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求解了两条相同的耦合Frenkel-Kontorova (FK) 链在低温、有限温和高温情况下的格波解及色散关系, 进而研究了耦合FK双链的晶格振动特点. 结果表明, 耦合FK双链的色散关系包含一个声学支和一个光学支, 两者的频谱范围和频率禁带与FK链的恢复力系数、链间耦合强度系数均有关联, 低温和有限温的情况还与外势深度有关系. 并且研究发现当链间耦合强度较小时, 不存在频率禁带; 当链间耦合强度逐渐增加到某一临界值后, 频率禁带出现, 且随着链间耦合强度增加, 频隙不断变大, 这是因为光学支随着链间耦合强度增加不断向高频方向移动. 此外, 还发现带隙结构出现的临界链间耦合强度始终为FK链恢复力系数的2倍, 并不受温度的影响. 本文还研究了给定链间耦合强度下温度对耦合FK双链色散关系的影响规律. 本研究内容可为分析链间界面耦合和温度对晶格的振动特点和物理性质的影响提供理论依据, 从而对于能量输运、热调控等实际应用发挥重要的指导作用.
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关键词:
- 耦合Frenkel-Kontorova双链 /
- 链间耦合 /
- 色散关系 /
- 温度 /
- 热传导
We obtain the lattice wave solution and the dispersion relation of the lattice vibration equation of the two identical coupled Frenkel-Kontorova (FK) chains, and we study the lattice vibration characteristics of the coupled FK chains. The results show that the dispersion relation of coupled FK chain contains an acoustic branch and an optical branch. The spectral range and frequency band gap are related to the coefficient of restoring force of each chain and the inter-chain coupling strength, and it is also related to the depth of the on-site potential for the low temperature case and finite temperature case. Moreover, it is found that there is no frequency band gap for weak inter-chain coupling. The frequency gap appears when the inter-chain coupling strength exceeds a critical value, and the frequency band gap will become bigger with the inter-chain interaction increasing. This is because the optical branch moves towards high frequency region with the inter-chain coupling increasing. We also find that the critical inter-chain coupling strength of frequency band gap is always twice the restoring force coefficient of FK chain, and it does not depend on temperature. In addition, we study the effect of temperature on the dispersion relationship of coupled FK chain with a fixed inter-chain coupling strength. These results provide a theoretical basis for analyzing the effects of inter-chain coupling and temperature on the vibrational characteristics and physical properties of lattice, and thus providing an important guide for the energy transport, thermal management and other practical applications.-
Keywords:
- two coupled Frenkel-Kontorova chains /
- inter-chain coupling /
- dispersion relation /
- temperature /
- heat conduction
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图 2 耦合FK双链在低温小振动近似情况下的色散关系 (
$ {k_1} = {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ) (a)${k_{\text{c}}} = 1$ ; (b)${k_{\text{c}}} = 2$ ; (c)${k_{\text{c}}} = 5$ Fig. 2. Dispersion relationship of the coupled FK chains under low temperature approximation with small vibration (
$ {k_1} = $ $ {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ): (a)${k_{\text{c}}} = 1$ ; (b)${k_{\text{c}}} = 2$ ; (c)${k_{\text{c}}} = 5$ .
图 3 耦合FK双链(
$ {k_1} = {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ )在有限温度(T = 0.05)下的色散关系 (a)${k_{\text{c}}} = 1$ ; (b)${k_{\text{c}}} = 2$ ; (c)${k_{\text{c}}} = 5$ Fig. 3. Dispersion relationship of the coupled FK chains (
$ {k_1} = $ $ {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ) under finite temperature (T = 0.05): (a)${k_{\text{c}}} = 1$ ; (b)${k_{\text{c}}} = 2$ ; (c)${k_{\text{c}}} = 5$ .
图 4 耦合FK双链在高温近似下的色散关系(
${k_1} = {k_2} = 1$ ,$ V = 1 $ ,$ m = 1 $ ) (a)${k_{\text{c}}} = 1$ ; (b)${k_c} = 2$ ; (c)${k_{\text{c}}} = 5$ Fig. 4. Dispersion relation of coupled FK chains under high temperature approximation (
$ {k_1} = $ $ {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ): (a)${k_{\text{c}}} = 1$ ; (b)${k_{\text{c}}} = 2$ ; (c)${k_{\text{c}}} = 5$ .
图 5 耦合FK双链在不同温度下的色散关系(
$ {k_1} = {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ,${k_{\text{c}}} = 1.0$ ) (a)$ T = 0.005 $ ; (b)$ T = 0.01 $ ; (c)$ T = 0.02 $ ; (d)$ T = 0.06 $ ; (e)$T = 0.10$ ; (f)$T = 0.20$ Fig. 5. Dispersion relation of coupled FK chains at different temperature (
$ {k_1} = {k_2} = 1 $ ,$ V = 1 $ ,$ m = 1 $ ,${k_{\text{c}}} = 1.0$ ): (a)$ T = 0.005 $ ; (b)$ T = 0.01 $ ; (c)$ T = 0.02 $ ; (d)$ T = 0.06 $ ; (e)$T = 0.10$ ; (f)$T = 0.20$ . -
[1] Nelson L A, Sekhon K S, Frita J E 1978 Proceedings of the 3rd International Heat Pipe Conference Palo Alto, CA, USA, May 22–24, 1978 p450
[2] Pop E 2005 Ph. D. Dissertation (Stanford, California: Stanford University)
[3] Krishnan S, Garimella S V, Chrysler G M, Mahajan R V 2007 IEEE Trans. Adv. Packaging 30 462
Google Scholar
[4] Shi L, Dames C, Lukes J R, Reddy P, Duda J, Cahill D G, Lee J, Marconnet A, Goodson K E, Bahk J H 2015 Nano. Micro. Thermophys. Eng. 19 127
Google Scholar
[5] 黄昆, 韩汝琦 1983 固体物理学 (北京: 高等教育出版社) 第93页
Huang K, Han R Q 1983 Solid-State Physics (Beijing: Higher Education Press) p93 (in Chinese)
[6] 陆栋, 蒋平 2011 固体物理学 (北京: 高等教育出版社) 第52—60页
Lu D, Jiang P 2011 Solid-State Physics (Beijing: Higher Education Press) pp52–60 (in Chinese)
[7] 李正中 2002 固体理论 (北京: 高等教育出版社) 第18—29页
Li Z Z 2002 Solid State Theory (Beijing: Higher Education Press) pp18–29 (in Chinese)
[8] Thomas J A, Turney J E, Iutzi R M, Amon C H, McGaughey A J 2010 Phys. Rev. B 81 081411
Google Scholar
[9] Zhu L, Li B 2014 Sci. Rep. 4 4917
[10] Su R X, Zhang X 2018 Appl. Thermal Eng. 144 488
Google Scholar
[11] Li N, Li B 2007 Phys. Rev. E 76 011108
Google Scholar
[12] Li N, Li B 2012 AIP Adv. 2 041408
Google Scholar
[13] Li N, Li B 2013 Phys. Rev. E 87 042125
Google Scholar
[14] Wang X W, Zhong Z R, Xu J 2005 J. Appl. Phys. 97 064302
[15] Yang D J, Zhang Q, Chen G, Yoon S F, Ahn J, Wang S G, Zhou Q, Wang Q, Li J Q 2002 Phys. Rev. B 66 165440
Google Scholar
[16] Han Z, Fina A 2011 Prog. Polym. Sci. 36 914
Google Scholar
[17] Cohen Y, Ya'akobovitz A 2021 Microelectr. Eng. 247 111575
Google Scholar
[18] Shahil K M F, Balandin A A 2012 Solid State Commun. 152 1331
Google Scholar
[19] Pettes M T, Jo I, Yao Z, Shi, L 2011 Nano Lett. 11 1195
Google Scholar
[20] Bae M H, Li Z, Aksamija Z, Martin P N, Xiong F, Ong Z Y, Knezevic I, Pop E 2013 Nature Commun. 4 1
Google Scholar
[21] Ni Y, Chalopin Y, Volz S 2013 Appl. Phys. Lett. 103 061906
Google Scholar
[22] Koh Y K, Bae M H, Cahill D G, Pop E 2010 Nano Lett. 10 4363
Google Scholar
[23] Frenkel Y, Kontorova T 1938 Teor. Fiz. 8 1340
[24] 田强, 洪馥男 2006 大学物理 25 17
Google Scholar
Tian Q, Hong F N 2006 College Phys. 25 17
Google Scholar
[25] Su R X, Yuan Z Q, Wang J, Zheng Z G 2016 J. Phys. A:Math. Theor. 49 255003
Google Scholar
[26] 穆亚男, 郭建中 2014 声学技术 33 4
Mu Y N, Guo J Z 2014 Technical Acoustics 33 4
[27] 张荣英, 姜根山, 王璋奇, 吕亚东 2006 声学技术 25 35
Google Scholar
Zhang R Y, Jiang G S, Wang Z Q, Lü Y D 2006 Technical Acoustics 25 35
Google Scholar
[28] 王学梅 2014 中国电机工程学报 34 371
Wang X M 2014 Proceedings of the CSEE 34 371
[29] 余振坤, 郑新 2007 微波学报 23 61
Google Scholar
Yu Z K, Zheng X 2007 J. Microw. 23 61
Google Scholar
[30] Yan X H, Xiao Y, Li Z M 2006 J. Appl. Phys. 99 124305
[31] Singh D, Murthy J Y, Fisher T S 2011 J. Appl. Phys. 110 044317
Google Scholar
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