deletions | additions       diff --git a/Consider_the_types_of_waves__.tex b/Consider_the_types_of_waves__.tex  index 4817126..847fd32 100644  --- a/Consider_the_types_of_waves__.tex  +++ b/Consider_the_types_of_waves__.tex 
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Consider the types of waves that graphene can support, in hopes to integrate this into waveguides. Plasmons are high-frequency, collective density oscillations of an electron liquid and occur in many metals and semiconductors [3]. Surface plasmons are the collective oscillations of charges at the surface of plasmonic materials and SPs coupled with photons form composite particles of surface plasmon polaritons [4]. It is found that two types of electromagnetic surface waves can exist in graphene, TE and TM modes [4]. The TE mode should come as a surprise. It is know that SPPs can exist at an interface between a dielectric and a metal for TM modes but TE modes don't exist since they don't satisfy boundary conditions at the interface [5]. Visually this is shown in figure 2 for a TM mode at a metal dielectric interface.  It is predicted that TE modes exist in graphene, a mode that does not exist in systems with parabolic electron dispersion [6]. Further it is shown that TE modes lie in the range $1.667 <\frac{\hbar\omega}{\mu}<2$ [6]. TM modes are supported when $0