Phonon-induced giant linear-in-$T$ resistivity in magic angle twisted bilayer graphene: Ordinary strangeness and exotic superconductivity

We study the effect of electron-acoustic phonon interactions in twisted bilayer graphene on resistivity in the high-temperature transport and superconductivity in the low-temperature phase diagram. We theoretically show that twisted bilayer graphene should have an enhanced and strongly twist-angle dependent linear-in-temperature resistivity in the metallic regime with the resistivity magnitude increasing as the twist angle approaches the magic angle. The slope of the resistivity versus temperature could approach one hundred ohm per kelvin with a strong angle dependence, but with a rather weak dependence on the carrier density. This higher-temperature density-independent linear-in-$T$ resistivity crosses over to a $T^4$ dependence at a low density-dependent characteristic temperature, becoming unimportant at low temperatures. This angle-tuned resistivity enhancement arises from the huge increase in the effective electron-acoustic phonon coupling in the system due to the suppression of graphene Fermi velocity induced by the flat-band condition in the moiré superlattice system. Our calculated temperature dependence is reminiscent of the so-called “strange metal” transport behavior except that it is arising from the ordinary electron-phonon coupling in a rather unusual parameter space due to the generic moiré flat-band structure of twisted bilayer graphene. We also show that the same enhanced electron-acoustic phonon coupling also mediates effective attractive interactions in $s,p,d$, and $f$ pairing channels with a theoretical superconducting transition temperature on the order of $\sim 5$ K near magic angle. The fact that ordinary acoustic phonons can produce exotic non-$s$-wave superconducting pairing arises from the unusual symmetries of the system.

Figure 1

(a) Schematic illustration of the moiré band structure (blue solid lines) in TBG. The dashed lines represent the pristine Dirac cones associated respectively with top and bottom graphene layers in the same valley. The interlayer hybridization reduces the velocity at Dirac points from the monolayer value $v_F$ to the renormalized value $v_F^{*}$. (b) $v_F^{*}/v_F$ as a function of $\theta$ from a full moiré band structure calculation as described in Appendix pp1. The horizontal dashed line indicates the phonon velocity $v_{\mathrm{ph}}/v_F$, which is assumed to remain the same when $\theta$ changes in our theory.

Figure 2

($\mathrm{a}\left)\rho \right(T)$ for three different twist angles. Electron density $n$ is $4\times {10}^{11}{\mathrm{cm}}^{-2}$, (b) $d\rho /dT$ as a function of $\theta$. High temperatures $T>T_{\mathrm{BG}}/4$ is assumed, so $d\rho /dT$ is independent of density based on Eq. (2). (c) and (d) $\rho \left(T\right)$ for different electron density $n$. $\theta$ is $1.1^{\circ }$ in (c) and $1.2^{\circ }$ in (d). Monolayer graphene phonon parameters (given in the text) are used.

Figure 3

Scattering rate $\hslash /\left(k_B\langle \tau \rangle \right)$ as a function of temperature $T$. Different solid lines are for different twist angles in (a) but for different density $n$ in (b) and (c). In (a), $n$ is $4\times {10}^{11}{\mathrm{cm}}^{-2}$. $\theta$ is $1.1^{\circ }$ in (b) and $1.2^{\circ }$ in (c). The black dashed line in each plot represents the Planckian scattering rate $\hslash /\left(k_B\langle \tau \rangle \right)=T$. Monolayer graphene phonon parameters (given in the text) are used.

Figure 4

[(a) and (b)] Log-log plot of $\rho \left(T\right)$ in TBG. At high temperature $T>T_L,\rho$ is linear in $T$ and essentially independent of density $n$. At low temperature $T<T_L,\rho$ scales as $T^4$. [(c) and (d)] The crossover temperature $T_L$, down to which linearity in $\rho \left(T\right)$ persists, as a function of density $n$. The dashed lines show $T_{\mathrm{BG}}/4$, while points on the solid lines represent $T_L$ read off from Figs. 2 and 2. $T_{\mathrm{BG}}/4$ provides an upper bound on $T_L$.

Figure 5

($\mathrm{a})d\rho /dT$ in the linear regime as a function of twist angle. The markers are experimental data of Ref. [33] for different TBG devices. The dashed line shows the theoretical value $d\rho /dT$ based on Eq. (2) with $D/v_{\mathrm{ph}}$ set to its value in monolayer graphene. The solid line shows the theoretical prediction with $D/v_{\mathrm{ph}}$ enhanced by a factor of 3. This figure is adapted from Ref. [33]. (b) The crossover temperature $T_L$, down to which $\rho \left(T\right)$ is linear in $T$, as a function of density $n$. The dots are experimental data extracted from Ref. [33]. The dashed line represents the theoretical estimation with $T_L=T_{\mathrm{BG}}/4$. The van Hove singularity occurring $\sim 2\times {10}^{12}{\text{cm}}^{-2}$ in TBG is not included in our continuum theory as discussed in the text.

Figure 6

Superconducting critical temperature $T_c$ in different channels (top) and the DOS per spin and per valley as a function of filling factor $n/n_M$ (bottom). $n/n_M$ is 1 when the flat bands are fully filled. The twist angle is $1.{05}^{\circ }$ in (a) and $1.1^{\circ }$ in (b).

Figure 7

Moiré band structure along high-symmetry directions in the moiré Brillouin zone for $+K$ valley and twist angle $\theta =1.2^{\circ }$. Dirac cones are centered at the moiré Brillouin zone corners ${\kappa }_{\pm }$

Figure 8

Resistivity $\rho$ as a function of temperature $T$. Twist angle $\theta$ is, respectively, $1.3^{\circ },1.1^{\circ },1.{09}^{\circ }$, and $1.{05}^{\circ }$ in (a), (b), (c), and (d). The Dirac velocity $v_F^{*}$ varies as a function of $\theta$. In each plot, solid lines are calculated based on Eq. (B7), while dashed lines are obtained from Eq. (2).