Chene Tradunsky & Or Cohen

with the great help of Ariel Amir

Square Lattice of Atoms

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…

ˆ

ˆ

B Bz

A Bxy

Energy Band in Various Magnitic Fields – Butterfly in Square Lattice

)(0

E

Be

h0

E0

E0-4t

E0+4t

Evolution of an eigen state

B

E

- Notice the edge states that don't exist for calculations infinite N

Evolution of an eigen state

Classical Explanation for Edge States

Magnetron Radius

Quantum Equivalent for Edge States

Hexagonal Lattice

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

ˆ

ˆ

B Bz

A Bxy

Hexagonal Butterfly

E0

E0-4t

E0+4t

)(0

Be

h0

1.00.80.60.40.2

E

Some physical explanationfor Low Magnetic Field

2

2

2tameff

222200)( 22)cos(2)cos(2 atkatkttEaktaktE yx

kyxk

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 :

)2

1(

2)2

1( n

cm

eBn

effL

Landau Levels In Square Lattice

B

E

What happens in hexagonal lattice ?

2E p

)3cos()cos(4)(cos41

)cos()3cos(4)2cos(2

2

0)(

akakak

akaktaktE

xyy

yxyk

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 :

nBE

Energy Levels In Hexagonal Lattice

B

E