Surface states

The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions.

A simplified model of the crystal potential in one dimension can be sketched as shown in Figure 1.

[3] 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.

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.

As a consequence, in the crystal these states are characterized by an imaginary wavenumber leading to an exponential decay into the bulk.

There is no strict physical distinction between the two types of states, but the qualitative character and the mathematical approach used in describing them is different.

In certain materials the topological invariant can be changed when certain bulk energy bands invert due to strong spin-orbital coupling.

More over, the surface state must have linear Dirac-like dispersion with a crossing point which is protected by time reversal symmetry.

[7] A simple model for the derivation of the basic properties of states at a metal surface is a semi-infinite periodic chain of identical atoms.

[1] In this model, the termination of the chain represents the surface, where the potential attains the value V0 of the vacuum in the form of a step function, figure 1.

The Shockley states are then found as solutions to the one-dimensional single electron Schrödinger equation with the periodic potential where l is an integer, and P is the normalization factor.

For z>0 the solution will be required to decrease exponentially into the vacuum The wave function for a state at a metal surface is qualitatively shown in figure 2.

It is an extended Bloch wave within the crystal with an exponentially decaying tail outside the surface.

The dipole perturbs the potential at the surface leading, for example, to a change of the metal work function.

The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors.

This leads to the following eigenvalues demonstrating the band splitting at the edges of the Brillouin zone, where the width of the forbidden gap is given by 2V.

The electronic wave functions deep inside the crystal, attributed to the different bands are given by Where C is a normalization constant.

It can be shown that the matching conditions can be fulfilled for every possible energy eigenvalue which lies in the allowed band.

As in the case for metals, this type of solution represents standing Bloch waves extending into the crystal which spill over into the vacuum at the surface.

The results for surface states of a monatomic linear chain can readily be generalized to the case of a three-dimensional crystal.

As a result, the surface states can be written as the product of a Bloch waves with k-values

Bulk energy bands that are being cut by these rods allow states that penetrate deep into the crystal.

These states exist in the forbidden energy gap only and are therefore localized at the surface, similar to the picture given in figure 3.

Such a state can propagate deep into the bulk, similar to Bloch waves, while retaining an enhanced amplitude close to the surface.

In the tight binding approach, the electronic wave functions are usually expressed as a linear combination of atomic orbitals (LCAO), see figure 5.

The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.

The energy levels of such states are expected to significantly shift from the bulk values.

[9] A naturally simple but fundamental question is how many surface states are in a band gap in a one-dimensional crystal of length

The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations.

Figure 1 . Simplified one-dimensional model of a periodic crystal potential terminating at an ideal surface. At the surface, the model potential jumps abruptly to the vacuum level (solid line). The dashed line represents a more realistic picture, where the potential reaches the vacuum level over some distance.
Figure 2 . Real part of the type of solution to the one-dimensional Schrödinger equation that corresponds to the bulk states. These states have Bloch character in the bulk, while decaying exponentially into the vacuum.
Figure 3 . Real part of the type of solution to the one-dimensional Schrödinger equation that corresponds to surface states. These states decay into both the vacuum and the bulk crystal and thus represent states localized at the crystal surface.
Figure 4 . Electronic band structure in the nearly free electron picture. Away from the Brillouin zone boundary the electron wave function has plane wave character and the dispersion relation is parabolic. At the Brillouin zone boundary the wave function is a standing wave composed of an incoming and a Bragg-reflected wave. This ultimately leads to the creation of a band gap.
Figure 5 . Atomic like orbitals of a Pt-atom. The orbitals shown are part of the double-zeta basis set used in density functional calculations. The orbitals are indexed according to the usual quantum numbers (n,l,m).