Phonon-Limited Mobility in $h$-BN Encapsulated $AB$-Stacked Bilayer Graphene

The weak acoustic phonon scattering in graphene monolayer leads to high mobilities even at room temperatures. We identify the dominant role of the shear phonon mode scattering on the carrier mobility in $AB$-stacked graphene bilayer, which is absent in monolayer graphene. Using a microscopic tight-binding model, we reproduce experimental temperature dependence of mobilities in high-quality boron nitride encapsulated bilayer samples at temperatures up to $\sim 200\mathrm{K}$. At elevated temperatures, the surface polar phonon scattering from boron nitride substrate contributes significantly to the measured mobilities of 15 000 to $20000{\mathrm{cm}}^2/\mathrm{Vs}$ at room temperature and carrier concentration $n\sim {10}^{12}{\mathrm{cm}}^{-2}$. A screened surface polar phonon potential for a dual-encapsulated bilayer and transferable tight-binding model allows us to predict mobility scaling with temperature and band gap for both electrons and holes in agreement with the experiment.

Figure 1

(a) Schematic illustration of the device structure. Carbon atoms are located at $\pm h_0/2$. The temperature dependence of the inverse (b) electron and (c) hole mobilities for various band gap values. The experimental data are shown by the marks and the BTE calculations by the lines, respectively. According to Matthiessen’s rule, the Coulomb-limited mobilities given in Table 1 were added to the calculated phonon-limited mobilities (see text).

Figure 2

(a) Electronic band structure of $AB$-stack bilayer graphene with a zero band gap. The band separation is depicted by ${\mathrm{\Delta }}_{12}$. (b) Phonon dispersion of the low-energy phonons (from bottom to top at $\mathrm{\Gamma }$ point): flexural phonon, transverse and longitudinal acoustic phonons, doubly degenerate shear phonons corresponding to sliding in lateral directions, out-of-plane breathing phonon mode. (c) Normalized spectra function $S(\hslash \omega )$ from Eq. (9) at room temperature for $n$-doped bilayer with a gap of $\mathrm{\Delta }=3.3\mathrm{meV}$ and concentration of $9\times {10}^{11}{\mathrm{cm}}^{-2}$. The integral of $S(\hslash \omega )$ gives the scattering rate, and integrals over specific energy windows give relative phonon contributions to scattering.

Figure 3

Phonon-limited mobility for (a) electrons and (b) holes as a function of band gap at room temperature and 150 K. The markers and solid curves represent experimental and theoretical mobilities, respectively. (c) and (d) depict room-temperature mobilities calculated for one type of scattering in the Boltzmann transport equation for electrons and holes, respectively.

Figure 4

(a) Overall experimentally measured and theoretically calculated electron mobilities as a function of temperature for $n$-doped bilayer with a gap of $\mathrm{\Delta }=3.3\mathrm{meV}$ and concentration $9\times {10}^{11}{\mathrm{cm}}^{-2}$. The intrinsic, SPP, shear, flexural, and residual phonon-limited mobilities are shown by the dashed curves from bottom to top (at room temperature), respectively. (b) and (c) show relative phonon contributions to the total mobility as a function of temperature for $\mathrm{\Delta }=3.3\mathrm{meV}$ and $\mathrm{\Delta }=42.4\mathrm{meV}$, correspondingly.