本研究主要以分子動力學法研究奈米材料的聲子性質，如色散關係與態密度等，並嘗試探討聲子性質與熱傳機制間的關係。首先以分子動力學模擬一維原子系統並進行分析方法測試，並與理論解相比較，確認採用原子速度進行傅立葉分析可以得到較好的色散曲線，且色散關係與態密度有所關連。接著對二維石墨烯薄膜進行模擬，嘗試建立起石墨烯的彈簧模型，以代表性單元晶格的概念並將碳原子間的鍵結以彈簧表示，並推導出動態方程式及色散關係式；此外以分子動力學模擬石墨烯薄膜，採用單位晶胞內的同一種原子進行分析，可以避免光學分支因為相位差的關係而相互抵消，並得知在不同波數方向的分析時，採用簡正座標的方法會較易進行分析，與彈簧模型的推導結果及文獻上的色散曲線相比對，發現彈簧模型中過度簡化原子間的鍵結而無法完整的描述石墨烯結構的振動行為。最後對奈米碳管進行分析，從色散曲線結果得知沿碳管軸向取單排原子進行分析，可以消除沿圓周方向產生的相位差。而在態密度的分析結果中，可發現使用速度自相關函數進行空間-時間傅立葉轉換的結果會較易於觀察，且與色散關係相呼應。
本研究成功建立一套健全的聲子色散關係與態密度的分析方法，分析石墨烯與奈米碳管的聲子性質，確立奈米薄膜與奈米管的分析方式，可提供往後奈米熱傳相關研究一個分析的依據。

This research studied phonon properties of nanomaterials, such as graphene and carbon nanotube, using equilibrium molecular dynamics simulation. We tried to establish the relationships between the behaviors of phonons and heat transfer mechanism. Firstly, one dimensional atomic system was simulated and analyzed. Both the dispersion relation and phonon density of states were extracted from molecular dynamics simulation. The phonon behaviors were compared with theoretical predictions to confirm the validity of analysis procedure. Next, two dimensional graphene film was analyzed. We tried to construct a spring model for graphene. The concept representative unit cell was employed and the bondings between carbon atoms were replaced by two sets of linear springs. The dynamic equation and dispersion relation for graphene were derived. It was observed that the optical branch of the dispersion curve was clear once the same atom within the unit cell was analyzed to avoid the cancellation of signal due to the phase difference. Meanwhile, it was easier to adopt normal coordinate method to obtain the dispersion relation along different reciprocal crystal directions. From the comparisons between the simulation results, literature and the prediction based on spring model, it was concluded that the spring was an over-simplification for atomic bond and further attention was needed for the equivalence of atomic bonds in graphene. Finally, we extracted the dispersion curve along the axial direction of the carbon nanotube. Only one row of atoms was needed for the analysis to aviod the phase difference in the circumferential direction. Meanwhile, it was noticed that it was easier to obtain phonon density of states using the space-time Fourier transformation of velocity autocorrelation function. The phonon density of states was consistent with the behavior of dispersion relation.
In this study, we successfully established the analysis procedure to extract phonon dispersion relation and density of states of nanomaterials from molecular simulation. This research could pave the way to correlate the thermal conduction mechanism and phonon behavior in the near future.

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