Download Solid State Physics: Exploring Band Gaps from Free Electron Model to Real Space and more Study notes Chemistry in PDF only on Docsity! seshadri@mrl.ucsb.edu MATERIALS 218/CHEMISTRY 227 ELECTRONIC II MATRL 218/CHEM 227: Class IX — More on electronic structure Ram Seshadri (seshadri@mrl.ucsb.edu) • In traditional solid state physics treatments (Kittel), we are taught that band gaps arise due to transla- tional periodicity, that there is Bragg reflection of free electrons at the edges of the Brillouin zone and this open up gaps. In 1D, the free electron wavefunction is: ψk(x) = exp(ik · x) These are plane wave solutions to the free electron Schrödinger equation. The energy and momentum are given by: ek = h̄2 2m k2; p = h̄k Instead of a free electron in 1D, consider a 1D lattice with lattice constant a. The Bragg condition for diffraction by waves of wavevector k is (k + G)2 = k2 where G is the reciprocal lattice vector, and G = 2πn/a for the 1D lattice where n is some integer. Therefore, the Bragg condition solves to k = ±1 2 G = ±nπ/a The first reflections and therefore, the first energy gaps occur at k = ±π/a. kk ε ε ∆ π/a−π/a On the left is a sketch of the free electron wave function and on the right is a sketch of the effect of imposing a lattice on the free electron wave function with lattice parameter a.1 At the k points k = ±π/a, an energy gap ∆ opens up. The second allowed band starts after the first one gives up. The second band is in the second Brillouin zone. • Such a description is limited in its applicability. What about a defect, a surface or an amorphous material that would not have translational periodicity, and therefore would not have Bragg reflection of electrons ? We know that such materials do have band gaps (window glass !) The solution is to look at real-space pictures and return to tight-binding models. • For the real space description, consider two energy levels, on two different orbitals A and B. When the orbitals are far apart, the energy levels are at the atomic limit. When they approach, bonding and 1This is referred to as the nearly free electron model. Note its resemblance to the kinds of dispersion relations we derived for s orbital bands. 1 seshadri@mrl.ucsb.edu MATERIALS 218/CHEMISTRY 227 ELECTRONIC II antibonding combinations form, splitting the two levels. The bonding energy level is the bottom of that particular DOS, and the antibonding level, the top. inverse distance en er gy A B antibonding bonding antibonding bonding Notice how the levels broaden. Somewhere in-between, there is some dispersion of the individual states, but there is still a gap. When the atoms approach very close, the gap vanishes. Such a picture is physically quite realistic. Many semiconductors (which have a gap) become metallic on being subject to hydrostatic pressure. • When states in a crystal are filled up, the rules are the same as what is required for filling up atomic orbital states. Start with the lowest energies, and pay heed to Pauli’s exclusion principle. The exclusion principle says that no band can have more than two electrons. EF metalsemi−metalinsulator Electrons in filled bands do not carry a current because they cannot move without violating Pauli’s ex- clusion principle. If they do move, their motion must be compensated by the motion of a hole in the same direction, or of an electron in the opposite direction. • The Peierls distortion: 1D Lattices with 1/2-filled (and indeed, 1/n filled bands) susceptible to distort in a special way that permits opening of a gap at the Fermi energy. This is called the Peierls distortion. Polyacetylene is a typical 1D system that undergoes such a distortion, as first suggested by Salem and Longuet-Higgins. The simple, tight-binding picture of such a gap opening up can be obtained by considering crystal or- bitals formed from a 1D lattice of s orbitals. At the center of the band, the number of bonds equals the number of antibonds. Two kinds of crystal orbitals can be envisioned, which are degenerate in energy: 2
