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
Evolution of an eigen state
B
E
- Notice the edge states that don't exist for calculations infinite N
Hexagonal Lattice
Same method – “Tight Binding”, putting in a matrix… but look what happens now !
ˆ
ˆ
B Bz
A Bxy
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
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