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### Inelastic electron tunneling spectroscopy

1. Introduction.-

Besides scanning, the STM can also be used to measure the differential conductance over selected points on a surface. This can be done at different vacuum gaps. Indeed, the tip can be approach as much as to press the surface and produce conductance measurements in contact. The STM is a very versatile measuring instrument that can give a lot of information on the atomic arrangement of unknown structures, their electronic properties and their conduction ones. The differential conductance, dI/dV (where I is the electronic current between the tip and the substrate, and V the applied bias), contains a lot of information.

When there is a vacuum gap between tip and substrate, the electrons form a tunneling current. In this conditions, the conductance is proportional to the spectral function. See the work by Tersoff and Hamann among others for some indications of how this can be obtained. The spectral function is also called the density of states, because it gives the number of states per unit energy. The idea is that the bias is related to energy through the charge of the electron, and the differential conductance is some sort of density of current per unit energy. In the Tersoff-Hamann picture, the changes in current are in part due to the changes of states in the sample.

The differential conductance is then proportional to the density of states. However the STM has the ability to focus the current into an area well below atomic dimensions. The density of states takes into account this by becoming the local density of states (LDOS). This is a density of states weighed by the spatial distribution of the wavefunctions corresponding to the states at the energy matching the bias. The spectral function is a generalization of the density of states. For the STM it is the local density of states.

And it contains all possible types of electronic states. Particularly, excitations of the system. The spectral function is mathematically defined by the function that describe how one electron propagates into a system of electrons in interaction with all possible degrees of freedom: vibrational, magnetic, etc. The conductance of the STM gives direct information of the system states that can be excited by an injected electron. The STM can then measure excitations. This gives rise to the Inelastic Electron Tunneling Spectroscopy (IETS) using the STM.

IETS was developed in the 60's for planar tunneling junctions using oxides or other insulators, in a thin film, between two metals (or superconductors). Dramatic signals of vibrational modes of impurities in the insulator were detected.

2. Theory of IETS.-

With the above physical picture in mind, we can then generalized the Tersoff and Hamann approximation to its many-body variant where all possible excitations of the system are included. To do this we just need to use the generalization of the LDOS. For that, the easiest way to go is to use the one-electron propagator in the system. The imaginary part of the propagator is proportional to the LDOS when it is evaluated in real space.

The propagator or one-electorn Green's function evaluates the probability of creating a particle in a point r at a given time t and finds it in another point r' at a later time t'. This is a retarded propagator because it follows the time evolution. We can mathematically express it by:

This is a very complex equation that contains all possible physics of the transport problem of an electron into a complex system. But if you have some way of calculating it, then you can have a perfect IETS simulator.
The equation tells you that you inject an electron by creating a particle with the field operator:

Here the annihilation operator c, destroys a particle in state lambda. When you add up over all possible states, each with its spatial distribution given by the wavefunction of the state and its time-dependent phase, you have all information you need on the local and time properties of the electron. The above Green's function contains an anti-commutator and a theta function that take care of the correct time evolution.
There are a number of treaties and web documents where correlators, propagators and Green's function in general are reall well explain. Check the one  by Odashima, Prado and Vernek.
Despite their complexity, the advantage of the above formalism is that it includes everything and then you can choose your controlled approximation to evaluate it as realistically as possible.

In order to evaluate the LDOS with IETS information, you need to Fourier transform the above propagator (assume invariance under time translation, or a t-t' dependence) and in the spirit of the  Tersoff and Hamann approximation evaluate it in real space at the position of the STM tip.

Let us assume you would like to compute the IETS of a graphene sheet due to the excitation of phonons. There are awesome experimental data on this (i.e. the paper by the Crommie group, or the paper by the Stroscio group). And it is very interesting to have quantitative simulations that permit us to clearlyu ascribed the different physical processes at play. One way to go is to evaluate the electronic structure of graphene with DFT, also the phonons, and then evaluate the electron-phonon couplings. Nowadays this can be done using the software developed by the quantum espresso research effort. A particularly interesting tool is the one developed by the Giustino group. The EPW code allows you to compute the electron-phonon coupling with very large precision. You can easily use this to evaluate your Green's functions. Its imaginary part will be proportional to the IETS conductance.