Tightbinding Method Molecular Physics Review! V = Matrix element of one electron Hamiltonian H between atomic orbitals on atoms A & B. Different V’s for different atomic orbitals (s, p etc.) & combinations. Detailed analysis shows that V’s depend on both atomic orbitals & on symmetry of resulting molecular orbital: V = Vij (i, j = atomic orbitals on A & B,  = resulting molecular orbital.

Tightbinding Method Real tetrahedral solid AB Do this same type of analysis. Assume nearest-neighbor interactions (V) only! Different V for every combination of s & p orbitals on atoms A & B & for every possible symmetry of molecular orbitals (, , etc.) In principle, can compute the V’s by doing integrals, using atomic orbitals In practice, treat as parameters to be fit to experiment!

Tightbinding Method Real tetrahedral solid AB By symmetry, for diamond structure materials (atom A = B & only s & p electrons): There are only 4 unique, non-zero, nearest-neighbor V’s: Vss  sA|H|sB = sB|H|sA Vsp  sA|H|pxB = sA|H|pyB = sA|H|pzB = pxA|H|sB = pyA|H|sB =pzA|H|sB Vpp  pyA|H|pxB Vpp  pyA|H|pyB = pzA|H|pzB Abbreviations: |sA |sA |pxA  |xA, etc. Note: if A  B (zincblende lattice) there are 8 V’s!

Tightbinding Method Real tetrahedral solid AB For diamond structure (Group IV) materials, 4 nearest-neighbor overlap interaction parameters: Vss , Vsp , Vpp , Vpp (zincblende materials: 8 V’s) Instead of calculating, treat as parameters fit to experiment. In tightbinding method, there are also diagonal matrix elements of H: sA|H|sA  sA = s-orbital “atomic energy” of atom A, etc. for p orbitals and atom B In practice, instead of calculating, or using experimental energies, treat as fitting parameters also. Sometimes, extend to 2nd & higher neighbors (with more parameters!).

Tightbinding Method Real tetrahedral solid AB Crystal structure effects: 4 nearest-neighbors  4 vectors di linking central atom: d1 = a(1,1,1)/4 , d2 = a(-1,-1,1)/4 d3 = a(-1,1,-1)/4, d4 = a(1,-1,-1)/4

Tightbinding Method Real tetrahedral solid AB From QM, atomic orbitals  Ylm (,) = spherical harmonic For example, s orbitals  Y00 (,) = constant p orbitals  Y1m ( ,) This must be considered, along with crystal geometry, in setting up tightbinding Hamiltonian for the solid!

Tightbinding Method Real tetrahedral solid AB Convenient to define: Vss  4Vss Vsp  - 4Vsp/(3)½ Vxx  4Vpp /3 + 8Vpp /3 Vxy  4Vpp /3 - 4Vpp These come from geometry of diamond lattice, plus 4 nearest neighbors. Instead of trying to calculate them from atomic functions, treat them, along with diagonal matrix elements of H (“atomic energies”) as fitting parameters.

LCAO Approximation Now, make tightbinding or LCAO approximation, just as in the model calculations discussed earlier (n = orbital label) H  R nHatn (R) + R,R nn Unn (R,R) Unn(R,R) = nearest-neighbor interaction Atomic solutions n(r), En known. k(r) = R n eikR bnn(r-R) (bn to be determined) Similar to models discussed, end up solving a k dependent determinant equation for bands.

Tightbinding Method Diamond lattice (zincblende has double number of parameters!) Consider  point: k = (0,0,0). g2 = g3 = g4 = 0; g1 =1 8 x 8 Hamiltonian matrix factors into four 2 x 2 matrices: one 2 x 2 matrix for s-like states three identical 2x2 matrices for p-like states s-like states: Es - E(0) Vss Vss Es -E(0) = 0  Es(0) = Es  |Vss| Es+(0) = Es + |Vss| (antibonding level = conduction band at  point) Es-(0) = Es - |Vss| (bonding level, below valence band top at  point)

Tightbinding Method Diamond lattice (zincblende has double number of parameters!) p-like states: Ep - E(0) Vxx Vxx Ep -E(0) = 0  Ep(0) = Ep  |Vxx| Ep+(0) = Ep + |Vxx| (antibonding level, above conduction band bottom at  point) Ep-(0) = Ep - |Vxx| (bonding level, top of valence band at  point) Bandgap at  point: Eg = Es - Ep + |Vss| - |Vpp| Note! This is the direct or optical gap, & not necessarily the fundamental gap!

Tightbinding Method Diamond lattice (zincblende has double number of parameters!) Harrison’s “Universal” tightbinding model See YC, Sect. 2.7.3 See also Harrison’s book! Fit bands from tightbinding approach to those of nearly free electron approach. For nearest-neighbor overlap energies get: Vl,l,m = l,l,m 2/(mod2) where: l,l = s, p, .., m = , , … d = nearest-neighbor distance l,l,m = a geometric factor, which depends only on lattice geometry.