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 40 rr A Assume a solution of the form. kexp 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 kexp 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 arH MO a r ta 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 Hk 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 Eca 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 Eca 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.