Electron-phonon scattering and in-plane electric conductivity in twisted bilayer graphene

We have surveyed the in-plane transport properties of the graphene twist bilayer using (i) a low-energy effective Hamiltonian for the underlying electronic structure, (ii) an isotropic elastic phonon model, and (iii) the linear Boltzmann equation for elastic electron-phonon scattering. We find that transport in the twist bilayer is profoundly sensitive to the rotation angle of the constituent layers. Similar to the electronic structure of the twist bilayer, the transport is qualitatively different in three distinct angle regimes. At large angles ($\theta >\approx {10}^{\circ }$) and at temperatures below an interlayer Bloch-Grüneisen temperature of $\approx 10$ K, the conductivity is independent of the twist angle, i.e., the layers are fully decoupled. Above this temperature the layers, even though decoupled in the ground state, are recoupled by electron-phonon scattering and the transport is different both from single-layer graphene as well as the Bernal bilayer. In the small-angle regime $\theta <\approx 2^{\circ }$, the conductivity drops by two orders of magnitude and develops a rich energy dependence, reflecting the complexity of the underlying topological changes (Lifshitz transitions) of the Fermi surface. At intermediate angles, the conductivity decreases continuously as the twist angle is reduced, while the energy dependence of the conductivity presents two sharp transitions, that occur at specific angle-dependent energies, and that may be related to (i) the well-studied van Hove singularity of the twist bilayer and (ii) a Lifshitz transition that occurs when trigonally placed electron pockets decorate the strongly warped Dirac cone. Interestingly, we find that, while the electron-phonon scattering is dominated by layer symmetric flexural phonons in the small-angle limit, at large angles, in contrast, it is the layer antisymmetric flexural mode that is most important. We examine the role of a layer perpendicular electric field finding that it affects the conductivity strongly at low temperatures, whereas this effect is washed out by Fermi smearing at room temperatures.

Figure 1

The Brillouin zones of the two rotated single layers with rotation angle $\theta ={15}^{\circ }$. The red squares in panel (a) depict all states ${\mathrm{k}}^{\prime }$ of layer $\mu =+1$ that are coupled to a given state $\mathrm{k}$ (blue $\times$) in layer $\lambda =-1$, according to the coupling condition (6). The black hexagon depicts the reciprocal twist bilayer unit cell defined by these coupling vectors, here centered at the unrotated $K$ point. Panel (b) displays the reciprocal SLG lattice vectors ${\mathrm{G}}_j^{\left[\lambda \right]}$ that are used in the first star approximation; $\lambda =\pm 1$ is the layer index and $j=0,\pm 1$ labels the three equivalent $K$ points. The vectors ${\mathrm{G}}_0^{\left[\lambda \right]}=0$ are not visible. The first star approximation restricts the interlayer hopping from a given state $\mathrm{k}$ to only three states ${\mathrm{k}}^{\prime }=\mathrm{k}+{\mathrm{G}}_j^{\left[\lambda \right]}-{\mathrm{G}}_j^{\left[\mu \right]}$ which is depicted in panel (c) for $\mathrm{k}$ (blue $\times$) in layer $\lambda =-1$ and ${\mathrm{k}}^{\prime }$ (red squares) in layer $\mu =+1$. The smaller symbols depict the vectors $\mathrm{k}+{\mathrm{G}}_j^{\left[\lambda \right]}$ and ${\mathrm{k}}^{\prime }+{\mathrm{G}}_j^{\left[\mu \right]}$, presented to show that the three allowed hopping sites ${\mathrm{k}}^{\prime }$ derive from the three equivalent $K$ points. With the support of phonons (d), interlayer scattering from state $\mathrm{k}$ to any state ${\mathrm{k}}^{\prime }$ is possible via three different processes $j=0,\pm 1$ with corresponding phonon vectors ${\mathrm{q}}_j$ [see also Eq. (24) of the text]. We show two states $\mathrm{k}$ (blue $\times$) and ${\mathrm{k}}^{\prime }$ (red square) and, with the smaller symbols, how the three phonon wave vectors ${\mathrm{q}}_j$ derive from the vectors $\mathrm{k}+{\mathrm{G}}_j^{\left[\lambda \right]}$ and ${\mathrm{k}}^{\prime }+{\mathrm{G}}_j^{\left[\mu \right]}$.

Figure 2

Phonon dispersions of the low-energy phonon modes that we consider in this work. See Eqs. (16, 17, 18) for the phonon dispersions and Table 1 for the relevant parameters.

Figure 3

Transport recoupling of the bilayer at large angles. The conductivity $\sigma$ as a function of twist angle for a Fermi energy of ${\epsilon }_F=50\mathrm{meV}$ and for temperatures $T=10$ K to $T=300$ K as indicated. Only at low temperatures does the transport become independent of the twist angle, as may be seen in the $T=10$ K panel for $\theta >{20}^{\circ }$. Thus, in contrast to the ground-state electronic structure, only at low temperatures does the conductivity of the twist bilayer become layer decoupled at large angles. The crossover between the layer decoupled and layer coupled transport behavior is determined by an angle-dependent interlayer Bloch-Grüneisen temperature, below which the phonon bath does not possess sufficient momentum to scatter between the two Dirac cones of each layer that are separated in momentum space by $\mathrm{\Delta }K=8\pi /\left(3a\right)sin\theta /2$. Note that in this calculation the temperature-dependent Fermi smearing is switched off by setting $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}=-\delta ({\epsilon }_F-{\epsilon }_{\mathrm{p}})$ in Eq. (50) of the main text, and thus the conductivity reflects only scattering processes occurring at the Fermi energy.

Figure 4

Particle-hole asymmetry of the transport in the twist bilayer. Shown is the contribution to the resistivity ${\sigma }_{\left[\eta \right]}^{-1}$ from the six distinct phonon modes $\eta =(\sigma ,\nu )$ that the bilayer possesses, plotted as a function of the Fermi energy ${\epsilon }_F$ for temperature $T=50$ K and a rotation angle of $\theta ={20}^{\circ }$. In contrast to the case in single-layer graphene, or the Bernal-stacked bilayer, the transport is asymmetric, a fact that arises as the moiré field itself does not possess particle-hole symmetry. Note that the temperature-dependent Fermi smearing $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}$ of the electron states is not included, and thus the resulting transport reflects only scattering processes that take place at the Fermi energy.

Figure 5

Band velocity dominated transport regime. At intermediate angles, the transport of the twist bilayer is dominated by band structure effects that result from the hybridization of the two Dirac cones from each layer. To see this, compare the band velocity averaged over the Fermi surface $\left|\mathrm{v}\right|$ [panel (c)] with the conductivity [panel (b)]. Clearly, the dramatic changes in conductivity that, for example, occur at 88 and 194 meV for the $\theta =3^{\circ }$ system are the result of corresponding changes in the Fermi surface averaged band velocity (and similar for all other angles presented). These sudden transitions result from topological changes in the Fermi surface that occur at these energies, and both produce noticeable features in the density of states [see panel (a)]. For details and explanation of the changes in Fermi surface topology, see Sec. 4b and Fig. 6. Note that the temperature-dependent Fermi smearing is switched off by setting $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}=-\delta ({\epsilon }_F-{\epsilon }_{\mathrm{p}})$ in Eq. (50) of the main text, and thus the conductivity reflects only scattering processes occurring at the Fermi energy.

Figure 6

Fermi surfaces of the twist bilayer with rotation angle $\theta =3^{\circ }$ shown for four different energies ${\epsilon }_1=87$ meV, ${\epsilon }_2=89$ meV, ${\epsilon }_3=193$ meV, and ${\epsilon }_4=195$ meV. The color encodes the band velocity $|{\mathrm{v}}_{\mathrm{p}}|$ in units of the SLG Dirac band velocity $v_{\text{SLG}}$ while the black hexagons depict the twist bilayer reciprocal space unit cell [see also Fig. 1]. The Fermi surfaces should be compared to the $\theta =3^{\circ }$ conductivity and average band velocity displayed in Fig. 5 and illustrate the topological changes that underlie the two pronounced features in the conductivity indicated by the arrows in Fig. 5.

Figure 7

Importance of the twist bilayer wave functions in the small-angle regime. At small twist angles the clear correspondence between the conductivity and the Fermi surface averaged band velocity seen at larger angles no longer holds. To see this, compare the Fermi surface averaged velocities [panel (b)] with the conductivities [panel (a)] in the highlighted region. While the average band velocity of the $\theta =1^{\circ }$ bilayer in this region is somewhat higher than that for the $\theta =2^{\circ }$ bilayer, the conductivity of the former is $\approx 5$ times lower than the latter. This indicates that the bilayer wave functions now play a dominating role in determining the transport, consistent with the fact that in the small-angle limit the twist bilayer wave functions become qualitatively different from those of single-layer graphene and show features of charge localization [2, 18]. Note that the temperature-dependent Fermi smearing is switched off by setting $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}=-\delta ({\epsilon }_F-{\epsilon }_{\mathrm{p}})$ in Eq. (50) of the main text, and thus the conductivity reflects only scattering processes occurring at the Fermi energy.

Figure 8

Resistivity ${\sigma }_{\left[\eta \right]}^{-1}$ due to particular phonon modes $\eta =(\sigma ,\nu )$ plotted vs Fermi energy ${\epsilon }_F$ for temperature $T=50$ K and rotation angle $\theta =2^{\circ }$. The temperature-dependent Fermi smearing $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}$ of the electron states is not included.

Figure 9

Conductivity $\sigma$ of the twist bilayer as a function of Fermi energy ${\epsilon }_F$ for a temperature $T=50$ K and a series rotation angles $\theta$ that encompass both large- and small-angle cases. The temperature-dependent Fermi smearing $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}$ of the electron states is included, which results in a smearing out of the data presented in Figs. 5 and 7, in which the Fermi smearing is not included. Note that the $y$-axis scaling changes at $\sigma =0.2$ A/V.

Figure 10

The reduction of conductivity $\sigma$ with decreasing rotation angle $\theta$ of the twist bilayer, shown for a series of Fermi energies ${\epsilon }_1=10$ meV, ${\epsilon }_2=50$ meV, ${\epsilon }_3=100$ meV, ${\epsilon }_4=150$ meV, and ${\epsilon }_5=200$ meV at temperature $T=50$ K. The inset shows the same data but calculated at $T=300$ K.

Figure 11

Temperature dependence of the conductivity $\sigma$ of the twist bilayer for a rotation angle of $\theta =4^{\circ }$. The main plot shows the inverse conductivity (resistivity) at fixed Fermi energies ${\epsilon }_1=50$ meV and ${\epsilon }_2=157$ meV. The lines are presented only as a guide to the eye. The inset panel shows the conductivity vs Fermi energy for different temperatures; note the logarithmic scaling of the $y$ axis. In both plots, circles (squares) and continuous (dashed) lines represent the results of a calculation including (not including) the temperature-dependent Fermi smearing $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}$.

Figure 12

Conductivity $\sigma$ (top) and average band velocity $v$ (bottom) plotted versus Fermi energy ${\epsilon }_F$ for temperature $T=50$ K and the two rotation angles ${\theta }_1=1.7^{\circ }$ (left) and ${\theta }_2=4^{\circ }$ (right). The different colors correspond to different strengths of the perpendicular electric field: $E_0=0, E_1=59.7\text{mV}/\text{Å}, E_2=119.4\text{mV}/\text{Å}$, and $E_3=179.1\text{mV}/\text{Å}$. The temperature-dependent Fermi smearing $\partial f_{\mathrm{p}}^0/\partial {\epsilon }_{\mathrm{p}}$ of the electron states is included.

Figure 13

Fermi surfaces at ${\theta }_1=1.7^{\circ }, {\epsilon }_1=40$ meV and at ${\theta }_2=4^{\circ }, {\epsilon }_2=200$ meV for different strengths of the perpendicular electric field: $E_0=0, E_1=59.7\text{mV}/\text{Å}, E_2=119.4\text{mV}/\text{Å}$, and $E_3=179.1\text{mV}/\text{Å}$. The color encodes the band velocity $|{\mathrm{v}}_{\mathrm{p}}|$ in units of the SLG Dirac band velocity $v_{\text{SLG}}$, the black hexagons depict the twist bilayer reciprocal unit cell.