Effect of interlayer exchange coupling interaction on topological phase of a bilayer honeycomb Heisenberg ferromagnet

Shi Hong-Chao; Tang Bing; Liu Chao-Fei

doi:10.7498/aps.73.20240437

Abstract

Layered magnetic topological materials are material systems that exhibit both magnetic ordering and topological properties in their smallest two-dimensional units. Studying these systems may lead to the observation of new physical properties and phenomena, which has attracted considerable attention from researchers. The effect of interlayer exchange coupling interactions on bilayer honeycomb Heisenberg ferromagnets with interlayer coupled topological phase is investigated by using linear spin wave theory. The influence of introducing two additional types of interactions, i.e. interlayer exchange coupling interaction and interlayer easy-axis anisotropy interaction, on the topological phase transition are also explored in this work. By calculating the magnon dispersion relations at various interlayer exchange coupling interaction intensities, it is found that the band gaps of high energy band and low energy band both close and reopen at the Dirac points when the system reaches the critical value of interlayer exchange coupling interaction. In magnon systems, such physical phenomena typically relate to topological phase transitions. When calculating the Berry curvature and Chern numbers for the bands in the aforementioned process, it is found that the sign of the Berry curvature reverses and the Chern numbers change when the critical value of interlayer exchange coupling interaction strength is reached, confirming that a topological phase transition occurs indeed. Introducing two other types of interlayer exchange coupling interactions in this process can lead various novel topological phases to occur in the system. The enhancement of interlayer easy-axis anisotropy interactions is likely to impede the topological phase transitions occurring in the system. We find that a major distinction between bilayer honeycomb ferromagnets and their single-layer counterparts lies in the fact that during a topological phase transition, the sign of the magnon thermal Hall coefficient does not change; on the contrary, abrupt shift in the thermal Hall coefficient curve occurs which can be seen as an indicator of topological phase transition of bilayer honeycomb ferromagnets, and is also reflected in the change in magnon Nernst coefficient. The research results of this work can provide theoretical support for developing novel spintronic devices with enhanced information transmission capabilities by using bilayer honeycomb ferromagnetic materials, and can also provide theoretical reference for studing other bilayer ferromagnetic systems.

Keywords:

Authors and contacts

1.
School of Physics and Mechanical & Electrical Engineering, Jishou University, Jishou 416000, China
2.
School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China

Corresponding author: Tang Bing, bingtangphy@jsu.edu.cn ; Liu Chao-Fei, liuchaofei@jxust.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12064011, 12375014, 11875149), the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 23A0404) and the School-level Graduate Scientific Research Project Foundation of Jishou University, China (Grant No. Jdy23052).

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References

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图 1 双层蜂窝状铁磁体晶格结构　(a)侧视图; (b)俯视图; (c)晶格矢量, 最近邻矢量$ {{\boldsymbol{\delta }}_n} $和次近邻矢量$ {{\boldsymbol{\varsigma }}_n} $分别用红色和蓝色箭头表示; (d) 第一布里渊区高对称路径$ { M} {\text{-}} { K}' {\text{-}} \varGamma {\text{-}} { K} {\text{-}} { M} $

Figure 1. Lattice structure of the bilayer honeycomb ferromagnet: (a) Side view; (b) top view; (c) the lattice vector, the nearest and next-nearest neighbor vectors, $ {{\boldsymbol{\delta }}_n} $ and $ {{\boldsymbol{\varsigma }}_n} $, are represented by red and blue arrows, respectively; (d) the high symmetric path $ { M} {\text{-}} { K}' {\text{-}} \varGamma {\text{-}} { K} {\text{-}} { M} $ in the first Brillouin zone.

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图 2 双层蜂窝状铁磁体能带结构　(a) $ {J_0} = 0.1 $; (b) $ {J_0} = 0.245 $; (c) $ {J_0} = 0.3 $; (d) $ {J_0} = 0.505 $; (e) $ {J_0} = 0.9 $, 其余参数设置为$ \varGamma '{=}0.1 $, $ {J_1} = {J_2} = 0 $; (f) 带隙图

Figure 2. Magnon band structures of the bilayer honeycomb ferromagnet: (a) $ {J_0} = 0.1 $; (b) $ {J_0} = 0.245 $; (c) $ {J_0} = 0.3 $; (d) $ {J_0} = $$ 0.505 $; (e) $ {J_0} = 0.9 $, the other parameters are set to $ \varGamma ' = 0.1 $, $ {J_1} = {J_2} = 0 $; (f) gaps as a function of $ {J_0} $.

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图 3 双层蜂窝状铁磁体最低能带对应的贝里曲率　(a) $ {J_0} = 0.2 $; (b) $ {J_0} = 0.5 $; (c) $ {J_0} = 0.51 $; (d) $ {J_0} = 0.8 $. 双层蜂窝状铁磁体最高能带对应的贝里曲率 (e) $ {J_0} = 0.15 $; (f) $ {J_0} = 0.24 $; (g) $ {J_0} = 0.25 $; (h) $ {J_0} = 0.8 $, 其余参数设置为$ \varGamma '{=}0.1 $, $ {J_1} = {J_2} = 0 $

Figure 3. Berry curvature of the lowest band in a bilayer honeycomb ferromagnet: (a) $ {J_0} = 0.2 $; (b) $ {J_0} = 0.5 $; (c) $ {J_0} = 0.51 $; (d) $ {J_0} = 0.8 $. Berry curvature of the highest band in a bilayer honeycomb ferromagnet: (e) $ {J_0} = 0.15 $; (f) $ {J_0} = 0.24 $; (g) $ {J_0} = $$ 0.25 $; (h) $ {J_0} = 0.8 $. Other parameters are set to $ \varGamma '{=}0.1 $ and $ {J_1} = {J_2} = 0 $.

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图 4 不同层间易轴各向异性相互作用强度下的陈数随$ {J_0} $强度变化曲线　(a)最低能带; (b)最高能带, 其余参数设置为$ \varGamma ' = 0.1, {J_1} = {J_2} = 0 $

Figure 4. Chern number as a function of the intensity of interlayer exchange coupling interaction $ {J_0} $ for the different intensity of interlayer easy-axis anisotropy interaction: (a) The lowest band; (b) the highest band, the other parameters are set to $ \varGamma ' = 0.1, $$ {J_1} = {J_2} = 0 $.

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图 5 陈数随层间交换耦合相互作用J₀和J₁强度变化图　(a)—(d)分别对应能量从高到低的4条能带, 其余参数设置为$ \varGamma ' = 0.1 $

Figure 5. Chern number as a function of the intensity of the interlayer exchange coupling interaction $ {J_0} $ and $ {J_1} $: (a)–(d) Correspond to four energy bands from high to low energy, the other parameters are set to $ \varGamma ' =0.1 $.

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图 6 陈数随层间交换耦合相互作用J₀和D强度变化图　(a)—(d)分别对应能量从高到低的4条能带, 其余参数设置为$ \varGamma ' = 0.1 $

Figure 6. Chern number as a function of the intensity of the interlayer exchange coupling interaction $ {J_0} $ and $ D $: (a)–(d) Correspond to four energy bands from high to low energy, the other parameters are set to $ \varGamma ' = 0.1 $.

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图 7 (a) 不同$ {J_0} $强度下的磁子热霍尔系数随温度变化曲线, 其他参数设置为$ \varGamma '{=}0.1, \;{J_1} = {J_2} = 0 $; (b) 磁子热霍尔系数随$ {J_0} $强度变化曲线

Figure 7. (a) Thermal Hall conductivity as a function of temperature under different intensity of interlayer exchange coupling interaction $ {J_0} $ with $ \varGamma '{=}0.1, \;{J_1} = {J_2} = 0 $; (b) thermal Hall conductivity as a function of different intensities of interlayer exchange coupling interaction $ {J_0} $.

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图 8 (a) 不同$ {J_0} $强度下的磁子能斯特系数随温度变化曲线, 其他参数设置为$ \varGamma '{=}0.1, \;{J_1} = {J_2} = 0 $; (b) 磁子能斯特系数随$ {J_0} $强度变化曲线

Figure 8. (a) Magnon Nernst conductivity as a function of temperature under different intensity of interlayer exchange coupling interaction $ {J_0} $ with $ \varGamma '{=}0.1, \;{J_1} = {J_2} = 0 $; (b) magnon Nernst conductivity as a function of different intensities of interlayer exchange coupling interaction $ {J_0} $.

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表 1 色散曲线对应的陈数

Table 1. Corresponding Chern numbers of magnon band structures.

参数	陈数
参数	能带1	能带2	能带3	能带4
$ {J_0} = 0.1, \;\varGamma '{=}0.1, \;{J_1} = 0.1, \;{J_2} = {J_3} = 0 $	–2	0	2	0
$ {J_0} = 0.245, \;\varGamma '{=}0.1, \;{J_1} = 0.1, \;{J_2} = {J_3} = 0 $	0	–2	2	0
$ {J_0} = 0.3, \;\varGamma '{=}0.1, \;{J_1} = 0.1, \;{J_2} = {J_3} = 0 $	0	–2	2	0
$ {J_0} = 0.505, \;\varGamma '{=}0.1, \;{J_1} = 0.1, \;{J_2} = {J_3} = 0 $	0	–2	0	2
$ {J_0} = 0.9, \;\varGamma '{=}0.1, \;{J_1} = 0.1, \;{J_2} = {J_3} = 0 $	0	–2	0	2

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表 2 能带对应的陈数

Table 2. Corresponding Chern numbers of magnon band structures.

序号	陈数
序号	能带1	能带2	能带3	能带4
①	0	–2	0	2
②	0	–2	2	0
③	1	–3	2	0
④	–1	–1	2	0
⑤	–3	1	2	0
⑥	–2	0	2	0
⑦	–2	2	0	0
⑧	–1	1	0	0
⑨	–1	0	1	0

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map

[1]	Zhang S Q, Xu R Z, Luo N N, Zou X L 2021 Nanoscale 13 1398 Google Scholar
[2]	Liu Z R, Hua C B, Peng T, Chen R, Zhou B 2023 Phys. Rev. B 107 125302 Google Scholar
[3]	张志东 2015 64 067503 Google Scholar Zhang Z D 2015 Acta Phys. Sin. 64 067503 Google Scholar
[4]	Xu M L, Huang C X, Li Y W, Liu S Y, Zhong X, Jena P, Kan E J, Wang Y C 2020 Phys. Rev. Lett. 124 067602 Google Scholar
[5]	MacDonald A H 2019 Physics 12 12 Google Scholar
[6]	Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Herrero P J 2018 Nature 556 43 Google Scholar
[7]	Tarnopolsky G, Kruchkov A J, Vishwanath A 2019 Phys. Rev. Lett. 122 106405 Google Scholar
[8]	Carr S, Fang S, Jarillo-Herrero P, Kaxiras E 2018 Phys. Rev. B 98 085144 Google Scholar
[9]	Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young A F, Dean C R 2019 Science 363 1059 Google Scholar
[10]	Ribeiro-Palau R, Zhang C, Watanabe K, Taniguchi T, Hone J, Dean C R 2018 Science 361 690 Google Scholar
[11]	Guerci D, Simon P, Mora C 2021 Phys. Rev. B 103 224436 Google Scholar
[12]	Feng H F, Li Y, Shi Y G, Xie H Y, Li Y Q, Xu Y 2022 Chin. Phys. Lett. 39 077501 Google Scholar
[13]	Cenker J, Huang B, Suri N, Thijssen P, Miller A, Song T, Taniguchi T, Watanabe K 2021 Nat. Phys. 17 20 Google Scholar
[14]	Kang S, Kim K, Kim B H, Kim J, Sim K I, Lee J U, Lee S, Park K, Yun S, Kim T, Nag A, Walters A, Garcia-Fernandez M, Li J, Chapon L, Zhou K J, Son Y W, Kim J H, Cheong H, Park J G 2020 Nature 583 785 Google Scholar
[15]	Zhang H, Feng X, Heitmann T, Kolesnikov A I, Stone M B, Lu Y M 2020 Phys. Rev. B 101 100405 Google Scholar
[16]	Zhang L C, Zhu F, Go D, Lux F R, dos Santos F J, Lounis S, Su Y, Blügel S, Mokrousov Y 2021 Phys. Rev. B 103 134414 Google Scholar
[17]	Ghader D, Khater A 2019 Sci. Rep. 9 15220 Google Scholar
[18]	Van Miert G, Smith C M 2016 Phys. Rev. B 93 035401 Google Scholar
[19]	Wang X S, Wang X R 2021 J. Appl. Phys. 129 151101 Google Scholar
[20]	王振宇, 李志雄, 袁怀洋, 张知之, 曹云姗, 严鹏 2023 72 057503 Google Scholar Wang Z Y, Li Z X, Yuan H Y, Zhang Z Z, Cao Y S, Yan P 2023 Acta Phys. Sin. 72 057503 Google Scholar
[21]	Stauber T, Low T, Gómez-Santos G 2018 Phys. Rev. Lett. 120 046801 Google Scholar
[22]	Ma J J, Wang Z Y, Xu S G, Gao Y X, Zhang Y Y, Dai Q, Lin X, Du S X, Ren J D, Gao H J 2022 Chin. Phys. Lett. 39 047403 Google Scholar
[23]	Hu J W, Zhu S Y, Hu Q Y, Wang Y H, Shen C M, Yang H T, Zhu X S, Huan Q, Xu Y, Gao H J 2024 Chin. Phys. Lett. 41 037401 Google Scholar
[24]	Li X F, Sun R X, Wang S Y, Li X, Liu Z B, Tian J G 2022 Chin. Phys. Lett. 39 037301 Google Scholar
[25]	Liu C F, Wang J 2022 Chin. Phys. Lett. 39 077301 Google Scholar
[26]	Lee J Y 2019 Nat. Commun. 10 5333 Google Scholar
[27]	Zhang Y H, Mao D, Cao Y, Jarillo-Herrero P, Senthil T 2019 Phys. Rev. B 99 075127 Google Scholar
[28]	Po H C, Zou L, Senthil T, Vishwanath A 2019 Phys. Rev. B 99 195455 Google Scholar
[29]	Hao Z, Zimmerman A M, Ledwith P, Khalaf E, Najafabadi D H, Watanabe K, Taniguchi T, Ashvin Vishwanath, Kim P 2021 Science 371 1133 Google Scholar
[30]	Nuckolls K P, Oh M, Wong D, Lian B, Watanabe K, Taniguchi T, Bernevig B A 2020 Nature 588 610 Google Scholar
[31]	Rademaker L, Mellado P 2018 Phys. Rev. B 98 235158 Google Scholar
[32]	Moon P, Koshino M 2014 Phys. Rev. B 90 155406 Google Scholar
[33]	Zhu X C, Guo H M, Feng S P 2021 Chin. Phys. B 30 077505 Google Scholar
[34]	Ghader D 2020 Sci. Rep. 10 16733 Google Scholar
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[1]	. Preface to the special topic: Two-dimensional magnetism and topological spin physics. Acta Physica Sinica, 2024, 73(1): 010101. doi: 10.7498/aps.73.010101
[2]	Xie Xiao-Jie, Sun Jun-Song, Qin Ji-Hong, Guo Huai-Ming. Pseudo-Landau levels of hexagonal lattice quantum antiferromagnets under bending strain. Acta Physica Sinica, 2024, 73(2): 020202. doi: 10.7498/aps.73.20231231
[3]	Liu En-Ke. Coupling between magnetism and topology: From fundamental physics to topological magneto-electronics. Acta Physica Sinica, 2024, 73(1): 017103. doi: 10.7498/aps.73.20231711
[4]	Zhang Shi-Hao, Xie Bo, Peng Ran, Liu Xiao-Qian, Lü Xin, Liu Jian-Peng. Novel electrical properties of moiré graphene systems. Acta Physica Sinica, 2023, 72(6): 067302. doi: 10.7498/aps.72.20230120
[5]	Liu Yi-Zhou, Zang Jiadong. Overview and outlook of magnetic skyrmions. Acta Physica Sinica, 2018, 67(13): 131201. doi: 10.7498/aps.67.20180619
[6]	Wang Yi-Jun, Liu Yang, Yu Guang-Hua. Effect of Pt spacers on interface exchange coupling in ferromagnetic/antiferromagnetic bilayers. Acta Physica Sinica, 2012, 61(16): 167503. doi: 10.7498/aps.61.167503
[7]	Liu Da-Yong, Chen Dong-Meng. Orbital ordering driven spin dimer state in double-layered antiferromagnet K3Cu2O7. Acta Physica Sinica, 2010, 59(10): 7350-7356. doi: 10.7498/aps.59.7350
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