Band structure Tight Binding Plane wave method bcc Muffin tin potentials, pseudopotentials

 Ze2 U( r )  inside a radius R and is constant outside 4 0r

Si Bachelor thesis Benedikt Tschofenig

http://www.quantum-espresso.org/ Tight binding

Tight binding does not include electron-electron interactions

2 2 2    ZA e HMO   Vr( )      2me 2 m eA 40 rr A Assume a solution of the form.          kexp ilka 1 mka 2 nka 3  c a  a  rla  1 ma 2 na 3  l,, m n a

atomic orbitals: choose the relevant valence orbitals

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tightbinding/tightbinding.php Tight binding

         kexp ilka 1 mka 2 nka 3  c a  a  rla  1 ma 2 na 3  l,, m n a

HMOk E kk 

aH MOk  E k  ak

     caaMOa H c maMOm  Hexp(( ihkajkalka 1 2 3 )) small terms nearest neighbors m

Ek c a a  a  small terms

There is one equation for each atomic orbital Tight binding, one atomic orbital

     caaMOa H c maMOm  Hexp(( ihkajkalka 1 2 3 )) small terms nearest neighbors m

Ek c a a  a  small terms

For only one atomic orbital in the sum over valence orbitals

     Eckaaa cH aaMOa  cH aaMOm  exp(( ihkajkalka 1   2   3 )) nearest neighbors m

one atomic orbital

  ik m Ek   te m       arH MO  a  r  ta rH MOa  r  m  Tight binding, simple cubic

  E  te ik m m ik a ik a E tee ikxx a  ik a  eey y  eeik zz a ik a 

 2t cos( kx a )  cos( k y a )  cos( k z a ) 

2  2 Effective mass m*   d2 E 2ta 2 dk 2 Narrow bands  high effective mass Density of states (simple cubic)

Calculate the energy for every allowed k in the Brillouin zone

E 2 t cos( kx a )  cos( k y a )  cos( k z a ) http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html E 2 t cos( kx a )  cos( k y a )  cos( k z a ) Tight binding, fcc

  E  te ik m m Density of states (fcc)

Calculate the energy for every allowed k in the Brillouin zone

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html Tight binding, fcc

Christian Gruber, 2008 Tight binding, fcc

http://www.phys.ufl.edu/fermisurface/ Graphene

C1 C2   3 1 C2 a C1c1 C2 a1  axˆ  ay ˆ 1 2 2 C1c1 C2c2 C1  3 1  a2  axˆ  ay ˆ C2 a 2 C1c1 C2 2 2 C1 C2

Two atoms per unit cell

Graphene has an unusual dispersion relation in the vicinity of the Fermi energy. 2 carbon atoms / unit cell

The standard guess for the wave function in the tight binding model is

         exp ijka  lka c  r  ja la c r  ja la k   1 2 a pz a 1 2  b p z b 1 2  j, l

For graphene, the valence orbitals are pz orbitals

Substitute this wave function into the Schrödinger equation b

H E  a k k 2 carbon atoms / unit cell

         exp ijka  lka c  r  ja la  c r  ja la  k   1 2 a pz a 1 2 b p z b 1 2  j, l

Hk E  k  Multiply by  * r and integrate pz a  

the orbital for the atom at j = 0, l = 0.

  ik m cHa a a cH b  a b  e  small terms m

  ik m  Eca a a  c b  a b  e  small terms  m 

1 0 m sums over the nearest neighbors 2 carbon atoms / unit cell

To get a second equation for ca and cb

 H E  * r Multiply k k by p z b   and integrate

the orbital for the atom at j = 0, l = 0.

  ik m cHa b a e cH b  b b  small terms m

  ik m  Eca b a ec  b  b b + small terms  m 

0 1 Write as a matrix equation Tight binding graphene

  ik  m  aHE a  a H b  e m  c a        0 ik  m c bHe a  b HE b   b  m 

m sums over the nearest neighbors.

There will be two eigen energies for every k.

N orbitals / unit cell results in N bands

 aH  a

t  a H  b