Bloch-Gr\"{u}neisen temperature and universal scaling of normalized resistivity in doped graphene revisited

TL;DR

This paper clarifies the Bloch-Grüneisen temperature in doped graphene using analytical and numerical methods, revealing a universal resistivity scaling and correcting previous estimates of the temperature.

Contribution

It introduces a corrected analytic relation for the BG temperature based on full inelastic scattering and demonstrates universal resistivity scaling in doped graphene.

Findings

The commonly used BG temperature is overestimated by a factor of 2.5 to 5.

The corrected BG temperature aligns well with experimental data.

Resistivity normalized by the BG temperature exhibits carrier density independence.

Abstract

In this work, we resolved some controversial issues on the Bloch-Gr\"{u}neisen (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 $\Theta^{LA}_{F} = 2\hbar v_{LA} k_F/k_B$. It is found that the commonly adopted BG temperature in graphene ($\Theta^{LA}_{F}$) 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 ($\mu$) numerically by finding the maximum of $\partial \rho(\mu,T)/\partial T$. Using the analytic expression of $\Theta_{BG,1}$ we can prove that the normalized resistivity defined as $R_{1}=\rho(\mu,T)/\rho(\mu,\Theta_{BG,1})$ plotted as a function of $(T/\Theta_{BG,1})$ 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.