1. Chene Tradunsky & Or Cohen with the great help of Ariel Amir

2. Square Lattice of Atoms Using "Tight Binding" method we created a matrix representing the Hamiltonian for the entire lattice ( Size - N 2 *N 2 ) After finding Eigen Values and Eigen States we got…

3. Energy Band in Various Magnitic Fields – Butterfly in Square Lattice E B E0E0 E 0 -4t E 0 +4t

4. Evolution of an eigen state B E - Notice the edge states that don't exist for calculations infinite N

5. Evolution of an eigen state

6. Classical Explanation for Edge States Magnetron Radius

7. Quantum Equivalent for Edge States

8. Hexagonal Lattice Same method – “Tight Binding”, putting in a matrix… but look what happens now !

9. Hexagonal Butterfly E0E0 E 0 -4t E 0 +4t B 1.00.80.60.40.2 E

10. Some physical explanation for Low Magnetic Field Dispersion in square lattice (B=0) : Behaves like free particle in 2D with effective mass ! Free particle in homogenous magnetic field receives extra energy – Landau Levels :

11. Landau Levels In Square Lattice B E

12. What happens in hexagonal lattice ? Dispersion in square lattice (B=0) : For certain K behaves like relativistic particle : A correction to the energy can be calculated which is similar to the Landau Levels :

13. Energy Levels In Hexagonal Lattice B E