Correlation and the density-matrix approach to inelastic electron holography in solid state plasmas

Abstract

It has been argued that collective excitations in solid state plasmas are a good candidate to measure correlation lengths of electrons. The method of choice seems to be energy filtered electron holography in the transmission electron microscope (TEM). The interference fringes contain information on the partial coherence of those electrons and thus on their correlation length. Previous experiments showed surprisingly high coherence. We calculate the density-density correlation function in the Al plasma excitation from the dynamic form factor and compare it to other estimates. For the Al plasma excitation, we find a small extension of 0.1 − 0.3 nm for the range over which the movement of charges is correlated, depending on the definition of that range. Using the density matrix formalism, the coherence length of plasmon scattered fast electrons in the TEM is calculated and found to agree with experiment. It is shown that the small correlation length does not contradict the coherence length of > 10 nm found for electrons having excited a plasmon in Al. The difference of nearly 2 orders of magnitude can be traced back to the long range Coulomb interaction between probe and target electrons. Two unexpected predictions ensue from the present approach: Inelastic holography experiments should show strongly increased contrast of interfeence fringes in vacuo, i. e. outside the specimen, and contrast inversion of fringes in inelastic holograms of very small particles.

Inelastic electron holography – first results with surface plasmons

“…It appears that the contrast corresponding to the resonance increases with increasing distance to the edge, which is a similar behaviour as predicted in [10] and already measured in [7]. A very interesting explanation for that in a way counterintuitive result can be found in [21] in terms of an interpretation of the famous double slit experiment.…”

“…In case of dipole transitions, some reasonable approximations confirm the experimental results for bulk plasmons satisfactorily [10][11][12][13]. In these cases,…”

“…The interpretation is that there is an outgoing inelastic wave, which, contrary to elastic scattering, is now spatially restricted to a so-called interaction sphere of a certain inelastic scattering event ( fig.11). This sphere is the dominating factor contributing to the region of coherence in the inelastically scattered wave field, as shown in [10,11]. Furthermore it was shown that the shape of the coherence region is given by the delocalized Coulomb field of the scattered electron and also determined by the physical properties of the object such as correlations in quasi-particle excitations, like plasmons, polaritons or phonons.…”

“…This means that the holographically measured coherence in the outgoing wave field is here determined rather by the Coulomb delocalization than by the correlation within the collective excitation [10]. Since, at the end, we are interested in correlations within quasi-particle excitations, we need to either reduce the Coulomb delocalization by decreasing beam energy or look for excitations, which provide a correlation length comparable to the interaction sphere extension.…”

“…Fortunately, by means of a biprism it is possible to shift off-diagonal elements into the diagonal, which makes them measurable by means of Off-Axis Electron Holography [11]. In terms of the density matrix approach, the hologram intensity for a certain shear d may be written as follows [10]:…”