Leading-Bloch-waves approximation

We start with the simplest case of a one-dimensional metal surface. Many of the basic concepts are demonstrated with this case. Then, we discuss several other types of surfaces that are frequently imaged by STM. 

Near the Fermi level, the surface wavefunctions in the vacuum region satisfy the Schrodinger equation in the vacuum  

The Bloch functions near the K points have a long decay length and contribute to the second term of Eq. (5.10). Following Eq. (5.12), in general, a surface Bloch function at that point has the form  

In addition to the term with n = 0, the term with n=-l have the same decay length, and thus have the same magnitude. Also, the Bloch function that generates the symmetric charge density must also be s metric. The lowest-order symmetric Fourier sum of the Bloch function near K is  

The charge density is proportional to 1 v jk I We then find the second constant in Eq. (5.10), 

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In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

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