Electron single flexural phonon relaxation, energy loss and thermopower in single and bilayer graphene in the Bloch–Gruneisen regime

Abstract

The flexural phonons serve as one of the important modes of interaction in graphene that can inhibit carrier mobility. For the estimation of scattering due to flexural phonons a two-phonon scattering process had been in place, as due to symmetry constraints out-of-plane deformations modulate electron hopping only in the second order. But recently it has been shown that electrostatic gating can break the planar mirror symmetry and activate single flexural phonon scattering processes (Gunst et al 2017 Phys. Rev. Lett. 118 046601). Motivated by this we perform single flexural phonon mechanism based analytical and numerical calculations of the electron phonon relaxation rate, energy loss rate and thermopower in single and bilayer graphene and obtain the power exponents of these quantities in the Bloch Gruneisen regime using the non-equilibrium Boltzmann transport equation. We find that the scattering by flexural phonons substantially changes the temperature dependencies from that observed due to in-plane phonons but the carrier concentration dependencies remain the same as of the in-plane phonons for all the three investigated quantities.

Energy Relaxation and Cooling in Impure Bilayer Graphene at Low Temperatures

“…where ρ and t are the coordinates and center-of-mass time (for the Wigner transformation), respectively, and ∈ ijk is the 3D antisymmetric tensor. From Equation (12), one gets that the collision integral is the combination of charge density distribution function…”

“…This leads to the observation of hot electron phenomena, one of the most vibrant areas in graphene research that has been intensively explored experimentally and theoretically. [3][4][5][6][7][8][9][10][11][12] Its significance springs from the fact that the thermal link between electrons and phonons in graphene is relatively weak, and this makes graphene an improved potential candidate for use in high-speed devices, [6] calorimetry, and bolometry. [1,2,9] For equilibration, the hot electrons begin to lose their energy through some cooling pathways of which the electronphonon (el-ph) interaction forms the principal channel of relaxation, as shown in Figure 1a.…”

“…[5][6][7][8][9] The energy loss can occur through the emission of several types of phonons depending on the temperature range. Longitudinal and transverse types of acoustic, [8,13] optic, [14][15][16] substrate surface, [17,18] piezoelectric, [19,20] and flexural phononic modes can appear as modes of el-ph scattering [12] under different environments (suspended, substrated, etc.) and different transport regimes (Bloch-Gruneisen, Equipartition, etc.).…”

“…This leads to the observation of hot electron phenomena, one of the most vibrant areas in graphene research that has been intensively explored experimentally and theoretically. [ 3–12 ] Its significance springs from the fact that the thermal link between electrons and phonons in graphene is relatively weak, and this makes graphene an improved potential candidate for use in high‐speed devices, [ 6 ] calorimetry, and bolometry. [ 1,2,9 ]…”

“…At low temperatures, acoustic phonons dominate other phononic modes and provide the main mechanism of electronic cooling. In the low‐temperature regime due to the unscreened deformation potential (DP) coupling, the temperature dependence of heat flux (cooling power) in single‐layer graphene (SLG) changes the exponent as a cube of temperature () for disorder limit as well as disorder‐assisted el–ph “supercollision” [ 10,21,22 ] and for clean limit, [ 9,12,23 ] and also the temperature‐dependent el–ph relaxation rate varies with T 2 in pure limit [ 8 ] and linear‐dependent ( T ) in impure limit [ 14 ] as well as disorder‐assisted el–ph “supercollision” in low‐temperature range at the Fermi surface in SLG. [ 10 ] A quantitative agreement has been achieved between calculated and measured values.…”