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

Hofstadter Butterfly. Chene Tradunsky & Or Cohen with the great help of Ariel Amir. 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…. Square Lattice of Atoms.

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

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

  2. Using "Tight Binding" method we created a matrix representing the Hamiltonian for the entire lattice ( Size - N2*N2) After finding Eigen Values and Eigen States we got… Square Lattice of Atoms

  3. Energy Band in Various Magnitic Fields – Butterfly in Square Lattice E0+4t E E0 E0-4t B

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

  5. Evolution of an eigen state

  6. Classical Explanation for Edge States Magnetron Radius

  7. Quantum Equivalent for Edge States

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

  9. Hexagonal Butterfly E E0+4t E0 E0-4t 0.2 0.4 0.6 0.8 1.0 B

  10. Some physical explanationfor 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 E B

  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 E B

  14. Thank you all !

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