本研究將針對過渡金屬釩氧化物材料進行第一原理之固態物理性質計算，由於釩擁有d軌域的電子組態，因而有多樣化的特性，應用領域極廣，而此論文將針對二氧化釩(VO2)及五氧化二釩(V2O5)此兩種釩氧化物進行第一原理計算，評估其熱力、熱傳、導電、導離子等特性，以及未來作為熱電材料及鋰電池陰極材料之可行性及功能分析。
研究步驟先建構釩化物之分子模型及相位，再進行數值收斂性測試且分析其結構最佳化之結果，確認其收斂之結果與實驗相符後，即搭配多電子模式密度泛函理論及自洽場運算其電子能帶結構、態密度等分析其導電、導熱功能。而熱力熱傳性能，則使用密度泛函微擾理論計算聲子傳播，搭配統計熱力學來計算熱容量、自由能、熵、及奈米尺度熱傳導係數等。而二氧化釩於室溫時即不遵守Wiedemann-Franz定理，相較於其他材料，其低溫時具有高導電低導熱性質，經週期性固態物理計算，初步評估其高導電率及高席貝克係數可大量增加熱電功率因子，但熱傳導係數降低不足，故熱電優值仍過低。
目前儲電研究為綠能發電最重要課題，鋰電池仍是具有最高功率密度與最大能量密度之儲能裝置，而新儲能材料(尤其陰極材料)用以繼續改進其儲電性能為關鍵課題。五氧化二釩由於其具有比惰性電極碳(或石墨)更高之還原電位¬，可作為鋰離子電池之陰極材料，且因分子結構為層狀結構，使得其易於嵌入半徑較小如鋰離子等，因而可儲存較高之鋰離子濃度而得到較高能量密度。此第一原理計算可藉由量子力學與電化學結果，推算五氧化二釩之鋰離子電池開路電壓，鋰離子嵌入過程等，而藉由各軸之鋰離子嵌入位置計算其各位置之穩態能量，最後求得最穩定之鋰離子嵌入路徑，本研究發現五氧化二釩為層狀結構且沿著其中一軸具有最小的能量障礙，另一軸為次小，而在通過層與層之間則有極大之能量障礙，結論為此LixV2O5作為鋰電池陰極材料具有層狀方向性與結構穩定性。

This thesis intends to predict the thermal (phonons) and electric (electrons) properties of vanadium oxides using first principles calculation to evaluate the thermoelectric performance. Also analysis of lithium intercalation in the LixV2O5 cathode materials was carried out to validate the energy storage performance of new material Li-ion batteries.
The research methodology starts from molecular structure set up with proper phase, then numerical test was performed to check the convergence of the algorithm and the accuracy with experimental results. Phonon transport simulation was carried out to evaluate the thermal properties of vanadium oxides using density functional perturbation theory (DFPT). And electron band structure was calculated to evaluate the electric properties of the material using density functional theory (DFT). Low thermal conductivity with high electric conductivity and great Seebeck coefficient were all derived and found that the VO2 may achieve a better thermoelectric power factor but the thermoelectric figure of merit is still too low for thermoelectric chip applications.
Another green energy application of the vanadium oxides is on the lithium ion battery. This thesis uses the same solid state physics technique to evaluate the lithium ion intercalation process at the LixV2O5 cathode material. Based on the band structure results from computational quantum mechanics and electrochemistry, we can calculate the open circuit voltage and the lithium ion intercalation along different axes. It is found that the migration of lithium ions along one of the three axes has the least diffusion barrier. The conclusion is that this divanadium pentaoxide material is stable and directional; that means it has to be positioned properly when it is used as the cathode material for a future Li-ion battery.

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 vi
表目錄 ix
符號定義與術語縮寫 x
第一章　緒論 1
1.1前言 1
1.2文獻回顧 4
1.3研究動機與目的 5
第二章 研究方法 7
2.1 聲子固態物理理論 (Solid State Physics- Phonon) 7
2.1.1倒晶格 (Reciprocal Lattice) 7
2.1.2不可分割第一布里淵區 (Irreducible First Brillouin Zone) 8
2.1.3雙原子線性彈簧模型 (Diatomic Linear Chain) 8
2.1.4光頻及聲頻聲子 (Optical and Acoustic Modes) 10
2.1.5縱向波及橫向波 (Longitudinal and Transverse Modes) 11
2.1.6 Einstein Model 11
2.1.7 Debye Model 12
2.1.8平面波 (Plane Wave) 12
2.1.9贋勢 (Pseudopotential) 13
2.2電子第一原理計算 (First principles Calculation- Electron) 14
2.2.1 Bloch Theory 14
2.2.2 Schrödinger Equations 14
2.2.3 Born-Oppenheimer Approximation 15
2.2.4 Hohenberg-Kohn Theorem 15
2.2.5 Kohn-Sham Equations 16
2.2.6局部密度近似法 (Local Density Functional Approximation) 16
2.2.7廣義梯度近似法 (Generalized Gradient Approximation) 17
2.2.8 Perdew-Wang 91 (PW91) 17
2.2.9 Perdew-Burke-Ernzerhof (PBE) 18
2.2.10自洽場 (Self-consistent Field Scheme) 18
2.3 諧和近似 (Harmonic Approximation) 19
2.3.1諧和震盪代入薛丁格方程式 19
2.3.2 Bose-Einstein Distribution 20
2.3.3熱容量 (Heat Capacity) 21
2.3.4自由能 (Free Energy) 22
2.3.5熵 (Entropy) 23
第三章 模擬方法 24
3.1 模擬流程 24
3.1.1 VASP輸入輸出檔案 25
3.1.2晶胞能量及收斂性測試 26
3.1.3電子性質 33
3.1.4諧和項熱力學性質 34
第四章 結果與討論 35
4.1結構最佳化之鍵長 35
4.2電子態密度 36
4.3 LixV2O5開路電壓計算 46
4.4鋰離子嵌入能量 47
4.5諧和項熱力學性質 49
第五章 結論與未來工作建議 52
5.1結論 52
5.2未來工作建議 52
參考文獻 53

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