Like the free-electron-gas model, the tight-binding model belongs in the independent-electrons framework.
This model is basically the only time that we see (at the level of the Struttura della Materia course) the actual calculation of an electronic band structure, that takes explicitly into account the presence of the periodic lattice potential.
Contrary to the free-electron picture, the tight-binding model describes the electronic states starting from the limit of isolated-atom orbitals.
This simple model gives good quantitative results for bands derived from strongly localized atomic orbitals, which decay to essentially zero on a radius much smaller than the next neighbor half-distance in the solid.
For the more interesting bands, the conduction bands, the results of tight-binding are usually in rather poor agreement with experiment. As we shall see, tight binding could be systematically improved by including additional levels/bands, so that the accuracy of the calculated bands increases, at the expense of the simplicity and transparency of the model.
Here we shall apply the general formalism to the simplest case of an isolated s band.
The starting point of this model is the decomposition of the total single-electron Hamiltonian into:
H = Hat + ΔU(r) ,where Hat contains the kinetic energy plus the potential of a single ion (the one placed at R=0, say), and ΔU(r) is the potential generated by all the others ions in the lattice except for the one already considered. The eigenfunctions of the atomic problem satisfy the Schrödinger equation
H φn(r) = En φn(r) ,where n represents collectively a full set of (orbital) atomic quantum numbers. We then expand a generic state localized around the atom as
φ(r) = n bn φn(r)[notice that this state is not quite generic, as the continuum states of the atom are left out of the sum]. We than make a combination of such localized states, with the symmetry of the lattice:
ψ(r) = R exp(i k·R) φ(r-R)[Exercise: verify that this state has the Bloch property ψ(r+R) = exp(i k·R) ψ(r) ].
For fixed k, we plug this state into the Schrödinger equation:
H ψ(r) = [Hat + ΔU(r)] ψ(r) = E(k) ψ(r)This differential equation maps to a matrix equation by multiplication on the left by φm*(r) and integration over the r variable:
n A(k)mn bn = E(k) n B(k)mn bnThis, for each fixed k, is a generalized eigenvalue problem for the matrix A(k), with a "metric" matrix B in place of the identity. E(k) represents the eigenvalue, corresponding to the eigenvector b (of course, also b is k-dependent, in general). The "energy" matrix above is:
A(k)mn = R exp(i k·R) φm*(r) H φn(r-R) drThe "overlaps" matrix above is:
B(k)mn = R exp(i k·R) φm*(r) φn(r-R) drwhere the sums over R extend over all the lattice points. This is conventiently rearranged (using the decomposition H = Hat + ΔU(r), the fact that φm*(r) are (left) eigenstates of Hat and the orthonormality of φn(r) and φm(r) ) into:
n C(k)mn bm = (E(k) - Em) n B(k)mn bnwhere the new matrix C(k) contains what is left of the Hamiltonian, i.e.
C(k)mn= R exp(i k·R) φm*(r) ΔU(r) φn(r-R) drNow the eigenvalue (E(k) - Em) of the generalized secular problem measueres the displacement of the "band" energy E(k) with respect to the original atomic value Em.
The size of this matrix eigenvalue problem is clearly as large as the number of eigenstates of the atomic problem, i.e. infinite. It is therefore necessary to do some approximation here. In particular, one could hope that all the off-diagonal matrix elements of the matrices at the right side of this eq. could be neglected for some given level m. This cannot work for atomic degenerate levels (such as p, d, f... orbitals) where the couplings between the degenerate levels form the main part of the Hamiltonian, the one that resolves the degeneracy: for the case of d-degenerate levels, one has to solve at least a d-dimensional matrix problem (at each value of k).
The only case where it sometimes makes sense forgetting the interaction with all levels with nm is with atomic s (l=0) nondegenerate levels. In this approximation, the matrix equation becomes a 1×1 trivial problem, for which one takes bm=1 and all the other bnm=0, and only the "m" energy equation:
C(k)mm = (E(k) - Em) B(k)mmremains, with solution:
E(k) = Em + C(k)mm/B(k)mmNow, in the infinite R-sums of the definitions of the B(k) and C(k) matrices, it is convenient to separate the contribution from R=0, the contribution from R in the first shell around the origin (first or nearest neighbors), the contribution of the second shell around the origin (second neighbors), etc. The R=0 contribution to B(k) is 1, and that to C(k) is
χ = ΔU(r) |φm(r)|² drwhich is a negative quantity, reflecting the attraction that the "other" nuclei produce on the band electron, which was not there when the atom was isolated. We shall indicate the R0 contributions to B(k) and C(k) as
φm*(r) φm(r-R) dr
γ(R)= φm*(r) ΔU(r) φm(r-R) dr
E(k) = Em + [χ + R0 cos(k·R) γ(R)] / [1 + R0 cos(k·R) β(R)]This E(k) gives the tight-binding band structure in terms of a set of parameters β(R), χ and γ(R). We also have an explicit recipe to compute these parameters in terms of overlap integrals at different sites.
Due to the exponential decay of the atomic wave functions at large distance, both the overlap integrals β(R) and the energy integrals γ(R) become exponentially small for large distance R between the centers of the atoms. It therefore makes sense to ignore all the integrals outside some Rmax, which would bring in only negligible corrections to the bandstructure E(k). One may obtain a band structure depending on a minimal number of parameters by making further rather radical approximations:
E(k) = Em + χ + γ R(NN) cos(k·R)where γ indicates the value of γ(R) for the nearest neighbors. In this approximation, the band is determined by 2 parameters only: Em + χ, which tunes the band mean energy, and γ, which sets the band width.
[Further details for this model can be studied, for example, in the Ashcroft-Mermin textbook "Solid State Physics" (Chap. 10).]
However, for p bands one can see qualitatively that the overlapping tails of the wavefunctions φm(r) and φm(r-R) can have opposite sign (the p wavefunction changes sign across some plane passing in the nucleus), thus γ(R)>0 in that case. As a consequence, a p band shows typically a band maximum at k=0, with negative curvature of E(k). This does not apply to the π-band of benzene in the exercise above, as the nodal plane where φm(r) changes sign is parallel to the line joining neighboring atoms.
Another standard elementary technique is the perturbative method: the starting point of the free-electron parabolic dispersion is perturbed by a periodic potential, assumed to be "weak". A qualitative understanding of how the bands change and how gaps open can be gathered by a perturbative analysis. Exact bands are then obtained if the plane-waves basis is used in a full diagonalization approach as in current electronic-structure codes.
Another popular toy bandstructure scheme is the Kronig-Penney model.
The best thing about these methods is that they all give the same qualitative result: a band spectrum. This is not a random coincidence: it reflects the discrete translational invariance of the periodic potential, the symmetry enforcing Bloch's theorem.
Comments and debugging are welcome!
|created: 11 Jan 2002||last modified: 16 Jan 2012 by Nicola Manini|