In situ determination of tip DOS

The following method, based on the modified Bardeen approach, allows the electronic states of the tip to be characterized. If the energy scale of feature in the DOS is larger than knT, the tunneling current is 

From Eq. (14.17), we observe that for V 0, the electrons, from the occupied states in the tip, tunnel to the empty states of the sample. The tunneling current has no relation to the empty states of the tip. On the other hand, if V 0, the tunneling current is determined by the occupied stutes ot the sample and the empty states of the tip. Therefore, the determinations of the 

These equations are the standard form of a well-studied class of integral equations, the Volterra equation of the second kind (see, for example, Brunner and van der Houwen, 1986). Before discussing the numerical method, we draw a few simple conclusions from those equations. Using a free-electron-metal tip (that is, if in the entire energy range of interest). 

In this case, the dynamic conductance equals the sample DOS up to a constant factor. Now, we measure the tunneling spectrum on the same sample using another tip of unknown DOS, and find a new dynamic conductance as a function of bias voltage, g(V). By solving Fq. (14.21), we obtain the relative DOS of the unknown tip. [Similarly, if the sample is a free electron metal, that is, 

This method is known as the marching method. The accuracy of the procedure and the correctness of the program can be verified by testing it with analytically soluble Volterra equations, for example, the test problems with nonsingular convolution kernels listed on pp. 505-507 of Brunner and van der Houwen s book (1986). 

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