1. Electronic Band Structure of Solids Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html

5. What are the energy levels? Sommerfeld: Bloch: For a given band index n, has no simple explicit form. The only general property is periodicity in the reciprocal space:

6. What is the velocity of electron? Sommerfeld: The mean velocity of an electron in a level with wave vector is: NOTE: Quantum mechanical definition of a mean velocity Bloch: The mean velocity of an electron in a level with band index and wave vector is: Conductivity of a perfect crystal:

7. What is the Wave function Sommerfeld: The wave function of an electron with wave vector is: Bloch: The wave function of an electron with band index and wave vector is: where the function has no simple explicit form. The only general property is its periodicity in the direct lattice (i.e., real space):

8. Sommerfeld vs. Bloch : Density of States Sommerfeld -> Bloch

9. Bloch : van Hove singularities in the DOS

10. Bloch : van Hove singularities in the DOS of Tight-Binding Hamiltonian

12. Sommerfeld vs. Bloch : Fermi surface in 3D Sommerfeld: Fermi Sphere Bloch: Sometimes sphere, but more likely anything else For each partially filled band there will be a surface reciprocal space separating occupied from the unoccupied levels -> the set of all such surfaces is known as the Fermi surface and represents the generalization to Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands are branches of the Fermi surface : for each n solve the equation in variable.

14. DOS of real materials: Silicon, Aluminum, Silver

15. Colloquial Semiconductor “Terminology” in Pictures ← PURE DOPPED ->

16. Measuring DOS: Photoemission spectroscopy Fermi Golden Rule: Probability per unit time of an electron being ejected is proportional to the DOS of occupied electronic states times the probability (Fermi function) that the state is occupied:

17. Measuring DOS: Photoemission spectroscopy Once the background is subtracted off, the subtracted data is proportional to electronic density of states convolved with a Fermi functions. We can also learn about DOS above the Fermi surface using Inverse Photoemission where electron beam is focused on the surface and the outgoing flux of photons is measured.

18. Fourier analysis of systems living on periodic lattice

19. Fouirer analysis of Schr ö dinger equation Potential acts to couple with its reciprocal space translation and the problem decouples into N independent problems for each within the first BZ.

22. Schr ödinger equation for “free” Bloch electrons Counting of Quantum States: Extended Zone Scheme: Fix (i.e., the BZ) and then count vectors within the region corresponding to that zone. Reduced Zone Scheme: Fix in any zone and then, by changing , count all equivalent states in all BZ.

23. “ Free” Bloch electrons at BZ boundary

27. Fermi surface in 2D for free Sommerfeld electrons

29. Fermi surface is orthogonal to the BZ boundary

30. Tight-binding approximation -> Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core electron dynamics and treat only valence orbitals localized in ionic core potential. There is another way to generate band gaps in the electronic DOS -> they naturally emerge when perturbing around the atomic limit . As we bring more atoms together or bring the atoms in the lattice closer together, bands form from mixing of the orbital states. If the band broadening is small enough, gaps remain between the bands.

31. Constructing Bloch functions from atomic orbitals

32. From localized orbitals to wave functions overlap

33. Tight-binding method for single s-band -> Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core electron dynamics and treat only valence orbitals localized in ionic core potential.

34. One-dimensional case -> Assuming that only nearest neighbor orbitals overlap:

35. One-dimensional examples: s-orbital band vs. p-orbital band

36. Wannier Functions -> It would be advantageous to have at our disposal localized wave functions with vanishing overlap : Construct Wannier functions as a Fourier transform of Bloch wave functions!

37. Wannier functions as orthormal basis set 1D example: decay as power law, so it is not completely localized!

39. Chemistry of Graphite: hybridization, covalent bonds, and all of that

41. Diagonalize 2 x 2 Hamiltonian

42. Band structure plotting: Irreducible BZ

44. Graphite band structure in pictures: Pseudo-Potential Plane Wave Method Electronic Charge Density: In the plane of atoms In the plane perpendicular to atoms

45. Diamond vs. Graphite: Insulator vs. Semimetal

49. CNT Band structure