Material Science I. p electron systems Kanoda II. d electron systems Fujimori
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1 Material Science I. p electron systems Kanoda II. d electron systems Fujimori
2 Download lecture note Download the lecture note prior to each class. Fujimori group home, Department of Physics, School of Science Courses Material Science URL: or 理 物理藤森研ホームページ 講義 物質科学 URL: NOTE: If you are not able to attend a class due to a conflicting official commitment, e.g., conferences, schools, business trips, machine times, etc., please inform Fujimori (fujimori@phys.s.u-tokyo.ac.jp) by . Your absence will be treated as an official leave.
3 References 1. 藤森淳 強相関物質の基礎 原子, 分子から固体へ ( 内田老鶴圃,2005 年 ) 2. N. Tsuda, K. Nasu, A. Fujimori, K. Siratori: Electronic Conduction in Oxides (Springer-Verlag, 2000) 津田惟雄, 那須圭一郎, 藤森淳, 白鳥紀一 電気伝導性酸化物 ( 改訂版 ) ( 裳華房,1993 年 ) 3. M. Imada, A. Fujimori, and Y. Tokura: Metal-Insulator Transitions, Review in Modern Physics 70, 1039 (1998) 4. 勝藤拓郎 基礎から学ぶ強相関電子系 ( 内田老鶴圃,2017 年 )
4 Electronic phase diagram of strongly correlated electron system at T = 0 K for n = 2, : spin Insulator diamagnetic : Electron chemical potential = Fermi level 2 electrons/atom in s level (e.g., He, Ne, Ar,..) T = 0 K
5 Electronic phase diagram of strongly correlated electron system at T = 0 K for n = 1, : spin Insulator diamagnetic : Electron chemical potential = Fermi level 1 electron/atom in s level T = 0 K
6 Electronic phase diagram of strongly correlated electron system at finite T, : spin 1 electron/atom in s level T = 0 K n 1
7 Electronic phase diagram of strongly correlated electron system for n > 1 many electrons/atom in p, d, or f level T = 0 K
8 Transition elements 1 st series: Incomplete 3d shell Shallow core levels Lanthanides Actinides KEK home page
9 Transition elements 2 nd series: Incomplete 4d shell Shallow core levels Lanthanides Actinides KEK home page
10 Transition elements 3 rd series: Incomplete 5d shell Shallow core levels Lanthanides Actinides KEK home page
11 Transition elements Shallow core levels Core levels Lanthanides Actinides Lanthanide series: Incomplete 4f shell KEK home page
12 Transition elements Shallow core levels Core levels Lanthanides Actinides Actinide series: Incomplete 5f shell KEK home page
13 Material Science II. d Electron systems 1. Electronic structure of transition-metal ions (June 12) 2. Crystal structure and band structure (June 19) 3. Mott insulators (June 26) 4. Metal-insulator transition (July 3) 5. High-temperature superconductivity (July 10) 6. Spin-related phenomena (July 17)
14 1.1 Atomic wave functions 1.2 Crystal-field splitting 1.3 Coulomb-exchange interaction 1.4 Multiplet splitting
15 1.1 Atomic wave functions Radial part Angular part (Spherical harmonics) Radial wave function of hydrogen atom Number of nodes = n l -1 (atomic units) 1 atomic unit = Bohr radius = nm
16 1.1 Atomic wave functions EuO, EuS Radial part Angular part (Spherical Wave functions harmonics) of O, S, and Eu Number of nodes = n l -1 Eu O/S O (2p 4 ) O 2- (2p 6 ) Eu (4f 7 6s 2 ) Eu 2+ (4f 7 ) Eu 1 atimic unit = Bohr radius = nm (atomic units) T. Kasuya, Kotai Butsuri 12,650 (1977)
17 1.1 Atomic wave functions s orbital p orbitals Radial part Angular part (Spherical harmonics) complex d orbitals
18 1.1 Atomic wave functions s orbital p orbitals Radial part Angular part (Spherical harmonics) real d orbitals
19 1.1 Atomic wave functions s orbital Spherical harmonics complex Cubic harmonics real p orbitals d orbitals
20 1.1 Atomic wave functions 1.2 Crystal-field splitting 1.3 Coulomb-exchange interaction 1.4 Multiplet splitting
21 1.2 Crystal-field splitting Giant magneto-resistive material (Perovskite structure) Magnetic semiconductor (Zinc Blende structure) Mn Ga, Cd As, Te Octahedral coordination Tetrahedral coordination
22 1.2 Crystal-field splitting Crystal fields from anion ligands (e.g., O 2-, F - ) Octahedral coordination Tetrahedral coordination : One-electron energy ( ): Degeneracy (inc. spin) 10Dq ~ ev
23 1.2 Crystal-field splitting Crystal fields from anion ligands (e.g., O 2-, F - ) Octahedral coordination Tetrahedral coordination Giant magneto-resistive material (Perovskite structure) Magnetic semiconductor (Zinc Blende structure) Mn Cu-oxide High-Tc superconductor (Layered perovskite str.) Fe-based superconductor Lowered symmetry
24 1.2 Crystal-field splitting Crystal fields from anion ligands (e.g., O 2-, F - ) Elongated octahedral coordination Elongated tetrahedral coordination (2) d x2-y2 d 3z2-r2 (4) d yz, d zx (2) (2) (4) d xy d yz, d zx (2) (2) (2) d xy d 3z2-r2 d x2-y2 : One-electron energy ( ): Degeneracy (inc. spin) 10Dq ~ ev
25 1.1 Atomic wave functions 1.2 Crystal-field splitting 1.3 Coulomb-exchange interaction 1.4 Multiplet splitting
26 1.3 Coulomb-exchange interaction Coulomb interaction Electron 1 Orbital part Spin part Electron 2 Orbital part Spin part X = (r s) r: Spatial coordinate s: Spin coordinate (=1,2) q =( ) : Orbital quantum number : Spin quantum number (=, )
27 1.3 Coulomb-exchange interaction Coulomb interaction Its expectation Hund s rule: value:u -J Spins want to align as much as allowed by Pauli principle. U U Coulomb integral Exchange integral ~ 4-8 ev ~ ev
28 1.3 Coulomb-exchange interaction Kanamori parameters (2 independent parameters) Coulomb integral Exchange integral ~ 4-8 ev ~ ev
29 1.1 Atomic wave functions 1.2 Crystal-field splitting 1.3 Coulomb-exchange interaction 1.4 Multiplet splitting
30 1.4 Multiplet splitting 4-electron system E: n-electron energy : one-electon energy J H U-U n = 4 3J H Kanamori parameters
31 1.4 Multiplet splitting Hund s rule: Spins want to align as much as allowed by Pauli principle. 5 ev K + Cu + Ca 2+ Sc 2+ Ti 2+ V 2+ Cr 2+ Mn 2+ Fe 2+ Co 2+ Ni 2+ Cu 2+ Zn 2+ Sc 3+ Ti 3+ V 3+ Cr 3+ Mn 3+ Fe 3+ Co 3+ Ni 3+ Cu 3+
32 1.4 Multiplet splitting Hund s rule: Spins want to align as much as allowed by Pauli principle. 5 ev K + Cu + Ca 2+ Sc 2+ Ti 2+ V 2+ Cr 2+ Mn 2+ Fe 2+ Co 2+ Ni 2+ Cu 2+ Zn 2+ Sc 3+ Ti 3+ V 3+ Cr 3+ Mn 3+ Fe 3+ Co 3+ Ni 3+ Cu 3+
33 1.4 Multiplet splitting 4-electron system 3J H -10Dq 3J H 10Dq n = 4 E: n-electron energy : one-electon energy
34 1.4 Multiplet splitting Ligand-field theory (Tanabe-Sugano diagram) Octahedral Mn 2+ (d 5 ) ion cm -1 B :Racah parameter~( )j H 6 = 2S+1 A 1 (spin multiplicity) High-spin ground state S=5/2 Low-spin ground state S=1/2 Kamimura, Sugano, Tanabe Ligand-field theory (Sho-ka-bo, 1969)
35 1.4 Multiplet splitting Ni O Optical absorption spectrum of octahedrally coordinated Ni 2+ (d 8 ) ion AIST, Japan E E(d 8 ) Optical absorption Ligand-field theory
36 1.4 Multiplet splitting Ligand-field theory explains: Optical spectra Color, Laser emission First Laser emission: Ruby (Cr 3+ in Al 2 O 3 ) R line Antiferromagnetic order of NiO T.H. Maiman et al., Phys. Rev. 123, 1151 (1961) Antiferromagnetism (T < T N ) Paramagnetism following the Curie-Weiss law (T > T N )
Material Science. I. p electron systems. Kanoda. II. d electron systems. Fujimori. Download lecture note
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