 Phys. Rev. B 62 5564 Google Scholar
[21]	Hector A L, Jura M, Levason W, Reid S D, Reid G 2009 New J. Chem. 33 641 Google Scholar
[22]	Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[23]	Blöchl P E 1994 Phys. Rev. B 50 17953 Google Scholar
[24]	Farooq M U, Hashmi A, Khan I, Hong J S 2017 Sci. Rep. 7 17101 Google Scholar
[25]	Liu C, Fu B T, Yin H B, Zhang G B, Dong C 2020 Appl. Phys. Lett. 117 103101 Google Scholar
[26]	Zhang J, Jia S, Kholmanov I, Dong L, Er D, Chen W B, Guo H, Jin Z H, Shenoy V B, Shi L, Lou J 2017 ACS Nano 11 8192 Google Scholar
[27]	He J J, Li S 2018 Comput. Mater. Sci. 152 151 Google Scholar
[28]	Xu C S, Feng J S, Prokhorenko S, Nahas Y, Xiang H J, Bellaiche L 2020 Phys. Rev. B 101 060404 Google Scholar
[29]	Fert A, Cros V, Sampaio J 2013 Nat. Nanotechnol. 8 152 Google Scholar

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