利用第一原理模擬計算La(Mg1/2,Ti1/2)O3材料，以密度泛函微擾理論及線性響應的方式，來研究微波介電材料La(Mg1/2,Ti1/2)O3在Γ點上的聲子行為，並以羅倫茲模型對於材料介電特性跟材料的晶格振盪的關係，來探討利用拉曼光譜實驗及紅外光光譜實驗於La(Mg1/2,Ti1/2)O3材料的結果輔以第一原理模擬計算所得到的資訊，來研究La(Mg1/2,Ti1/2)O3材料的聲子行為與介電特性的關係。

We use first-principles to begin the simulation on the microwave dielectric material La(Mg1/2,Ti1/2)O3. In our calculations, we use the Density-Functional Perturbation Theory and linear response method to get the phonons at Γ point. 
    And then, we make a few experimental observations like Raman spectrum and FTIR spectrum on the microwave dielectric material La(Mg1/2,Ti1/2)O3. According to our calculations, the calculated eigenvalues and corresponding eigenvectors of each vibration mode at 
Γ point provide a great help on assignment of our experimental observations.
    Finally, we use the Lorentz model to study the relation between vibration modes and dielectric properties on the microwave dielectric material La(Mg1/2,Ti1/2)O3.

第一章  緒論..............................................1
    1.1 研究動機..........................................1
    1.2 論文架構..........................................3
第二章  計算理論及模擬方法................................4
    2.1密度泛函理論(Density Functional Theory)............4
        2.1.1 Hohenberg-Kohn 理論.........................4
        2.1.2 Kohn-Sham Equation..........................6
        2.1.3 交換相干能..................................7
        2.1.4 週期邊界條件................................8
        2.1.5 k點取樣.....................................8
        2.1.6  虛位勢.....................................9
    2.2  密度泛函微擾理論................................10
         2.2.1 線性響應及晶格動力學......................11
         2.2.2 振動模式介電貢獻..........................13
    2.3 介電機制.........................................13
    2.4 拉曼(Raman) 與(傅力葉轉換紅外光吸收光譜) FTIR 光    
        譜...............................................18
         2.4.1 拉曼(Raman)光譜...........................18
         2.4.2  傅力葉轉換紅外光吸收光譜(FTIR)光譜.......20
第三章  實驗方法與步驟...................................22
   3.1 La(Mg1/2,Ti1/2)O3 材料製備........................22
         3.1.1 固態反應法................................22
         3.1.2 檸檬酸鹽法................................23
   3.2 特性分析..........................................26
         3.2.1 X-Ray 分析................................26
         3.2.2 Raman 光譜分析............................30
         3.2.3 FTIR 紅外光光譜分析.......................31
         3.2.4 SEM微結構觀察.............................34
第四章  計算結果與討論...................................36
    4.0 計算軟體介紹及計算流程...........................36
    4.1 La(Mg1/2,Ti1/2)O3晶格結構介紹....................38
    4.2 電子結構的計算...................................41
    4.3 La(Mg1/2,Ti1/2)O3材料在Γ點上的聲子計算...........48
第五章  結論.............................................77
參考文獻................................................79

圖表目錄
圖1-1 立方晶系鈣鈦礦結構的氧化物結構圖…………………………2
圖2-1 虛位勢示意圖…………………………………………………..10
圖2-2 物質中常見的四種極化………………………………………..17
圖2-3 介電常數實部對頻率之頻譜圖………………………………..17
圖2-4 史托克斯散射、反史托克斯散射及瑞立散射示意圖………..20
圖3-1 固態反應法製備LMT材料流程圖...........................................23
圖3-2 檸檬酸鹽法製備LMT材料流程圖...........................................25
圖3-3 固態反應法LMT粉末X-Ray繞射圖...........................................27
圖3-4 固態反應法LMT塊材X-Ray繞射圖...........................................28
圖3-5 檸檬酸鹽法P.H.值為7之LMT粉末X-Ray繞射圖..................28
圖3-6 檸檬酸鹽法P.H.值為9之LMT粉末X-Ray繞射圖..................29
圖3-7 檸檬酸鹽法P.H.值為11之LMT粉末X-Ray繞射圖..................29
圖3-8經1550℃燒結4小時的LMT塊材X-Ray繞射圖.......................30
圖3-9 La(Mg1/2,Ti1/2)O3塊材的Raman光譜圖.....................................31
表3-1 Raman光譜所得La(Mg1/2,Ti1/2)O3材料之active g mode振動模..............................................................................................................31
圖3-10 La(Mg1/2,Ti1/2)O3塊材的FTIR光譜圖....................................32
圖 3-11 La(Mg1/2,Ti1/2)O3塊材由FTIR光譜計算所得介電常數(a)實部與(b)虛部............................................................................................33
表3-2 FTIR光譜所得La(Mg1/2,Ti1/2)O3材料之active u mode振動模
..................................................................................................................34
圖3-12 以固態法製備之LMT材料樣品SEM表面微結構圖...............35
圖3-13 以檸檬酸鹽法製備之LMT材料樣品SEM表面微結構圖.........35
圖4-1 第一原理基態計算流程圖……………………………………..37
圖4-2聲子計算流程圖…………………………………………………38
圖4-3 unit cell 中的LMT結構圖(a)1 unit cell (b) (b)跟鄰近晶格中O
原子所構成八面體示意圖……………………………………………..40
表4-1 來自文獻上的LMT晶格結構…………………………………41
圖4-4 結構穩定性的測試……………………………………………..43
圖4-5 鬆弛(relax)後的LMT結構(a)單一個unit cell (b) Top view of 3x3 unit cell…………………………………………………………….44
圖4-6 LMT結構(a)電子態密度(electronic DOS) (b)以VASP軟體算出來的電子態密度(c)p-DOS圖………………………………..………...45
圖4-7 LMT能帶結構計算在布里淵區走的路徑由Z經Γ、A、B、D至E點…………………………………………………………………..46
圖4-8 LMT材料的能帶結構—在各能量上的分布…………………..48
圖4-9 La(Mg1/2,Ti1/2)O3材料的FTIR光譜……………………………54
圖4-10 La(Mg1/2,Ti1/2)O3材料的Raman光譜………………………...54
圖4-11 以固態法製備的La(Mg1/2,Ti1/2)O3材料的FTIR光譜………55
表4-2 計算得到之g mode頻率與Raman實驗所得頻率…………...55
表4-3 計算所得之u mode頻率與FTIR實驗所得頻率…………….56
表4-4 計算所得之mode effective charge……………………….........57
表4-5 每個模式相對介電貢獻………………………………………..59
表4-6 計算所得之各振動模式……………………………………60~74
表4-7 利用FTIR光譜及第一原理計算所得之介電常數……………75
圖4-12 利用FTIR光譜算出之固態法與檸檬鹽法製備樣品之Qxf   
       值………………………………………………………………75
表4-8 以FTIR光譜得到介電常數虛部之頻率、強度、半高寬及介電  
      常數……………………………………………………………..75

[1]R.D. Richtmyer, 'Dielectric Resonators', J. Appl. Phys. 10(1039)391-398
[2]P. Hohenbergand W. Kohn,"Inhomogeneous Electron Gas ", Phys. Rev. B136, 864(1964)
[3]W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects, Phys. Rev. 140,1133 (1965)
[4] John P. Perdew and Wang Yue, ’Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation’, Phys. Rev. B 33, 8800 (1986)  
[5]M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D.
Joannopouios,“Iterative minimization techniques for ab initio total energy
calculations molecular dynamics and conjugate gradients” Phys. Rev.
Mod. 64:1045,(1992)
[6]Kittel, ‘Introduction to Solid State Physics’ ,John Wiely & sons 7th ed.(1996) 
[7]P.Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni 'Ab initio calculation of phonon
dispersions in semiconductors' Phys. Rev. B43, 7231-7242 (1991)
[8]A. V. Hipple ,'Dielectrics and waves, 2nd edition', Atrech House, London, 1-86(1996)
[9]G. Burns,'Solid state physics, Academic press', Florida,450-486(1985)
[10]羅吉宗,'薄膜科技與應用',全華科技圖書,chapter8(2004)
[11]李匡邦,許東明,何東英,'光譜化學分析',揚智文化事業,chapter 11,(1997)
[12]Eugene Hecht, Optics, Addison Wesley, 3rd ed., New York, 1998.
[13]Girish Harse, A.S. Bhalla and L.E. Cross, 'Synthesis dielectric properties of a cubic perovskite La(Mg1/2,Ti1/2)O3' ,Materials Letters 18 (1994) 173-175
[14]M.P. Seabra ,V.M. Ferreira, 'Synthesis of La(Mg0.5,Ti0.5)O3 ceramics for microwave applications' ,Materials Research Bulletin 37 (2002) 255-262
[15]S. Irusta, M. P. Pina, M. Menendez, and J.Santamara, “Catalytic
Combustion of Volatile Organic Compounds over La-Based
Perovskites”, J. of Catalysis 179, 400-412(1998)
[16]C. Marcilly, P. Courty and B. Delmon, “Preparation of Highly
Dispersed Mixed Oxides and Oxide Solid Solutions by Pyrolysis of
Amorphous Organic Precursors”, J. Am. Ceram. Soc., 53(1), 56-57
(1970)
[17]W.G. Spitzer,R.C. Miller,D.A. Kleinman and L.E. Howarth , "Far infrared dielectric dispersion in BaTiO3,SrTiO3,and TiO2" ,Phys. Rev. 126,1710-1721(1962)
[18]I. Levin, T. A. Vanderah, T. G. Amos, and J. E. Maslar,’ Structural Behavior and Raman Spectra of  Perovskite-Like Solid Solutions (1-x)La(Mg0.5,Ti0.5)O3-xLa2/3TiO3’, Chem. Mater. 17, 3273-3280(2005)
[19]D.Y. Lee,S.J.Yoon,J.H. Yeo,'crystal structure and microwave dielectric properties of
La(Mg1/2,Ti1/2)O3 ceramics',J. Mater. Sci. Lett. 19,131-134(2000)
[20]Rousseau, D. L.,Bauman, R. P. and Porto, S. P. S.,'Normal mode determination in crystals',J. Raman Spectroscopy,10 253-290(1981)
[21]E.Cockayne ,and B. P. Burton,'Phonons and static dielectric constant in CaTiO3 from 
first principles',Phy. Rev. B62, 3735(2000)
[22]Eric Cockayne,'First-principles calculations of the dielectric properties of perovskite-type materials',J. Eur. Ceram. Soc. 23,2375-2379(2003)
[23]J. W. Bennett, I. Grinberg,and A. M. Rappe,'Effect of symmetry lowering on the dielectric response of BaZrO3', Phy. Rev. B73,180102(2006)
[24]http://www.abinit.org
[25] http://opium.sourceforge.net/
[26]M. Born and K. Huang, ’Dynamical Theory of Crystal Lattices’, Oxford University Press, Oxford.(1954)