Modelling relativistic effects in momentum-resolved electron energy loss spectroscopy of graphene
Published in Radiation Effects and Defects in Solids, 2018
K. Lyon, D. J. Mowbray, Z. L. Miskovic
We adopt here an approach to this problem developed in (12), which gives analytical expressions for distributions describing the total energy loss of the external charged particle, , the so-called ohmic energy loss that is deposited in graphene, , and the radiative energy loss that is emitted in the form of transition radiation in the far field regions, . By using independent physical definitions for those three processes, it was shown explicitly in (12) that energy conservation is maintained in the form . It should be stressed that all those expressions take complex-valued, in-plane conductivity of graphene, , as an independent input function, which may be available in an analytic form from empirical models, or tabulated from ab initio calculations (6). Moreover, it is often sufficient to take the so-called optical limit (i.e.) by using the frequency-dependent conductivity of graphene, (9). This limit is expected to work very well for the experiment in (11) owing to a rather small range of the wavenumber transfers, , used in that work.