利用第一原理計算研究多鐵氧化物Cu3Mo2O9的磁性,電子態及鐵電性質 - 政大學術集成

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(1)國立政治大學理學院應用物理研究所碩士論文 Graduate Institute of Applied Physics, College of Science National Chengchi University Master Thesis. 利用第一原理計算研究多鐵氧化物 Cu3Mo2O9 的磁性, 電子態及鐵電性質 Ab Initio Studies of The Magnetic, Electronic and Ferroelectric Properties of Multiferroic Oxide Cu3Mo2O9. 蕭逸修 Yi-Hsiu Hsiao. 指導教授：郭光宇博士 Advisor : Guang-Yu Guo, Ph.D.. 中華民國一○一年九月 September, 2012.

(2) Abstract In this thesis, we used the ab initio method to study a multiferroic oxide Cu3 Mo2 O9 . The correlations of electrons must be considered in this system so that a reasonable energy gap can be obtained. Due to the geometric frustration of magnetic structure caused by crystal structure, the ground state spin configuration in this system still has not been determined experimentally. We found some spin configurations similar to the non-collinear anti-ferromagnetic spins configuration suggested by Vilminot et al.. Competition between exchange interactions and spin-orbit coupling effect determines the canting of spins on Cu atoms. The calculated exchange parameters agree with the experimental results well. By using Berry phase calculations, we obtained the theoretical value of spontaneous electric polarization. The strength of polarization in our results is in the same order of results of experiments. However, the direction of electric polarization we found (along b-axis) is different from the experimental measurements (along c-axis). We have found a spin configuration that the theoretical electric polarization of the state agrees with the experimental results. However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer. Since there still is no ab initio calculation studying this oxide, we hope that our studies can help those who are also interested in this material.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(3) 中文摘要在此論文中，我們利用第一原理計算研究多鐵材料 Cu3Mo2O9 的磁性、電子態及多鐵性質。我們發現在此系統中，電子與電子間的庫倫排斥力必須被考慮，以致於導帶與價帶間能隙能夠被良好地描述。由於晶體結構所導致的幾何不穩定性，系統的磁結構尚未在實驗測量中被確定。在我們的理論計算當中得到的磁結構與 Vilminot 等研究人員根據實驗結果猜測出的非線性反鐵磁結構類似。交換作用與自旋軌道耦合間的爭競決定了電子自旋方向的傾斜。計算所得到的交換作用係數與實驗結果吻合良好。利用 Berry’s phase 計算，我們得到了系統自發電極化的理論值，其強度與實驗量測值在同一個數量級。然而，在我們計算中得到的電極化方向(平行於 b 軸)與實驗(平行於 c 軸)不符。此外，我們發現一磁結構之理論電極化方向與實驗相符，然而其磁結構之對稱性與實驗不符。目前，尚未有第一原理計算研究此氧化物，我們希望此論文能夠對同樣有興趣研究此材料的研究人員有所幫助。. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(4) Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 7. 1. Introduction. 8. 2. Density Functional Theory 2.1 Born-Oppenheimer approximation . . 2.2 Thomas-Fermi Theory . . . . . . . . 2.3 Density Functional Theory . . . . . . 2.3.1 Hohenberg-Kohn Theorem . . 2.3.2 Kohn-Sham Equation . . . . . 2.3.3 Exchange-Correlation Energy 2.4 Mott Insulators . . . . . . . . . . . . 2.4.1 Hubbard Model . . . . . . . . 2.4.2 Beyond DFT : DFT+U . . . .. n. 4. Ch. Crystal Field Theory engchi 3.1 Atomic Orbitals . . . . . . . . . . . . . . 3.2 Crystal Field Theory . . . . . . . . . . . 3.3 High Spin and Low Spin . . . . . . . . . 3.4 Crystal Field Stabilization Energy . . . . 3.5 Jahn-Teller Theorem . . . . . . . . . . . 3.6 Colors of Transition Metal Complexes . . Multiferroics 4.1 Introduction . . . . . . . . . 4.2 Symmetry . . . . . . . . . . 4.3 Geometric Frustration . . . . 4.4 Multiferroics . . . . . . . . 4.4.1 Type-I Multiferroics 4.4.2 Type-II Multiferroics. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 1. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . . . .. i n U . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . . . .. y. . . . . . . . .. sit. . . . . . . . .. er. io. 3. . . . . . . . .. ‧. Nat. al. . . . . . . . .. 學. ‧ 國. 立. 政治大 . . . . . . . . . . . . . . . . . . . . . .. v. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. 9 9 10 11 12 13 14 15 15 17. . . . . . .. 18 18 19 21 22 23 23. . . . . . .. 25 25 26 27 27 27 30.

(5) . . . . . .. 33 33 36 40 40 49 58. Summary and Conclusions Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61 66. 立. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 政治大. 學 ‧. io. sit. y. Nat. n. al. er. 6. Calculated Physical Properties of Cu3 Mo2 O9 5.1 Introduction . . . . . . . . . . . . . . . . . 5.2 Crystal Structure and Computational Details 5.3 Magnetic Structure . . . . . . . . . . . . . 5.4 Exchange Interactions . . . . . . . . . . . . 5.5 Electronic Structure . . . . . . . . . . . . . 5.6 Spontaneous Electric Polarization . . . . .. ‧ 國. 5. Ch. engchi. 2. i n U. v. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . ..

(6) List of Figures. ‧. y. n. 4.1. 20 20. er. io. 3.5. 19. 學. Nat. 3.3 3.4. 立. 政治大. sit. 3.2. d-orbitals - For d x2 − y2 , the four lobes lie on the x and y axes, and for d z2 , there are two lobes on the z axes and there is a donut shape ring that lies on the xyplane around the other two lobes. The four lobes of d xy orbital lie in-between the x and the y axes, and so on. [Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/] . . . . . . . . . . . . . . . . . . . . . . Structure of complexes - (a) The octahedral complex, where six ligands attached to the central transition metal along x, y and z axes, forming an octahedron. (b) The tetrahedral complex, four ligands form a tetrahedron around the central ion. (c) The square-planar complex, which can be seen as the simplification of the octahedral complex, where the two ligands along the z-axis are removed. . . . . Common crystal field splittings of d-orbitals . . . . . . . . . . . . . . . . . . . The low-spin and high-spin cases in the octahedral crystal field - In the left case, the low-spin case, the strong-field ligands lead to a large ∆oct , which is larger than the pairing energy so that the lower energy orbitals will completely be filled before occupation ofa the l Chigher orbitals occurs.nIni vthe right case, the high-spin case, the weak-field ligandshlead e ntoga small h i∆octU, which is smaller than the pairing c energy so that it is easier to put the remaining electrons into the higher levels of eg set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . color wheel - A sample absorbs visible light of color of the specific wavelength and reflects the rest, in appears the complementary color of light. Find the color that is absorbed, then move directly across the wheel to the other side to get the complement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ‧ 國. 3.1. Geometric frustration - (a) Ising antiferromagnets on the triangle lattice. The remaining one spin can no longer point in a direction oppsite to both two spins. (b) Scheme of water ice molecules where hollow circles are oxygen atoms and filles circles are hydrogen atoms. For each oxygen atom, two of the neighboring must reside in the far position and two of them in the near position. . . . . . . .. 3. 21. 23. 27.

(7) 4.2. 4.3. 4.4. Perovskite Structure - The general stoichiometry ABX3 , where A (blue particles located at the corners of the cube) and B (orange particle located in the centre of the cube) are cations and X (red particles located in the face-centred positions of the cube) are anions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YMnO3 - (a) The centrosymmetric structure above the critical temperature. (b) The ferroelectric structure below the critical temperature where practically rigid MnO5 tilt breaking the spatial inversion symmetry. . . . . . . . . . . . . . . . Different types of spin structures relevant for TbMnO3 - (a) T N2 < T < T N1 . Sinusoidal spin density wave. Every spins point to the same direction but different magnitude of local moment. The magnitude of these moments vary periodically. This structure is centrosymmetic and consequently not ferroelectric. (b) T < TN2 . The cycloidal spiral magnetic structure with the wave vector Q = Qx and spins rotating in the xz-plane. This structure breaks the spatial inversion symmetry so that the spontaneous electric polarization appears. (c) In a so-called ”proper screw” the spins rotate in a plane perpendicular to Q. Here the inversion symmetry is broken, but most often it does not produce polarization. . Ising chain composed of Co2+ and Mn4+ - Ising chains with the up-up-downdown spin order and alternating ionic order, in which electric polarization is induced through symmetric exchange striction. The two possible magnetic configurations leading to the opposite polarizations are shown. The atomic positions in the undistorted chains are shown with dashed circles. . . . . . . . . . . . . .. 立. ‧ 國. y. sit. 32. er. Structure of Cu3 Mo2 O9 - (a) Three crystallographically inequivalent Cu2+ sites a i v The quasi-one-dimensional (Cu1, Cu2 and Cu3) lareCincluded in the system. n U two neighboring Cu atoms are h e n g cand h ieach chain is formed by Cu4 tetrahedra, bridged by an oxygen ion. (b) The quasi-one-dimensional chains are connected by bridging MoO4 which is not drawn in the figure for simplifying. Four formular units (f.u.) are included in a primitive unit cell. . . . . . . . . . . . . . . . . Guessed spin configurations - (a) The symmetric analysis of the neutron diffraction experiments gave a guessed spin configuration with all spins lying on the ac-plain. Only spins on Cu1 and Cu3 are presented. (b) Hamasaki et al. suggested a rather different spins configuration. Only spins on Cu1 are presented, pointing almost along the b-axis and being canted by the DM interaction. While spins on Cu2 and Cu3 are considered forming singlet dimers which weakly interact with Cu1 on the antiferromagnetic chains. . . . . . . . . . . . . . . . . . Ferroelectricity of Cu3 Mo2 O9 - The spontaneous electric polarization apperrs along c-axis (Pkc '500 µC/m2 ). When applying the magnetic field along caxis, the direction of the electric polarization changes to the a-axis (Pka '800 µC/m2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. n. 5.3. io. 5.2. 31. ‧. Nat. 5.1. 29. 政治大. 學. 4.5. 28. 4. 34. 35. 35.

(8) 5.5. 5.6. 立. 政治大. 學. 5.15 5.16 5.17. sit. er. n. 5.14. io. 5.13. 42. 42. y. Nat. 5.8 5.9 5.10 5.11 5.12. 37. ‧. 5.7. Crystal Structure of Cu3 Mo2 O9 - We used the atomic structure data from XRD experiments. Their are three crystallographically inequivalent Cu sites (Cu1, Cu2 and Cu3) and seven crystallographically inequivalent O sites (O1, O2, ..., O7). Each Cu ion is surrounded by O ions causing the square-planar-like crystal field. Take Cu1 and its surrounding O ions for example, the distance between ˚ The contributions of O2 ions Cu1 and O1 (O2, O3) ions is 1.86 (2.3, 2.13) A. affecting on Cu1 can be neglected. O1 and O3 ions surrounding the Cu1 ions form a parallelogram, causing the square-planar-like crystal field. . . . . . . . . Calculated spin configurations are similar to the results of neutron diffraction experiments. All spins lie in the ac-plain almost. The spins on Cu1 form the AFM chain, and the spins on Cu2 and Cu3 form singlet dimers. We rotated the spins on Cu1 around the b-axis, while the spins on Cu2 and Cu3 remain fixed. The most stable noncollinear AFM spin configuration is that of an angle θ = 107◦ or 287◦ between a-axis and direction of moments on Cu1. . . . . . . . . . The energy versus θ of the spins on Cu1 in the system. The most stable noncollinear AFM spin configuration is that of an angle θ = 107◦ or 287◦ approximately between a-axis and direction of the spins on Cu1. . . . . . . . . . . . . All spins lie in the ac-plane in the most stable noncollinear AFM spin configuration, i.e. an angle θ = 107◦ or 287◦ between a-axis and direction of spins of Cu1. When the spins on Cu1 rotate to another angle, the spins cant away from the ac-plane occur, especially on Cu1. . . . . . . . . . . . . . . . . . . . . . . The NM electronic band structure. . . . . . . . . . . . . . . . . . . . . . . . . a l electronic densities of states Total and site-projected i v of the NM state. . . . . . . n C NM orbital- and site-projected densities h eelectronic i U of states of Cu1 atoms. . . . h n c g NM orbital- and site-projected electronic densities of states of Cu2 and Cu3 atoms. The electronic band structure of the collinear FM state from the collinear spinpolarized GGA+U calculations. . . . . . . . . . . . . . . . . . . . . . . . . . Total and site-projected electronic densities of states of the collinear FM state from the collinear spin-polarized GGA+U calculations. . . . . . . . . . . . . . The electronic band structure of the collinear AFM-t state from the collinear spin-polarized GGA+U calculations. . . . . . . . . . . . . . . . . . . . . . . . Total and site-projected electronic densities of states of the collinear AFM-t state from the collinear spin-polarized GGA+U calculations. . . . . . . . . . . . . . The electronic band structure of the collinear AFM-s state from the collinear spin-polarized GGA+U calculations. . . . . . . . . . . . . . . . . . . . . . . . Total and site-projected electronic densities of states of the collinear AFM-s state from the collinear spin-polarized GGA+U calculations. . . . . . . . . . . . . .. ‧ 國. 5.4. 5. 47 49 50 51 52 53 53 55 56 56 57.

(9) 5.18 The electronic band structure of the most stable non-collinear AFM state (θ = 107◦ or 287◦ approximately) we found from GGA+U calculation with spin-orbit coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.19 Total and site-projected electronic densities of states of the most stable noncollinear AFM state we found from GGA+U calculation with spin-orbit coupling. 59. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 6. i n U. v.

(10) List of Tables 3.1 3.2. Quantum numbers of atomic orbitals . . . . . . . . . . . . . . . . . . . . . . . Colors of cobalt (III) complexes . . . . . . . . . . . . . . . . . . . . . . . . .. 5.1. NM and collinear spin-polarized calculations (a) without and (b) with considering the electron-electron correlations. . . . . . . . . . . . . . . . . . . . . . . Non-collinear spin-polarized calculations without considering the Hubbard U. The spin-orbit coupling has not been considered in (a) and (b), and has been considered in (c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-collinear spin-polarized GGA+U calculations . . . . . . . . . . . . . . . Non-collinear spin-polarized GGA+U calculations without the spin-orbit coupling Non-collinear spin-polarized GGA+U calculations with the spin-orbit coupling A series of collinear spin-polarized GGA+U calculations . . . . . . . . . . . . Exchange Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A spin configuration that the theoretical electric polarization of the state agrees with the experimental a results. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. ‧. ‧ 國. 學. io. sit. y. Nat. 5.3 5.4 5.5 5.6 5.7 5.8. n. er. 5.2. 政治大. iv l C n hengchi U. 7. 18 24. 38. 39 41 43 45 48 48 60.

(11) Chapter 1 Introduction As a relatively new course, the computational physics can be traced back to 1960s. At that time, extensive applications of quantum mechanics and discovery of complicated moleculars like double helix structure of DNA motivated scientists using computers as a tool to study interactions and mechanism in the macromolecules. With the developement of technology of computer, objects of study transit from atomic and molecular levels to the large-scale condensed matter systems. The computational physics becomes a new branch of physical studies in addition to theoretical and experimental physics up to now. Here we only introduce the ab initio method in this thesis. The so-called ”ab initio” means that there needs no any empirical parameters. The calculation starts directly at the level of established laws of physics, i.e. the quantum mechanics. What we have to do is to input the position of atoms in an unit cell. Though there are phenomena which cannot be described, the ab initio method still gains lots of achievements a lin the material can be explained irrefutably that some behaviors i v or predicted. n C h e2On9,gwith An interesting multiferroics, Cu3 Mo i U magnetoelectric coupling has attractted c hstrong our attention. The origin of the electricity of this material may probably be a novel mechanism. A small spin cluster model has been suggested by Kuroe et al. which is very different from the magnetic superlattice. Since there still has no ab initio calculation to study it, we performed a series of calculations based on spin density functional theory. The traditional DFT and the DFT+U method consisting of in a Hubbard-like correction which gives a better description of electronic correlation will be introduced in chapter 2. The crystal field theory introduced in chapter 3 helps us a lot analysing the electronic structure of Cu3 Mo2 O9 . It can describe the way of the splitting of d-orbitals to explain colors, strength of magnetic moment and structural distortion of transition metal complexes. The overview of multiferroics in chapter 4 describes many mechanisms of ferroelectricity, type-I (weak or no magnetoelectric coupling) and type-II (strong magnetoelectric coupling) multiferroics. In chapter 5, we discuss results in our calculations including the magnetic structure, electronic structure and spontaneous electric polarization of the system. Finally, summary and conclusions will be given in chapter 6.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. 8.

(12) Chapter 2 Density Functional Theory The density functional theory (DFT) is one of the most successful approach describing the properties of matters in the ground state. It was introduced in two seminal papers by HohenbergKohn (1964) [1] and Kohn-Sham (1965) [2]. Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998. The main idea of DFT is that it describes a many-electron system as a single electron moving in the pseudopotential, i.e. wavefunction of electrons represented as a functional of the electron density depends on spatial position. Only three degrees of freedom makes it possible to simulate the more complex system (larger number of particles) than before. It acquires the considerable achievement in material calculation science that some difficult question could be answered and some precursory material could be predicted.. 立. 政治大. ‧. ‧ 國. 學. er. io. sit. y. Nat. Born-Oppenheimer approximation a v. n. 2.1. i l C n h e n gmechanics, Thanks to the developement of quantum c h i U we can describe those behaviors in the. microscopic scale that Newton’s mechanics cannot explain. To solve a problem of condensed matter physics, we write down the Hamiltonian correspond to the system, ZI ZJ e2 h¯ 2 2 1 N ∇I + ∑ 2 I6=J 4πε0 |RI − RJ | I=1 2MI N. H =−∑. h¯ 2 n 2 1 n e2 1 n N ZI e2

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(14) − ∇i + ∑ − ∑ ∑ ∑ |ri − RI | , 2m i=1 2 i6= j 4πε0

(15) ri − r j

(16) 4πε0 i=1 I=1. (2.1). where N and n are numbers of nuclei and electrons respectively. M and m are masses of nucleus and electron respectively. Ze and e are charges of nuclei and electrons respectively. R and r are positions of nucleus and electron respectively, and h¯ is the reduced Planck constant and ε 0 is the vacuum permittivity. The Hamiltonian above includes the kinetic energy of nuclei and electrons, and the Coulomb potential energy of nucleus-nucleus, electron-electron and nucleuselectron interactions. Now we can solve the Schrodinger equation to obtain the wavefunction of. 9.

(17) the system, Hψ(r1 , r2 , ..., rn , R1 , R2 , ..., RN ) = Eψ(r1 , r2 , ..., rn , R1 , R2 , ..., RN ). (2.2). where ψ is the eigenfunction of the operator H corresponds to energy E. However, it is diffcult to deal with such complex equation including N nuclei and n electrons components. A preliminary method to simplify the problem is the Born-Oppenheimer approximation proposed in 1927 [3], also called the adiabatic approach, describing the seperation of electronic motion, nuclear vibrations, and molecular rotation. Since any nucleus is much more massive than an electron, it must accordingly have much smaller velocities, about two orders of time scale larger than electrons. The electrons that move around the nuclei in the system have enough time to reach the ground state as the nuclei undergo a small perturbation. Therefore, the system now can be seen that electrons moving in the arrangement of stationary nuclei. The Hamiltonian H then becomes. 政治大 H = H +H. 立. n. (2.3). e. ‧ 國. 學. where H n is the nucleus part including the first two terms of equation 2.1, and H e is the electron part including the remaining three terms. The wavefunction seperated into two parts,. ‧. ψ(r1 , r2 , ..., rn , R1 , R2 , ..., RN ) = ψn (R1 , R2 , ..., RN ) × ψe (r1 , r2 , ..., rn ),. sit. y. Nat. and. (2.4). er. io. (Hn + He )ψn (R1 , R2 , ..., RN ) × ψe (r1 , r2 , ..., rn ). n. = (Ena+ Ee )ψn (R1 , R2 , ..., RN ) × ψe (rv1 , r2 , ..., rn ),. (2.5). i l C n U then the wavefunction of electrons can However, although the problem h ebendiscussed g c h i separately. has been simplified, it is still too complicated to solve. Some theories proposed for simplifying the problem further. In the Thomas-Fermi theory, the precursory idea that the wavefunction of electrons is represented as a functional of the electronic density, which is proposed by Llewellyn Thomas and Enrico Fermi in 1927 [4] and will be discussed in the next section. And in the Hartree-Fock theory proposed in 1930, every electron moves in the mean field caused by other electrons. The Hamiltonian of the many-body system is decomposed into a sum of Hamiltonians of several one-body systems, and this theory will not be discussed in the following.. 2.2. Thomas-Fermi Theory. The Thomas-Fermi theory [4], which was the forerunner of the DFT, suggests a precursory idea that wavefunction of electrons is represented as a functional of the electronic density, Z. ρ(r) = ρ(r1 ) = n. ψ ∗ (r1 , r2 , ..., rn )ψ(r1 , r2 , ..., rn )dr2 ...drn ,. 10. (2.6).

(18) where ρ, n and ψ are the density, number and wavefunction of electrons respectively. To realize the theory, we firstly consider a box of stationary and uniform electrons. States must be occupied in turn from the ground state to the highest occupied state correspond to the wavenumber kF which is related to the density, 3π 2 ρ = kF3 , (2.7) and the kinetic energy of the electrons is T=. Ω h¯ 2 2 k , π 2 10m F. (2.8). where Ω is the volume of the box. Now we go back to the real system. A non-uniform system is seen locally uniform. The density and wavenumber are functions of the position. Thus the kinetic energy of electrons within the Thomas-Fermi theory is. 立. 政Z 治 3 h¯ k (r) 大ρ(r). T [ρ] = d r 2 2 F. 3. 5. (2.9). 2m. Nat. and the total energy is. Z. 0. ρ(r)ρ(r ) 3 3 0

(19)

(20)

(21) r − r0

(22) d rd r ,. (2.10). y. Unu−e [ρ] =. 1 ρ(r)Vnu (r)d r and Ue−e [ρ] = e2 2 3. er. io. sit. Z. ‧. ‧ 國. 學. Here we must notice that this kinetic term is associated with the non-interacting electrons. The Coulomb potential energy of the nucleus-electron and the electron-electron interactions are. n. = T [ρ] +Unu−e [ρ] +Ue−e [ρ]. (2.11) aE[ρ] v i l C can successfully be nsimplifed A many-body Schrodinger equation into a one-body Schrodinger hengchi U equation. It is a large step of achievement that the equation is simple enough to be solved. However, the accuracy of the method above is limited. Because the kinetic energy term is oversimplified that it consider the electrons as the non-interacting Fermi gas, the exchange-correlation of electrons are completely neglected. There are interactions between electrons respect to spin called the exchange interaction. It lowers the total energy further, since it makes electrons more apart. The physist Dirac tried to add an exchange term into the equation, but still failled because of that the kinetic term which see electrons as a uniform gas is conceptually a big error, and the correlation of electrons is completely neglected.. 2.3. Density Functional Theory. Although the concept of the functional of the electronic density can be traced back to the Thomas-Fermi theory in 1927 [4], the confirmed theoretical foundation, the Hohenberg-Kohn Theorem, is not proposed till 1964 by Hohenberg and Kohn [1]. However, it does not provide a way of finding the electronic density in the ground state. By employing the Kohn-Sham equation 11.

(23) proposed in 1965 [2], the ground state density is able to be found and the density functional theory gains the incomparable achievement in physics and chemistry from this moment on. The complete discussion will be in the following subsections.. 2.3.1. Hohenberg-Kohn Theorem. There are two theorems proposed and proved by Hohenberg and Kohn as follows. The First Hohenberg-Kohn theorem The first theorem demonstrates that the Hamiltonian and all the properties of the system are uniquely determined by the electron density. The proof is derived in the following. Assume that there are two different external potentials generated by a set of nuclei, V 1 (r) and V 2 (r), we would have two different Hamiltonians, H 1 and H 2 , each leads to the same ρ(r) for its ground state, but the two corresponding wavefunctions, ψ1 (r) and ψ2 (r), are different. Now we take ψ2 as a trial wave function for the problem of the Hamiltonian H 1 ,. 立. 政治大. ‧ 國. 學. E1 < hψ2 |H1 | ψ2 i = hψ2 |H2 | ψ2 i + hψ2 |H1 − H2 | ψ2 i = E2 +. Z. ρ(r)[V1 (r) −V2 (r)]dr, (2.12). ‧. sit. y. Nat. where E 1 and E 2 are eigenenergies of H 1 and H 2 respectively. Similarly, we take ψ1 as a trial wave function for the problem of the Hamiltonian H 2 ,. a. ρ(r)[V2 (r) −V1 (r)]dr. (2.13). er. io. E2 < hψ1 |H2 | ψ1 i = hψ1 |H1 | ψ1 i + hψ1 |H2 − H1 | ψ1 i = E1 +. Z. n. v l would obtain a contradictory Adding Eq. (2.12) and (2.13), we n i relation, Ch. engchi U. E1 + E2 < E2 + E1 .. (2.14). The first Hohenberg-Kohn theorem is proved that no two different external potentials can give the same electron density for the ground state. The total energy is a unique functional of the electron density which can be written as E[ρ] = T [ρ] +Un−e [ρ] +Ue−e [ρ],. (2.15). where T and U n−e are kinetic energy and nucleus-electron Coulomb potential energy. And the U e−e here that differs from the term of Thomas-Fermi theory is the energy related to the electronelectron interaction including the electron-electron Coulomb repulsion, exchange interaction and Coulomb correlation.. 12.

(24) The Second Hohenberg-Kohn Theorem As follows , the second Hohenberg-Kohn theorem, the variation principle, enable us to make sure that the ground state density is what we are looking for. Recall that any trial electron density ρ1 defines its own Hamiltonian H 1 and hence its own wavefunction ψ1 . The wavefunction can now be taken as the trial wavefunction for the Hamiltonian H 0 generated by the external potential V ext . Thus, hψ1 |H0 | ψ1 i = T [ρ1 ] +Unu−e [ρ1 ] +Ue−e [ρ1 ] = E[ρ1 ] ≥ E[ρ0 ] = hψ0 |H0 | ψ0 i = E0 ,. (2.16). the functional E[ρ] has its minimum relative to variations δ ρ of the density at the equilibrium density ρ0 , δ E[ρ] |ρ=ρ0 = 0. (2.17) δρ. 2.3.2. 政治大 Kohn-Sham Equation 立. ‧. ‧ 國. 學. The kinetic energy has a large contribution to the total energy. The failure of the ThomasFermi theory is due to that it makes the wrong approximation of the term of kinetic energy. The density functional theory, therefore, was ignored until Kohn and Sham introduced a method which treated the majority of the kinetic energy exactly. The theory begins by considering the non-interacting reference system that all electrons move in an effective potential with no interaction, in other words, a mean field single particle picture. The ground state is a Slater determinant of orthogonal orbitals ψi which satisfies the equations. er. io. sit. y. Nat. n. a. l C h¯ 2 ni (r) = Ei (r)ψi (r) Uwith [− ∇2 +Ve f f (r)]ψ ih engchi 2m. v. n. ∑ |ψi(r)|2 = ρ(r),. (2.18). i=1. where V e f f is the effective potential including the nucleus-electron, electron-electron Coulomb term and the ill-defined exchange-correlation term, and n is the number of electrons. Due to the Hohenberg-Kohn theorem, the kinetic energy and total energy are given by n Z. T [ρ] = ∑. i=1. ψi∗ (r)(−. h¯ 2 2 ∇ )ψi (r)dr and E[ρ] = T [ρ] + 2m. Z. ρ(r)Ve f f (r)dr.. (2.19). The functional T [ρ] is just a particular case, the ground state electron density now can equivalently be obtained by the solution of th Euler-Lagrange equations, Z δ δ T [ρ] 0= E[ρ] − µs ρ(r)dr = +Ve f f (r) − µs , δ ρ(r) δρ. (2.20). where the Lagrange multiplier µs satisfies the constraint that the density integrates to the correct number of electrons. It is very important to realize that if the exact form of exchange-correlation term in the effective potential can be defined, the Kohn-Sham strategy would lead to the exact 13.

(25) energy. This will be briefly discussed in the next section.. 2.3.3. Exchange-Correlation Energy. Since the work of Kohn and Sham, the remaining question is the available form of the exchange-correlation term of the effective potential, Z. Ve f f (r) = Vext (r) +. 0. ρ(r ) 0 δ EXC [ρ(r)]

(26)

(27) dr +VXC (r) and VXC (r) = , 0

(28) r − r

(29) δ (r). (2.21). where E XC is the exchange-correlation energy. Some successful methods of exchange-correlation approximation like the local density approximation (LDA) [5] and generalized gradient approximation (GGA) [6] are briefly introduced in the following. By the way, the Perdew-BurkeErnzerhof exchange-correlation functional (PBE) [7] is used in our calculations.. 立. 政治大. The Local Density Approximation (LDA). Z. sit. Nat. LDA EXC [ρ(r)] =. y. ‧. ‧ 國. 學. Local density approximation [5], the basis of all approximate exchange-correlation functionals, is the simplest and the most widely used one. The central idea of this approximation is the well-defined limit of exchange-correlation energy of the uniform electron gas. A non-uniform system can be seen as a locally uniform system, and the exchange-correlation energy is ρ(r)εXC [ρ(r)]dr,. io. er. (2.22). n. where εXC [ρ(r)] is the exchange-correlation energy per electron al v of an uniform electron gas of i n C h can be furtherUseperated density ρ(r). The quantity εXC [ρ(r)] into two parts, exchange and engchi correlation terms, εXC [ρ(r)] = εX [ρ(r)] + εC [ρ(r)]. (2.23) The exchange part was originally derived by Bloch and Dirac where 3 3ρ(r) 1/3 ) . εX [ρ(r)] = − ( 4 π. (2.24). However, there has no explicit form for the correlation part, but the numerical quantum MonteCarlo simulations of the uniform electron gas with good accuracy. It is able to introduce the spin to the LDA (LSDA), and the form becomes LSDA EXC [ρ↑ (r), ρ↓ (r)] =. Z. ρ(r)εXC [ρ↑ (r), ρ↓ (r)]dr.. (2.25). The accuracy of the L(S)DA is limited. Only materials with slowly varying electron density can be describe well. However, some bulk and surface of solids exist the rapid varying density. For some materials that have strongly correlated d or f electrons, L(S)DA even predicts them to 14.

(30) be metallic instead of insulating ground state. It tends to underestimate the ground state energy, band gap and the ionization energies, but overestimates the bonding energy. The Generalized Gradient Approximation (GGA) To promote the accuracy, not only ρ(r) but also the gradient of ρ(r) are considered to account the exchange-correlation energy since the fact that the electron density of a real system is non-uniform. The form of exchange-correlation energy in the generalized gradient approximation [6] can be expressed as GGA EXC [ρ↑ (r), ρ↓ (r)] =. Z. ρ(r)εXC [ρ↑ (r), ρ↓ (r), ∇ρ↑ (r), ∇ρ↓ (r)].. (2.26). The GGA has reduced the errors of LDA, but the variation of electron density still must be slow enough.. Mott Insulators. ‧ 國. 學. 2.4. 立. 政治大. ‧. Though density functional theory successfully describes various properties of materials, there still exist some systems like Mott insulators, named after N. F. Mott, cannot be explained. In 1937, J. H. de Boer and E. J. W. Verwey indicated that a variety of transition metal oxides predicted to be conductors by band theory are insulators [8]. Also in 1937, N. F. Mott and R. Peierls tried to explain that this anomaly can be the result of that band theory did not consider the on-site electron-electron Coulomb repulsion. Since the density functional theory is a mean-field a l describe the many-bode i v correlations, there must be some single particle theory, which cannot n Ch modifications. The Hubbard model and e nthegimproved c h i Udensity functional theory, the so-called DFT+U, will be introduced in the following subsections.. n. er. io. sit. y. Nat. 2.4.1. Hubbard Model. The Hubbard model, originally proposed in 1963 [9, 10], is an approximate model which can be used to solve the anomaly of Mott insulators. First, we regard the nuclei in the system as the fixed array, i.e. without considering the lattice vibrations. For simplicity, every site of the array has only one energy level, thus only two electrons can occupy an energy level (spin up and spin down) due to the Pauli principle. Electrons move around the array of nuclei interact via a screened Coulomb interaction, the biggest interaction will be the Coulomb repulsion of the two electrons occupy the same site. For simplicity, There is no interaction between electrons on different sites. The on-site interactions are modeled by a term which is zero if the atom is empty of electrons or has only a single electron on it, but has the value of repulsion energy U if the atom has two electrons. The competitive term, the kinetic term, is introduced allowing electrons to move from one site to another. The energy scale t which governs this ’hopping’ will 15.

(31) be determined by the overlap of two wavefunctions. Since wavefunctions decay exponentially, we can consider that the hopping only occurs between the neighboring atoms. Now, we can write down the Hubbard Hamiltonian. To begin with, we recall the creation and annihilation operators which deal with the harmonic oscillator. We will spend some time reviewing their properties, which parallel those of the operators in the Hubbard model in many ways, in this more familiar setting. r ab =. mω xb+ i 2¯h. r. 1 pb 2mω h¯. and. ab+ =. r. mω xb− i 2¯h. r. 1 pb, 2mω h¯. (2.27). where pb and xb are momentum and position operators. From [ pb, xb] = −i¯h, it is easy to show that these operators obey the commutation relations,. 政治大 + ab, ab = 1,. 立1. and the Hamiltonian is. (2.28). 1 1 (2.29) pb2 + mω 2 xb2 = h¯ ω(b a+ ab + ). 2m 2 2 The Hubbard model is also written in terms of ’fermion’ creation and annihilation operators. However, there are several different respects. The fermion operators in the Hubbard model are not introduced in terms of familiar position and momentum operators. They are distinguished biσ , where by attaching indices j and σ (spin up or spin down) which can be written as cb+ iσ and c + cbiσ is the operator which creates an electron of spin σ on lattice site i, cbiσ is the operator which biσ is the number operator. As a annihilates an electrons of spin σ on lattice site i, and niσ = cb+ iσ c a consequence, the occupation number by a single number n, as iv l C states are no longerncharacterized h e n gbyca hcollection for a single harmonic oscillator, but instead i U of occupation numbers niσ . We can write a state as | n1↑ , n2↓ , n3↓ , ... >. Because these operators are used to describe fermions, they must satisfy the following relations to agree with the Pauli principle,. ‧. ‧ 國. 學. b= H. n. er. io. sit. y. Nat. {b ciσ , cb+jσ 0 } = δi, j δσ ,σ 0 b+jσ 0 } = 0 {b c+ iσ , c. (2.30). {b ciσ , cbjσ 0 } = 0, b B} bBb + BbA b (the anticommutation relation). The maximum occupation of a parb =A where {A, ticular site with a given spin is 1. The Hubbard Hamiltonian is then, H = −t. ∑. hi, jiσ. bjσ +U ∑ nbi↑ nbi↓ − µ ∑(b cb+ ni↑ + nbi↓ ). iσ c i. i. (2.31). The first term is the kinetic term. It describes the annihilation of an electron of spin σ on site j and its creation on site i (or vice-versa). The symbol <i, j> emphasizes that hopping is allowed. 16.

(32) only between two neighboring sites. The second term , the correlation term, is the Coulomb repulsion energy, an energy U will be added if there is doubly occupied on j-site. The competition between the kinetic term and the correlation term characterize the system. As t/U >> 1, the energy is minimized making the kinetic term as small as possible through delocalization (little price is paid on the occupied atomic sites to overcome repulsion U). As t/U << 1, the kinetic energy of electrons is not large enough to overcome the on-site repulsion. Electrons undergo a Mott localization. The last term is a chemical potential term which controls the filling. We refer to the situation where the filling is one electron per site as ’half-filling’ since the lattice contains half as many electrons as the maximum number (two per site). Studies of the Hubbard model often focus on the half-filled case because it exhibits a lot of interesting phenomena (Mott insulators, anti-ferromagnetic order, etc.). 政治大 Due to the insufficiency of立 density functional theory describing the strong correlated system,. 2.4.2. Beyond DFT : DFT+U. ‧. ‧ 國. 學. several attempts like the self-interaction correction (SIC) method [11], the Hartree-Fock (HF) method [12] and the GW approximation (GWA) [13] for improving the DFT were proposed. However, any of them has its inadequacy. From 1990 to 1995, the LDA+U method consists in a Hubbard-like correction to the LDA energy functional was introduced and developed by Anisimov and coworkers, which gives a better description of electronic correlations. A rotationally invariant version was introduced by Liechtenstein et al. [14], and the simplified approach was introduced by Dudarev et al. [15],. n. er. io. sit. y. Nat. a. l C= ELSDA + ELSDA+U. v 2 (U − J) im,σ (n − nm,σ ), n ∑ 2i U σ. hengch. where U and J are screened Coulomb and exchange parameters.. 17. (2.32).

(33) Chapter 3 Crystal Field Theory Crystal Field Theory (CFT) is an ionic theory that describes the breaking of symmetries of orbital states, usually d or f -orbitals, in the presence of surrounding ligands. It was proposed by the physicist Hans Bethe in 1929 [16]. Subsequent modifications were proposed by J. H. Van Vleck [17] to allow for some covalency in the interactions to be successfully used to explain and predict some magnetic, spectral and thermodynamic properties of transition metal complexes.. 立. ‧. ‧ 國. 學. 3.1. 政治大. Atomic Orbitals. y. Nat. n. er. io. sit. An atomic orbital is the probability distribution of electrons around the nucleus of atom. The shape of the orbital mainly depends on the quantum numbers (n, the principle quantum number, l, the orbital quantum number, and m, the angular momentum quantum number) associated with al i v an orbital can be occupied by two the particular energy state. Each set of n C hof quantum numbers en electrons, spin up and down, as listed in Table h i ToUunderstand CFT, it is important to have g c3.1. Table 3.1: Quantum numbers of atomic orbitals are listed, where n (n∈N) is the principle quantum number, l (l = 0, 1, ..., n−1) is the orbital quantum number and m (m = 0, ±1, ..., ±l) is the angular momentum quantum number. orbital n=1 n=2 n=3 n=4. s l=0 m=0 m=0 m=0 m=0. p l=1. d l=2. f l=3. m=0, ±1 m=0, ±1 m=0, ±1. m=0, ±1, ±2 m=0, ±1, ±2. m=0, ±1, ±2, ±3. a clear picture of the shapes (angular dependence functions) of the d-orbitals. Consider a free transition metal atom. There are five d-orbitals in two spatial groups, three of the five d-orbitals (d xy , d yz and d zx ) are collectively referred to as t 2g , means that all three (t) orbitals have the 18.

(34) 政治大. Figure 3.1: d-orbitals - For d x2 − y2 , the four lobes lie on the x and y axes, and for d z2 , there are two lobes on the z axes and there is a donut shape ring that lies on the xy-plane around the other two lobes. The four lobes of d xy orbital lie in-between the x and the y axes, and so on. [Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/]. 立. ‧ 國. 學. ‧. reflection asymmetry (2) and inversion symmetry (g). Take d xy for example, the four lobes lie in-between the x and the y axes as can be seen in Figure 3.1, and so on. The remaining two (d z2 and d x2 − y2 ) as eg , means that both two (e) orbitals have the reflection symmetry (1) and inversion symmetry (g). For d x2 − y2 , the four lobes lie on the x and y axes, and for d z2 , there are two lobes on the z axes and there is a donut shape ring that lies on the xy-plane around the other two lobes. All d-orbitals a lare degenerat when the atom v is free. However, in the presence i n Energy of t 2g and eg split into two Cofh d-orbitals are broken. of surrounding ligands, symmetries U i e h g ccomplex splittings occur in other cases and will levels in octahedral and tetrahedral cases.n More be discussed in the following sections.. n. er. io. sit. y. Nat. 3.2. Crystal Field Theory. As mentioned before, CFT is an ionic theory. The ligands are regarded as point charges, the interactions between the metal ion and the ligands are purely electrostatic. All covalent bonding effects are ignored. Here we firstly look at the most common case, the octahedral complex. Six ligands attached to the central transition metal along x, y and z axes, forming an octahedron as shown in Figure 3.2. We can see that symmetries of orbitals d xy , d yz and d zx do not change in the presence of the ligands, and remain in the t 2g group. Orbitals d z2 and d x2 − y2 also remain in the eg group. While orbitals of t 2g set are farther from the ligands than eg set that lobes of orbitals directly lie along the x, y and z axes, which will have lower energy due to experience less Coulomb repulsion from ligands. Now the energy of d-orbitals of the atom split into a threefold degenerate level and a two-fold degenerate level with an energy difference, the crystal-field 19.

(35) Figure 3.2: Structure of complexes - (a) The octahedral complex, where six ligands attached to the central transition metal along x, y and z axes, forming an octahedron. (b) The tetrahedral complex, four ligands form a tetrahedron around the central ion. (c) The square-planar complex, which can be seen as the simplification of the octahedral complex, where the two ligands along the z-axis are removed.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. i n U. Ch. v. Figure 3.3: Common e crystal h i splittings of d-orbitals n g c field splitting parameter (∆oct ). Likewise, the other cases like tetrahedral, square-planar cases and even the more complex cases can all be discussed in the same way. As shown in Figure 3.3, in the tetrahedral case, the d-orbitals also split into two levels (t 2 and e) with energy difference (∆tet ), while the more stable orbitals are d z2 and d x2 − y2 which are oppsite to the octahedral case. And we can find that the energy difference of the tetrahedral case is smaller than the octahedral case. This is because that a tetrahedral complex has fewer ligands, and ∆tet is roughly equal to 4/9 ∆oct . The square planar case can be considered as an extension of the octahedral case, more complex energy levels of the d-orbitals could be obtained by discussing in the same way. Except for the arrangement of the ligands around the metal ion that can affect the splitting way of energy levels, the oxidation of the atom can dominate the magnitude of the energy splitting of the d-orbitals. A higher oxidation state of the atom leads to a larger splitting because that the ligands are closer to the atom. The nature of the ligands surrounding the metal ion also affects the magnitude of the energy splittings. The following series is the part of so-called spectrochemical. 20.

(36) Figure 3.4: The low-spin and high-spin cases in the octahedral crystal field - In the left case, the low-spin case, the strong-field ligands lead to a large ∆oct , which is larger than the pairing energy so that the lower energy orbitals will completely be filled before occupation of the higher orbitals occurs. In the right case, the high-spin case, the weak-field ligands lead to a small ∆oct , which is smaller than the pairing energy so that it is easier to put the remaining electrons into the higher levels of eg set.. 立. 政治大. series, an empirically series, the ability of ligands to lead to the splitting of energy are listed.. ‧ 國. 學. − − − − − CO > CN − > NO− 2 > NH3 > SCN > H2 O > OH > F > SCN > Cl. (3.1). ‧. Ligands that lead to a large splitting, at the left side of the series, of energy are called strongfield ligands. Those at the right side are called weak-field ligands. Interestingly, the series above against the CFT where CO, NH3 and H2 O are neutral ligands rather than anions, and CO is found to be one of the strongest ligands. The reasons behind this can be explained by ligand field theory (LFT), which delivers insight into the process of chemical bonding in transition al v metal complexes, will not be discussed ni C here.. n. er. io. sit. y. Nat. hengchi U. 3.3. High Spin and Low Spin. Considering the octahedral complex, where energy split into two levels of t 2g and eg sets. One may worry about how electrons occupy these orbitals. When there are four to seven electrons in the d-orbitals of the atom, two possible diagrams of distribution of electrons can occur, namely high-spin or low-spin. Take a four-electrons complex for example. The first three elections occupy the orbitals of t 2g set with no doubt, and the remaining one electron occupies the d-orbitals in two ways as shown in Figure 3.4. The right diagram refers to as the high-spin case. The weak-field ligands lead to a small ∆oct so that it is easier to put the remaining electron into the orbitals of eg set than put into the orbitals of t 2g set to form a pair. This is because that two electrons put into the same orbial would repel each other due to the Coulomb repulsion, and the energy cost is larger than ∆oct . Therefore, electrons tend to occupy each of the five d-orbitals before any pairing occur in accord with Hund’s rule. Conversely, a large ∆oct caused by the strong-field ligands lead to the low-spin case. The remaining one electron tends to form a pair in 21.

(37) the orbitals of t 2g set because the energy cost of pairing is smaller than ∆oct . The lower energy orbitals will completely be filled before occupation of the higher orbitals occurs according to the Aufbau principle. By the way, as mentioned before, the crystal field splitting energy for tetrahedral complex is refer to as ∆tet , and is roughly equal to 4/9∆oct , which is typically small enough to tend to the high-spin case. The two diagrams can be used to explain or predict some magnetic properties of some compounds like magnetic moment and number of unpaired electrons. A compound has unpaired electrons will lead to the paramagnetic property and be attracted by magnetic field, while a compound with all electrons paired will lead to the diamagnetic property and be weakly repeled by magnetic field. By using the simple spin-only formula, we can predict the magnetic moment, p (3.2) µ = 4S (S + 1), where S is the spin quantum number (1/2 for each unpaired electron) and the unit of the moment is Bohr magneton (µ B ). We can also determine the number of unpaired electrons by measuring the magnetic moment and analysing the nature of the compound.. 立. ‧ 國. 學. 3.4. 政治大. Crystal Field Stabilization Energy. ‧. n. er. io. sit. y. Nat. A compound in the ground state means that electrons distribute in the most stable way in the system. Any excited system tends to go through a transition to the state with the best stability. There is a parameter for counting the stability of a state of the system called crystal field stabilization energy (CFSE) in CFT. It is easy for us taking an example to discuss the so-called CFSE. For an octahedral complex, a l the energy levels split into v the lower t 2g level and the higher i eg level, and the energy differenceC is ∆oct . An electron innthe more stable t 2g orbital is treated as hengchi U contributing −0.4∆oct , whereas an electron in the higher energy eg orbital contributes to a destabilisation of +0.6∆oct to CFSE. The more stable of the state to the system, the smaller the value of CFSE. If the electron pairing (put two electrons into the same orbital) energy P is smaller than ∆oct , it is the low-spin case. Conversely, it is the high-spin case. As shown in Figure 3.4, four d-electrons occupy the t 2g orbitals in the low-spin case, CFSE = (−0.4∆oct ) × 4 + P = −1.6∆oct + P.. (3.3). CFSE = (−0.4∆oct ) × 3 + 0.6∆oct = −0.6∆oct .. (3.4). For high-spin case,. The use of CFSE enable us to predict or explain the structure of some complexes. In fact, structures of many d 8 complexes are square-plannar can be explained by CFSE. Also, crystal field stabilization can be applied to analyse transition-metal complexes of all geometries.. 22.

(38) Figure 3.5: color wheel - A sample absorbs visible light of color of the specific wavelength and reflects the rest, in appears the complementary color of light. Find the color that is absorbed, then move directly across the wheel to the other side to get the complement.. 3.5. Jahn-Teller Theorem. 政治大. The Jahn-Teller theorem (named after Hermann Arthur Jahn and Edward Teller), also called Jahn-Teller distortion often, was published in 1937 [18]. The crystal distortion occur due to the asymmetry of distribution of electrons in the system, which mostly be observed in octahedral system. In the octahedral d 9 case (nine electrons occupy the d-orbitals), the ninth electron may occupy either d z2 or d x2 − y2 orbitals. If it occupies the d z2 orbital, there will be more Coulomb repulsion along the z-axis so that the compound tends to elongate along the z-axis. Conversely, elongation along x and y axes if the ninth electron occupies the d x2 − y2 orbital. Likewise, the distortion can occur theoretically in almost all cases except d 3 , d 8 , d 10 , high-spin d 5 and lowspin d 6 , since the distributions of electrons of these cases are symmetric. Considerable distortion are usually observed in high spin d 4 , low spin d 7 and d 9 cases, since the unpaired electron v to the more distortion. occupies the orbital of eg set, athe l more Coulomb repulsion ilead. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. Ch. 3.6. n engchi U. Colors of Transition Metal Complexes. Here we discuss the colors of some complexes that can be explained by CFT in this section. As we know, if a sample absorbs all wavelength of visible light, none reflects and reaches our eyes from that sample, it appears black. Conversely, if the sample do not absorb any wavelength of visible light, it appears white. Since the sample absorbs visible light of color of the specific wavelength and reflects the rest, in appears the complementary color of light. The color wheel shown in Figure 3.5 can demonstrate which color a sample will appear. Find the color that is absorbed, then move directly across the wheel to the other side to get the complement. For example, if the sample absorbs the red-light, it appears cyan, and if the sample absorbs the bluelight, it appears yellow. Now we get back to the subject, as mentioned before, energy of the dorbitals of the transition metal atom will split into two or more levels in the presence of ligands. An electron occupies the lower energy d-orbital would jump to the higher unoccupied d-orbital as it absorbs light of wavelength correspond to the certain energy ( the crystal field splitting).. 23.

(39) Fortunately, the energy (the crystal field splitting) often corresponds to the visible region of the spectrum. Therefore, we can explain or predict colors of the transition metal complexes. In Table 3.2, a number of cobalt (III) complexes are listed [19], various ligands (including the strong-field Table 3.2: Cobalt (III) complexes, colors and corresponding absorbed lights are listed. Co3+ complex [CoF6 ]3− [Co(H2 O)6 ]3+ [Co(NH3 )5 Cl]2+ [Co(NH3 )5 H2 O]3+ [Co(NH3 )6 ]3+ [Co(CN)6 ]3−. absorbed light red orange yellow blue-green blue ultraviolet. color seen green blue violet red yellow-orange pale yellow. 政治大 and weak-field ligands) combine with the cobalt ion and present different colors. The results 立 agree with CFT well. Take [Co(CN) ] for example, CN is a strong-field ligand so that it will 6. −. 3−. ‧ 國. 學. ‧. lead to a large crystal-field splitting. Thus the energy an electron needs for transition must be high enough (ultraviolet light). Conversely, F− is a weak-field ligand, which will lead to a small splitting. The energy of red light is large enough to excite the transition.. n. er. io. sit. y. Nat. al. Ch. engchi. 24. i n U. v.

(40) Chapter 4 Multiferroics 4.1. Introduction. 政治大 Multiferroics is a material simultaneously having more than one ferroic order parameter in 立 a phase. These parameters are ferromagnetism, ferroelectricity and ferroelasticity (Nowadays. ‧ 國. 學. ‧. what most people mean by multiferroic material predominantly applies to the coexistence of magnetism and ferroelectricity). Some coexistent order parameters in the material has the cross coupling. Take a material involving the coexistance of ferromagnetism and ferroelectricity with cross coupling for example, it means that we can control the electric order in the system by applying a magnetic field and vice versa. This behavior is called the magnetoelectric effect [20]. Such material has the potential of technological applications that it provides opportinities for designing the better electronic devices in the future. As is known, the common data storage a iv device like the hard disk drive isl devised using the ferromagnetism of the material. And also the n Ch U e n g cdevised common data storage device, the flash memory, h i using the ferroelectricity of the material. The two storage devices have their respective pros and cons. If the cross coupling of a discovered multiferroic material can be strong enough, the large degrees of freedom of its properties can probably be used to devise some better electronic devices including the advantages of both hardisk and flash memory, or to create some multifunctional electronic components like a new types of 4-state logic (i.e., with both up and down polarization and up and down magnetization). The studies of multiferroic material can be traced back to 1960s [21]. However, it has been considered that the strong enough coupling between ferromagnetism and ferroelectricity is impossible. The early studies represent that the conditions to cause the ferromagnetism and ferroelectricity usually interfere each other. For example, a condition for causing ferroelectricity requires empty d-orbitals (diamagnetic ions), while a condition for ferromagnetism requires partially filled d-orbitals (paramagnetic ions). Therefore, the interesting field has been forgotten for a while. Since the precursory works on thin films of BiFeO3 [22], TbMnO3 [24] and TbMn2 O5 [23] in 2003, multiferroic materials attract much attention once again. Considerable effort has been devoted to search for new compounds with good multiferroic properties and even strong. n. er. io. sit. y. Nat. 25.

(41) enough magnetoelectric coupling. The advancements of the ab initio calculation and experimental techniques make it possible to predict or design the system, and to realize the mechanics of the coexistence of the order parameters and the cross coupling of them. There are generally two types of the multiferroic materials distinguished by considering the microscopic source of the ferroelectricity. The first type, called the type-I multiferroics, contains those materials with weak or no cross coupling of ferromagnetism and ferroelectricity. Conversely, the type-II multiferroics appears the strong cross coupling between ferromagnetism and ferroelectricity. The two types of multiferroics will be discussed in the following.. 4.2. Symmetry. Symmetry, has been generalized to mean invariance in physics, is that the state remain invariant after undergoing a transformation. This definition can be applied to not only concrete but also abstractive problems. Here we only treat the perspective of physics, including aspects of space and time. The behavior under spatial and time version can characterize orders in the system, since ferroelectricity, ferromagnetism and ferroelasticity are closely related to the breaking of the spatial or time inversion symmetry.. 立. ‧. ‧ 國. 學. Time reversal. 政治大. y. Nat. n. er. io. sit. Time reversal is that we replace the expression for time with its negative in formulas or equations. It does not mean that we really turn back the clock, but reverse the motions. For example, an electron doing thea circular motion around the z-axis clockwise, the centripetal force iv l C acting on the electron point to the center of the circle, andnthe magnetic field is induced along the h e n gtransformation c h i U on the electron, the circular motion positive z-axis. Now we have a time reversal become counterclockwise, the centripetal force acting on the electron still point to the center of the circle. However, the magnetic field is induced along the oppsite direction, the negative z-axis. We can say that the classical mechanism of the circular motion of an electron agree with the time inversion symmetry. However, the magnetic field induced by an electron requires the breaking of time reversal symmetry. Also, the condition for ferromagnetism requires the spontaneous breaking of time reversal symmetry. Spatial reversal Likewise, spatial reversal is that we replace the expression for position with its negative in formulas or equations, which we also call the parity transformation. Considering a neutral atom, the symmetric center is located at the nucleus. The system without any perturbation agrees with the spatial inversion symmetry. When putting the atom in an electric field, the distribution of electrons in the surroundings of the nucleus deviate along the oppsite direction of the electric field. The breaking of the spatial inversion symmetry occurs producing an electric polarization in 26.

(42) Figure 4.1: Geometric frustration - (a) Ising antiferromagnets on the triangle lattice. The remaining one spin can no longer point in a direction oppsite to both two spins. (b) Scheme of water ice molecules where hollow circles are oxygen atoms and filles circles are hydrogen atoms. For each oxygen atom, two of the neighboring must reside in the far position and two of them in the near position.. 政治大 the atom. The condition for ferroelectricity requires the spontaneous breaking of spatial reversal 立 symmetry.. ‧ 國. 學. 4.3. Geometric Frustration. ‧. n. er. io. sit. y. Nat. The concept of geometric frustration [25, 26] is important in some system. It means that there is no single ground state in the system. A nonzero entropy remains in the system even at zero temperature. It can be traced back to 1950 [27]. An Ising antiferromagnet on the triangle lattice, when two of three spins are pointed in oppsite directions to satisfy their antiferromagnetic al i v oppsite to both two spins. That is interaction, the remaining one can no longer point in a direction n C h eminimize i Uenergy of all interactions. The geometric n g c h the to say, it is impossible to simultaneously frustration is not a phenomenon occurs unusually. In fact, it occurs in the ordinary ice which we contact almost everyday [28]. Four oxygen atoms form a tetrahedral structure, and the hydrogen atoms locate between two oxygen atoms being closer to one of the two. Every oxygen atom is surrounded by four hydrogen atoms. For each oxygen atom, two of the neighboring must reside in the far position and two of them in the near position, so-called ’Ice rules’.. 4.4 4.4.1. Multiferroics Type-I Multiferroics. The sources of ferromagnetism and ferroelectricity of type-I multiferroic materials are different, and weak or no magnetoelectric effect appears in such a system. The critical temperature of appearance of ferroelectricity is often higher than magnetism, and the spontaneous electric polarization P (of order 10 - 100 µC/cm2 ) is often rather large. Four different subclasses de-. 27.

(43) Figure 4.2: Perovskite Structure - The general stoichiometry ABX3 , where A (blue particles located at the corners of the cube) and B (orange particle located in the centre of the cube) are cations and X (red particles located in the face-centred positions of the cube) are anions.. 政治大. pending on the mechanism of ferroelectricity will be discussed as follows.. 立. Perovskites. ‧ 國. 學. ‧. Firstly, we introduce the perovskite structure which has the general chemical formula ABX3 , named originating from the perovskite mineral (CaTiO3 ), where A and B are cations and X is an anion (often the oxygen atom). In an ideal perovskite structure, the A cations (blue) is located at the corners of the cube, the B cations (orange) in the centre and the X anions (red) in the face-centred positions. There are many magnetic and ferroelectric materials with the perovskite structure. What scientists want to find is a material simultaneously has the ferromagnetism and ferroelectricity with strong cross a l coupling. But, early studies v represent that the conditions to i cause the ferromagnetism and ferroelectricity usually interfere n each other. For example, a conCh U i e h g c d-orbitals, while for ferromagnetism requires dition for causing ferroelectricity requiresnempty partially filled d-orbitals. It is considered that the coexistence of ferromagnetism and ferroelectricity is impossible. However, out of speculation, some materials with perovskite structure do simultaneously have the ferromagnetism and ferroelectricity [29, 30], for example, RMnO3 (R=Tb, Dy, Ho). This is the so-called d 0 -dn problem. A possible way to explain this problem is to make ”mixed” perovskite with d 0 and dn ions. It means that the source of the ferromagnetism and ferroelectricity originate from different ions. The d 0 transition metal ions locate on B-cites, the empty d-orbitals hybridized with p-orbitals of the surrounding one or three oxygen atoms forming the strong covalent bond and causing the off-centered shifts which break the spatial inversion symmetry, i.e. causing the spontaneous electric polarization. And the d n ions locate on A-cites contributing to the ferromagnetism. There seens to be no contradiction in this way. Unfortunately, this kind of multiferroic materials have weak or no cross coupling between ferromagnetism and ferroelectricity since the sources are different.. n. er. io. sit. y. Nat. 28.

(44) 政治大. Figure 4.3: YMnO3 - (a) The centrosymmetric structure above the critical temperature. (b) The ferroelectric structure below the critical temperature where practically rigid MnO5 tilt breaking the spatial inversion symmetry.. 立. ‧ 國. 學. Lone Pairs. ‧. There is another way oppsite to the above one to explain the coexistence of ferromagnetism and ferroelectricity, where ferroelectricity is caused by the A-cite cations, and the B-cite cations contribute to the ferromagnetism. This way usually occurs on those materials with perovskite structure having active ns2 electrons, called lone-pair, on the cations located on A-cite. Take BiFeO3 for example, the outer Bi-6s2 lone-pair cause the empty Bi-6p-orbital to come closer in energy to the O-2p-orbital. This a l lead to the hybridization ofi v Bi-6p and O-2p orbitals and drive n the d n transition metal ions locate the off-centered shifts resulting inCthe And U h eferroelectricity. i h n g cNo contradiction occurs. However, the magnetoon B-cites contributing to the ferromagnetism. electric effect in this kind of system is none or weak, due to different sources of ferromagnetism and ferroelectricity.. n. er. io. sit. y. Nat. Structured Distortion The mentioned two above are related to bondings that lead to the off-centered shifts of atoms. Here we introduce a mechanism of ferroelectricity caused by the relatively complicated deformation of the crystal structure. Take YMnO3 [32]for example, a tilting of the practically rigid MnO5 in the crystal occurs as belowing the critical temperature. The asymmetric change of distance of Y-O bonds in the system lead to the ferroelectricity. And the ferromagnetism comes from the magnetic Mn3+ ions.. 29.

(45) Charge Ordering In the typical ferroelectric materials, the main source of ferroelectricity is the deformation of cations and anions in the crystal as above. Recently, it is found that the ferroelectricity can occur due to the charge ordering [31]. In some strong correlation systems, due to the strong interaction between electrons, charges are localized on different sites leading to a disproportion and an ordered superlattice breaking the spatial inversion symmetry. This is often observed in transition metal oxides, especially those formally containing transition metal ions with different valence. For example, the magnetite Fe3 O4 is a mixed-valence oxide where the iron atoms have a statistical distribution of Fe3+ and Fe2+ above the critical temperature. Below the critical temperature, the combination of Fe2+ and Fe3+ species arrange themselves in an ordering pattern, causing the ferroelectricity.. 4.4.2. Type-II Multiferroics 政. 治. 大立 The theme of multiferroics which gets the most attention recently is the type-II multiferroics. ‧. ‧ 國. 學. The discovery of strong magnetoelectric effect in the materials excite scientists to devote rather considerable effort to search for new compounds and theoretical models. They can be separated into two groups with different mechanisms of multiferroics, the spiral and collinear magnetic structures in the materials as follows.. sit. y. Nat. Spiral Magnetic Structure. er. io. n. The first group is the spiral magnetic structure in the material which appears below the a i vthe type-II multiferroics known to critical temperature along with lthe Mostnof Cferroelectricity. h e n g c hwork i Uon TbMnO3 for example [24]. Below date belong to this group. Take the pioneering T N1 , the magnetic structure is a sinusoidal spin-density wave, and every spins point to the same direction but different magnitude of local moment. The magnitude of these moments varies periodically as shown in Figure 4.4 [33]. This structure is centrosymmetic and consequently not ferroelectric. As the temperature decreases below T N2 , the cycloidal spiral with the wave vector Q = Qx and spins rotating in the xz-plane appears. The spontaneous electric polarization along the z-axis appears since the spiral magnetic structure break the spatial inversion symmetry. By applying the strong enough magnetic field along the y-axis, the direction of electric polarization will change from z-direction to x-direction. The strong magnetoelectric effect inspire scientists to find more new compounds and study the theoretical models. A microscopic approach [34] and a phenomenological approach [35] are proposed to describe the mechanism of the electric polarization. The former, which is the so-called inverse effect of (relativistic) DzyaloshinskiiMoriya interaction, gives a relation between the ferroelectricity and the arrangement of spins, P ∝ ri j × [Si × S j ],. 30. (4.1).

(46) Figure 4.4: Different types of spin structures relevant for TbMnO3 - (a) T N2 < T < T N1 . Sinusoidal spin density wave. Every spins point to the same direction but different magnitude of local moment. The magnitude of these moments vary periodically. This structure is centrosymmetic and consequently not ferroelectric. (b) T < TN2 . The cycloidal spiral magnetic structure with the wave vector Q = Qx and spins rotating in the xz-plane. This structure breaks the spatial inversion symmetry so that the spontaneous electric polarization appears. (c) In a so-called ”proper screw” the spins rotate in a plane perpendicular to Q. Here the inversion symmetry is broken, but most often it does not produce polarization.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. where ri j is the vector pointing from spins Si to S j . The magnitude of the polarization is proportional to the spin-orbit coupling constant. In a particular case, the spiral magnetic structure as shown in Figure 4.4(c) breaks the spatial inversion symmetry but appears no ferroelectricity most often, although in certain a l cases it might. And the later v also gives a relation obtained by i analysing the symmetries of the ferroelectricity and the magnetism of the system, n C. hengchi U. P ∝ [(M · ∇)M − M(∇ · M)].. (4.2). Since the ferroelectricity occurs relating to the magnetism, it is not surprised that the ferroelectricity is strongly affected by the magnetic field. Collinear Magnetic Structure In the second group, the collinear magnetic structure in the material strongly affects the ferroelectricity. All spin moments aligned along the same direction, and the spin-orbit coupling in the system is neglected. Since the strength of exchange striction of ferromagnetic and antiferromagnetic arrangements of two neighbor spins (↑↑ and ↓↓) are different, the crystal distortion may occur breaking the spatial inversion symmetry, i.e. lead to the spontaneous electric polarization. When the magnetic field is applied in the certain direction, direction of the electric polarization changed relating to the transformation of spin arrangement. Take Ca3 CoMnO6 for example, the crystal has the inversion symmetry at high temperature. However, below the crit31.

(47) Figure 4.5: Ising chain composed of Co2+ and Mn4+ - Ising chains with the up-up-down-down spin order and alternating ionic order, in which electric polarization is induced through symmetric exchange striction. The two possible magnetic configurations leading to the opposite polarizations are shown. The atomic positions in the undistorted chains are shown with dashed circles.. 政治大. ical temperature, Co2+ and Mn4+ ions alternating along the simplified Ising chains exhibit an up-up-down-down (↑↑↓↓) magnetic order. Two possible crystal distortion due to the exchange striction are shown in Figure 4.5. The two possible magnetic configurations leading to the opposite polarizations are shown. In this case, valences of the two transition metal ions are different (Co2+ and Mn4+ ), and thus lead to the different strength of exchange striction. Interestingly, the same effect occur even for same magnetic ions despite the fact that the exchange interaction in transition metal oxides usually occurs via intermediate oxygens and depends on both the distance between the metal ions and the metal-oxygen-metal bond angle.. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 32. i n U. v.

(48) Chapter 5 Calculated Physical Properties of Cu3Mo2O9 Introduction. 立. 學. ‧ 國. 5.1. 政治大. ‧. The magnetoelectric multiferroics [36] is the material which involve the coexistance of magnetic and electric order with cross coupling. It means that we can control the electric order in the system by applying a magnetic field and vice versa, i.e., the so-called magnetoelectric effect [20]. Such materials provide opportunities for devising the multifunctional electronic devices in the future. However, it has been considered that strong enough coupling between magnetic and electric order is impossible in 1960s, and hence such interesting field has been forgotten for a while. Since the precursory work on TbMnO3 [24], multiferroics attracted much attention once al v again. Considerable effort has been C hdevoted to searchUforn inew compounds with strong magnee n gmaterial toelectric coupling. Recently, an interesting c h i Cu3Mo2O9 has been found by Kuroe et al. [37] that it has remarkable magnetoelectric effect being as strong as TbMnO3 . What is the most special is that the origin of the electricity of this material probably be a novel mechanism. A small spin cluster model has been suggest by Kuroe et al. which is very different from the magnetic superlattice. Such interesting system motivated us to have some theoretical simulations to study it. Cu3 Mo2 O9 was prepared as a brown-orange powder by Vilminot et al. [38], and the space group is orthorhombic Pnma measured using X-ray powder diffraction. There are three crystallographically inequivalent Cu sites (Cu1, Cu2 and Cu3) as shown in Figure 5.1 [39]. The quasi-one-dimensional chains formed by Cu4 tetrahedra are connected by bridging MoO4 which is not presented in the figure for simplifying, and each two neighboring Cu atoms are bridged by an oxygen ion. Four formular unit (f.u.) are included in a primitive unit cell. The magnetism is dominated by Cu ions, each has spin 21 , while other ions are nonmagnetic. Since each Cu chain is a segment of the pyrochlore structure, geometric frustration possibly plays an important role in this system, i.e. the system has degenerate ground states. Therefore, the magnetic struc-. n. er. io. sit. y. Nat. 33.