Large power dissipation of hot Dirac fermions in twisted bilayer graphene

Recent pioneering experimental discoveries in twisted bilayer graphene (tBLG) by Cao et al Cao et al. (2018a, b), have created great interest in the study of their electronic properties and has ushered in a new era in the condensed matter physics Wu et al. (2018); Yankowitz et al. (2019); Lu et al. (2019); Sharpe et al. (2019); Tomarken et al. (2019); Roy and Juričić (2019); Serlin et al. (2020); Polshyn et al. (2019); Cao et al. (2020); Wu et al. (2019); Das Sarma and Wu (2020). Among the discoveries, the existence of correlated insulating phases and superconductivity at low temperatures and a highly resistive linear-in-temperature $T$ resistivity $\rho$ at high temperature, are remarkable and exciting Cao et al. (2018a, b); Yankowitz et al. (2019). Very recently, the observation of a quantum anomalous Hall effect in twisted bilayer graphene aligned to hexagonal boron nitride has been reported in tBLG Serlin et al. (2020). In tBLG a small twist angle $\theta$, near the magic angle $\theta_{m}$, between the two layers plays the most significant role and acts as one of the tunable parameters, similar to the carrier density $n_{s}$ and temperature $T$, of the samples in limiting their electronic properties Cao et al. (2018a, b); Yankowitz et al. (2019); Wu et al. (2019); Das Sarma and Wu (2020). The transport results of Cao et al Cao et al. (2020) establish magic angle bilayer graphene as a highly tunable platform to investigate ‘strange metal’ behavior. Because of the twist between the layers the band structure is a moiré flat band with the twist angle dependent suppressed Fermi velocity ${v_{F}}^{*}(\theta$) $<v_{F}$, the bare Fermi velocity in monolayer graphene, and the large density of states $D(E_{k})$ near $\theta_{m}$ at which ${v_{F}}^{*}(\theta$) = 0 Wu et al. (2019); Das Sarma and Wu (2020); Bistritzer and MacDonald (2011). The strongly enhanced electrical resistivity $\rho$, near $\theta_{m}$, with linear-in-temperature behavior has been observed for $T>\sim$ 5 K Polshyn et al. (2019); Cao et al. (2020).

Theoretically, the electrical resistivity has been investigated in tBLG, at higher temperature and away from the moiré miniband edge, by considering the effect of electron- acoustic phonon (el-ap) interaction Polshyn et al. (2019); Wu et al. (2019); Das Sarma and Wu (2020). It is shown that the phonon limited resistivity $\rho_{p}$ = $\rho$ (T, $\theta$) is strongly enhanced in magnitude, twist-angle dependent and linear-in- $T$ occurring for $T>T_{L}$, where $T_{L}$ (on the order of few kelvins) is the temperature above which linearity in $\rho(T,\theta)$ develops. This linear-in- $T$ is observed for $T_{L}=T_{BG}/8$ Das Sarma and Wu (2020), where $T_{BG}=2\hbar v_{s}k_{F}/k_{B}$ is the Bloch- Grüneisen (BG) temperature, $v_{s}$ is the acoustic phonon velocity, and $k_{F}=\sqrt{\pi n_{s}/2}$ is the Fermi wave vector in tBLG. The enhancement in $\rho(T,\theta)$, about three orders of magnitude greater than that in monolayer graphene (MLG) at $T\sim$ 10 K, is shown to arise from the large increase in the effective el-ap scattering in tBLG due to the suppression of $v_{F}$ induced by the moiré flat band. In the metallic regime i.e. for $T>T_{m}(<T_{L})$, where $T_{m}$ is the metallic temperature, above which $d\rho(T,\theta)/dT>0$, and it is $n_{s}$ and $\theta$ dependent. The $\rho(T,\theta$) is found to increase with increasing $T$ as the twist angle $\theta$ approaches $\theta_{m}$. The linear dispersion taken for the Dirac fermions in tBLG is an approximation that is valid for Fermi energy near the Dirac point and hence its transport study is limited to the $n_{s}\leq 10^{12}$ cm^-2. Interestingly, it is also shown that the same enhanced el-ap interaction can also produce superconductivity with $T_{c}\sim 1K$ in s, p, d and f orbital pairing channels Wu et al. (2018, 2019).

The theory of Wu et al Wu et al. (2019); Das Sarma and Wu (2020) explains the available experimental data of $\rho$ well for $T>$ 5K Polshyn et al. (2019); Cao et al. (2020). In their theory, all the effects of disorder, impurities and defects are ignored assuming that the system is extremely clean and the Fermi energy is slightly away from the Dirac point. However, the hot electron relaxation is an important transport property which is controlled by only electron-phonon interaction and independent of disorders and impurities.

The electron system in samples subject to large electric fields or photoexcitation establishes its internal thermal equilibrium at an electron temperature $T_{e}$ greater than the lattice temperature $T$ because electron-electron interaction occurs at the time scale of several femtoseconds which is much smaller than the electron-phonon scattering time. Consequently, the electron system is driven out of equilibrium with the lattice. In steady state, these electrons will relax towards equilibrium with the lattice by dissipating energy with phonons as the cooling channels. The study of hot electron power loss $P$ is important as it affects thermal dissipation and heat management which are key issues in nanoscale electronics device. Moreover, it is crucial for applications in variety of devices such as calorimeters, bolometers, infrared detectors, ultrafast electronics and high speed communications. Hot electron cooling has been extensively studied theoretically and experimentally in MLG Kubakaddi (2009); Tse and Das Sarma (2009); Bistritzer and MacDonald (2009); Viljas and Heikkilä (2010); Betz et al. (2012); Baker et al. (2012); Low et al. (2012); Baker et al. (2013); Somphonsane et al. (2013); Laitinen et al. (2015) and conventional bilayer graphene (BLG) Viljas and Heikkilä (2010); Katti and Kubakaddi (2013); Bhargavi and Kubakaddi (2014); Huang et al. (2015).

In the present work, we investigate the effect of enhanced el-ap coupling on the power dissipation $P$ of the hot electrons in moiré flat band in tBLG. It is studied as a function of twist angle, electron temperature and electron density. We show that the twist angle $\theta$ acts as one of the strong tunable parameters of $P$. Additionally, a relation between power loss and phonon limited mobility $\mu_{p}$ is brought out in BG regime.

Wu et al Wu et al. (2019) have used the effective Dirac Hamiltonian with a renormalized velocity for electron energy spectrum, in order to obtain their analytical results. In moiré flat band, the electron energy spectrum is assumed to be Dirac dispersion $E_{k}=\hbar{v_{F}}^{*}|k|$, which is an approximation that is valid for near Dirac point, with an effective Fermi velocity ${v_{F}}^{*}\equiv{v_{F}}^{*}(\theta)$. Because of this approximation our theory will be limited to the carrier density $n_{s}\leq 10^{12}$ cm^-2. The density of states is $D(E_{k})=g(E_{k})/[2\pi{(\hbar{v_{F}}^{*})}^{2}]$ with the degeneracy $g$ = $g_{s}$ $g_{v}$ $g_{l}$, where $g_{s}$, $g_{v}$ and $g_{l}$ are, respectively, spin, valley and layer degeneracy each with the value of 2. We consider electron-acoustic phonon interaction within the deformation potential approximation with the longitudinal acoustic (LA) phonons of energy $\hbar\omega_{q}$ and wave vector q interacting with the tBLG Dirac electrons in the moiré miniband. The LA phonons in tBLG are assumed to be unaffected by the tBLG structure and are taken to be the same as the MLG phonons. In MLG the experimental observations of electrical conductivity Efetov and Kim (2010) and power loss Betz et al. (2012); Baker et al. (2012, 2013) are very well explained by the electron interaction with only LA phonons, without screening. Wu et al Wu et al. (2019); Das Sarma and Wu (2020) have explained the linear-in-T resistivity data in tBLG with only electron-LA phonon interaction. We use the modified ordinary el-ap matrix element Wu et al. (2019) ${|M(q)|}^{2}=[(D^{2}\hbar qF(\theta))/(2A\rho_{m}v_{s})][1-(q^{2}/4k^{2})]$ where $D$ is the first-order acoustic deformation potential coupling constant, $A$ is the area of the tBLG, $\rho_{m}$ is the areal mass density and $v_{s}$ is the LA phonon velocity. The detailed tBLG moiré wave function gives rise to the form factor function $F(\theta)$ which modifies the el-ap interaction matrix element in tBLG as compared with the MLG Wu et al. (2019). It is shown to be between 0.5 and 1.0 and being nearly parabolic for 1^∘ $<$ $\theta<$ 2^∘ in the neighborhood of a minimum at $\theta$ = $\sim$ 1.3^∘ Das Sarma and Wu (2020). Following the Refs. Kubakaddi (2009); Manion et al. (1987); Kaasbjerg et al. (2014), and taking care of additional layer degeneracy, we obtain an expression for the electron power loss in tBLG and it is given by

$$P=-\frac{gD^{2}F(\theta)}{4\pi^{2}n_{s}\rho_{m}\hbar^{5}{v_{s}}^{3}{{v_{F}}^{*}}^{3}}\int_{0}^{\infty}d(\hbar\omega_{q}){(\hbar\omega_{q})}^{2}\int_{\gamma}^{\infty}dE_{k}\frac{(E_{k}+\hbar\omega_{q})}{{[1-{(\gamma/E_{k})}^{2}]}^{1/2}}\times G(E_{q},E_{k})[N_{q}(T_{e})-N_{q}(T)][f(E_{k})-f(E_{k}+\hbar\omega_{q})],$$

(1)

where $n_{s}$ is the electron density, $\gamma=(E_{q}/2)$, $E_{q}=\hbar{v_{F}}^{*}q$, $N_{q}(T)={[exp(\hbar\omega_{q}/k_{B}T)-1]}^{-1}$ is the Bose-Einstein distribution at lattice temperature $T$ and $G(E_{q},E_{k})=[1-{(\gamma/E_{k})}^{2}]$, is due to the spinor wave function of the electron in the electron -phonon matrix element, in the quasi-elastic approximation Kubakaddi (2009). By setting F($\theta$) =1, $g_{l}$ = 1 and ${v_{F}}^{*}(\theta)=v_{F}$ in Eq.(1), we regain the equation that is applicable to MLG Kubakaddi (2009) and silicene Kubakaddi and Phuc (2020), similar to the acoustic phonon induced resistivity in tBLG Polshyn et al. (2019); Wu et al. (2019). The twist angle dependence of ${v_{F}}^{*}\equiv{v_{F}}^{*}(\theta)$ is shown to be very well approximated by Polshyn et al. (2019); Das Sarma and Wu (2020)

$${v_{F}}^{*}\approx 0.5|\theta-\theta_{m}|v_{F},$$

(2)

which clearly indicates that twist angle effect is very large for $\theta$ closer to $\theta_{m}$. We use this relation while computing $P$ for different twist angles.

In the Bloch-Grüneisen (BG) regime $T$, $T_{e}$ $<<T_{BG}$, $q<<$ 2$k_{F}$, the power loss is given by

$$P=\Sigma({T_{e}}^{4}-T^{4})/{n_{s}}^{1/2},$$

(3)

where $\Sigma=\Sigma_{0}(D^{2}/{v_{s}}^{3})$ and $\Sigma_{0}=(g\pi^{5/2}{k_{B}}^{4}F(\theta))/(60\sqrt{2}\rho_{m}\hbar^{4}{{v_{F}}^{*}}^{2})$. Hence, in BG regime $P\sim T^{4}$, ${n_{s}}^{-1/2}$ and ${{v_{F}}^{*}}^{-2}$.

We obtain the following numerical results of power loss in tBLG using the parameters Wu et al. (2019); Das Sarma and Wu (2020): $\rho_{m}=7.6\times 10^{-8}$ gm/cm², $\theta_{m}$ = 1.02^∘, $v_{s}$ = 2$\times$10⁶ cm/s, $v_{F}$ = 1$\times$10⁸ cm/s and $D$ = 20 eV Kubakaddi (2009); Bistritzer and MacDonald (2009); Baker et al. (2012); Efetov and Kim (2010); DaSilva et al. (2010); Hwang and Das Sarma (2008), noting that Polshyn et al Polshyn et al. (2019) and Wu et al Wu et al. (2019) have used $D$ = 25 ± 5 eV. In order to bring out the angular dependence of the power loss, we confine our illustrations for $\theta$ =1.1^∘, 1.2^∘ and 1.3^∘ which are closer to magic angle $\theta_{m}$ =1.02^∘. For these angles, the effective Fermi velocity ${v_{F}}^{*}$ = 4, 9 and 14$\times$10⁶ cm/s ($>$ 1.5 $v_{s}$ Wu et al. (2019)), respectively, which are much smaller than the bare $v_{F}$, and the effect of ${v_{F}}^{*}$ on the transport coefficients will be very large. For, further increase of $\theta$, ${v_{F}}^{*}$ tends to $v_{F}$ at about 3.0^∘. The values of the function $F(\theta)$ for different $\theta$ are taken from figure 3 of Das Sarma et al Das Sarma and Wu (2020), and because of its value between 0.5 and 1, it will have smaller influence on $P$ than ${v_{F}}^{*}$. We have presented the calculations for lattice temperature $T$ = 0.1 K, and $n_{s}=0.1-1n_{0}$, with $n_{0}$ =1$\times$10¹² cm^-2, which keeps us slightly away from the Dirac point and within the linear region of moiré flat band. For $n_{s}=Nn_{0}$, $T_{BG}=38.3\sqrt{N}$ which is smaller by a factor of $\sqrt{2}$ compared to MLG.

First we explore the dependence of power loss $P$ on electron temperature $T_{e}$ for twist angles $\theta$ =1.1^∘, 1.2^∘ and 1.3^∘. In figure 1a, $P$ is presented as a function of $T_{e}$ (1-50 K) for $n_{s}=n_{0}$. For all the $\theta$, we observe the generic nature of the behavior, where in at very low $T_{e}$ power loss increases rapidly then slows down at higher temperature. For the temperatures $T_{e}$ $<<$ $T_{BG}$, the rapid increase may be attributed to the increasing number of phonons as their wave vector $q\approx k_{B}T_{e}/\hbar v_{s}$ increases linearly with $T_{e}$. For $\theta$ =1.1^∘, the power law $P\sim{T_{e}}^{4}$ is found to be obeyed for $T_{e}<\sim$ 2.5 K, which is about $T_{BG}$/15. The exponent 4 of $T_{e}$ is manifestation of two-dimensional phonons with unscreened electron-phonon coupling. In order to see the effect of $\theta$ on the range of validity of the power law, we have plotted $P/{T_{e}}^{4}$ vs $T_{e}$ in figure 1b. It is found that, as $\theta$ increases the range of $T_{e}$ in which power law is obeyed marginally increases. For example, for $\theta$ =1.2^∘ and 1.3^∘, power law is found to be satisfied for $T_{e}$ up to about 3 and 3.5 K, respectively, although $T_{BG}$ is same. This happens because as $\theta$ increases ${v_{F}}^{*}$ also increases and tends towards $v_{F}$. In the BG regime, in which $P\sim$ ${{v_{F}}^{*}}^{-2}$, we find $\Sigma$ = 2.66$\times$10${}^{-15}/\sqrt{N}$ W/K⁴-cm, 5.13$\times$10^-16/$\sqrt{N}$ W/K⁴-cm and 2.1$\times$10^-16 /$\sqrt{N}$ W/K⁴, for $\theta$ =1.1^∘, 1.2^∘ and 1.3^∘, respectively, as compared to 5.23$\times 10^{-18}/\sqrt{N}$ W/K⁴-cm in MLG. In the higher temperature region of $T_{e}$ = 10 $-$ 50 K (30 $-$ 50 K), $P\sim{T_{e}}^{\delta}$ with $\delta$ $\sim$ 2.0 $-$ 2.2 ($\sim$ 1.7 $-$ 2.0), for all $\theta$s, as compared to the resistivity which is found to show linear-in-temperature for temperature $\geq$ $T_{BG}$/8 Das Sarma and Wu (2020).

In figure 1c, $T_{e}$ dependence of $P$ is shown for $n_{s}$ = 0.5 $n_{0}$ and the same behavior is observed as in figure 1a. However, for the same $\theta$, in the low temperature region $P$ is found to be marginally larger (smaller) at lower (higher) $T_{e}$ than that for $n_{s}$ = $n_{0}$. We observe that the temperature below which the power law $P\sim{T_{e}}^{4}$ is obeyed shifts to lower side for smaller $n_{s}$. For example, for $\theta$ =1.1^∘ power law is obeyed for $T_{e}<\sim$2 K, which may be attributed to the lower $T_{BG}$ = 27.0 K. More importantly, from the figures 1a and 1c, we find that $\theta$ acts as a tunable parameter of power loss, in addition to $T_{e}$ and $n_{s}$. The influence of $\theta$ on $P$ is very much large compared to $n_{s}$. For $\theta$ =1.1^∘ and 1.3^∘, for the $T_{e}$ range considered, $P$ is in the range $\sim 10^{4}-10^{9}$ eV/s and $\sim 10^{3}-10^{8}$ eV/s, respectively. These values are comparable to those in monolayer MoS₂ Kaasbjerg et al. (2014) but about three and four orders of magnitude greater than those in GaAs heterojunction Ma et al. (1991) and Si-inversion layer Fletcher et al. (1997), respectively.

In order to compare the power loss in tBLG with that in MLG and conventional BLG, $P$ dependence on $T_{e}$ is depicted in figure 2, for $n_{s}$ = $n_{0}$ with $P$(tBLG) taken for $\theta$ =1.1^∘. We find that the power loss in tBLG is very large ($\sim 2\times 10^{4}-2\times 10^{9}$ eV/s) compared to that in MLG ($\sim 4\times 10^{1}-6\times 10^{6}$ eV/s) and BLG ($\sim 9\times 10^{1}-3\times 10^{7}$ eV/s) Bhargavi and Kubakaddi (2014). Defining a ratio $R_{p}$ = $P$(tBLG)/ $P$(MLG), it is found that $R_{p}$ = $\sim$ 450, 260 and 300, respectively, at 1, 10 and 50 K, and $R_{p}$ is expected to be smaller for larger $\theta$. This enhancement is attributed to the significantly reduced ${v_{F}}^{*}$. This may be compared with the $\rho$ enhancement in tBLG, which is of three orders of magnitude greater than that in MLG at $\sim$10 K Wu et al. (2019); Das Sarma and Wu (2020). It is also noticed that the range of $T_{e}$ in which power law is obeyed is much larger in MLG than in tBLG.

We have presented in figure 3 the electron density (=0.1-1.0 $n_{0}$) dependence of the power dissipation for two electron temperatures $T_{e}$ = 1 K (figure 3a) and 5 K (figure 3b). For $n_{s}$ = 0.1(1.0) $n_{0}$ the $T_{BG}$ = 12.1 (38.3) K. From figure 3a, we see that $P$ is found to decrease with increasing $n_{s}$, as found in MLG Kubakaddi (2009); Baker et al. (2013), with power law $P$ $\sim$ ${n_{s}}^{-1/2}$ being followed at larger $n_{s}$ and small deviation occurring at lower $n_{s}$. This is due to the fact that $T_{BG}$ goes on decreasing with decreasing $n_{s}$. The $P$ $\sim{n_{s}}^{-1/2}$ dependence in tBLG is in contrast to the $P$ $\sim{n_{s}}^{-3/2}$ dependence in conventional BLG Bhargavi and Kubakaddi (2014); Huang et al. (2015). On the other hand for $T_{e}$ = 5 K (figure 3b), $P$ increases (flattens) with increasing $n_{s}$ in the low (high) $n_{s}$ region, because we are moving away from the $T_{e}$ $<<$ $T_{BG}$ region.

The energy relaxation time $\tau_{e}$ is another important quantity studied in the hot electron relaxation process, as it determines the samples suitability for its applications in optical detectors (bolometer, calorimeter and infrared detectors) and .high speed devices. For a degenerate electron gas it is given by $\tau_{e}$ = $[(p+1){(\pi k_{B})}^{2}({T_{e}}^{2}-T^{2})/(6E_{F}P)]$, where $p$ is the exponent of energy in density of states and $E_{F}$ is the Fermi energy Baker et al. (2012); Kubakaddi and Biswas (2018). In BG regime, since $P$ $\sim{T_{e}}^{4}$ and ${n_{s}}^{-1/2}$, we find $\tau_{e}$ $\sim{T_{e}}^{-2}$ and independent of $n_{s}$ (as $E_{F}$ $\sim$ ${n_{s}}^{1/2}$). In figure 4, $\tau_{e}$ is presented as a function of $T_{e}$, for $\theta$ = 1.1^∘, 1.2^∘ and 1.3^∘, in tBLG along with the $\tau_{e}$ in MLG for $n_{s}$ = $n_{0}$. In both tBLG and MLG, $\tau_{e}$ is found to decrease with increasing $T_{e}$ and the decrease is rapid at lower temperature ($<\sim$10 K). It is found that $\tau_{e}$ in tBLG, for $\theta$ =1.1^∘, is an order of magnitude smaller than that in MLG and this difference decreases with increasing $\theta$. The ratio $\tau_{e}$ (MLG)/ $\tau_{e}$ (tBLG), for $\theta$ =1.1^∘, is found to be 10.0, 6.6 and 6.9, respectively, for $T_{e}$ = 5, 10 and 20 K. This ratio is not as large as the ratio of $P$’s, because of the product $E_{F}P$ in the denominator of the expression for $\tau_{e}$, noting that for the same $n_{s}$, the $E_{F}$ in tBLG is much smaller than that in MLG. By increasing $\theta$ the $\tau_{e}$ increases significantly, indicating that twist angle is an important tunable parameter for $\tau_{e}$ also. It may be noted that samples with faster energy relaxation (i.e. smaller $\tau_{e}$) find applications in ultrafast electronics and high speed communications. On the other hand, samples with longer energy relaxation time are preferred in photodetectors and energy harvesting devices like hot carrier solar cells.

Finally, in BG regime, we bring out a simple relation of $P$ with phonon limited mobility $\mu_{p}$ and resistivity $\rho_{p}$ in tBLG. In this regime, $P$, $\mu_{p}$ and $\rho_{p}$ are sensitive measures of the el-ap coupling. While $P$ is determined by the energy relaxation through el-ap interaction, $\mu_{p}$ and $\rho_{p}$ involve momentum relaxation through the same mechanism. A relation between these measurable properties is expected because of the same underlying mechanism. This kind of relation between $P$ and $\mu_{p}$ is listed for different electron systems in Ref Kubakaddi (2017). In tBLG, the equation $\rho_{p}(T)=AT^{4}$ for the phonon limited resistivity is obtained from Min et al Min et al. (2011) with suitable replacement of $g_{s}g_{v}$ by $g_{s}g_{v}g_{l}$, $k_{F}=\sqrt{(\pi n_{s}/2)}$ and inserting $F(\theta)$ in the numerator in their Eq. (8) for A. There by, using the relation $\mu_{p}(T)=1/(n_{s}e\rho_{p}(T))$, the phonon limited mobility is found to be $\mu_{p}(T)=[15(ge\hbar^{4}\rho_{m}{v_{s}}^{5}{{v_{F}}^{*}}^{2}{n_{s}}^{1/2}T^{-4})][16\sqrt{2}\pi^{5/2}D^{2}{k_{B}}^{4}F(\theta)]$, where $e$ is the electron charge. Expressing Eq. (3) as $P=F_{e}(T_{e})-F_{e}(T)$ Kubakaddi (2009); Ma et al. (1991), where $F_{e}(T)=\Sigma T^{4}/{n_{s}}^{1/2}$ and $P=F_{e}(T_{e})$ for $T_{e}>>T$, we obtain a very simple relation $F_{e}(T)\mu_{p}(T)=(e{v_{s}}^{2}/2)$, which is exactly same as that of MLG Kubakaddi (2017). This relation is analogous to Herring’s law Herring (1954), which relates phonon-drag thermopower $S^{g}$ and $\mu_{p}$. Alternatively, power loss can be related to $\rho_{p}$ by the formula $F_{e}(T)=(n_{s}e^{2}{v_{s}}^{2}/2)\rho_{p}(T)$. The advantage of these relations is, if $F_{e}(T)$ is measured then $\mu_{p}(T)$ and $\rho_{p}(T)$ can be determined or the vice-versa, and the measurements of power loss may be preferred as it is independent of lattice disorders and impurities. From our calculated value of $P=F_{e}(T)$ = 4.45$\times 10^{-14}$ W at 2 K for $\theta$ =1.1^∘ and $n_{s}$ = $n_{0}$, we estimate $\rho_{p}(T)$ = 0.8 $\Omega$, which is nearly agreeing with the value obtained by Wu et al (see figure 4a of Wu et al. (2019)), and $\mu_{p}(T)$ = 7.5$\times 10^{6}$ cm²/V$-$s.

We would like to make the following remarks. In the literature the values of $\theta_{m}$ given are varying between 1.02^∘ to 1.1^∘ Cao et al. (2018a); Roy and Juričić (2019); Wu et al. (2019); Das Sarma and Wu (2020). However, we believe that our findings and analysis with $\theta_{m}$ = 1.02^∘ Wu et al. (2019); Das Sarma and Wu (2020) hold good for the $\theta$ values closer to any chosen $\theta_{m}$. We want to emphasize that, our analytical results will be of great help to experimental researchers and secondly can be used to determine $D$ as the measurements of $P$ are independent of lattice disorder and impurities, unlike resistivity.

We have studied the hot electron power loss $P$ due to the simple acoustic phonon interaction, via deformation potential coupling, in tBLG of low electron density $n_{s}\leq$ 10¹² cm^-2 for small twist angles $\theta$ and for $T_{e}$ $\geq$ 1 K. For $\theta$ closer to the magic angle, $P$ is enhanced by a few hundred times that in MLG due to the great suppression of the Fermi velocity ${v_{F}}^{*}$ leading to the strong el-ap scattering. Consequently, twist angle emerges as an additional important tunable parameter of $P$. Although BG regime power law $P\sim{T_{e}}^{4}$ is obeyed in low $T_{e}$ region, $P$ vs $T_{e}$ behavior still remains super linear at higher $T_{e}$ where acoustic phonon limited resistivity $\rho_{p}$ is linear-in-temperature. For a given $n_{s}$, although $T_{BG}$ is independent of $\theta$, the range of $T_{e}$ in which $P\sim{T_{e}}^{4}$ is obeyed increases marginally with increasing $\theta$. The energy relaxation time $\tau_{e}$, is found to be smaller by an order of magnitude than in MLG and decreasing with increasing $T_{e}$. As $\theta$ approaches $\theta_{m}$ the $\tau_{e}$ decreases significantly indicating that $\theta$ can be used as an important parameter to tune $\tau_{e}$ also. Finally, simple and useful relations of $P$ with $\mu_{p}$ and $\rho_{p}$ are obtained in the BG regime. From the relation between $P$ and $\rho_{p}$, using our calculated $P$, the estimated value is closer to the $\rho_{p}$ of Wu et al Wu et al. (2019). Experimental observations may test the validity of our predictions.