Semiclassical Boltzmann transport theory for graphene multilayers

We calculate the conductivity of arbitrarily stacked multilayer graphene sheets within a relaxation time approximation, considering both short-range and long-range impurities. We theoretically investigate the feasibility of identifying the stacking order of these multilayers using transport measurements. For relatively clean samples, the conductivities of the various stacking configurations depend on the carrier density as a power law for over two decades. This dependence arises from a low-density decomposition of the multilayer band structure into a sum of chiral Hamiltonians. For dirty samples, the simple power-law relationship no longer holds. Nonetheless, identification of the number of layers and stacking sequence is still possible by careful comparison of experimental data to the results presented here.

Figure 1

(a) Schematic illustration of three types of stacking arrangements, labeled A, B and C. The honeycomb lattice of a single sheet has two triangular sublattices, labeled $\alpha$ and $\beta$. (b) Each added layer cycles around this stacking triangle in either the right-handed or the left-handed direction. Reversals of the direction of this rotation tend to increase the number of low-energy pseudospin doublets.

Figure 2

Graphene conductivity assuming short-range scatterers. For clean samples, ($n_{\mathrm{imp}}\lesssim {10}^{10}\hspace{0.5em}{\mathrm{cm}}^{-2}$), the conductivity ${\sigma }_{\mathrm{EMT}}(n)$ has about two decades of power-law dependence. (a) Conductivity values (solid lines) for monolayer, bilayer, and ABC trilayer graphene each follow different power laws (dashed lines). (b) Conductivity values (solid lines) for the different stacking sequences for tetralayer graphene also have different power laws (dashed lines). This indicates that for short-range impurities, in most cases transport measurements can be used to identify the number of graphene layers. (c, d) For dirty samples, where $n_{\mathrm{imp}}={10}^{12}\hspace{0.5em}{\mathrm{cm}}^{-2}$, the density dependence is no longer given by a power law. Solid lines show the conductivity after using effective medium theory to average over the disorder-induced carrier density fluctuations. Dashed lines are the results before such averaging. Although the conductivity does not show power-law dependence on carrier density, the transport properties still depend strongly on the number of layers and their relative stacking order. Therefore, transport measurements could still be used to identify the type of graphene multilayer.

Figure 3

Graphene conductivity (solid lines) assuming charged impurity scattering. (a) For clean samples, long-range Coulomb scatterers show a similar power-law dependence (dashed lines) of conductivity on carrier density for monolayer, bilayer, and ABC trilayer graphene. (b) ABCA and ABAB stacking orders of tetralayer graphene show a power-law dependence (dashed lines) similar to that of monolayer and bilayer graphene. We conclude that the low-density transport properties for many graphene multilayers look quite similar under long-range scattering, making it difficult to distinguish between them in a transport measurement. (c) For dirty samples, the transfer curves for monolayer, bilayer, and trilayer graphene are each quite different and can be easily distinguished. Note the logarithmic scale on the $y$ axis.) (d) For dirty tetralayer graphene, the conductivity depends strongly on the stacking sequence, and transport measurements could distinguish between the various types of stacking.

Figure 4

Effective medium theory result (solid lines) for the conductivity assuming short-range disorder as a function of dimensionless density $n/n_{\mathrm{imp}}$ for the $J$-chiral Hamiltonian [see Eq. (B9)] for $J=2$, 3, and $4$. The results assume an impurity density of $n_{\mathrm{imp}}={10}^{11}$ cm${}^{-2}$. Note that $\sigma /{\sigma }_{\min }$ is independent of $d_{\mathrm{sc}}$ because $d_{\mathrm{sc}}$ appears as a multiplicative factor in Eq. (B5). Dashed lines show the minimum conductivity (i.e., $\sigma /{\sigma }_{\min }=1$) and the Boltzmann conductivity [Eq. (B5)] without performing the EMT average. Note that the puddle regime (marked with arrows) gets larger with increasing $J$.

Figure 5

Chirality factor $F(\varphi )=|\langle s,\mathbf{k},\varphi =0|s,{\mathbf{k}}^{\text{'}},\varphi \rangle |{}^2$ is determined by the wave-function overlap between initial and final states. (Top) ABA stacked trilayer graphene and (bottom) ABCB stacked tetralayer graphene for $E_{\mathrm{F}}=0.1\hspace{0.5em}\mathrm{eV}$. The bands are labeled according to the $J$-chiral decomposition reported in Table . For intraband scattering, the chirality factor at a low Fermi energy is given by Eq. (B3). However, the interband chirality factor is identically 0 for scattering between the $J=1$ and the $J=2$ bands of the ABA trilayer and strongly suppressed for scattering between the $J=1$ and the $J=3$ bands of the ABCB tetralayer. As discussed in the text, we believe that this suppression is unphysical for generic short-range impurities. Insets: Tight-binding band structure determined numerically by solving Eq. (1) as discussed in the text; dashed lines indicate the Fermi energy.