1. 1 Crystal = Lattice + Basis a1a1 a2a2 a1a1 a2a2 1,2: Primitive unit vectors and cell 1 2 3: Not a primitive one (Conventional one)  Primitive unit cell: not unique  Multiple ways to draw them Wigner-Seitz unit cell a1'a1' a2'a2'  Mathematical concept: Voronoi cell  spans the entire direct space without leaving any gaps or holes-Tesselation  Brillouin zone: reciprocal space (Mainly)

2. Reciprocal Lattice 2  Time dependent periodic function  Crystals have spatial periodicity 1D case (For example potential energy)

3. Reciprocal lattice (RL)  3d case: For example, Charge distribution, n(r)  With any translational lattice vector, R it should be n(r+R)=n(r) 3 Fourier space lattice vector One point in reciprocal space  Very abstract concept  RL can easily mapped out with diffraction experiment Required condition

4. Brag condition 4  Periodic structure X-ray or Electron wave (TEM) Detectable condition:

5. 5 D S  Each atom scatters the incident wave Output at detector Periodic (3d) Expansion of periodic function in fourier space Whole crystal

6. Electron energy states in crystals 6  Energy levels of single atoms: Typically discrete  Solids: Wave functions overlap and form new wave function and energy levels One-Dimensional periodic potential: (Periodic arrangement of atoms) U a+b UoUo -b 0 a a+b E = Energy of electrons (0,a) (-b,0) BC: Continuity(x=0): A+B=C+D Derivative (x=0): iK(A-B)=Q(C-D)

7. More BC 7 Wave vector We got four homogeneous equations

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9. Born-von Karman Boundary condition 9  Deals with end points of crystal Momentum conservation law

10. One electron inside a periodic potential  Electron does not belong to individual atom  Wave function extends over the whole crystal  Electron mean free path, d can be as high as thousands of angstroms  Number of atoms in ~d 3 is 10 6 ~10 8  Electron can zigzag through these atoms  Electron gas model 10

11. Absorption coefficient of Graphene 11 X direction Y direction Z direction ω(eV)

12. Thank you 12

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