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Bloch-Grüneisen temperature and universal scaling of normalized resistivity in doped graphene revisited

In this work, we resolved some controversial issues on the Bloch-Grüneisen (BG) temperature in doped graphene via analytical and numerical calculations based on full inelastic electron-acoustic-phonon (EAP) scattering rate and various approximation schemes. Analytic results for BG temperature obtained by semi-inelastic (SI) approximation (which gives scattering rates in excellent agreement with the full inelastic scattering rates) are compared with those obtained by quasi-elastic (QE) approximation and the commonly adopted value of $Θ^{LA}_{F} = 2\hbar v_{LA} k_F/k_B$. It is found that the commonly adopted BG temperature in graphene ($Θ^{LA}_{F}$) is about 5 times larger than the value obtained by the QE approximation and about 2.5 times larger than that by the SI approximation, when using the crossing-point temperature where low-temperature and high-temperature limits of the resistivity meet. The corrected analytic relation based on SI approximation agrees extremely well with the transition temperatures determined by fitting the the low- and high-$T$ behavior of available experimental data of graphene's resistivity. We also introduce a way to determine the BG temperature including the full inelastic EAP scattering rate and the deviation of electron energy from the chemical potential ($μ$) numerically by finding the maximum of $\partial ρ(μ,T)/\partial T$. Using the analytic expression of $Θ_{BG,1}$ we can prove that the normalized resistivity defined as $R_{1}=ρ(μ,T)/ρ(μ,Θ_{BG,1})$ plotted as a function of $(T/Θ_{BG,1})$ is independent of the carrier density. Applying our results to previous experimental data extracted shows a universal scaling behavior, which is different from previous studies.