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

Bloch-Grüneisen temperature ($\Theta_{BG}$) in doped graphene has been extensively discussed in the literature [1-31]. $\Theta_{BG}$ is ususally defined as the crossing point in temperature between the low-temperature (LT) limit expression and high-temperature (HT) limit expression for resistivity as functions of temperature [1-10]. $\Theta_{BG}$ is an important characteristic temperature for designing graphene-based devices in applications such as optical detectors, bolometers Yan2012 ; Betz2013 and in cooling pathways and supercollisions Bistritzer2009 ; Betz2013 ; McKitterick2016 ; Song2012 ; Ma2014 ; Tikhonov2018 ; Kong2018 . If one considers separate contributions in $\Theta_{BG}$ due to longitudinal-acoustic (LA) and transverse-acoustic (TA) phonon scatterings, simple analytic expressions for $\Theta_{F}^{a}$ ($a=LA,TA$) can be obtained based on the LT- and HT-limit expressions for resistivity. The current general consensuses is that $k_{B}\Theta_{F}^{a}=2\hbar v_{a}k_{F}$ ($a=LA,\,TA$) through analyses based on quasi-elastic scattering approximation [1-27]. In the LT limit, it can be shown that $\rho^{LT}_{a}(\mu,T)\propto T^{4}$ and in the HT regime $\rho^{HT}_{a}(\mu,T)\propto T$Hwang2008 ; Efetov2010 .

Critical inconsistencies exist in Figs. 2b and 3b of Efetov2010 and in Fig. 2 of DSouza2017 . The crossing points found in both studies are $\sim$ 5 - 6 times smaller than the theoretical values predicted by $\Theta_{F}^{LA}$ above. To remedy this inconsistency an artificial scaling parameter of $\zeta=0.2$ was introduced in Efetov2010 to make the BG resistivity $\rho(\mu,\zeta\Theta_{F}^{LA})$ fall within the experimentally accessible range. Using the same definition of $\Theta_{F}^{a}$, it was shown that the BG transition occurs for $T<0.2\Theta_{F}^{LA}$ Yan2012 , $T\lesssim 0.15\Theta_{F}^{a}$ Sohier2014 , $T\lesssim 0.25\Theta_{F}^{a}$ McKitterick2016 ; Ansari2017 , and $1/\tau^{LT}_{a}(\mu,T)\propto T^{4}$ for $T\lesssim 0.2\Theta_{F}^{a}$ as implied in Ref. Mariani2010 . Our analysis based on quasi-elastic approximation also gives the same conclusion, i.e. $k_{B}\Theta_{BG,0}=2\hbar v_{LA}k_{F}/5$. More detailed analyses are given in supplemental material (SM)Nguyen2020SM .

Contrary to the above, the analyses in Refs.Song2012 ; Ma2014 ; Tikhonov2018 ; Kong2018 suggest that $k_{B}\Theta_{BG}=\hbar v_{LA}k_{F}$. Note that in Ref. Betz2013 $\Theta_{BG}=\Theta_{F}^{LA}=2\hbar v_{LA}k_{F}/k_{B}$ was used in the arguments and calculations but their results are compared not only with those from Refs. Bistritzer2009 ; Kubakaddi2009 ; Viljas2010 using the same $\Theta_{F}^{LA}$ but also with those from Ref. Song2012 using $\Theta_{BG}=\hbar v_{LA}k_{F}/k_{B}$. To resolve these controversies a more careful analysis of the BG temperature and a revisit of the universal scaling of the normalized resistivity $R(T/\Theta_{BG})$ in doped graphene is needed.

The resistivity in doped graphene can be calculated according toAshcroft1976 ; Nguyen2020

where $\sigma$ is the conductivity. In common practice, $-df(\epsilon_{k})/d\epsilon_{k}$ is approximated by $\delta(\epsilon_{k}-\mu)$ since $\tau(\epsilon_{k})|\epsilon_{k}|$ is slow varying over the range of $k_{B}T$. With this approximation we haveNguyen2020

where $D^{p}_{a}(\theta)\approx B^{2}T^{p}_{a}(\theta)$ (when $E_{1}$ is neglected) and $T^{p}_{a}(\theta)$ describes the angular dependence of the net electron-acoustic-phonon (EAP) scattering strength with $a=LA,TA$ and $p=\pm$ for phonon absorption or emission ($T^{p}_{a}(\theta)$ is depicted in Fig. 1 of Nguyen2020 ). $\hbar\Omega^{p}_{a}$ is the corresponding phonon energy. Throughout the paper, we only consider the $n$-doped case. Due to electron-hole symmetry in the Dirac Hamiltonian, the behavior of $p$-doped case will be identical. We use $v_{F}=1.0\times 10^{6}$ (m/s), $v_{LA}=2.0\times 10^{4}$ (m/s), $v_{TA}=1.3\times 10^{4}$ (m/s), $\rho_{m}=7.6\times 10^{-7}$ (Kg/m²) Nguyen2020 ; Kumaravadivel2019 , $g_{0}$ = 20 (eV) and $\beta=3$ Castro2010 ; Nguyen2020 .

Using the quasielastic (QE) approximation for scattering rates Hwang2008 ; Efetov2010 and setting $\rho_{QE}^{LT}(\mu,T)=\rho_{QE}^{HT}(\mu,T)$ gives Nguyen2020SM

where $\Theta_{F}^{LA}=2\hbar v_{LA}k_{F}/k_{B}$ is a characteristic temperature, which is often used as the BG temperature in the literature [1-27]. The more appropriate BG temperature within quasielastic approximation should be $\Theta_{BG,0}$, although it is still quite different from the results derived from the full calculation. The transferred acoustic phonon energy is determined by $\hbar\omega_{a}^{p}=\hbar v_{a}q_{a}^{p}\approx 2\hbar v_{a}k\sin(\theta/2)$ since $v_{a}/v_{F}\ll 1$ Nguyen2020 . Thus, we obtain $\left<\hbar\omega_{a}^{p}\right>_{p}=2\hbar v_{a}k_{F}\sin(\theta/2)$ for $\epsilon_{k}=\mu$. Therefore, $\max\left(\left<\hbar\omega_{a}^{p}\right>_{p}\right)=2\hbar v_{a}k_{F}=k_{B}\Theta^{a}_{F}$ has the physical meaning of the maximal transferred acoustic phonon energy.

In Refs. Fuhrer2010 ; Betz2013 it is suggested that $k_{B}\Theta_{BG}$ is close to the maximum transferred phonon energy.

Next, we consider a semi-inelastic (SI) approximation, which gives results very close to the full inelastic scattering calculation Nguyen2020 . In the SI approximation, the LT and HT limits of $\tau^{-1}(\mu,T)$ are given by

respectively, where $E_{1}=g_{0}/\epsilon(q)$ is the screened deformation potential for $LA$ phonon, $B={3\beta\gamma_{0}}/{4}$ denotes electron-phonon coupling strength due to unscreened gauge fields for both $LA$ and $TA$ phonons, and $\beta_{v}=v_{LA}/v_{TA}$ with $v_{LA}\,(v_{TA})$ being the sound velocity of $LA\,(TA)$ phonon. Since $E_{1}^{2}/B^{2}\ll 1$, it is a good approximation to neglect the $E_{1}$ contribution.

By setting $\rho_{SI}^{LT}(\mu,T)=\rho_{SI}^{HT}(\mu,T)$ we obtain Nguyen2020SM

($a=LA,TA$) denote separate contributions from deformation potential (labeled by $d$) and unscreened LA and TA phonon scatterings. And $b_{a}=\sqrt[3]{{\pi}/{6}}(\delta_{a,LA}+\beta_{v}\delta_{a,TA})\sqrt[3]{\frac{1+\beta_{v}^{2}}{1+\beta_{v}^{5}}}$.

Because the resistivity $\rho_{\mu}(\mu,T)$ is proportional to $T^{4}$ ($T$) in the low-$T$ (high-$T$) limit, $\Theta_{BG}$ should be near the maximum of $\partial\rho_{\mu}(\mu,T)/\partial T$. Using the semi-inelastic scattering rate at $\epsilon_{k}=\mu$, we haveNguyen2020

where $\Lambda=\frac{3v_{F}B^{2}}{\rho_{m}\hbar^{2}v_{LA}^{5}}$, $c_{0}=16.5,\,c_{1}=65.7$, and $\alpha_{a}=T/\Theta^{a}_{F}$ Nguyen2020 .

Using Eq. (7) we can solve the equation

analytically and get the BG temperature at the peak of $\partial\rho_{\mu}(T)/\partial T$ (See Sec. III in SM Nguyen2020SM for derivations )

with $\tilde{b}_{a}=(\delta_{a,LA}+\beta_{v}\delta_{a,TA})\sqrt[3]{\frac{3(1+\beta_{v}^{-1})}{c_{0}(1+\beta_{v}^{2})}}$. Note that the $E_{1}$ term has been neglected in Eq. (9a). We found

The theoretical results described by Eqs. (3), (4), (6), and (9) are plotted in Fig. 1 for comparison.

Above we have shown that for the special case of $\epsilon_{k}=\mu$, $\Theta_{BG,1}$ and $\Theta_{BG,2}$ obtained by two different approaches can differ by about 30%. For full considerations including contribution from all possible $\epsilon_{k}$’s, it is impossible to determine the temperature dependence of resistivity analytically. However, it is possible to calculate $\rho(\mu,T)$ according to Eq. (1) numerically with the full inelastic scattering rate without fixing $\epsilon_{k}$ at $\mu$. We then take the derivatives of the full $\rho(\mu,T)$ to determine $\Theta_{BG}$. The results are displayed by the red squares in Fig. 1 at five densities ranging from $13.6-108\times 10^{12}cm^{-2}$ corresponding to samples studied in Efetov2010 and we see that $\Theta_{BG}$ so determined falls between $\Theta_{BG,1}$ and $\Theta_{BG,2}$.

To compare with experimental results, we extract the experimental data of resistivities in n-doped graphene from Efetov2010 at five carrier densities and plot them as colored dots in Fig. 2 for $n=13.6\times 10^{12}\,cm^{-2}$ (black), $28.6\times 10^{12}\,cm^{-2}$ (red), $46.5\times 10^{12}\,cm^{-2}$ (green), $68.5\times 10^{12}\,cm^{-2}$ (blue), and $108\times 10^{12}\,cm^{-2}$ ( magenta). We can fit these data well by using Eq. (1) with the full inelastic scattering rates as solid curves in Fig. 2 which essentially go through the data points with slight deviation at the high-temperature end. To fit the data, a residual scattering rate beyond the acoustic phonon scattering mechanisms is added for a given $n$; that is, $1/\tau_{0}$ = 9.4, 7.9, 6.6, 5.95, and 5.5 $THz$ for the five respective carrier densities. The values of other constants adopted ($E_{1},B,\beta_{v}$) are the same as those used in Nguyen2020 . We can also fit these data at high- and low-$T$ limits by using Eq. (5) with semi-inelastic scattering rates as shown in dash-dotted and dashed curves, respectively. The crossing points between those curves determine $\Theta_{BG,1}$, which are shown as blue circles in Fig. 1 and they fall perfectly on the theoretical curve (blue solid). For comparison, $\Theta_{BG,0}$ determined by using the low- and high-$T$ quasielastic scattering rates Hwang2008 ; Efetov2010 as given in Eq. (3) as a function of density is shown as the black solid curve.

The total doping-dependent Bloch-Grüneisen temperatures $\Theta_{BG}$ (shown by the red squares in Fig. 1 at five densities) are determined from the peak values of $\partial\rho_{F}(n,T)/\partial T$ plotted in Fig. 3. Note that, $\rho_{F}(n,T)$ are obtained by using the full calculation described in Eq. (1) with the inelastic scattering rate plus a correction term ${1}/{\tau_{0}}$ which takes into account scattering mechanisms beyond the acoustic-phonon scattering. For comparison, we also show $\partial\rho_{\mu}(n,T)/\partial T$ as dash-dotted curves in Fig. 3. It should be noted that adding a constant correction term $1/\tau_{0}$ to the scattering rate will have no effect on $\partial\rho_{\mu}(n,T)/\partial T$. For a given $n$, $\Theta_{BG,2}$ shifts to the right of $\Theta_{BG}$ since the ratio of $\rho_{\mu}(n,T)$ to full $\rho(n,T)$ falls between $0.7$ and $1$Nguyen2020 . This comes from the fact that $-df(\epsilon)/d\epsilon$ can be approximated by $\delta(\epsilon-\mu)$ to obtain $\rho_{\mu}(n,T)$ only when $|\epsilon|\tau(\epsilon_{k})$ is slow varying over $k_{B}T$, which is not quite satisfied in graphene Nguyen2020 .

Finally, we consider the universal scaling of normalized resistivity by using the well justified SI approximation and the full inelastic scattering rate for $\epsilon_{k}=\mu$. The normalized resistivity is defined asZiman

In the SI approximation, $\rho(\mu,T)$ is given by Eq. (7) and we have

where $\alpha_{a,i}=\Theta_{BG,i}/\Theta^{a}_{F}$ with $i=1,2$.

From Eq. (5), we see that $\rho_{SI}^{LT}(\mu,\Theta_{BG,i})\propto\frac{(k_{B}\Theta_{BG,i})^{4}}{|\mu|^{3}}\propto|\mu|$ and similarly $\rho_{SI}^{HT}(\mu,\Theta_{BG,i})\propto{(k_{B}\Theta_{BG,i})}\propto|\mu|$. Therefore, the normalized resistivity of doped graphene is independent of $|\mu|$ and that is why one can get a universal curve for $R_{i}(T/\Theta_{BG,i})$ regardless of doping level of the sample. This feature comes out naturally from our approach by using the semi-inelastic scattering rate. The normalized resistivity $R_{i}$ is proportional to $\left({T}/{\Theta_{BG,i}}\right)^{4}$ in the LT limit and ${T}/{\Theta_{BG,i}}$ in the HT limit.

Now instead of the SI approximation, we use the resistivity given in Eq.(2) with full inelastic scattering rate

where $Q_{a}^{p}(\theta)=q_{a}^{p}/k$ is the ratio of the phonon momentum to electron momentumNguyen2020 . Our numerical results indicate that $R^{in}$ is almost identical to $R_{1}$ and $R_{2}$. Obviously, as $T\rightarrow\Theta_{BG,i}$, $R_{i}$ and $R^{in}$ should approach 1.

The results for $R_{1}$ together with the results taken from Ref. Efetov2010 are demonstrated in Fig. 4. We found a significant difference between our results and those from Efetov2010 , especially for $T/\Theta_{BG,i}<1$. It is noted that in Ref. Efetov2010 the normalized resistivity is defined as $R_{0}\left(\frac{T}{\Theta_{BG}}\right)={\rho(\mu,T)}/{\rho(\mu,\xi\Theta_{BG})}$ with $\xi=0.2$ instead of 1. Since $\Theta_{BG}$ adopted in Ref. Efetov2010 is $2\hbar v_{LA}k_{F}/k_{B}=\Theta^{LA}_{F}$, which makes $0.2\Theta^{LA}_{F}\approx\Theta_{BG,0}$, the BG temperature determined by QE approximation given in Eq. (3). Although the universal scaling or behavior of $R_{i}$ as a function of the normalized temperature $T/\Theta_{BG,i}$ does not depend on $\mu$, for the same $T$ range of investigation, the heavier graphene gets doped (i.e. the larger $|\mu|$ induces the larger $\Theta_{BG,i}$), the narrower the range of $T/\Theta_{BG,i}$ becomes.

In conclusion, we have clarified the issues of BG temperatures in graphene via analytical and numerical calculations based on full inelastic EAP scattering rate and various approximation schemes. We found that the commonly adopted BG temperatures in graphene ($k_{B}\Theta_{F}^{LA}=2\hbar v_{LA}k_{F}$) Hwang2008 ; Bistritzer2009 ; Kubakaddi2009 ; Efetov2010 ; Fuhrer2010 ; Viljas2010 ; Castro2010 ; Mariani2010 ; Min2011 ; Sarma2011 ; Yan2012 ; Chen2012 ; Cooper2012 ; Munoz2012 ; Fong2012 ; Somphonsane2013 ; Fong2013 ; Betz2013 ; Park2014 ; Sohier2014 ; McKitterick2016 ; Meunier2016 ; Ansari2017 ; Rani2017 ; DSouza2017 ; Gunst2017 ; Ansari2018 need to be corrected by a factor around 2.5, when using the same criterium [$\rho^{LT}_{\mu}(\mu,T)=\rho^{HT}_{\mu}(\mu,T)$]. The BG temperature induced by the in-plane EAP scattering in semi-inelastic approximation is uncovered as $\Theta_{BG,1}\approx[(\Theta_{LA,1}^{3}+\beta_{v}^{5}\Theta_{TA,1}^{3})/(1+\beta_{v}^{5})]^{1/3}$ with $\Theta_{a,1}=(\pi/6)^{1/3}\hbar v_{a}k_{F}/k_{B}$. The corrected analytic relation 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 Efetov2010 . We also show that Refs. Mariani2010 ; Yan2012 ; Sohier2014 ; McKitterick2016 ; Ansari2017 well agree with the quasi-elastic (QE) prediction. When the inelastic EAP scattering rate and the deviation of electron energy from the chemical potential ($\mu$) are fully taken into account, the resistivity $\rho(\mu,T)$ can only be described numerically. For this case we determine the BG temperature by the point where $\partial\rho(\mu,T)/\partial T$ is a maximum and thus $\partial^{2}\rho(\mu,T)/\partial T^{2}=0$ (criterium 2). If we also apply criterium 2 to find the BG temperature in the SI approximation, we get $\Theta_{BG,2}\approx[(\Theta_{LA,2}^{3}+\beta_{v}^{2}\Theta_{TA,2}^{3})/(1+\beta_{v}^{2})]^{1/3}$ with $k_{B}\Theta_{a,2}=(16/c_{0})^{1/3}\hbar v_{a}k_{F}$, which happen to be very close to the value $\hbar v_{a}k_{F}$ deduced in Refs. Song2012 ; Ma2014 ; Tikhonov2018 ; Kong2018 . We found that the BG temperature determined by the full numerical calculation with criterium 2 falls between the values obtained via the SI appoximation with criterium 1 ($\Theta_{BG,1}$) and criterium 2 ($\Theta_{BG,2}\approx 1.35\Theta_{BG,1}$). These values are about a factor 2 higher than the BG temperature ($\Theta_{BG,0}$) obtained with the oversimplified QE approximation and a factor 2-2.5 lower than the commonly adopted value of $\Theta^{LA}_{F}$.

Finally, the resistivity normalized to its value at $T=\Theta_{BG,i}$ [$R_{i}=\rho(\mu,T)/\rho(\mu,\Theta_{BG,i})$] plotted as a function of the normalized temperature $T/\Theta_{BG,i}$ displays a universal scaling behavior, which is independent of the carrier densityEfetov2010 . Applying our results to the experimental data extracted from Ref. Efetov2010 does show such a universal scaling behavior, which obeys the relation $R_{i}(1)=1$.