#tight Binding Approximation To Electronic Bandstructure 2009

CALIFORNIA INSTITUTE OF TECHNOLOGY APh 114b Solid State Physics Lecture 5 H.A. Atwater Winter, 2009

Tight Binding Approximation to Electronic Bandstructure We now take a very different approach to energy band structure, relative to that seen in the nearly free electron approximation. We recognize that the low lying core levels of a free atom are strongly localized in space. When free atoms are brought together, the core electronic levels remain strongly localized, and thus we don’t anticipate large changes in the energy eigenstates as we go from atom to solid. Thus we might expect that the electronic wavefunction in the solid can be expressed as a superposition of atomic wavefunctions, or linear combination of atomic orbitals (LCAO) as it is often termed. Let’s begin by assuming that solutions for the atomic eigenstates and eigenfunctions, i.e., the solutions to H at r − R n  i r − R n   E i  i r − R n  are known. We then assume that these atoms form a solid where the atoms are arranged on a lattice with lattice vectors R n . The total potential of this Hamiltonian, which is the Hamiltonian of the solid expressed as a perturbed atomic Hamiltonian, 2 H solid  H at  H ′  −  ∇ 2  V at r − R n   H ′ r − R n  2m

The perturbation H ′ is in this case the difference between the atomic potential and the crystalline solid potential. For an atom localized at R n , H ′ r − R n  

∑ V at r − R m  m≠n

Of course, what we ultimately seek are solutions to the Schrodinger equation in a crystalline solid which are Bloch waves,  k r, so that H solid  k r  Er k r But multiplying by  ∗k and integrating yields Ek

  ∗k H solid  k d 3 r Ek    ∗k  k d 3 r

The variational principle states that if one uses instead of the true wavefunction  k , a trial wavefunction  k , then one obtains an approximate energy E ′ k that is always bigger than Ek. IN this case, we try as an approximation solution a linear combination of atomic eigenfunctions k ≃ k 

∑ a n  i r − R n  n

where the coefficient a n are chosen so as to construct a Bloch wave: a n  e ikR n This can be seen by noting  kG 

∑ e ikR

n

e iGR n  i r − R n    k

n

Now let’s find E ′ k

1

  ∗k H solid  k d 3 r E k    ∗k  k d 3 r ′

The denominator is:

  ∗k  k d 3 r  ∑ e ikR −R    ∗i r − R m  i r − R n d 3 r n

m

n,m

We now make our first physical approximation: we note that  i r − R n  only has significant value near R m ; thus we only retain terms with n  m, so

  ∗k  k d 3 r  ∑  ∗i r − R n  i r − R n d 3 r  N n

where N is the number of atoms in the solids. Since we already know the atomic eigenstate E i , we write Ek ≃ E ′ k  1 ∑ e ikR n −R m    ∗i r − R m E i  H ′ r − R n  i r − R n d 3 r N n,m

where in the sums on n, m to form H ′ we sum over nearest neighbors for simplicity, and as before, for the term in E i nearest neighbor overlap is neglected.

In a simple case where  i is a spherically symmetry s-orbital, we can write down the nearest neighbor tight binding band structure as Ek  E i − E ′io − E ′inn

∑

e ikr n −r m 

mnearest neighbors

where E ′io  −   ∗i r − R n H ′ r − R n  i r − R n d 3 r E ′inn  −   ∗i r − R n H ′ r − R n  i r − R n d 3 r Note that E ′io  0 since H ′ r − R n   0. Now let’s specialize to a simple cubic lattice with r n − r m  a, 0, 0 0, a, 0 0, 0, a

For the simple cubic lattice, Ek is then Ek ≃ E i − E ′io − 2E ′inn cos k x a  cos k y a  cos k z a