Asymmetric Electron Energy Loss in Drift-Current Biased Graphene

Abstract

The electric drift-current bias was recently introduced as a new paradigm to break the Lorentz reciprocity in graphene. Here, we study the impact of the nonreciprocal response in the energy extracted from a beam of swift charges traveling in the vicinity of a graphene sheet with drifting electrons. It is demonstrated that the drift bias leads to an asymmetric electron-energy-loss spectrum that depends on the sign of the charge velocity. It is found that when the drift and electron beam velocities have comparable values but opposite signs, the energy loss is boosted resulting in a noncontact friction-type effect. In contrast, when the drift and electron beam velocities have the same sign, the energy loss is negligible. Furthermore, it is shown that different theoretical models of the drift-biased graphene conductivity yield distinct peaks for the energy-loss spectrum, and thereby electron beam spectroscopy can be used to test the validity of the different theories.

Appendix

The function G(z) in Eq. (6c) is given by:

$$ G(z)=z\;\mathrm{sq}\left(z-1\right)\mathrm{sq}\left(z+1\right)-\left[\ln \left(\left(z+\mathrm{sq}\left(z-1\right)\mathrm{sq}\left(z+1\right)\right){e}^{i{\theta}_0}\right)-i{\theta}_0\right] $$

(10)

where ln represents the standard logarithm function with a branch cut in the negative real axis and θ₀ =  − π/4. The function sq(ω) with ω = ω'  + iω'' is determined by

$$ \mathrm{sq}\left(\omega \right)=\Big\{{\displaystyle \begin{array}{c}-\sqrt{\omega },\kern1.25em \mathrm{if}\kern0.5em {\omega}^{\hbox{'}}<0\kern0.15em \mathrm{and}\kern0.15em {\omega}^{\hbox{'}\hbox{'}}<0\\ {}\sqrt{\omega },\kern2em \mathrm{otherwise}\end{array}}\operatorname{} $$

(11)

where √ is the standard square root function with a branch cut in the negative real axis.