臺灣博碩士論文加值系統

近幾年來，擁有強電子相關(electronic correlated)以及準低維度(quasi-low dimension)特性的材料，引起了廣泛的注意以及討論。這些材料中有趣的複鐵電(multiferroic)性質已被研究而且在科技產業的應用上有相當的潛力。強烈結構形變的準二維perovskite氧化物，如BiCoO3 和PbVO3，正是這樣的材料。從X光粉末繞射(X-ray powder diffraction)之數據指出其巨大的結構形變，因而預測這些材料中有強大的自發電偶極化以及非線性光學係數。雖然這兩者的晶體結構以及部份電子特性方面已經被報導，但是仍然缺乏在磁交互作用以及光學性質上的研究。
本篇論文中，BiCoO3 和PbVO3 的電子特性、磁交互作用、線性以及非線性光學性質將透過第一原理(ab initio)系統性的研究，藉由密度泛函理論(density function theory)與使用projector augmented-wave(PAW)的generalized gradient approximation(GGA)。透過GGA + Harbbard U 近似材料中強烈的電子庫倫力交
互作用。自旋基態計算中包括五種不同的Nèel spin state 組態：FFF(鐵磁態)、FFA(A-type 反鐵磁)、AAF(C-type 反鐵磁)、AAA(G-type 反鐵磁)、和AFF。首先，我們將探討BiCoO3 和PbVO3 的能帶(energy band)以及態密度(density of states)理論計算結果，而這些計算可以幫助我們瞭解完整的電子特性及磁性來源。同時亦討論自洽電荷密度(charge density)的結果，其顯示出強烈的結構形變以及巨大的電偶極化效應原因。第二，藉由計算不同自旋組態的總能量之後套用海森堡模型(Heisenberg model)可以求出自旋互異耦合(spin-exchange coupling)能量。此結果指出兩者皆擁有反鐵磁基態趨勢。
利用獨立電子近似可以計算理論線性光學性質。虛部的界電函數可以經由能帶躍遷及費米黃金法則(Fermi golden rule)求出，之後透過Kramer-Krònigrelation 可以算出實部的界電函數。一旦界電函數確定之後，所有的線性光學性質即可決定。而藉由單光子及複光子響應，可以建立非線性和線性光學之間的簡單物理圖像。期望本文中線性以及非線性光學上的計算在之後的相關實驗可以提供有用的資訊。

In recent years, the multiferroic materials, especially the signi cantly electron correlations and low-dimension systems such as quasi-two-dimension perovskites, PbVO3 and BiCoO3, have been attracted enormous interests in wide technological applications. From X-ray powder di raction data, the giant structural distortion has been found, which implies the strong polarization as well as signicant second-harmonic generation. Although the crystal structures and the electronic properties of these two perovskites have been reported, the knowledge on the magnetic exchange interaction and optical properties are still absent.
In this thesis, a systematic ab initio study of the electronic structure and magnetic interaction as well as the linear and nonlinear optical properties of the PbVO3 and BiCoO3 has been preformed within the density function theory with the generalized gradient approximation using the projector augmented-wave method. The on-site Coulomb
interaction has been included in generalized gradient approximation with Hubbard U scheme. The ground-state spin con gurations of the ve di erent N eel spin states such
as FFF (FM), FFA (AAFM), AAF (CAFM), AAA (GAFM), and AFF were considered. Firstly, the theoretical band structures, and density of states in both PbVO3 and BiCoO3 are calculated. The former can help us to nd the electronic properties, and the latter shows the origin of the magnetism of both perovskites. Secondly, the self-consistent charge density was displayed and shows the reasons of the signi cantly structural distortion and strong polarization. Finally, the values of the spin-exchange coupling are determined by mapping the calculated total energies of various N eel spin states onto Heisenberg model. The result shows that the both perovskites are anti-ferromagnetic.
The linear optical spectra were calculated by the independent-particle approximation. The imaginary part of dielectric function of these two perovskites can be evaluated from direct interband transitions by Fermi golden rule, and the real part can obtained by Kramer-Kronig relation. Once the dielectric function are known, the linear optical properties such as linear absorption, refractive index, and electron energy loss spectra can be determined. It is found that signi cant anisotropic optical spectra and second-harmonic generation coe cients exist in both perovskites. The prominent structures in
the spectra of χ(2)(2ω;ω;ω) of PbVO3 and BiCoO3 have been successfully correlated with the features in the corresponding linear optical dielectric function ε"(ω) in terms of single- and double-photon resonances. The linear and nonlinear optical properties would provide useful information for next research.

1 Introduction 6
2 Theoretical tools 8
2.1 Density Function Thoery and Approximations . . . . . .8
2.1.1 Born Oppenheimer approximation . . . . . . . . . . .8
2.1.2 Hartree-Fork approximation . . . . . . . . . . . . 9
2.1.3 Density Function Theory . . . . . . . . . . . . . .11
2.1.4 Local Density Approximation (LDA) and Generalized Gradient
Approximation (GGA) . . . . . . . . . . . . . . . . . . .12
2.1.5 Band calculation and Projector Augmented Wave (PAW) method 14
2.1.6 Rotational invariant LDA plus U method . . . . . . 16
2.2 Optical formulas . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Results of PbVO3 21
3.1 Material and input parameters . . . . . . . . . . . 21
3.2 Electronic structure and magnetism . . . . . . . . . 22
3.2.1 Density of states . . . . . . . . . . . . . . . . 22
3.2.2 Band structure . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Charge density and polarization . . . . . . . . . 32
3.2.4 Heisenberg model and exchange energy . . . . . . . 32
3.3 Optical properties . . . . . . . . . . . . . . . . . 37
3.3.1 Linear optical properties . . . . . . . . . . . . 37
3.3.2 Second-harmonic generation coe cients . . . . . . 43
4 Results of BiCoO3 46
4.1 Material and input parameters . . . . . . . . . . . 46
4.2 Electronic structure and magnetism . . . . . . . . . 47
4.2.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.2 Band structure . . . . . . . . . . . . . . . . . . 50
4.2.3 Charge density and polarization . . . . . . . . . 50
4.2.4 Heisenberg model and exchange energy . . . . . . . 55
4.3 Optical properties . . . . . . . . . . . . . . . . . 59
4.3.1 Linear optical properties . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Second-harmonic generation coe cients . . . . . . .60
5 Summary 70
6 Bibliography 72

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