耦合Frenkel-Kontorova双链的格波解及其色散关系

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

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 onsite 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.