Surface states

Surface states are electronic states found at the surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the electronic band structure from the bulk material to the vacuum. In the weakened potential at the surface, new electronic states can be formed, so called surface states. Surface states are electronic states found at the surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the electronic band structure from the bulk material to the vacuum. In the weakened potential at the surface, new electronic states can be formed, so called surface states. As stated by Bloch's theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves Here u n k ( r ) {displaystyle u_{n{oldsymbol {k}}}({oldsymbol {r}})} is a function with the same periodicity as the crystal, n is the band index and k is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions. The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected. A simplified model of the crystal potential in one dimension can be sketched as shown in Figure 1. In the crystal, the potential has the periodicity, a, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in Figure 1 is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image charges and the formation of surface dipoles and it rather looks as indicated by the dashed line. Given the potential in Figure 1, it can be shown that the one-dimensional single-electron Schrödinger equation gives two qualitatively different types of solutions. The first type of solution can be obtained for both metals and semiconductors. In semiconductors though, the associated eigenenergies have to belong to one of the allowed energy bands. The second type of solution exists in forbidden energy gap of semiconductors as well as in local gaps of the projected band structure of metals. It can be shown that the energies of these states all lie within the band gap. As a consequence, in the crystal these states are characterized by an imaginary wavenumber leading to an exponential decay into the bulk. In the discussion of surface states, one generally distinguishes between Shockley states and Tamm states, named after the American physicist William Shockley and the Russian physicist Igor Tamm. However, there is no real physical distinction between the two terms, only the mathematical approach in describing surface states is different. All materials can be classified by a single number, a topological invariant; this is constructed out of the bulk electronic wave functions, which are integrated in over the Brillouin zone, in a similar way that the genus is calculated in geometric topology. In certain materials the topological invariant can be changed when certain bulk energy bands invert due to strong spin-orbital coupling. At the interface between an insulator with non-trivial topology, a so-called topological insulator, and one with a trivial topology, the interface must become metallic. More over, the surface state must have linear Dirac-like dispersion with a crossing point which is protected by time reversal symmetry. Such a state is predicted to be robust under disorder, and therefore cannot be easily localized. SEE http://rmp.aps.org/abstract/RMP/v82/i4/p3045_1