Phonon-limited resistivity of multilayer graphene systems

We calculate the theoretical contribution to the doping and temperature ($T$) dependence of electrical resistivity due to scattering by acoustic phonons in Bernal bilayer graphene (BBG) and rhombohedral trilayer graphene (RTG). We focus on the role of nontrivial geometric features of the detailed, anisotropic $\mathrm{k}·\mathrm{p}$ band structures of these systems—e.g., Van Hove singularities, Lifshitz transitions, Fermi surface anisotropy, and band curvature near the gap—whose effects on transport have not yet been systematically studied. We find that these geometric features strongly influence the temperature and doping dependencies of the resistivity. In particular, the band geometry leads to a nonlinear $T$ dependence in the high-$T$ equipartition regime, complicating the usual $T^4$ to $T$ Bloch-Grüneisen crossover. Our focus on BBG and RTG is motivated by recent experiments in these systems that have discovered several exotic low-$T$ superconductivity proximate to complicated hierarchies of isospin-polarized phases. These interaction-driven phases are intimately related to the geometric features of the band structures, highlighting the importance of understanding the influence of band geometry on transport. While resolving the effects of the anisotropic band geometry on the scattering times requires nontrivial numerical solution, our approach is rooted in intuitive Boltzmann theory. We compare our results with recent experiments and discuss how our predictions can be used to elucidate the relative importance of various scattering mechanisms in these systems.

Figure 1

We depict the lattice structure of Bernal bilayer graphene (top) and rhombohedral trilayer graphene (bottom). The left side of the image shows top-down views of the $xy$ plane, labeling atoms with their layer number ($\{1,2\}$) and sublattice index ($\{A,B\}$). The right side of the figure shows the stacking from a cross-section view.

Figure 2

Overview of phonon scattering in Bernal bilayer and rhombohedral trilayer layered graphene systems. The top row [(a) and (e)] depicts qualitatively distinct kinematically allowed scattering manifolds for different Bloch states at the same energy. In these figures, the black curve depicts a Fermi surfaced (near the hole-doped VHS), the pink dot denotes a reference Bloch state, and the colored points denote the points in $\mathrm{k}$ space that the reference state can scatter too while conserving conservation of energy and momentum. These are the “scattering manifolds” (Sec. 3b), which depend on the geometry of the system. The coloring of the scattering manifolds encodes the transitions rates from the reference state. These plots demonstrate the nontrivial kinematics and geometry at play in scattering in these systems. The middle layers [(b), (c), (f), and (g)] shows the density of states of the two systems, with labels showing how the Fermi surface geometry changes as the sample is doped. The bottom layer [(d) and (h)] gives the central results of this work, the scaling of the resistivity with temperature in various regions of $n\text{−}T$ space. This is captured by a heat map of $dln\left[\rho \right(n,T\left)\right]/dlnT.$ We emphasize the clear connection between the scaling behavior and the geometric features in the density of states. The doted line gives the naive $T_{\mathrm{BG}}^{*}$ calculated with Eq. (1) for an isotropic system. We see that at sufficiently large dopings, the color contours begin to follow the $\sqrt{\left|n\right|}$ profile traced out by the black dotted $T_{\mathrm{BG}}^{*}$ line, reflecting the fact that at large dopings, the Fermi surface becomes roughly circular. However, our system exhibits a surprising suppression of $T_{\mathrm{BG}}^{*}$ as doping is decreased and the Fermi surface qualitatively changes. These results are calculated with an interlayer potential of $\mathrm{\Delta }=0.07\text{eV}$, and should be compared with Fig. 3, which treats the zero-field case. The analogous results for a simple Dirac cone (gapped and ungapped) are given in Fig. 4 for further comparison.

Figure 3

Resistivity scaling with $T$ due to phonon scattering in ungapped BBG and RTG, as in the absence of the interlayer potential, to be compared with Fig. 2. As in Fig. 2, the top layers [(a), (b), (d), and (e)] plot the density of states of the system for various doping levels, labeled with Fermi surface geometries. The bottom row [(c) and (f)] provides a heat map of $dln\left[\rho \right]/dlnT$ over $n\text{−}T$ space, mapping out the various regimes of resistivity scaling. As in the case of the gapped systems (Fig. 2), we see that the BG transition is strongly affected by the band geometry. Though this is more subtle without the applied field, we still see the effects clearly in the RTG case, which still exhibits an annular Fermi surface over a small doping window. As before, the dotted line gives the expected BG crossover for an isotropic system; for the ungapped systems plotted here, this estimate is quite accurate as long as the sample is sufficiently doped.

Figure 4

Resistivity scaling with $T$ due to phonon scattering in simple gapped (right) and ungapped (left) Dirac cones, to be compared with Figs. 2 and 3. As in those figures, the bottom row provides a heat map of $dln\left[\rho \right]/dlnT$ over $n\text{−}T$ space. The top row plots the density of states of the system for various doping levels. The dotted line gives $T_{\mathrm{BG}}^{*}$ [Eq. (1)].

Figure 5

We plot a heat map of $ln\left[\rho \right]$ over doping density and temperature for the BBG (left), RTG (middle), and Dirac (right) systems for both the gapless (top) and gapped (bottom) cases. For BBG and RTG, the gapped systems correspond to a displacement field $\mathrm{\Delta }=0.07\text{eV}$, and the gapped Dirac system has $M=0.05\text{eV}$. Several features are prominent. The dark blue in the lower corners shows the universal features of the BG transition. All systems but the ungapped Dirac cone display density dependence throughout the high-$T$ regime, with a significant increase in resistivity near charge neutrality. This is especially prominent for the gapped RTG and BBG systems, where the resistivity spikes to $\rho \approx 3000\mathrm{\Omega }$.

Figure 6

Accurate approximate calculations of resistivity in the equipartition regime can be made efficiently with the protocol discussed in Sec. 3c. Here we plot these approximate results over a large range of $T$, extending the scope of Fig. 5. We see that the large spike in resistivity near charge neutrality is a relatively low-$T$ behavior and that the resistivity decreases at higher temperatures, as $T$ get high enough to excite carriers in the conduction band. We emphasize that even at high-$T$, the resistivity does not return to simple linear scaling, but instead asymptotes to a constant value. (See also Figs. 7 and 16.)

Figure 7

Resistivity data for hole-doped Bernal bilayer (top) and rhombohedral trilayer (bottom) graphene stacks, evaluated with the displacement field at $\mathrm{\Delta }=0.07\text{eV}$. Resistivity is given in ohms on a linear scale. The leftmost two columns give the results of our full numerical calculation for the resistivity of the two systems at various doping levels up to 30 and $120\text{K}$, respectively. The third column gives the same resistivity curves extended to $800\text{K}$, making use of the equipartition assumption discussed in Sec. 3d. The far right columns indicate the doping levels of the curves shown in each row. From the full low-$T$ results (left) we may extract the effective BG crossover. We see that for large dopings, $\rho \left(T\right)$ exhibits a BG-EP crossover temperature $T_{\mathrm{BG}}^{*}$ as high as $40\text{–}60\text{K}$; however, there is a sharp drop in $T_{\mathrm{BG}}^{*}$ to around $20\text{K}$ upon the Lifshitz transition to the annular Fermi surface. The $T_{\mathrm{BG}}^{*}$ continues to drop as we approach charge neutrality, dropping as low as $5\text{–}10\text{K}$. We note that there is no sharp or discontinuous behavior at the Van Hove singularities. The high-$T$ equipartition results (center) show how band curvature effects lead to nonlinearity in $\rho (n,T)$ and how the effects can complicate the BG crossover. We see also that for dopings close to charge neutrality, we should expect a large spike in resistivity at moderate temperatures $\left(100\text{–}300\text{K}\right)$.

Figure 8

We plot the kinematically allowed scattering manifolds for phonon scattering on a Dirac cone, given by Eq. (10). The Fermi surface at $\mu =-0.25\text{eV}$ is plotted in black and a reference point on the Fermi surface is identified with a pink dot. The colored points mark the set of k-space points that the reference point can scatter too while conserving energy and momentum. The color coding on the scattering manifold are proportional to the scattering rate between the two points. The figure is shown for descending temperatures: 300, 100, 30, and $10\text{K}$. The Bloch-Grüneisen transition is demonstrated by the fact that the 300 and $100\text{K}$ figures (top) only differ quantitatively by the scale of the color bar, while they are qualitatively distinct from the lower-temperature versions (bottom).

Figure 9

We plot the logarithm of the relaxation lengths $\left\{l_{\mathrm{k}}\right\}$ for a Dirac cone band structure for various temperatures. We fix the chemical potential at $\mu =-0.25\text{eV}$ and perform the calculation at $T=10$ (top), 30 (middle), and $100\text{K}$ (bottom). The far left column shows the relaxation lengths plotted over energy, while the central and rightmost columns give a heat map of the relaxation lengths in momentum space for the bottom (center) and top (right) bands, respectively. We emphasize that at low T, states near the Fermi level become long-lived. Further, states near the Dirac point are always long-lived due to a vanishing scattering manifold.

Figure 10

We plot heat maps of the relaxation length in $k$ space for the Bernal bilayer band structure. We set $\mu =-0.066$ (left) and $-0.058\text{eV}$ (right). For each $\mu$, we do the calculation for $T=10$ (top), 30 (middle), and $100\text{K}$ (bottom). and We see that at low $T$, the specific geometry of the Fermi surface is very important to relaxation, but that this information tends to get washed out at higher $T$.

Figure 11

We plot the equipartition “length constants” over a wide range of energy for the various systems under study. The top row gives Bernal bilayer graphene, the middle row gives rhombohedral trilayer graphene, and the bottom row gives standard Dirac cone graphene. Gapless systems are on the left and gapped systems are on the right. With these values stored we can efficiently compute the resistivity of these systems up to very high temperatures using Eq. (19), though our results will miss the low-$T$ BG physics, as discussed in Sec. 3c. This is how we generate the high-$T$ results in Figs. 7 and 6. We emphasize that the ungapped Dirac cone graphene, which has diverging $c_{\mathrm{k}}$ near charge neutrality due to a vanishing scattering manifold, is the outlier here. All other systems we consider have band curvature effects near charge neutrality that more than compensate for the vanishing scattering manifolds and suppress the divergence of $c_{\mathrm{k}}$.

Figure 12

We plot the “$g$ function,” $\left[g\right(\varepsilon \left)\right]$, defined in Eqs. (21) and (22) for Bernal bilayer (top), rhombohedral trilayer (center), and Dirac cone graphene (bottom). We plot $g\left(\varepsilon \right)$ for both ungapped (left) and gapped (right) cases. These graphs demonstrate clearly why single ungapped Dirac cone graphene has such a robust high-$T$ linear resistivity and why all the other systems display nonlinear resistivity effects at high-$T$.

Figure 13

We plot heat maps of $dln\left[\rho \right(n,T\left)\right]/dlnT$ for the various systems using the resistivities calculated in the high-$T$ equipartition calculation (the data from Fig. 6). This should be compared with Figs. 2, 3, 4. Since this is based on resistivity data from the equipartition regime, it does not contain any BG physics, and any deviations from linear scaling are due to band curvature effects. We emphasize that these figures show linear-in-$T$ scaling for dopings away from charge neutrality, but then demonstrate a flattening of the $\rho \left(T\right)$. We further emphasize that the flattening seen near the van Hove singularities is present here as well, indicating it is an effect of band geometry in the thermal averaging and not a transition in the nature of the scattering.

Figure 14

We plot resistivity data over temperature for hole-doped and electron-doped Bernal bilayer systems in the absence of a displacement field ($\mathrm{\Delta }=0$). This should be compared with the Bernal bilayer data in Figs. 7 and 16. As with Fig. 7, the leftmost two columns give the results of our full numerical calculation for the resistivity of the two systems at various doping levels up to 30 and $120\text{K}$, respectively. The third column gives the high-$T$ results in the EP regime. The far-right column denotes the doping values corresponding to the resistivity curves.

Figure 15

We plot resistivity data over temperature for hole-doped and electron-doped rhombohedral trilayer systems in the absence of a displacement field ($\mathrm{\Delta }=0$). This should be compared with the rhombohedral trilayer data in Figs. 7 and 16. As with Fig. 7, the leftmost two columns give the results of our full numerical calculation for the resistivity of the two systems at various doping levels up to 30 and $120\text{K}$, respectively. The third column gives the high-$T$ results in the EP regime. The far-right column denotes the doping values corresponding to the resistivity curves.

Figure 16

We plot resistivity data over temperature for electron-doped samples, complementing the hole-doped data presented in Fig. 7. We emphasize that the results are analogous to the hole-doped side. As with Fig. 7, the leftmost two columns give the results of our full numerical calculation for the resistivity of the two systems at various doping levels up to 30 and $120\text{K}$, respectively. The third column gives the high-$T$ results in the EP regime. The far-right column denotes the doping values corresponding to the resistivity curves.