搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

耦合Frenkel-Kontorova双链的格波解及其色散关系

苏瑞霞 黄霞 郑志刚

引用本文:
Citation:

耦合Frenkel-Kontorova双链的格波解及其色散关系

苏瑞霞, 黄霞, 郑志刚

Lattice wave solution and its dispersion relation of two coupled Frenkel-Kontorova chains

Su Rui-Xia, Huang Xia, Zheng Zhi-Gang
PDF
HTML
导出引用
  • 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.
    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.
      通信作者: 郑志刚, zgzheng@hqu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 51706118, 11875135)、中央高校基本科研业务费专项资金(批准号: 2022YQLX03, 2019QS05)和泉州市科技计划(批准号: 2018C085R)资助的课题.
      Corresponding author: Zheng Zhi-Gang, zgzheng@hqu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 51706118, 11875135), the Fundamental Research Funds for Central Universities, China (Grant Nos. 2022YQLX03, 2019QS05), and the Quanzhou Science and Technology Plan, China (Grant No. 2018C085R).
    [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 462Google 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 127Google 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 081411Google Scholar

    [9]

    Zhu L, Li B 2014 Sci. Rep. 4 4917

    [10]

    Su R X, Zhang X 2018 Appl. Thermal Eng. 144 488Google Scholar

    [11]

    Li N, Li B 2007 Phys. Rev. E 76 011108Google Scholar

    [12]

    Li N, Li B 2012 AIP Adv. 2 041408Google Scholar

    [13]

    Li N, Li B 2013 Phys. Rev. E 87 042125Google 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 165440Google Scholar

    [16]

    Han Z, Fina A 2011 Prog. Polym. Sci. 36 914Google Scholar

    [17]

    Cohen Y, Ya'akobovitz A 2021 Microelectr. Eng. 247 111575Google Scholar

    [18]

    Shahil K M F, Balandin A A 2012 Solid State Commun. 152 1331Google Scholar

    [19]

    Pettes M T, Jo I, Yao Z, Shi, L 2011 Nano Lett. 11 1195Google 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 1Google Scholar

    [21]

    Ni Y, Chalopin Y, Volz S 2013 Appl. Phys. Lett. 103 061906Google Scholar

    [22]

    Koh Y K, Bae M H, Cahill D G, Pop E 2010 Nano Lett. 10 4363Google Scholar

    [23]

    Frenkel Y, Kontorova T 1938 Teor. Fiz. 8 1340

    [24]

    田强, 洪馥男 2006 大学物理 25 17Google Scholar

    Tian Q, Hong F N 2006 College Phys. 25 17Google Scholar

    [25]

    Su R X, Yuan Z Q, Wang J, Zheng Z G 2016 J. Phys. A:Math. Theor. 49 255003Google 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 35Google Scholar

    Zhang R Y, Jiang G S, Wang Z Q, Lü Y D 2006 Technical Acoustics 25 35Google 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 61Google Scholar

    Yu Z K, Zheng X 2007 J. Microw. 23 61Google 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 044317Google Scholar

  • 图 1  耦合FK双链模型示意图

    Fig. 1.  Schematics of the coupled FK chains model.

    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 462Google 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 127Google 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 081411Google Scholar

    [9]

    Zhu L, Li B 2014 Sci. Rep. 4 4917

    [10]

    Su R X, Zhang X 2018 Appl. Thermal Eng. 144 488Google Scholar

    [11]

    Li N, Li B 2007 Phys. Rev. E 76 011108Google Scholar

    [12]

    Li N, Li B 2012 AIP Adv. 2 041408Google Scholar

    [13]

    Li N, Li B 2013 Phys. Rev. E 87 042125Google 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 165440Google Scholar

    [16]

    Han Z, Fina A 2011 Prog. Polym. Sci. 36 914Google Scholar

    [17]

    Cohen Y, Ya'akobovitz A 2021 Microelectr. Eng. 247 111575Google Scholar

    [18]

    Shahil K M F, Balandin A A 2012 Solid State Commun. 152 1331Google Scholar

    [19]

    Pettes M T, Jo I, Yao Z, Shi, L 2011 Nano Lett. 11 1195Google 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 1Google Scholar

    [21]

    Ni Y, Chalopin Y, Volz S 2013 Appl. Phys. Lett. 103 061906Google Scholar

    [22]

    Koh Y K, Bae M H, Cahill D G, Pop E 2010 Nano Lett. 10 4363Google Scholar

    [23]

    Frenkel Y, Kontorova T 1938 Teor. Fiz. 8 1340

    [24]

    田强, 洪馥男 2006 大学物理 25 17Google Scholar

    Tian Q, Hong F N 2006 College Phys. 25 17Google Scholar

    [25]

    Su R X, Yuan Z Q, Wang J, Zheng Z G 2016 J. Phys. A:Math. Theor. 49 255003Google 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 35Google Scholar

    Zhang R Y, Jiang G S, Wang Z Q, Lü Y D 2006 Technical Acoustics 25 35Google 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 61Google Scholar

    Yu Z K, Zheng X 2007 J. Microw. 23 61Google 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 044317Google Scholar

  • [1] 李冀, 陈亮, 冯芒. 基于离子阱中离子晶体的热传导的研究进展. 物理学报, 2024, 73(3): 033701. doi: 10.7498/aps.73.20231719
    [2] 钱黎明, 孙梦然, 郑改革. α相三氧化钼中各向异性双曲声子极化激元的耦合性质. 物理学报, 2023, 72(7): 077101. doi: 10.7498/aps.72.20222144
    [3] 曹义刚, 付萌萌, 杨喜昶, 李登峰, 王晓霞. 热传导对横截面不同的直管道中Kelvin-Helmholtz不稳定性的影响. 物理学报, 2022, 71(9): 094701. doi: 10.7498/aps.71.20211155
    [4] 李文秋, 赵斌, 王刚. 电子温度对螺旋波等离子体中电磁模式能量沉积特性的影响. 物理学报, 2020, 69(21): 215201. doi: 10.7498/aps.69.20201018
    [5] 李文秋, 赵斌, 王刚, 相东. 螺旋波等离子体中螺旋波与Trivelpiece-Gould波模式耦合及线性能量沉积特性参量分析. 物理学报, 2020, 69(11): 115201. doi: 10.7498/aps.69.20200062
    [6] 赵顾颢, 毛少杰, 赵尚弘, 蒙文, 祝捷, 张小强, 王国栋, 谷文苑. 双旋光双反射结构的温度-辐射自稳定性原理和实验研究. 物理学报, 2019, 68(16): 164202. doi: 10.7498/aps.68.20190429
    [7] 王德鑫, 那仁满都拉. 耦合双泡声空化特性的理论研究. 物理学报, 2018, 67(3): 037802. doi: 10.7498/aps.67.20171805
    [8] 敖宏瑞, 陈漪, 董明, 姜洪源. 基于多物理场的TFC磁头热传导机理及其影响因素仿真研究. 物理学报, 2014, 63(3): 034401. doi: 10.7498/aps.63.034401
    [9] 蒋中英, 张国梁, 马晶, 朱涛. 磷脂在膜结构间的交换:温度和离子强度的影响. 物理学报, 2013, 62(1): 018701. doi: 10.7498/aps.62.018701
    [10] 袁宗强, 褚敏, 郑志刚. Fermi-Pasta-Ulam β 格点链系统能量载流子研究. 物理学报, 2013, 62(8): 080504. doi: 10.7498/aps.62.080504
    [11] 任益充, 范洪义. 不变本征算符方法求解含不同在位势的一维双原子链的色散关系. 物理学报, 2013, 62(15): 156301. doi: 10.7498/aps.62.156301
    [12] 王冠宇, 宋建军, 张鹤鸣, 胡辉勇, 马建立, 王晓艳. 单轴应变Si导带色散关系解析模型. 物理学报, 2012, 61(9): 097103. doi: 10.7498/aps.61.097103
    [13] 刘文, 刘德胜, 李海宏. 二维链间扩展的极化子动力学研究. 物理学报, 2010, 59(9): 6405-6411. doi: 10.7498/aps.59.6405
    [14] 宋建军, 张鹤鸣, 戴显英, 胡辉勇, 宣荣喜. 应变Si价带色散关系模型. 物理学报, 2008, 57(11): 7228-7232. doi: 10.7498/aps.57.7228
    [15] 赵国伟, 徐跃民, 陈 诚. 等离子体天线色散关系和辐射场数值计算. 物理学报, 2007, 56(9): 5298-5303. doi: 10.7498/aps.56.5298
    [16] 陈国庆, 吴亚敏, 陆兴中. 金属/电介质颗粒复合介质光学双稳的温度效应. 物理学报, 2007, 56(2): 1146-1151. doi: 10.7498/aps.56.1146
    [17] 周桂耀, 侯峙云, 潘普丰, 侯蓝田, 李曙光, 韩 颖. 微结构光纤预制棒拉制过程的温度场分布. 物理学报, 2006, 55(3): 1271-1275. doi: 10.7498/aps.55.1271
    [18] 高 琨, 付吉永, 刘德胜, 解士杰. 链间耦合对聚合物中双激子态反向极化的影响. 物理学报, 2005, 54(2): 665-668. doi: 10.7498/aps.54.665
    [19] 秦 颖, 王晓钢, 董 闯, 郝胜智, 刘 悦, 邹建新, 吴爱民, 关庆丰. 强流脉冲电子束诱发温度场及表面熔坑的形成. 物理学报, 2003, 52(12): 3043-3048. doi: 10.7498/aps.52.3043
    [20] 范植开, 刘庆想. 谐振腔链色散关系及场分布的解析研究. 物理学报, 2000, 49(7): 1249-1255. doi: 10.7498/aps.49.1249
计量
  • 文章访问数:  2846
  • PDF下载量:  35
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-12-22
  • 修回日期:  2022-03-16
  • 上网日期:  2022-07-19
  • 刊出日期:  2022-08-05

/

返回文章
返回