Ba(B’1/3B”2/3)O3之材料特性研究與在天線方面應用

全文

(1)國立台灣師範大學物理系研究所博士論文. Ba(B’1/3B”2/3)O3 material property investigation and its application in antenna. 指導老師：賈至達博士(Prof. Chih-Ta Chia) 陳穎叡博士(Prof. Yiing-Rei Chen). 研究生：陳美瑜 (MeiYu Chen)撰中華民國一百零三年七月.

(2) 摘要本論文主要分成兩部分，主要目的在探討 Ba(B’1/3B”2/3)O3 鈣鈦礦結構行為。 Ba(B’1/3B”2/3)O3 為工業上主要的微波介電材料，其中 Ba(Mg1/3Ta2/3)O3 更是以低介電損失而聞名。本文中將利用第一原理模擬計算去推測完美的 Ba(B’1/3B”2/3)O3 晶體其體積模量(bulk modulus), 包含拉曼與紅外線吸收的聲子振動行為，以及由聲子與電子貢獻的介電常數。另一方面利用 Ba(Mg1/3Ta2/3)O3 材料去設計一個介電共振天線。首先，利用第一原理模擬出 Ba(Mg1/3Ta2/3)O3 與 Ba(Mg1/3Nb2/3)O3 的體積模量分別為 156 GPa 與 258 GPa。Ikawa et al 教授的論文(1998)中顯示了實驗 Ba(Mg1/3Ta2/3)O3 的體積模量為 154 GPa，這實驗結果與本論文中模擬體積模量相當接近。可惜的是 Ba(Mg1/3Nb2/3)O3 材料並沒有實驗的體積模量數據可供比較。第一原理也可分析並提供 Ba(B’1/3B”2/3)O3 簡正振動模的頻率與行為，包含了九個拉曼聲子，十六個紅外聲子以及三個無法激發的聲子振動模。詳細的分類與振動型式可見附錄一。經由第一原理推測出的 Ba(B’1/3B”2/3)O3 單晶理論頻率與參考論文中的陶瓷多晶樣品實驗頻率相當的接近。在計算介電常數上，Born and Huang 提供了有效電荷模型可計算由聲子提供的介電貢獻。在微波範圍中，只要考慮聲子與電子的介電貢獻，經由計算後，模擬 Ba(Mg1/3Ta2/3)O3 的聲子與電子介電貢獻分別為 23.4 與 4.14。此結果與紅外吸收實驗所得知結果相當接近 (εr(phonon)=23.3 and εr(electron)=4.4). 而 Ba(Mg1/3Nb2/3)O3 的理論推測值結果也相當符合實驗。而有四個對介電貢獻貢獻重大的聲子分別為 2Eu, 2A2u, 4Eu, 與.

(3) 3A2u，其中兩個聲子(2Eu and 2A2u) 為鋇離子與其他離子的相對運動；而另外兩個聲子(4Eu, 3A2u) 為 B” 離子與氧的的相對運動。經由辨別各聲子的運動行為，我們可以解釋參雜不同雜質的 Ba(B’1/3B”2/3)O3 聲子改變行為，例如 SrxBa1-x(Mg1/3Ta2/3)O3 (x< 0.5)。在參雜量小於 0.5 ，拉曼光譜並無相變，兩個 A1g 特徵模(420 cm-1 and 800 cm-1)卻有不同的頻率改變行為。這可以合理推測鍶離子偏好位於鋇離子與鎂離子的位置，而非鉭離子的位置。而且因為鍶偏好佔據鋇離子位置上而且鍶離子具有較小的質量與較大的 Born 有效電荷導致隨著參雜濃度增加，樣品的介電常數也隨之增加。在 Ba(B’1/3B”2/3)O3 的應用上，本文使用 Ba(Mg1/3Ta2/3)O3 材料設計一個介電共振天線，其主要應用在無線通訊方面，共振頻率在 2.4 GHz 至 2.484 GHz。此天線在 2.44 GHz 中有最小的回饋損失 (Return loss) -34.67 (dB)，與最強的效率(68 %)與天線增益(5.13)。在 3D 的輻射途中，天線的 xy 平面具有類似全方面的輻射圖，但是在 y-z 與 x-z 平面表現出定向的輻射。關鍵字: 鈣鈦礦, 第一原理, 介電共振天線.

(4) Abstract In this paper, there are two approaches to investigate Ba(B’1/3B”2/3)O3 perovskite ceramics. First study focus on ab-initio simulation of Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3. The properties of Ba(B’1/3B”2/3)O3 ideal single crystal can be predicted or compared with measured results, such as bulk modulus, normal vibration motions, and permittivity values contributed by phonon and electrons. The other practical study is to design an dielectric resonator antenna of Ba(Mg1/3Ta2/3)O3. The estimated bulk modulus of Ba(Mg1/3Ta2/3)O3 is around 156 (GPa) while measured value is 154 (GPa).The normal vibrational modes of Ba(B’1/3B”2/3)O3, including 9 Raman phonons, 16 IR modes and 3 silent modes, are classified and illustrated in Appendix I. These frequencies of calculated phonon modes are not only quite close to measured frequencies, but also for the permittivity values. The calculated permittivity of Ba(Mg1/3Ta2/3)O3 due to phonon and electron obtained the value of 23.4 and 4.14, respectively;. that. are. consistent. with. measured. value. of. from. IR. analysis(εr(phonon)=23.3 and εr(electron)=4.4). The same conclusion also applies for the results of Ba(Mg1/3Nb2/3)O3. In addition, the behaviors of the four dominant modes to permittivity, 2Eu, 2A2u, 4Eu, and 3A2u, were figured out. 2Eu and 2A2u mode are the vibration of Ba atoms against other atoms while 4Eu, and 3A2u mode refer to the relative motion of Ta/Nb atoms and oxygen atoms. Through identifying the actions of each vibration mode, the variation of phonon mode in substituted.

(5) SrxBa1-x(Mg1/3Ta2/3)O3 system (x< 0.5) can be explained. Two A1g mode (420 cm-1 and 800 cm-1) which have the same vibration atoms perform different behaviors. It indicates that Sr atoms prefer locate on Ba and Mg site, instead Ta site. Furthermore, Sr substituted at Ba site would lead to higher permittivity values because of smaller mass and larger Born effective charge of Sr atoms. The last chapter depicts an attempt to utilize Ba(Mg1/3Ta2/3)O3 ceramics in antenna. The permittivity value was applied in design a dielectric resonator antenna of Ba(Mg1/3Ta2/3)O3 for WLAN applications (2.4 GHz to 2.484 GHz). The measured return loss (S11) have lowest point of -34.67 dB at 2.44 GHz. The efficiency and gain of antenna both peaked at 2.44GHz, being 68 % and 5.13, respectively. Radiation patterns in x-y plane perform omni-directional but the y-z and x-z plane of radiation patterns illustrate the directional radiation.. Keyword: perovskite, first principle, dielectric resonator antenna.

(6) ACKNOWLEDGEMENT 對於這個致謝,我有許多的話想說卻又說不清楚,對於終於要博士畢業的當口,畢業即失業的壓力真是苦樂交雜,但是也對自己想說終於完成這件事了第一個最該感謝的是賈至達老師,自碩士在老師的門下之後,老師總是對我多加包容,苦口婆心地給我指導,雖然有時候我並沒有聽得進去,但老師仍是秉持著不能不管我的態度繼續的教導我,並且提供出國機會讓我去見識世界,努力的寫合作計畫讓我能夠申請芬蘭的博士班學程, 讓我能在芬蘭與 Heli 學習並工作。第二個要感謝的是陳穎叡老師,本文中的理論計算都是向她學習而來,因為我不懂寫程式,理論基礎又很差,所以陳穎叡老師花了很多心力幫我,可是我卻沒有完成介電損失的理論計算,讓老師您失望了。另外也感謝系上的 cluster,雖然常常斷電讓我的計算失敗，但是至少還是有些東西出來。第三個要感謝的是許志雄老師，許老師並不是我的導師，但是認識他之後在他身上學到很多做人做事的道理，他比賈老師還要諄諄教誨，但是為人卻又很隨和，很願意幫助別人，現在我在芬蘭的學習也常常去徵詢許老師的意見。還有遠在芬蘭的 Heli Jantunen，謝謝她願意出計畫錢養我。Heli 雖然很忙，但是在投稿論文的時候也是一句一句的面對面告訴我怎麼改與怎樣把枯燥的數據轉化成會讓審稿委員認為是有趣的故事。至於在助教期間跟陸健榮老師手下管理普物實驗室也是很難得的經驗，陸老師的認真負責，處理眉眉角角的細節讓粗枝大葉的我感到佩服也學習到很多。還有謝.

(7) 謝宜君學姊幫我當妹妹看待，我有很多論文都有參考宜君學姊的資料，雖然不常見面，但是每次相見都很愉快。接下來是在本論文上曾經幫助我的人，陳穎叡老師實驗室的李欣翰，芬蘭方面的 Vamsi Palukuru 幫我設計 BMTO 的介電共振天線,Geotemi 教我基本天線常識還有一起在拉曼實驗室分享甘苦的黃同慶學長，最好的朋友碧容，杰翰，益豪，詠謙，鈴鈞，黃捷等，同為普物助教的藝丰，學弟布丁，還有系辦的助教們，謝謝你們的幫助，也希望你們的近況一切平安順利。最後最後要謝謝我的家人，感謝你們能夠接受我到三十多歲都還沒有賺錢養你們，而且不常回家，不常打電話，一個人跑到遙遠的芬蘭讓你們擔心，我的一切全為你們給予的，也希望有一天你們能我以為榮。.

(8) Contents Introduction .............................................................................................. 6 Background ............................................................................................ 6 Structure characteristics ....................................................................... 11 Raman scattering .................................................................................. 14 Infrared Reflectance Spectroscopy ...................................................... 17 Theoretical First principle (ab-initio) calculations .............................. 19 The background of dielectric resonator antenna .................................. 20 Summary .............................................................................................. 20 Reference ............................................................................................. 21 Theoretical calculations of Ba(Mg1/3Nb2/3)O3 and Ba(Mg1/3Ta2/3)O3 ...... 23 Background .......................................................................................... 23 Basic concept of Density Functional Theory ....................................... 24 History development ............................................................................ 24 Exchange-correlation energy functional and two approximation ........ 28 Pseudopotential. between. norm-conversing. and. ultra-soft. pseudopotential .................................................................................................... 29 The. ground. state. calculation. of. Ba(Mg1/3Ta2/3)O3. and. Ba(Mg1/3Nb2/3)O3 ................................................................................................. 32 Vibrational normal modes .................................................................... 37 1.

(9) Permittivity from phonon and electrons .............................................. 40 Summary .............................................................................................. 43 Reference ............................................................................................. 43 The comparison of measured modes with the first principle investigation of Ba(Mg1/3Nb2/3)O3 and Ba(Mg1/3Ta2/3)O3 ................................................................. 45 Ba(Mg1/3Ta2/3)O3 & Ba(Mg1/3Nb2/3)O3 ................................................ 45 SrxBa1-x(Mg1/3Ta2/3)O3 ......................................................................... 52 Conclusion ........................................................................................... 55 Reference ............................................................................................. 55 Antenna design of Ba(Mg1/3Ta2/3)O3........................................................ 58 Dielectric resonator antenna ................................................................ 58 Design, Measurements and Fabrications ............................................. 64 Results and analysis ............................................................................. 66 Conclusion ........................................................................................... 69 Reference ............................................................................................. 69 Summary .................................................................................................. 71 Appendix I ................................................................................................................... 73. 2.

(10) List of Tables TABLE 1-1 THE NORMAL VIBRATION MODES OF ABO3 GROUP OF PEROVSKITE ............... 13. TABLE 2-1 THE AXIS VECTOR AND POSITION OF ATOMS OF BA(MG1/3TA2/3)O3. ................... 33. TABLE 2-2 THE EIGENVECTORS AND EIGENVALUES OF NORMAL MODES OF BMTO AND. BMNO .......................................................................................................................................... 38. TABLE 3-1 THE RAMAN NORMAL MODES AND THEIR ASSIGNMENTS OF BMTO AND. BMNO .......................................................................................................................................... 46. TABLE 3-2 THE IR PHONON AND THEIR PERMITTIVITY CONTRIBUTIONS OF BMTO: ΔΕ. REFER TO THE DIELECTRIC PERMITTIVITY CONTRIBUTION FROM IR PHONON; TOX IS FOR OPTICAL PHONON FREQUENCY (CM-1); TOΓ IS FOR WIDTH OF OPTICAL PHONONS (CM-1)........................................................................................................................ 48. TABLE 3-3 THE IR PHONON AND THEIR PERMITTIVITY CONTRIBUTIONS OF BMNO ....... 49. TABLE 3-4 THE MEASURED AND CALCULATED PERMITTIVITY CONTRIBUTION FROM. ELECTRONS AND PHONONS .................................................................................................. 51. TABLE 3-5 THE MICROWAVE PROPERTIES OF SRXBA1-X(MG1/3TA2/3)O3 WITH X=0, 1/8, 2/8, 3/8,. 4/8 ................................................................................................................................................. 54. 3.

(11) Lists of Figures FIG. 1-1 SCHEME OF (A) MICROWAVE CIRCUITS, (B) EQUIVALENT CIRCUIT AND (C). IMPEDANCE ............................................................................................................................... 10. FIG. 1-2 (A) UNIT CELL STRUCTURE F A(B’1/3B”2/3)O3 (B) THE 1:2 ORDERING STRUCTURE OF A(B’1/3B”2/3)O3. ....................................................................................................................... 13. FIG. 1-3 THE ENERGY LEVEL OF RAYLEIGH AND RAMAN SCATTERING. ............................. 15. FIG. 1-4 THE SETUP OF FTIR SPECTROMETER. ............................................................................ 19. FIG. 2-1 SCHEME OF KOHN-SHAM TRANSFORMATION. ............................................................ 28. FIG. 2-2 THE TOTAL ENERGY OF BA(MG1/3TA2/3)O3. ..................................................................... 35. FIG. 2-3 THE TOTAL DENSITY OF ELECTRON STATES OF BA(MG1/3TA2/3)O3 AND. BA(MG1/3NB2/3)O3. ...................................................................................................................... 35. FIG. 2-4 THE BULK MODULUS CALCULATION OF BA(MG1/3TA2/3)O3. ...................................... 36. FIG. 2-5 THE BULK MODULUS CALCULATION OF BA(MG1/3NB2/3)O3. ..................................... 37. FIG. 3-1 RAMAN SPECTRA OF SRXBA1-X(MG1/3TA2/3)O3 WITH X=0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8,. 1 .................................................................................................................................................... 53. FIG. 4-1 GEOMETRY FOR THE DIELECTRIC RESONATOR ANTENNA MODEL. ...................... 60. FIG. 4-2 ANTENNA INPUT IMPEDANCE MODEL........................................................................... 62. FIG. 4-3 SCHEMATIC LAYOUT OF PROPOSED DRA (DIMENSIONS IN MM) ............................ 65. FIG. 4-4 SATIMO STARLAB SYSTEM FOR THE MEASUREMENT OF RADIATION. 4.

(12) PROPERTIES OF THE DRA ANTENNA. .................................................................................. 66. FIG. 4-5 MEASURED RETURN LOSS OF DRA. ............................................................................... 67. FIG. 4-6 EFFICIENCY OF DRA OF BMT. ........................................................................................... 67. FIG. 4-7 MEASURED RADIATION TE-MODES PATTERNS AT F = 2.45 GHZ. (A) X-Y PLANE (Φ=0O); (B) Y-Z PLANE (Φ=90O) (C) X-Z PLANE (Θ=90O)...................................................... 68. 5.

(13) Introduction. Background Historians found that humans started using ceramics to manufacture of ceramic wares since the Neolithic age. "Ceramic" means pottery and porcelain, which is evolved from the Greek word “keramos”, refers to the use of clay by firing in the 900-1200 oC temperature range. The meaning of ceramics change with time. Nowadays, the higher purity of chemical ingredients broadens the meaning of ceramics. Usually, ceramics are composed as polycrystalline, inorganic, non-metallic materials that acquire their mechanical strength through a firing and sintering process. Unlike metals, which transformed phases with changing temperature, ceramics exhibited the resistance of high temperature, high stability and corrosion. The daily use ceramics, such as tiles, ceramic ware, bath, or a beautiful Jingdezhen ceramic art, are known as "traditional ceramics". Beyond traditional ceramics, the fine ceramics required high purity of chemicals and have more and more applications in modern society.. Many special characteristics of fine ceramics, such as piezoelectric,. ferroelectric, pyroelectric and electro-optic behaviors can be applied into various technology. Nowadays, traditional crafts and art which originally dominated the evolution of traditional ceramics gradually became the fine ceramics of today's 6.

(14) science and tech. Fine ceramics is refined as high-purity inorganic raw materials, using a variety of chemical or physical methods precise control of the composition and uniformity, then dry pressing, slip casting, injection molding or the like method after forming, sintering, and the precise reaction after steps to microstructure and physical properties and resistance to a certain standard, and then processed into finished products with superior function. It has a hard, wear-resistant, pressure resistance, heat, acid, alkali and other long-lasting features, and has a very excellent optical, electromagnetic, thermal features and biocompatibility. Range based on the use of fine ceramics can be divided into three categories: (1) electronic ceramics, (2) structural ceramics and (3) Biomedical ceramics. This article is to discuss the most important electronic ceramic dielectric ceramics. Dielectric ceramics can be sorted to three types: Class I, Class II, and Class III,1 which are classified by permittivity values. In this study, Class I dielectrics is what we concerned. Class I dielectrics are sorted as medium-permittivity ceramics, which have permittivity between 15 to 100 and loss tangent lower than 0.003. The main applications for Class I dielectrics are high power transistor capacitors at 0.5 – 50 MHz, capacitor for 1kHz-100MHz, and microwave resonant device for 0.5 GHz- 50 GHz. The first microwave resonant devices were designed by Rychtmyer in 1939.2 A 7.

(15) cylindrical dielectric material was bent into an annular end to end to form a waveguide, which can be radiated specific microwave frequency and regarded as a resonator or antenna. However, after next 20 years, there was no obviously breakthroughs due to the lack of the resonator cavity and detectors.3 Around 1970, two findings enhanced the dielectric materials developments on microwave region. First, microwave circuit analysis was built, so that the microwave properties of manufactured samples can be measured. The other breakthrough was that titanium dioxide (TiO2) to be used as resonator ceramic materials. Through continuous improvement, Ba2Ti9O20 and perovskite-like ceramics have been found. Rutile and perovskite structure ceramics are two dominant branches for microwave materials. Oxygen octahedral structures characteristics in rutile and perovskite ceramics affect the microwave dielectric properties, such like permittivity, loss and temperature coefficients of frequency. In sum, many Class I dielectrics play a pivotal role in various way of modern technology such as smart phones, satellite communications, base-stations, General Packet Radio Service (GPRS) and so on. The requested properties of microwave communication materials are high dielectric permittivity, low loss and stable temperature coefficients. . High permittivity: the main reason why one microwave resonator request high permittivity is that sizes of dielectrics are inversely proportional to the square 8.

(16) root of their permittivity. When electromagnetic plane-waves propagate in the dielectric materials, the boundary condition was fulfilled. While the ends of circuits are short-circuited or open-circuited, the length of dielectrics are L. n. . 2.  2. , ,. 3 2. ,....... ,. where L refer to size of dielectrics and λ represent the. microwave wavelength in dielectric media. E  E0 e.  .  jkx. 2. . Re{ f. 0 Re{  r }. .  (1.1.1).  }. Light speed in vacuum c.  . . 1. . k. . , k  k'-jk"     Re    j Im  .  0 f . C f Re{  r }. L. 1.  , when μr~ 1,. L. 1 R e { r }. (1.1.2). Therefore, higher permittivity of dielectrics enable smaller required sizes of devices. . There are two important factors for resonators: resonant frequency and quantity factor. The quantity factor (Q) is a parameter which relate with energy consumption and storage capacity of resonant circuit. In other word, it can be defined as inversion of loss tangent value, tanδ=1/Q3; Loss tangent is defined as the ratio of unit circle consumption energy and total storage energy in the devices.. e jt 9.

(17) I. I. ～. Dielectrics. 1 Z. ωC. ～. R. C. δ. 1 R. θ. Fig. 1-1 Scheme of (a) Microwave circuits, (b) Equivalent circuit and (c) Impedance. A typical microwave resonant circuit was demonstrated in Figure 1-1(a), and the equivalent circuit was shown in Figure 1-1(b).. V0e jt  jCV0e jt R 1 1 V Then, I 0  0    jC Z Z R I (t )  I 0e jt . (1.1.3) (1.1.4). The total impedance of circuits (Z) is a complex number, shown in Fig. 1-1(c) The dielectric is regarded as an ideal capacitor, thus R→∞，θ=π/2、δ=0 and no energy loss. However, the real situation in circuit is that R≠∞，θ<π/2, due to metallization of dielectrics and radiation loss. The average power of energy consumption was shown in (1.1.5).. Power . V02 1   cos   sin  2R R. (1.1.5). For a low loss materials, δ→0 and sin   tan  . The tanδ, was also called dissipation factor, is an important factor in microwave circuits. The request of loss tangent have to be lower than 1/5000. . Temperature. coefficient:. dielectric 10. properties. of. all. materials. are.

(18) temperature-dependent, not only the permittivity but resonant frequency. 1    1  f    ，     f  T    T . f . (1.1.6). The relationship between each other is.    2( f  T ). (1.1.7). where αT is the thermal expansion coefficient. The variations of resonant frequency have to be smaller than 1-3 ppm/oC. 4. T .  2.   f  0  1 ~ 3 ppm/ o C. (1.1.8). Dielectric materials have a wide application in microwave regions. High permittivity and low permittivity have different application region. However, no matter what materials were applied in circuit due to their characteristics of materials, the low loss is dominant for all types of applications.. Structure characteristics Perovskite ABO3 structure were found after twenty century, their extraordinary dielectric, ferroelectric, piezoelectric properties attracted scientists to investigate this type of material. One of complex perovskite group contain several cations ordering which contained A-site ordering, or B-site ordering, such as A(B’1/2B”1/2)O3 、 A(B’1/3B”2/3)O3、(A’1/2A”1/2) (B’1/2B”1/2)O3、(A’1/2A”1/2) (B’1/3B”2/3)O3. These types of complex perovskite have a widely application in ferroelectrics and dielectrics.4 The . symmetry of A(B’1/3B”2/3)O3 belong to P 3 m1  D33d 11. 5-6. , and the structure of unit cell.

(19) is shown in Fig. 1-2(a).6 F. Galasso et al. found that A(B’1/3B”2/3)O3 have hexagonal structure. From zero point to c axis direction, 1:2 ordering structure of B’ and B” atoms can be observed. If the B’ and B” atoms are misplaced, the symmetry is broken, thus the symmetry group would transform to Pm3m  Oh1 , which are belonged to the ABO3 group. The normal modes of the complex perovskite group were calculated by group theory and shown in Table 1-1.7. O. c. z y. Ba x. B’ ” B” a b. 12.

(20) Fig. 1-2 (a) unit cell structure f A(B’1/3B”2/3)O3 (b) the 1:2 ordering structure of A(B’1/3B”2/3)O3.. Table 1-1 The normal vibration modes of ABO3 group of perovskite. (A’1/2A”1/2) A(B’1/2B”1/2) Compound. (A’1/2A”1/2). ABO3. (B’1/2B”1/2)O O3. A(B’1/3B”2/3)O3. BO3 3. Space. Pm3m  Oh1. Fm3m  Oh5. Fm3m  Oh5. . F 4 3m  Td2. . P 3 m1  D33d. group. A1g(O) Raman. －. Eg(O). A1(O). 4A1g(A,B”,O). E(O). 5Eg(A,B”,O). F2g(B). 2F2g(A,O) 13.

(21) 9Eu(A,B’,B”,O) 4F1u(A,B’,B”, 5F1u(A’,A”,B’ infrared. －. 3F1u(A,B,O) O). 7A2u(A,B’,B”,. B”,O) O) 6F2(A’,A”,B. Raman and －. －. －. －. ’,B”,O). IR. Silent. F2u(O). F1g(O)+F2u(O). 3F2u(O). 2F1(O). A2g(O)+2A1u(O). Acoustic. F1u. F1u. F1u. F2. A2u+Eu. The unit cell of A(B’1/3B”2/3)O3 has 15 atoms, 45 degree of freedom. Table 1-1 show the normal vibration modes of A(B’1/3B”2/3)O3. Through group theory calculation, 9 Raman modes, 16 IR modes, and 3 silent modes and 3 acoustic modes are revealed.. Raman scattering The interaction of light and materials, there are many interaction mechanisms, such as scattering, transmission, reflection, refraction, and emission. The scattering effect between light and materials can be classified as elastic and inelastic scattering. The dipoles which are excited by the light would emit the electromagnetic waves in all directions. These waves are not in phase with each other, thus the condition of constructive interference cannot be met. This kind of scattering is called Rayleigh 14.

(22) scattering. In non-absorbing medium, the more defect of the object exist, the higher the Rayleigh scattering intensity excite. In addition, Rayleigh scattering usually is accompanied by Raman scattering whose energy differ from the incident ray. The scattering causes from the coupling between incident light and phonons (lattice vibrations) and generate a higher frequency of the incident light (anti-Stokes Effect) or a lower frequency than the initial frequency (Stokes Effect), as shown in Fig. 1-3. These characteristics are closely related with the electric, physical, and chemical properties of materials.8 Rayleigh scattering. Stokes effect. Anti-stokes effect. Vibrational energy. The ground state of electrons Fig. 1-3 The energy level of Rayleigh and Raman scattering.. The phonons of materials can be generated by light or heat energy. However, not all of phonon can be excited by light and be detected by spectrometers because 15.

(23) phonon is very sensitive to the symmetry of crystal structures, especially for ferroelectric perovskite strucutre. To study ferroelectric materials, Raman scattering is one of effective way. The phase transitions with temperature result in the spectral changes.. 9. Raman measurement is one of the best tools in this topic because it is. non-contact, non-destroying technique, being also sensitive for detecting the local structure of materials. In addition, it is also sensitive to microstructural properties, especially to non-crystallized structure system. Micro-Raman measurements can be used on the determination of structure and their heterogeneity. Especially Raman scattering method is useful to obtain the parameters of crystal. These parameters can truly reflect the changes of structure as a function of e.g. temperature, pressure, electric field or stress. High-frequency vibration frequency changes correspond to changes of the crystal lattice, such as bond strength of the tetrahedral or octahedral structure. When the strengths of bonds increase, the corresponding modes go to higher frequency. Low frequency modes change correlate to whole movement of atoms, such as the electric dipole strength. For Raman intensity, the relative intensity of the Raman modes can reflect changes of vibrational modes; when the relative intensity of a specific vibrational mode decrease compared other vibrational modes strength, this vibrational mode associated with the atomic positions inside the crystal has changed. In other word, the atoms are not in the position of lattice. Therefore, no specific 16.

(24) phonon can be produced, thus the relative intensity becomes weaker. Raman signals were collected in this paper by DILOR XY-800 spectroscopy at room temperature, and detected by CCD dector which were cooled down by liquid nitrogen. The Ar+ ionic laser is as excited source of 514.5 nm. The resolution of Raman spectrum was obtained to 0.5 cm-1.. Infrared Reflectance Spectroscopy Infrared reflectance spectroscopy can reveal the relation of relative permittivity of material with frequency. The models of dielectric permittivity are Drude model and Lorentzian model. The Drude model can explain the free electrons behaviors in metal. The electric and thermal conductivity of simple metal behave this model. For the dielectric materials, Lorentzian model can be perfectly described. The electrons behave as a oscillator that bounded in the atoms. If an external field from excitation that applied to a bounded electron, the electron behavior can be regarded as a damping oscillator applied a periodic oscillation force. The equation of motion was written as: 𝑑2 𝑥. 𝑑𝑥. 𝑚 𝑑𝑡 2 + 𝛾𝑚 𝑑𝑡 + 𝑚𝜔02 𝑥 = −𝑒𝐸. (1.4.1). where 𝜔0 refer to resonant frequency, 𝐸 = 𝐸0 𝑒 −𝑖𝜔𝑡 , 𝑥 = 𝑥0 𝑒 −𝑖𝜔𝑡 𝑥0 =. 𝑒𝐸. 1. (1.4.2). 𝑚 (𝜔 2 −𝜔02 )+𝑖𝛾𝜔. then dipole moment can be written as 𝑃0 = 𝑛𝑒𝑥0 = 𝜒𝑒 𝐸0 , where 𝜒𝑒 is susceptibility 17.

(25) of electrons. In addition, 𝜀 = 1 + 4𝜋𝜒𝑒 (𝜔), 𝑥0 = −. 𝑛𝑒 2. 𝜀 =1−. 1. (1.4.3). 𝑚 (𝜔 2 −𝜔02 )+𝑖𝛾𝜔 4𝜋𝑛𝑒 2. 1. 𝑚. (𝜔2 −𝜔02 )+𝑖𝛾𝜔. (1.4.4). The relative permittivity is a complex number, the real and imaginary part of permittivity are shown. 𝜀𝑟 = 1 +. 2 (𝜔 2 −𝜔2 ) 𝜔𝑝 0. (1.4.5). (𝜔02 −𝜔2 )2 +𝛾2 𝜔2 2 𝛾𝜔 𝜔𝑝. 𝜀𝑖 = (𝜔2−𝜔2 )2 +𝛾2 𝜔2. (1.4.6). 0. where ω2𝑝 =. 4𝜋𝑛𝑒 2 𝑚. The experiments were measured by Fourier Transform Infrared ray spectrometer (FTIR, Bruker IFS66v/S, MA, USA) in the lab of Prof. Liu(NTNU). The spectrometer were designed and worked as a Michelson interferometer in Fig. 1-4. The reflectance spectroscopy can obtain many information of materials, such as refractive index, conductivity and permittivity.. 18.

(26) Fig. 1-4 The setup of FTIR spectrometer.. Theoretical First principle (ab-initio) calculations Ab-initio is to mean “from the beginning”. Usually it is applied to the theoretical calculations based on quantum mechanics and totally deduced from the theories without any experimental results and empirical or half-empirical formulas. The behavior of electrons dominate most of properties of materials.10 In condensed matter region, the chemical bonds between atoms and their properties of materials can obtained by solving Schrödinger equations of electrons. Many researches provide remarkable breakthroughs in order to solve not only the extremely difficult many-body electrons Schrödinger equations but also the interactions energy between electrons. Nowadays, many successful predictions and results of first principle, such as molecules or solid state materials, were studied. The different software of first principle calculation have their own advantage and disadvantage; for example, Vienna 19.

(27) Ab initio Simulation Package (VASP) have extremely soft pseudo-potential that only require less calculation loading. However, the non-linear responds functions have not been developed yet. For ABINIT software, although it is one of most useful and open free source, the norm-conserving pseudopotential of ABINIT would cause heavy calculation loading. The more detail for first principle theory would be described in Chapter 2.. The background of dielectric resonator antenna Ba(Mg1/3Ta2/3)O3 is well known and usually applied in resonator due to its ultra-low microwave loss. However, Ba(Mg1/3Ta2/3)O3 is used to applied in antenna are rarely found. Chapter 4 describes an attempt of designing an dielectric resonator antenna by using the characteristics of Ba(Mg1/3Ta2/3)O3. The brief introductions of dielectric antenna are also introduced in section 4.1.. Summary To summarize chapter 1, the structure properties of Ba(B’1/3B”2/3)O3, the experiment methods to investigated and possible antenna application were brief introduced. Ba(B’1/3B”2/3)O3 is one of most famous complex perovskite due to ultra-low loss in microwave region. Raman and FTIR measurement are very efficient tools to investigate the vibration modes which reveal the structural characteristics and IR result can deduce the permittivity from phonon and electron contributions. However, 20.

(28) the assignment of Raman and IR spectra sometimes are incorrect, thus First principle provide a direct set of normal vibration modes, including vibrational frequencies, oscillation styles of atoms.. Reference 1. J. Moulson and J. M. Herbert, Electroceramics: Materials, Properties, Applications; p. 277. John Wiley & Sons Ltd, England, 2003. 2. R. D. Richtmyer, Dielectric Resonators, J. Appl. Phys., 10(1939), 391. 3. Terrell A. Vanderah, Talking Ceramics, Science, 298 (2002), pp. 1182-1184. 4. Ebbe Nyfors, “CYLINDRICAL MICROWAVE RESONATOR SENSORS FOR MEASURING MATERIALS UNDER FLOW”, PhD thesis, Department of Electrical and Communications Engineering, Helsinki University of Technology, Finland, 2000. 5. F. Galasso and J. Pyle, “Ordering in Compounds of the A(B’0.33Ta0.67)O3 Type” Inorg. Chem., 2 (1963), 482. 6. Hiroshi Tamura, Lattice vibrations of Ba(Zn1/3Ta2/3)O3 crystal with ordered perovskite structure, Jpn. J. Appl. Phys. 25 (1986) ,787. 7. I. G. Siny, R. S. Katiyar, Cation arrangement in the complex perovskites and vibration spectra, J. Raman spectroscopy, 29 (1998), pp385-390. 8. G Lucazeau, L Avello, “Raman spectroscopy in solid state physics and materical sciene. Theory. Techniques and applications”, Analusis, 23(1995), pp.301-311. 21.

(29) 9. R. Loudon, “The Raman effect in crystals”, ADVANCE IN PHYSICS, 50 (2001), pp. 813-864. 10. M. D. Segall, P. J. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clark, and M. C. Payne, “First-principles simulation: ideas, illustrations and the CASTEP code” J. Phys: Condens. Matter.,14 (2002),2717-2744.. 22.

(30) Theoretical. calculations. of. Ba(Mg1/3Nb2/3)O3 and Ba(Mg1/3Ta2/3)O3 Background From 1926, Schrodinger developed the quantum mechanics which motion of equation can be described by a complete set of wave-functions. Based the principle of energy conservation, the equation can be depicted by a set of wave functions of time and positions 𝜑(𝑟, 𝑡), as V is potential. 𝜕. ℏ2. iℏ 𝜕𝑡 𝜑(𝑟, 𝑡) = (− 2𝑚 ∇2 + 𝑉)𝜑(𝑟, 𝑡). (2.1.1). For most of materials, such as molecular, polymer and solid-state, larger than 1023 electrons have been concerned and each of them have their own equations of motions. To simplify the complicatedness of calculations, the usages of approximations are necessary such as simpler functions or integrate methods. Born-Oppenheimer approximation is one of most useful approximations for ab-initio calculation. It can predigest the Schrodinger’s equations and divided two independent parts between these equations. The motions equations of electrons and nucleus can calculated separately. However, it is not enough to solve the problems. It is impossible to solve this amount of calculation loading. Therefore, it is essential to get suitable method and simplify the problem by other approximations or assumptions. However, 23.

(31) it is not enough to solve the problems until 1960. Prof. Walter Kohn proposed the theory called “Density Functional Theory”, shorted as DFT. DFT is a theory of correlated electrons many-body system. DFT approximation provides an acceptable description about the environment of ground states and makes the simulations of ab-initio calculations become a reliable method to approach the physical properties of materials. In recently year, DFT have widely applications in physical and chemistry, especially molecules and solid states. The remarkable successes of local density approximation (LDA) and Generalized-gradient approximation (GGA) functional approximation with Kohn-Shan approach led the interest and a trustful result compared to experimental data.1. Basic concept of Density Functional Theory History development In 1964, a famous paper written by P. Hohenberg and W. Kohn establish the foundation of DFT. The density of particle play a dominate role in ground state of a quantum many-body system. Kohn regards that “all” properties of the system can be considered to be functionals of the ground state density. Another important paper, which are written by W. Kohn and L. J. Sham, provide the effective method to describe the form of potentials and interacting force. The Hohenberg-Kohn theorems, which are an exact theory of many-body 24.

(32) systems, content two important concepts. First theorem interprets that all properties of system are determined only by the ground state density as the variable. The second theorem proves that the ground state energy 𝐸0 (𝑟) is lowest value of energy functionals of system, as well as the density 𝑛(𝑟)minimizes at ground state density 𝑛0 (𝑟). Many interacting particles behave in an external potential 𝑉(𝑟) . The Hamiltonian can be written 2. 2. ̂ = − ℏ ∑𝑖 ∇𝑖 2 + ∑𝑖 V𝑒𝑥𝑡 (𝑟𝑖 ) + ∑𝑖≠𝑗 𝑒 𝐻 |𝑟 −𝑟 2𝑚 𝑒. 𝑖. 𝑗|. (2.2.1). Theorem 1 are said that potential 𝑉(𝑟)is determined by the density of ground state particles 𝑛0 (𝑟) in the system of the interacting particles and external potential. In other word, since the Hamiltonian is fully determined, except constant shift of energy, all (ground and excited) the states of electrons are resolved. Thus, all properties of system are controlled totally by only the ground state density 𝑛0 (𝑟). Theorem 2 determines the energy variable 𝐸[𝑛] can determined the exact ground state energy and density. However, the properties of excited states need to take statistical ensemble in account. For example, the specific heat of systems are decided by free-energy functional of the density, which are reported from the work of Mermin.2 Hohenberg-Kohn theorems define and prove the consistency that the density of 25.

(33) electrons grounds states behaves the basic physical quantity of many-body systems. But the feasible forms of potential and kinetic energy in an interacting particles system have been not found yet. Kohn and Shan3 solved this problem in 1965. They replace the original many-body problem by an auxiliary independent-particle problem. The approach of Kohn and Sham assumes that electrons density of interacting systems is equal to that of some specific non-interacting systems. Thus, the ground state density can be represented by the ground state density of non-interacting particles system. The Hamiltonian can be written to the typical kinetic energy form 𝑇𝑠 𝜎 and an effective local potential 𝑉𝑒𝑓𝑓 (𝑟) acting on an electron of spin σ at r position. (See (3.2.1)). In addition, the density of state is given by sums of square of the orbitals for each spin. 𝜎. 𝜎 2 n(r) = ∑𝜎 𝑛(𝑟, 𝜎) = ∑ ∑𝑁 𝑖=1|𝜓𝑖 (𝑟)|. (2.2.2). Thus, independent-particle kinetic energy is given by 1. 𝜎. 𝜎 2 𝜎 2 𝑇𝑠 = − 2 ∑ ∑𝑁 𝑖=1|⟨𝜓𝑖 |∇ |𝜓𝑖 ⟩|. (2.2.3). The main part of potential term are attributed to classical Coulomb interaction energy of electron density n(𝑟) interacting with itself (the Hartree energy); 1. 𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 [𝑛] = 2 ∫ 𝑑3 𝑟𝑑3 𝑟′. 𝑛(𝑟)𝑛(𝑟 ′ ) |𝑟−𝑟′|. (2.2.4). Therefore, Kohn-Sham assumption can rewrite (3.2.1) for ground state energy 26.

(34) functional. 1. 𝐸𝐾𝑆 = 𝑇𝑠 [𝑛] + 2 ∫ 𝑑𝑟𝑉𝑒𝑥𝑡 (𝑟)𝑛(𝑟) + 𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 [𝑛] + 𝐸𝐼𝐼 + 𝐸𝑥𝑐 [𝑛]. (2.2.5). 𝑉𝑒𝑥𝑡 (𝑟) is the external potential due to nuclei and other external field and 𝐸𝐼𝐼 is the interaction between the nuclei. All effect of exchange and correlation are collected into the exchange-correlation energy 𝐸𝑥𝑐 . In order to solve Kohn-Sham equation, the minimization with respect to density n(𝑟, 𝜎) can figure out the solution of Kohn-Sham equations. The solution of independent-particle Kohn-Sham problem determines all properties of full many-body system. Through deriving variational Kohn-Sham equation (2.2.5), the total energy and electrons density of system can be obtained. Self-consistent calculation method is used to obtain the density of states. Fig. 2-1 illustrates the relation of Kohn-sham system and many-body system. The left side of Fig. 2-1 depicts that the process of generation of wave-functions and density. The external potential can decide the wave-functions of excited and ground states. Through this, we can determine the density of states. However, the difficulty to apply into real system is many-body effect of electrons. The right side of Fig. 3-1 illustrates that the self-consistent field method(SCF) is how to solve this difficult problems through KS assumption. First, the initial electrons density has been assumed through specified method. Through Hamiltonian equation, the new density can be generated by the eigenvectors and eigenvalues of (2.2.5). Finally, the convergent density 𝑛0 27.

(35) and total energy 𝐸0 of system can be obtained.. 𝑉𝑒𝑥𝑡 (𝒓). 𝑛0 (𝒓). 𝑛0 (𝒓). 𝑉𝐾𝑆 (𝒓). KS. Ψ𝑖 ( 𝒓 ). ψ𝑖=1,𝑁𝑒 (𝑟). Ψ0 ( 𝒓 ). ψ𝑖 (𝒓). Fig. 2-1 Scheme of Kohn-Sham transformation.. Exchange-correlation energy functional and two approximation The Kohn-Sham equation successful separate out the non-interacting particle kinetic energy and long-term Hartree terms, the other unknown terms are classified to exchange-correlate energy. Exchange energy ( 𝐸𝑥 ) obey Pauli principle while correlation energy causes from the correlation between orbital electrons. It is reasonable to approximate the Exc as a local or nearly local functional of the density. Therefore, energy Exc can be expressed in the form. 𝐸𝑥𝑐 [𝑛] = ∫ 𝑑𝑟 𝑛(𝑟)𝜀𝑥𝑐 ([𝑛], 𝑟). (2.2.5). where 𝜀𝑥𝑐 ([𝑛], 𝑟) is an energy per electron at point r that depends only upon the density 𝑛(𝑟, 𝜎) in some neighborhood of point r. Due to the exchange-correlation 28.

(36) energy cannot be obtained, an approximation has to be applied. While the density of electrons varies slightly, electron density at one point is close to density of electrons in the vicinity. This approximation is called as Local Density approximation (LDA). Therefore, (2.2.5) can be written as 𝜕𝜀. 𝛿𝐸𝑥𝑐 [𝑛] = ∑𝜎 ∫ 𝑑𝑟[𝜀𝑥𝑐 + 𝑛 𝜕𝑛𝑥𝑐𝜎 ]𝑟,𝜎 𝛿𝑛(𝑟, 𝜎). (2.2.6). If one electrons system change rapidly with position, LDA simulation of interacting electrons behavior is not enough accurate. The effect of density variation cannot be ignored. Another approximation, which is called generalized gradient approximation (GGA), has been applied to the more complex interacting electrons system. Actually, GGA is widely adopted by the chemistry community. A functional of the magnitude of the gradient of the density is introduced into (2.2.6), which are shown in (2.2.7) 𝜕𝜀. 𝜕𝜀. 𝛿𝐸𝑥𝑐 [𝑛] = ∑𝜎 ∫ 𝑑𝑟[𝜀𝑥𝑐 + 𝑛 𝜕𝑛𝑥𝑐𝜎 + 𝑛 𝜕∇𝑛𝑥𝑐𝜎 ∇]𝑟,𝜎 𝛿𝑛(𝑟, 𝜎). (2.2.7). This means exchange-correlation energy not only correlates the density of electrons, but also relates to their density fluctuation.. Pseudopotential. between. norm-conversing. and. ultra-soft pseudopotential The pseudopotential is applied in electronic structure to replace strong Coulomb potential of nucleus and tightly bound core electrons. The Orthogonalized plane wave 29.

(37) (OPW) approach can transform a eigenvalue problem to a smooth part of valence function|Ψ𝑝𝑠 ⟩ plus core function|Ψ𝑐 ⟩. Where b𝑐 = ⟨Ψ𝑐 |Ψ𝑝𝑠 ⟩ |Ψ⟩ = |Ψ𝑝𝑠 ⟩ + ∑𝐶 b𝑐 |Ψ𝑐 ⟩. (2.2.8). The wavefunction is used in Schrodinger equation and rewrite the form 𝐻|Ψ𝑝𝑠 ⟩ = 𝜀|Ψ𝑝𝑠 ⟩ − ∑ b𝑐 (𝜀 − 𝐸𝑐 )|Ψ𝑐 ⟩ = 𝜀|Ψ𝑝𝑠 ⟩ − ∑(𝜀 − 𝐸𝑐 )|Ψ𝑐 ⟩⟨Ψ𝑐 |Ψ𝑝𝑠 ⟩ 𝐶. 𝐶. (𝐻 + ∑𝐶(𝜀 − 𝐸𝑐 )|Ψ𝑐 ⟩⟨Ψ𝑐 |)Ψ𝑝𝑠 ⟩ = 𝜀|Ψ𝑝𝑠 ⟩. (2.2.9). Therefore, pseudopotential can be written 𝑉𝑝𝑠 = 𝑉 + ∑𝐶(𝜀 − 𝐸𝑐 )|Ψ𝑐 ⟩⟨Ψ𝑐 |. (2.2.10). There are four common important properties for difference pseudopotential methods (1) The new Schrödinger equations have the same eigenvalues with untransformed equations. (2) This pseudopotential is the function of eigenvalues; each quantum state do not necessary to have the same pseudopotial (3) The difference between pseudopotential and real potential distribute in the area of inner electrons, not in the outer area which are sensitive to bonding and activity. In addition, the pseudo-wavefucntion and real wavefunction also have this characteristic. 30.

(38) (4) The pseudo-wavefuctions |Ψ𝑝𝑠 ⟩ are not orthogonal with wavefuctions of inner electrons and much smoother than original wavefuctions. Scientists regarded the interaction between valence electrons and inner electrons and nuclei as a classical scattering problem of quantum mechanism. A pseudo core rc was introduced. The pseudopotential outside of rc equal to columb potential of nuclei, but when r< rc, the pseudowavefucntions that have no node existence and can be decomposed to few plane waves are generated. In order to find suitable way to describe pseudopotential, Hamnann, Schluter, Chang (HSC) give a requirement to produce accurate, transferable pseudopotential: The total charge of modified wavefuctions have equal to that of unmodified wavefuctions when r< rc. They proved the variation rate of phase shift of wavefuction which were influenced by scattering core with incident energy only relative to the total charge of scattered wavefuctions at scattering core. The non-local part(r< rc.) of pseudopotential have to be double integration which are heavy loading to calculation. Kleinman and Bylander use an approximation to simplify non-linear part and the calculation only require one integration. This is much simple for calculation and save time. However, although norm-conserving method is beneficial to obtain soft pseudopotenial, the atoms which have electron state of 1s, 2p , 3d are “hard” to obtain suitable pseudopotential that have no nodes and conversation of charge in Rc. 31.

(39) Therefore, Vanderbilt introduced a kind of ultra-soft pseudopotential whose are not norm-conserving anymore, but augment charge to satisfy the generalized norm-conversing condition. In general, although ultra-soft pseudopotential enable saving much calculation time, norm-conversing pseudopotential of different atoms are still widely used. Ba(Mg1/3Ta2/3)O3(Abbreviated as BMTO) is the lowest dielectric loss and highest quantity factor in microwave dielectric materials. However, due to high sintering temperature and second phase in calcined process, BMTO material has fewer industrial applications. Ba(Mg1/3Nb2/3)O3 (shorted as BMNO) are one of alternative to replace Ta ions to achieve cheaper cost. Therefore, the topic of this thesis would focus on the comparison between BMTO and BMNO. In this chapter, we would like to discuss the theoretical calculation of pure Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3 single crystal using VASP which provide fast-calculation speed and accurate pseudo-potential to get better results of crystal constant and linear respond calculation results.. The ground state calculation of Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3 In this part, VASP software is applied to study the basic physical and optical properties such as bulk modulus and vibration modes. The proper puesdo-potentials 32.

(40) and valence electrons numbers for this calculation have been concerned. The positions of atoms and relative parameters are listed in Table 2-1. Initial crystal constant data is referred from XRD result. The grids of k points should be decided carefully. With higher wave-function numbers and fine k-point grids result in lower and precise results; however, there is always dilemma for precision and time consuming. Usually, the smaller lattice sizes request fine grids. For BMTO compound, 4x4x4 Monhorst grids are used; The NGx, NGy and NGz control the number of grid-points in the FFT-mesh into the direction of the three lattice-vectors. The requested NGx, NGy and NGz for Ba(Mg1/3Ta2/3)O3 crystal are 42,42, 52,respectively. Table 2-1 The axis vector and position of atoms of Ba(Mg1/3Ta2/3)O3.. vector a. 5.7736. 0.0000. 0.0000. b. -2.8868. 5.0000. 0. c. 0.0000. 0.0000. 7.0932. Atom position x. Y. Z. Atom. 0.000. 0.000. 0.000. Ba. 1/3. 2/3. 0.6656. Ba. 2/3. 1/3. 0.3344. Ba. 33.

(41) 0.000. 0.000. 0.3344. Mg. 1/3. 2/3. 0.1775. Ta/Nb. 2/3. 1/3. 0.8225. Ta/Nb. 0.171. 0.829. 0.334. O. 0.171. 0.342. 0.334. O. 0.658. 0.829. 0.334. O. 0.829. 0.171. 0.666. O. 0.342. 0.171. 0.666. O. 0.829. 0.658. 0.666. O. 0.500. 0.000. 0.000. O. 0.000. 0.500. 0.000. O. 0.00000. 0.00000. 0.500. O. In order to get best crystal parameters, several new parameters need to be introduced. The best volume and crystal constant can be obtained by relaxing the positions of atoms and their crystal constant. The self-consistent calculations of electrons are preceded. Therefore, a conjugate-gradient algorithm is used to relax the ions into their instantaneous ground state. Fig. 2-2 depict the total energy with various length of a axis and c/a ratio. The most stable energy locate on the crystal parameter 34.

(42) of a=5.714, c=7.0208, for Ba(Mg1/3Ta2/3)O3 and a=5.7145, c=7.0145 for Ba(Mg1/3Nb2/3)O3, respectively.. Fig. 2-2 The total energy of Ba(Mg1/3Ta2/3)O3.. 40. BMTO BMNO. Total DOS. 30 20 10 0 -30 -25 -20 -15 -10 -5 Energy (eV). 0. 5. 10. Fig. 2-3 The total density of electron states of Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3.. The density of states of Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3 are shown in Fig. 2-3. The main difference between BMTO and BMNO exist on the energy larger than zero. The DOS peaks on energy of -25 eV and -10 eV are mainly attributed to Mg and 35.

(43) Ba atoms, while peak at -15 eV and peaks from -5 to 0 eV cause from oxygen atoms. The DOS peaks of Nb/Ta atoms distribute around 2.5-5 eV.. 0,04. -132,4. dTOTEN/dV. Total energy (eV). 0,06. -132,5 -132,6 -132,7. y=-0.97507+0.00489x. 0,02 0,00 -0,02 -0,04 -0,06. 185. 190. 195. 200. 205. 190. 210. 3. Volume(A ). 195 200 205 3 Volume (A ). 210. Fig. 2-4 The bulk modulus calculation of Ba(Mg1/3Ta2/3)O3.. The theoretical bulk modulus can be calculated through total energy curves which is the function of the volume of unit cell. The bulk modulus is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. B=. −𝜕𝑃 𝜕𝑉 𝑉. 𝜕2 𝐸. = 𝑉 𝜕2 𝑉. (2.3.1). Fig. 2-4 show the minimum total energy of BMTO distribute on volume of unit cell of 199.4 Å. The the slope in Fig. 2-4 (b) refer to 2nd derivate value of total energy. Therefore, the bulk modulus of BMTO can be calculated. B=𝑉. 𝜕 2𝐸 𝑒𝑉 𝐽 = 199.4 ∗ 0.00489 = 0.975 3 = 1.56 × 1011 ( 3 ) = 156 𝐺𝑃𝑎 2 𝜕 𝑉 𝑚 Å. The bulk modulus of BMNO can be obtained in Fig. 2-5. The unit cell volume of BMNO is 198.4 Å. Thus, the bulk modulus of BMNO is 258 (GPa).. 36.

(44) B=𝑉. 𝜕 2𝐸 𝑒𝑉 𝐽 11 = 198.4 ∗ 0.00815 = 1.616 = 2.58 × 10 ( ) = 258 𝐺𝑃𝑎 𝜕 2𝑉 𝑚3 Å3. -128.190. E Polynomial Fit of Data1_E. Total energy (eV). -128.195 -128.200. dTOTEN/dV. -128.205 -128.210 -128.215 -128.220. 0.02. y=-1.61876+0.00815x. 0.00. -128.225 -128.230. 196. 197. 198. 199. 200. 201. -0.02 195 196 197 198 199 200 201 202 3 Volume (A ). 202. 3. Volume(A ). Fig. 2-5 The bulk modulus calculation of Ba(Mg1/3Nb2/3)O3.. The experimental result of BMTO have been reported by H. Ikawa et al.. the experimental value of bulk modulus is around 153 GPa, which is very closed to our theoretical result. However, no experimental result of BMNO can be found. Compared Fig. 2-4 and Fig. 2-5, the bulk modulus of BMNO is much larger than that of BMTO. In general speaking, the bulk modulus of BMTO is average to that of steel (160 GPa) and BMNO have much larger bulk modulus.. Vibrational normal modes There are two methods to calculate normal vibration modes in this paper: selective dynamic and frozen phonon. The selective dynamic method enables to calculate the vibration modes quickly, but frozen phonon method exhibits the merit although each displacement of atoms need to be mutually calculated. The force matrix can deduce Born effective charge which can calculate the permittivity caused from phonon. 37.

(45) The phonon calculation results were calculated by selective dynamic method, and obtained 45 modes which exhibit 3 acoustic modes, 9 Raman modes(4A1g+5Eg), 16 IR modes(7A2u+9Eu) and 3 silent modes (2A1u+A2g).. Table 2-2 The eigenvectors and eigenvalues of normal modes of BMTO and BMNO. Eigenvalues Normal vibrational Modes BMTO. BMNO. 105. 106. Eg(Ba). 108. 107. A1g(Ba). 162. 177. Eg(Ta/Nb). 216. 273. A1g(Ta/Nb). 260. 295. Eg(Ta/Nb,O). 376. 378. Eg(Ta,Nb,O). 423. 428. A1g(Ta/Nb,O). 593. 585. Eg(O). 800. 790. A1g(O). 101. 104. A2u (all atoms). 115. 116. Eu (all atoms). Raman. IR. 38.

(46) Silent modes. 139. 148. Eu (Ba). 138. 149. A2u (Ba). 189. 176. Eu(Mg, O). 236. 219. Eu(Ta, O). 271. 258. Eu(O). 279. 287. A2u(O). 328. 337. A2u(Mg, O). 332. 336. Eu(Mg, O). 404. 399. Eu(O). 436. 427. A2u(O). 548. 506. Eu(O). 596. 587. A2u(O). 648. 636. Eu(O). 838. 800. A2u(O). 179. A2g(O). 259. A1u(O). 168. Alu(O). The detailed normal vibration modes are listed in Appendix I.. 39.

(47) Permittivity from phonon and electrons For permittivity at microwave region, there are two main sources: electrons, and phonons. The permittivity from phonon can be estimated from Born effective charge and corresponding eigenvalues. The permittivity from electrons contribution can be obtained by the plane-wave basis. In this section, the basic formulas were illustrated. First of all, Born and Huang 5proved the dielectric susceptibility tensor and dielectric tensor ∗ ∗ ̅̅̅̅̅̅ ̅̅̅̅̅̅ 𝑍𝜇𝛼 𝑍𝜇𝛽. ε𝛼𝛽 = (ε∞ )𝛼𝛽 + ∑𝜔𝜇2 ≠0 𝑉𝜀. (2.5.1). 2 0 𝑚0 𝜔𝜇. Born effective charge are obtained through 1. ∗ ̅̅̅ (𝑍 𝜇 )𝛼. =. ∗ ∑𝑖𝛾 𝑍𝑖𝜇𝛾. 𝑚 2 ( 𝑚0 ) (𝑎𝜇 )𝑖𝛾 𝑖. (2.5.2). μ is one specified normal mode, Z𝛼 represent the effective charge in direction α. m𝑖 refers to the ion mass, while. m0 = 1, and 𝑎μ is the eigenvector of μ mode.. Through these two formulas, the dielectric matrix of BMTO and BMNO can be obtained. In addition, the dielectric tensor is diagonal and permittivity in polycrystalline ceramics can be approximately average of diagonal values of three directions.6-7 For BMTO. 0 0   24.6    phonon   0 24.7 0    phonon  23.4  0 0 20.9   40. (2.5.3).

(48) For BMNO. 0 0   36.4    phonon   0 36.2 0    phonon  32.6  0 0 25.4  . (2.5.4). The average permittivity value of BMTO and BMNO are 23.4 and 32.6, respectively. In next chapter these theoretical data would be compared to the experimental result. In addition, it provide information to other relative compounds. Regarding the electron contribution to permittivity, the vector potential A and coulomb gauge (∇ ∙ 𝐀 = 0) have been concerned. When a EM wave propagate in the media, the Hamiltonian is that, 1. 𝑒. ℋ = 2𝑚 (𝒑 + 𝑐 𝑨)2 + 𝑉(𝑟) = ℋ0 + ℋ𝑒𝑅 𝑷2. (2.5.5). 𝑒. where ℋ0 = 2𝑚 + 𝑉(𝑟), and ℋ𝑒𝑅 ≅ 𝑚𝑐 𝑨 ∙ 𝒑. (2.5.6). In addition, vector potential can be written as 𝐀 = A𝜖̂(𝑒 𝑖(𝒒∙𝒓−𝜔𝑡 ) + 𝑐. 𝑐. ). (2.5.7). The transition probability R R= 2𝜋 ℏ. 𝑒. 2𝜋 ℏ. ∑𝑘|⟨𝜓𝑐 |ℋ𝑒𝑅 |𝜓𝑣 ⟩2 |𝛿(𝐸𝑐 (𝑘) − 𝐸𝑣 (𝑘) − ℏ𝜔) = 𝐸(𝜔) 2. (𝑚𝜔)2 |. 2. | ∑𝑘|⟨𝜓𝑐 |𝒑|𝜓𝑣 ⟩2 |. (2.5.8). The loss of EM wave were attributed to absorption in the media and equal to Rℏω 𝑑𝐼. 𝑑𝐼. 𝑑𝑥. 𝑐. loss = − 𝑑𝑡 = − (𝑑𝑥) ( 𝑑𝑡 ) = 𝛼𝐼 𝑛 =. 𝜀𝑖 𝜔𝐼. where I(𝒓𝟐 ) = 𝐼(𝐫𝟏 )𝑒 −𝛼(𝒓𝟐 −𝒓𝟏 ) , α =. 𝑛2 4𝜋𝜅 𝜆0. = 𝑅ℏ𝜔 , 𝑛̃ = 𝑛 + 𝑖𝜅, ε(ω) = 𝑛̃2. 41. (2.5.9).

(49) Therefore, the imagery and real part of permittivity from electrons can be written as: 2𝜋𝑒. 𝜀𝑖 (𝜔) = (𝑚𝜔 )2 ∑𝑘|⟨𝜓𝑐 |𝒑|𝜓𝑣 ⟩2 |𝛿(𝐸𝑐 (𝑘) − 𝐸𝑣 (𝑘) − ℏ𝜔) 𝜀𝑟 (𝜔) = 1 +. 4𝜋𝑒 2 𝑚. ∑𝑘 (. 2 𝑚ℏ𝜔𝑐𝑣. ). |⟨𝜓𝑐 |𝒑|𝜓𝑣 ⟩2 |. (2.5.10) (2.5.11). 2 −𝜔2 𝜔𝑐𝑣. From upper formula, permittivity from electrons can be calculated if wavefunctions can be read. In addition, the wave vector of excited photon q is much larger than Brillouin zone, the transition are considered direct and k 𝑣 = k 𝑐 = 𝑘 𝜓𝑐,𝑘 (r) = ∑𝑮 𝑐𝐺 (𝑐, 𝒌)𝑒 𝑖(𝒌+𝑮)∙𝒓 and 𝜓𝑣,𝑘 (r) = ∑𝑮 𝑐𝐺 (𝑣, 𝒌)𝑒 𝑖(𝒌+𝑮)∙𝒓 ⇒ ⟨𝜓𝑐,𝑘 |𝒑|𝜓𝑣,𝑘 ⟩ = ∑𝑮 𝑐𝑮∗ (𝑐, 𝒌)𝑐𝑮 (𝑣, 𝒌)ℏ(𝒌 + 𝑮). (2.5.12). Therefore, to summarize all coefficients of plane wavefunctions, the real permittivity from electrons can be obtained. For BMTO. 0 0   4.17    e   0 4.17 0    e  4.14  0 0 4.09  . (2.5.13). For BMNO. 0 0   4.80    e   0 4.80 0    e  4.76  0 0 4.69  . (2.5.14). 42.

(50) Summary In this chapter, the crystal parameter, bulk modulus, and normal vibrational modes (3 acoustic, 9 Raman-active modes, 16 IR-active modes, and 3 silent modes) at 0 K of Ba(Mg1/3Ta2/3)O3 and Ba(Mg1/3Nb2/3)O3 were studied. In addition, the permittivities from phonon and electron contribution also were calculated will be compared with measured values and have a discussion in next chapter.. Reference 1.. R. M. Martin, Electronic structure: Basic theory and Practical Methods; p.. 119,206. Cambridge University Press, 2004. 2.. N. David Mermin, “Thermal properties of the inhomogeneous electron gas”,. Phys. Rev. 137: A1441-1443, 1965. 3.. K. Andersen, Phys. Rev. B, 12, 3060(1975).. 4.. H. Ikawa et al, Paraelectric and elastic properties of ceramics with nominal. composition (A1-xAx')(BB')O3(A,A'=Ba:Sr,Ca) (1998),pp529-532. 5.. M. Born and K. Huang, Dynamical Theory of Crystal Lattices; p286.. Oxford University Press, 1954. 6.. Eric Cockayne and Benjamin P. Burton, “Phonons and Static Dielectric. Constant in CaTiO3 from First Principles”, Phys. Rev. B, 62 (2000) 3735–3743 43.

(51) 7.. Y. Dai, G. Zhao, L. Guo, H. Liu, “First-principles study of the difference in. permittivity between Ba(Mg1/3Ta2/3)O3and Ba(Mg1/3Nb2/3)O3”, Solid State Communications, 149 (2009) 791–794. 44.

(52) The comparison of measured modes with the first principle investigation of Ba(Mg1/3Nb2/3)O3 and Ba(Mg1/3Ta2/3)O3. Ba(Mg1/3Ta2/3)O3 & Ba(Mg1/3Nb2/3)O3 Many researchers had ever studied the vibrational spectra for BMTO and BMNO. First I. G. Siny1 et al. found the vibration modes of BMTO which relate to 1:2 ordering structure. Then several research groups1,4-6 tried to investigate the properties of 1:2 ordering complex perovskite by XRD, Raman, IR or EXAFS methods. Chia et al.. 2-3. also. found. the. highest. frequency. Raman. A1g(O). mode. of. xBa(Mg1/3Ta2/3)O3-(1-x)Ba(Mg1/3Nb2/3)O3 cearmics are correlated the microwave properties. Chen6 et al found the distances of oxygen octahedral structure are strongly correlated to the permittivity and dielectric loss in microwave region. Other relative research, such as BMTO doping different additives BaTiO37 or nickel6, also found A1g(O) mode performance can be correlated to dielectric properties of materials. In recently year, due to the rapidly development of computing calculation, some first calculations of perovskite were investigated. Gotthard Saghi-Szabo et al. revealed the structural and ferroelectric properties change of PbTiO3 and PbZr1/2Ti2/2O3. The 45.

(53) theoretical piezoelectric stress moduli were obtained. 8 T. Takahashi found the radius and electronegativity differences have no influence on the phase stability of Ba(B1/3'2+B2/3''5+)O3 (B'2+=Co, Mg, Mn, Ni, Zn; B''5+=Nb, Ta), and B-site cation ordering can enhance cell distortion and keep more stable phase9. The first principle calculation of BMTO and BMNO were researched by Dai et al.. 10 In this chapter, we will use Dai et al. reference to compare our experiment and calculated results. Chia et al. tried to assign the Raman and IR modes of BMTO and BMNO. However, due to lack of directly evidence, few assignments was corrected comparing the first principle results10. Table 3-1 shows the Raman peaks from experiment results and the symmetry modes (g mode) of BMTO and BMNO. All of eigenvalues of modes are close to the experiment result, it indicate the good correspond to the experiment results. There are 6 differences for the assignment of Chia et al.2 and Chen et al.3 results. The first correction are Eg(Ta/Nb) at 162 cm-1 for BMTO whose vibration is shown in mode 33 of Appendix I. Then A1g(Ta/Nb) and Eg(Ta/Nb) should be change to each other according to calculation results. Phonon of 385 cm-1 should be classified as Eg mode, not the degenerate modes(Eg+Alg(O)); The mode of 423 cm-1 should be A1g mode, instead Eg(O) mode. An unassigned mode at 593 cm-1 from the study of Chia et al. was confirmed as Eg mode.. Table 3-1 The Raman normal modes and their assignments of BMTO and BMNO 46.

(54) Eigenvalues. BMTO. Normal. Assignment (Chia et. vibrational. al. 2003). assignment. BMNO Modes. BMTO. BMNO. 1045. 106. Eg(Ba). 104. 105. Eg(Ba). 108. 107. A1g(Ba). 106. 105. A1g(Ba). 162. 177. Eg(Ta/Nb). 158. 175. Eg(O)*. 216. 273. A1g(Ta/Nb). 211. 263. Eg(Ta)*. 260. 295. Eg(Ta/Ng,O). 264. 296. A1g(Ta)*. 376.3. 378. Eg(Ta,Nb,O). 385. 385. Eg(O). 385. 385. A1g(O)*. 432. 437. Eg(O)*. 797. 788. A1g(O). 423. 428. A1g(Ta/Nb,O). 593. 585. Eg(O). 800. 790. A1g(O). The Raman modes can be clearly identified and applied to other similar crystals to deduce the structural change via Raman measurement. However, the Raman vibrations do not have significant contribution to permittivity. The phonons that measured by FTIR are dominate sources for permittivity. Table 3-2 reveals that the results of K-K transformation analysis from IR measurements and theoretical results 47.

(55) of BMTO and BMNO(Chapter 2 and report from Dai et al.10 ) Table 3-2 The IR phonon and their permittivity contributions of BMTO: Δε refer to the dielectric permittivity contribution from IR phonon; TOx is for optical phonon frequency (cm-1); Toγ is for width of optical phonons (cm-1). K-K transformation from. The calculation result from. The calculation result. IR measure. Dai et al.(GGA) 10. from chapter 2. Δε. TO xc. TOγ. xc. Δε. xc. Δε. 0.19. 103. 6.1. 1A. 101. 0.43. 1A. 101. 0.65. 0.05. 114. 5. 1E. u. 109. 0.28. 1E. u. 115. 0.09. 4.5. 140. 2.62. 2A. 2u. 129. 4.88. 2A. 2u. 138. 6.69. 0.52. 149. 16.8. 2E. 132. 3.81. 2E. 139. 5.17. 0.01. 174. 15.9. 3E. 176. 0. 3E. 189. 0.01. 8. 228. 19.2. 4E. 222. 11.42. 4E. 236. 14.7. 5.1. 245. 19.1. 1. 258. 9.88. 5E. u. 264. 1.12. 5E. u. 271. 1.12. 2.25. 275. 10.9. 3A. 2u. 280. 7.06. 3A. 2u. 279. 10. 0.55. 314. 10.2. 4A. 327. 0.12. 4A. 328. 0.68. 0.005. 412. 12.9. 6E. 328. 0.47. 5E. 332. 0.98. modes. 2u. u. u. u. 2u. u. 48. modes. 2u. u. u. u. 2u. u.

(56) 0.015. 434. 9.71. 7E. u. 409. 0.02. 7E. u. 404. 0.007. 0.001. 466. 14.4. 5A. 2u. 430. 0.15. 5A. 2u. 436. 0.07. 0.2. 523. 21.8. 8E. u. 517. 0.34. 8E. u. 548. 0.94. 0.3. 536. 30.7. 6A. 2u. 579. 0.8. 6A. 2u. 596. 1.53. 0.35. 607. 37.0. 9E. u. 643. 0.38. 9E. u. 648. 0.52. 0.25. 624. 23.8. 7A. 2u. 817. 0. 7A. 2u. 838. 0. Table 3-3 The IR phonon and their permittivity contributions of BMNO. K-K transformation from. The calculation result from. The calculation result. IR measure. Dai et al.(GGA) 10. from chapter 2. Δε. TO xc. TOγ. xc. Δε. xc. Δε. 1.25. 80.5. 15.7. 1A. 98.7. 2.73. 1A. 104. 2.32. 1.67. 104. 7.98. 1E. 111. 0.51. 1E. 116. 0.07. 9.4. 149. 3.06. 2E. u. 137. 14.82. 2E. u. 148. 12.5. 0.05. 168. 19.4. 2A. 2u. 140. 6.14. 2A. 2u. 149. 9.08. 3.72. 213. 8.53. 3E. 153. 1.31. 3E. 176. 1.13. 11.9. 226. 24.9. 4E. 192. 12.78. 4E. 219. 16.7. 3.21. 246. 18.5. 5E. 241. 1.57. 5E. 258. 3.32. modes. 2u. u. u. u. u. 49. modes. 2u. u. u. u. u.

(57) 3A. 275.1. 7.15. 3A. 288. 11.8. 7.7. 4A. 353. 0.12. 4A. 337. 0. 380. 55.0. 6E. 357. 0.17. 6E. 336. 0.66. 0.01. 406. 5.4. 7E. u. 389. 0.07. 7E. u. 399. 0.02. 0.01. 431. 5. 5A2. 418. 0.26. 5A. 2u. 427. 0.12. 0.28. 500. 24.2. 8E. u. 489. 0.36. 8E. u. 506. 0.66. 0.41. 586. 50.6. 6A. 2u. 574. 1.17. 6A. 2u. 587. 1.99. 0.91. 612. 29.1. 9E. u. 639. 0.62. 9E. u. 636. 1.09. 0.04. 672. 35.6. 7A. 2u. 795. 0. 7A. 2u. 800. 0. 7.5. 275. 12.9. 0.52. 316. 10.9. 0.15. 331. 0.00. 2u. 2u. u. u. 2u. 2u. u. Table 3-2 and Table 3-3 depict that permittivity of BMTO and BMNO are mainly attributed to 4 phonons: 2Eu, 2A2u, 4Eu, and 3A2u. These modes in Appendix I are correspond 2Eu to mode 34 and mode 35 to, 2A2u to mode 36, 4Eu to mode 25 and mode26, and 3A2u to mode 19. 2Eu mode represents that Ba atoms have relative movements to other atoms at x and y axis while 2A2u refers to same kind of movement at z axis. In addition, the movements of 4Eu mode are that Ta/Mg atoms move against oxygen atoms at x and y axis; the vibration of 3A2u are the same movement but at z axis. There is one reference 50.

(58) to support the modes of Ta/Nb atoms against oxygen are the dominant to permittivity and losses, but a few mistake the reference made.11 Wang et al.. 11. reported FTIR. measurement were analyzed by Four parameter semi-quantum model(FPSQ) and found that two modes (Eu~241 cm-1 and Eu~220 cm-1) which are the vibration of high-valence cation Ta5+ in TaO6 octahedra play the most important role for dielctric constant and loss . In my opinion, Eu(241 cm-1) was mis-assigned and should be corrected to A2u mode. Table 3-4 shows the permittivity contributions from phonon and electron in chapter 2 and Dai’s report. Table 3-4 The measured and calculated permittivity contribution from electrons and phonons. The calculation. The calculation. result from Dai et. result from chapter. al.(GGA) 10. 2. The experiment data. ionic. electronic. ionic. electronic. ionic. Electronic. BMT. 23.3. 4.4. 16.4. 6.1. 23.4. 4.1. BMN. 30.9. 4.3. 27.3. 6.4. 32.6. 4.8. Comparing the experiment and first principle calculation result, the results in chapter 2 is close to the result of IR experiment. However, values obtained from Dai’s report 51.

(59) are close to the experiment values (22-25 for BMTO and 31-33 for BMNO) in microwave region.. SrxBa1-x(Mg1/3Ta2/3)O3 The study of first principle of BMTO is also beneficial to understand the behavior of SrxBa1-x(Mg1/3Ta2/3)O3 with the Sr concentration (x=0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 1). This series of samples were analyzed by the master thesis of I. H Chang.10 The Raman spectra of SrxBa1-x(Mg1/3Ta2/3)O3 with x=0.0 -1.0 are shown in Fig. 3-1. We will only discuss x<0.5 samples due to the Raman spectra of x < 0.5 samples show the same phase with pure BMTO. No extra peak appearance and large Raman shift variation with different Sr content represent that the micro-structural properties change without phase transition.. 52.

(60) F. E. D. C. B. A. Intensity (arb. units). x=0 x=0.125 x=0.25 x=0.375 x=0.5 x=0.625 x=0.75 x=0.875 x=1. 0. 200. 400 600 800 -1 Raman Shift (cm ). 1000. Fig. 3-1 Raman spectra of SrxBa1-x(Mg1/3Ta2/3)O3 with x=0, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 1. The obvious blueshift of A1g(800 cm-1) mode and Eg(570 cm-1) mode were observed with x increase, while other modes have no obvious frequency change. It can be explained through the movement of each vibration mode. Interestingly, A1g modes at 420 cm-1(mode 11) and 800 cm-1(mode 2) have the same vibration atoms along z axis: 3 O atoms and 1 Ta atom. But the A1g (800 cm-1) go higher frequency with x increase, but A1g (430 cm-1) have no obvious change. The reason for that may be Sr atom site on Mg site. There is supportive evidence from EXAFS spectrum of Sr core, which exhibit complex peak that cannot be analyzed. But why Sr atoms locate on Mg site not Ta site? That’s because that, if Sr atom site on Ta site, the reduced 53.

(61) mass of both A1g modes (430 cm-1 and 800 cm-1) decrease and the vibration frequency increase. But the Raman shift of A1g at 430 cm-1 were not affected by Sr doping. Therefore, with Sr content increase, Sr atom prefer site on Ba site or Mg site, but not Ta site. Table 3-5 The microwave properties of SrxBa1-x(Mg1/3Ta2/3)O3 with x=0, 1/8, 2/8, 3/8, 4/8. Sr Concentration. Relative Density(%)α. Permittivity. Q× f Value (GHz). 0. 97.7. 24.4. 9.9x104. 0.125. 97.2. 25.8. 11.9x104. 0.25. 96.4. 26.3. 7.7x104. 0.375. 95.9. 27.9. 7.9x104. 0.5. 95.1. 28.1. 5.9x104. In addition, the permittivity values increase with x content, shown in Table 3-5. There are two possible reason for higher permittivity. As far as we know, there are 4 dominant IR modes (2Eu, 2A2u, 4Eu, and 3A2u.) that determine the value of permittivity. While Sr concentration increase, the Born effective charge of 2A2u and 2Eu become larger and permittivity would increase due to the smaller mass of Sr atoms.. 54.

(62) Conclusion This chapter shows that many experiment results can be well explained through first principle calculation. The phonon assignments, the permittivity value at microwave region, were discussed for BMTO and BMNO. Four important phonons (2Eu, 2A2u, 4Eu, and 3A2u.) dominate the permittivity value of BMTO and BMNO. In addition, the theoretical analysis of pure BMTO is helpful to explain the substituted SrxBa1-x(Mg1/3Ta2/3)O3 system. When x < 0.5, Sr atoms prefer site on Ba and Mg site, instead Ta site. Sr substitution at Ba site would lead to higher permittivity values because of smaller mass and larger Born effective charge of Sr atoms compared Ba atoms.. Reference 1.. I. G. Siny, Ruiwu Tao, R. S. Katiyar, Ruyan Guo, A. S. Bhalla, “RAMAN SPECTOSCOPY OF Mg-Ta ORDER-DISORDER IN Ba(Mg1/3Ta2/3)O3”, J. Phys. Chem Solids, 59 (1998), pp181-195.. 2.. C.-T Chia, Y. C. Chen, H. F. Cheng,and I. N. Lin, “Correlation of Microwave Dielectric. Properties. and. Normal. Vibration. Modes. of. xBa(Mg1/3Ta2/3)O3-(1-x)Ba(Mg1/3Nb2/3)O3 ceramics: I. Raman spectroscopy”, J. Appl. Phys.,Vol. 94, No. 5, 3360-3364 (2003). 3.. Y. C. Chen, H. F. Cheng, H. L. Liu, and C.-T Chia, “Correlation of Microwave 55.

(63) Dielectric. Properties. and. Normal. Vibration. Modes. of. xBa(Mg1/3Ta2/3)O3-(1-x)Ba(Mg1/3Nb2/3)O3 ceramics: II. Infrared spectroscopy”, J. Appl. Phys.,Vol. 94, No. 5, 3365-3370 (2003). 4.. A. Dias, V.S.T. Cininelli, “Raman scattering and X-ray diffraction investigations on hydrothermal barium magnesium niobate ceramics”, J. Eur. Ceram. Soc., 21 (2001) 2739-2744. 5.. R. L. Moreira, F. M. Matinaga, A. Dias, “Raman-spectroscopic evaluation of the long-range order in Ba(B’1/3B”2/3)O3 ceramics”, Appl. Phys. Let., 78 (2001), No.4, 428. 6.. M. Y. Chen, P. J. Chang, C. T. Chia, Y. C. Lee, I. N. Lin, L. J. Lin, J. F. Lee, H. Y. Lee, and T. Shimada, “Extended X-ray absorption fine structure, X-ray diffraction and Raman analysis of nickel-doped Ba(Mg1/3Ta2/3)O3”, J. Eur. Ceram. Soc., 27 (2007) 2995-2999. 7.. C. T. Chia, P. J. Chang, M. Y. Chen, I. N. Lin, H. Ikawa, L. J. Lin, “Oxygen-octahedral phonon properties of xBaTiO3+(1−x)Ba(Mg1/3Ta2/3)O3 and xCa(Sc1/2Nb1/2O3+(1−x) Ba(Sc1/2Nb1/2O3 microwave ceramics”, J. Appl. Phys., 101 (2007) 084115.. 8.. Gotthard Sághi-Szabó, Ronald E. Cohen, and Henry Krakauer, “First-principles study of piezoelectricity in tetragonal PbTiO3 and PbZr1/2Ti1/2O3,” Phys. Rev. 56.

(64) B, 59 (1999) 12771. 9.. T. Takahashi, “First-principles Investigation of the Phase Stability for Ba(B1/3’2+B2/3”5+)O3 Microwave Dielectrics with the complex Perovskite Structure ”, Jpn. J. Appl.Phys., 39 [9B](2000) 5637.. 10. Y. Dai, G. Zhao, L. Guo, H. Liu, “First-principles study of the difference in permittivity between Ba(Mg1/3Ta2/3)O3and Ba(Mg1/3Nb2/3)O3”, Solid State Communications, 149 (2009) 791–794 11. C. H. Wang, G. H. Liu, X. P. Jing, G. S. Tian, X. Lu, and J. Shao, “First-Principle Calculation and Far Infrared Measurement for Infrared-Active Modes of Ba(Mg1/3Ta2/3)O3”, J. Am. Ceram. Soc., 93 (2010) 3782-3787. 12. I. H. Chang, “Spectroscopic Study of xSr(Mg1/3Ta2/3)O3-(1-x)Ba(Mg1/3Ta2/3)O3 Microwave Materials”, Master Thesis. National Taiwan Normal University, Taiwan, 2009.. 57.