Phonon-limited carrier mobility and resistivity from carbon nanotubes to graphene

Under which conditions do the electrical transport properties of one-dimensional (1D) carbon nanotubes (CNTs) and 2D graphene become equivalent? We have performed atomistic calculations of the phonon-limited electrical mobility in graphene and in a wide range of CNTs of different types to address this issue. The theoretical study is based on a tight-binding method and a force-constant model from which all possible electron-phonon couplings are computed. The electrical resistivity of graphene is found in very good agreement with experiments performed at high carrier density. A common methodology is applied to study the transition from one to two dimensions by considering CNTs with diameter up to 16 nm. It is found that the mobility in CNTs of increasing diameter converges to the same value, i.e., the mobility in graphene. This convergence is much faster at high temperature and high carrier density. For small-diameter CNTs, the mobility depends strongly on chirality, diameter, and the existence of a band gap.

Figure 1

Comparison of the resistivity of graphene vs temperature for different hole densities. (a) Present work (solid lines) and experimental data (dotted lines), which correspond to the temperature-dependent part of the resistivity measured in Ref. [24]. (b) Present work (solid lines) and DFT calculations [23].

Figure 2

Phonon-limited mobility in CNTs with different chirality [armchair $(n,n)$, zigzag $(n,0)$, chiral $(2n,n)$, and chiral $(4n,n)]$ compared to graphene (green line) vs (a) diameter at 300 K and $n_{2\text{D}}={10}^{12}{\mathrm{cm}}^{-2}$, (b) carrier density at 300 K and $d\approx 3.5$ nm, and (c) temperature at $n_{2\text{D}}=13.6\times {10}^{12}{\mathrm{cm}}^{-2}$ and $d\approx 3.5$ nm.

Figure 3

Ratio between the mobility calculated by considering phonons with energy up to $\hslash \omega$ and the mobility calculated with all phonons (a) for armchair CNTs; (b) for zigzag, $(2n,n),(4n,n)$ metallic CNTs, and graphene; and (c) for zigzag, $(2n,n)$ semiconducting CNTs. $d\approx 3.5$ nm, $n={10}^{12}{\mathrm{cm}}^{-2}$.

Figure 4

Ratio between the mobility calculated by considering phonons with energy below 130 meV and the mobility calculated with all phonons vs temperature. $d\approx 3.5$ nm, $n={10}^{12}{\mathrm{cm}}^{-2}$.

Figure 5

Phonon-limited mobility in CNTs vs diameter for $T=100$ K (dashed lines), 300 K (dash-dotted lines), and 500 K (dotted lines). Carrier density: (a) ${10}^{12}{\mathrm{cm}}^{-2}$, (b) $13.6\times {10}^{12}{\mathrm{cm}}^{-2}$. The horizontal lines indicate the mobility for graphene. The horizontal scale is smaller in (b) than in (a), highlighting the faster convergence of all the mobilities at high carrier density.

Figure 6

Phonon dispersion of graphene. The red lines are the frequencies calculated using the 4NN force constant model, and gray lines are DFT-LDA calculations using abinit.

Figure 7

Comparison between our simulations (green dotted lines) and experimental data [37] (red markers) for the resistivity of graphene at carrier density varying from 1 (top) to $3\times {10}^{12}{\mathrm{cm}}^{-2}$ (bottom) with a step of $0.5\times {10}^{12}{\mathrm{cm}}^{-2}$. The blue horizontal dotted lines represent the impurity-limited resistivity ${\rho }_0$.

Figure 8

${\rho }_{\text{ph}}$ (blue makers) and ${\rho }_{\text{SO}}$ (red markers) vs temperature at electron densities 1 (empty markers) and $13.6\times {10}^{12}{\mathrm{cm}}^{-2}$ (filled markers).

Figure 9

Band structures of metallic armchair (a)–(d), metallic zigzag (e)–(h), and semiconducting zigzag (i)–(l) CNTs with diameters of $3.5$ [(a), (e), and (i)], 6 [(b), (f), and (j)], 10 [(c), (g), and (k)], and 16 nm [(d), (h), and (l)]. The 2D effective carrier density is fixed at ${10}^{12}{\mathrm{cm}}^{-2}$. The horizontal line represents the Fermi level at 300 K.

Figure 10

Evolution of the Fermi level at 300 K in the band structure of armchair (a)–(d), metallic zigzag (e)–(h), and semiconducting zigzag (i)–(l) with a diameter of $3.5$ nm and 2D effective carrier density of 1 [(a), (e), and (i)], 6 [(b), (f), and (j)], 13 [(c), (g), and (k)], and $20\times {10}^{12}{\mathrm{cm}}^{-2}$ [(d), (h), and (l)].

Figure 11

Mobility $\left(\mu \right)$ multiplied by the carrier density $\left(n_{2\text{D}}\right)$ vs $n_{2\text{D}}$ in graphene and CNTs ($T=300$ K, $d\approx 3.5$ nm). The same data but presented as $\mu$ vs $n_{2\text{D}}$ are shown in Fig. 2.