Tunneling an elementary model

In this section, we discuss the concept of tunneling through an elementary one-dimensional model. In classical mechanics, an electron with energy E moving in a potential U z) is described by 

Consider the case of a piecewise-constant potential, as shown in Fig. 1.3. In the classically allowed region, E U,Eq. (1.2) has solutions 

By applying a bias voltage V, a net tunneling current occurs. A sample state i ) with energy level E lying between Ef—eV and Ef has a chance to tunnel into the tip. We assume that the bias is much smaller than the value of the work function, that is, eV . Then the energy levels of all the sample states of interest are very close to the Fermi level, that is, E — cj). The probability w for an electron in the nth sample state to present at the tip surface, z = W, is 

In an STM experiment, the tip scans over the sample surface. During a scan, the condition of the tip usually does not vary. The electrons coming to the tip surface, z = VF, have a constant velocity to flow into the tip. The tunneling 

If V is small enough that the density of electronic states does not vary significantly within it, the sum in Eq. (1.10) can be conveniently written in terms of the local density of states (LDOS) at the Fermi level. At a location z and energy E, the LDOS ps(z, E) of the sample is defined as 

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