Figure 4: Phase space for two-body collisions in graphene.

(a) Allowed and (b) forbidden process in which two electrons with momenta k₁ and k₂ scatter into k₃ and k₄ (k′₃ and k′₄). The total momentum Q=k₁+k₂ is conserved in both panels. Intra-band (inter-band) scattering can be represented on an ellipse (hyperbola) with distance between foci equal to v|Q|, where v is the Fermi velocity, and the major axis equals the total energy E (E′). E=v(|k₁|+|k₂|) is conserved in a, which represents intra-band scattering. On the contrary, in b, E′=v(|k'₃|−|k'₄|) of the outgoing particles is smaller than E. This forbidden scattering process represents a collision between two incoming particles in the same band, and two outgoing particles in different bands, i.e. an Auger process, as for Fig. 3c. Energy conservation implies that Auger processes can only take place in the ‘degenerate limit’, i.e. when the vertices of the two confocal conical sections coincide with the foci. In this limit the ellipse and hyperbola collapse onto a segment and half-line, respectively, and all the momenta are collinear. (c) The 3d solid represents the allowed values of |Q| in units of E/(v), plotted as a function of energies ε₁ and ε₃ (both in units of total energy E) of an incoming and outgoing particle, respectively. Region I corresponds to intra-band processes; region II to inter-band processes; region III to Auger processes. No phase space is available for CM and Auger recombination for MDFs in 2d.