1. Solid State Physics Yuanxu Wang School of Physics and Electronics Henan University wyxhenu@gmail.com 双语教学示范课程 1

2. Page  2 Chapter 5 Band theory  It can distinguish the conductor, Insulator, and semiconductor  Analyze the features of bonding between the atoms in solid  Analyze the appearing of surface state Band theory —— The main theoretical foundation for researching electronic motion in solids Band theory —— Qualitatively clarified the universal characteristics of electrons motion in crystals

3. Page  3 (i) Rigid lattice approximation. The nuclei are taken as fixed at their equilibrium positions; the large difference between the masses of the electrons and of the nuclei is the basic justification for this. (ii) One-electron approximation in local form. In essence, the complicated many-body electron problem is simplified to a single one-electron problem with an appropriate local potential. A conceptual scheme for this is provided by the Hartree-Fock theory, with a local approximation to the exchange potential, or by the density functional theory (DFT). (iii) Relativistic effects are neglected. Whenever necessary, one should replace the Schrödinger equation (5.1) with the Dirac equation, all the techniques developed for the former equation being generalizable to the latter. Often one includes relativistic terms of interest by perturbation theory. Purpose: Solving the energy of electron in crystal (5.1)

4. Page  4 Band Engineering on TiO 2 : as a promising photocatalytic compound Why? Energy and environmental problem Photocatalytic: 1) production of hydrogen 2) degradation of organic and inorganic pollutants in water and air. Steps of photocatalytic reaction: 1)Carrier producing by irradiation of light in catalyst 2)Effectively separating of photogenerated holes and electrons 3)Suppressing recombination of photogenerated electrons and holes 4)Carrier transporting to surface of catalyst 5)Chemically reacting on surface

5. Page  5 TiO 2 ? Only absorption ultraviolet light due to its large band gap 3.2 eV 1)Increasing the absorption of visible light: decreasing the band gap by doping or codoping, such Cr-C codoped TiO 2 2)Suppressing recombination of electrons and holes: impurity state and impurity band 3)Effectively separating of photogenerated holes and electrons

6. Page  6 Band structure of TiO 2 and TiO 2 (Mo+C) Passivated ( Mo+C)-doped TiO 2 is a strong candidate for PEC hydrogen production through water splitting, because it reduces the band gap to the ideal visible-light region, but does not affect much of the position of CBM. Yanqin Gai et al. Phys. Rev. Lett. 102, 036402 Doping may introduce a impurity state which will become a carrier recombination center

7. Page  7 Thermoelectric: Zintl compound with one-dimensional chain Ca 5 Al 2 Sb 6 Band Structure

8. Page  8 —— Why the electron mean free path is far larger than the distance of the atoms in a crystal? ——Band theory provides the basis for analyzing the theory of semiconductor, which has promoted the development of semiconductor technology. —— With the development of computer technology, the study of band theory from development of qualitative universal to the complex band structure computation of specific material. Band theory is the one-electron approximation theory —— Considering each electron transport independently in the field with an equivalent potential.

9. Page  9 The one-electron approximation —— the earliest method for study of the atoms with multi electrons: the Hartree-Fock self-consistent field method. The starting point of band theory —— Electrons in solids are not bound to individual atoms, but rather movement in the entire solid ------ electrons sharing. The state of the sharing electrons —— If the atomic core in its equilibrium position, the influence of the atomic core deviate from the equilibrium position as perturbation. An ideal crystal —— the lattice with periodicity, and the equivalent lattice potential field has a periodicity.

10. Page  10 The electrons in crystal move in the equivalent potential field of lattice periodicity Wave function: Periodic lattice potential

11. Page  11 To deal with the motion of a single electron in a periodic potential in the one- dimensional crystal. The first simplification —— adiabatic approximation: The larger mass of ion than electron, which may lead to the slower motion speed of ion; if discuss the problems of electron, we think ion is fixed at the instantaneous position. The second simplification —— Using the Hartree-Fock self-consistent field method, many electrons’ problem is simplified into a single-electron problem, and each electron motion in the fixed ionic potential field and the average potential field of other electrons. The third simplification —— all ionic potential field and the average potential field of other electrons are the periodic potential field.

12. Page  12 Processing the motion of a single electron in three-dimension crystal motion in a periodic potential field Calculation of the energy eigenvalues —— Select a complete set with the form of the Bloch function, then the wavefunction of electron states in a crystal can be expanded according to this function set. Calculation of the electron wavefunction ——Determine the coefficients of the electron wave function expansion according to each eigenvalue, then obtain the specific wave function. —— The electronic wavefunction is put into the Schrödinger equation to determine the secular equation which can satisfy the coefficient of expansion, then we can obtain the energy eigenvalues by solving the secular equation ( 久期方程 ).

13. Page  13 OsN 2

14. Page  14 IrN 2

15. Page  15 PtN 2 Pyrite

16. Page  16

17. Page  17 Surface and bulk bands of BaZrO3 ZrO 2 -termiantedBulk