Theory of remote phonon scattering in top-gated single-layer graphene

We extend the theory of interfacial plasmon-phonon scattering to top-gated single-layer graphene. As with bottom-gated graphene, interfacial plasmon-phonon (IPP) modes are formed from the coupling between the graphene plasmon and the surface polar phonon modes in the top and bottom oxides. We study the effect of the top oxide thickness on dynamic screening and electron-IPP coupling. The remote phonon-limited electron mobility ${\mu }_{\mathrm{RP}}$ and electron scattering rates in a HfO${}_2$-covered, SiO${}_2$-supported single-layer graphene are computed at various electron densities $n$ for different dielectric thickness $t_{\mathrm{ox}}$. We find that ${\mu }_{\mathrm{RP}}$ is much more dependent on $t_{\mathrm{ox}}$ at low $n$. They also agree with the experimentally estimated room temperature ${\mu }_{\mathrm{RP}}$ from Zou et al. [Phys. Rev. Lett. 105, 126601 (2010)]. The electron density dependent ${\mu }_{\mathrm{RP}}$ is predicted to be between 5500 and $24200$ cm${}^2$V${}^{-1}$s${}^{-1}$ from $n={10}^{12}$ to ${10}^{13}$ cm${}^{-2}$ at 300 K.

Figure 1

Basic model used in our calculation. We assume that the graphene is an infinitely thin layer suspended in the middle of the vacuum gap between a semi-infinite bottom oxide layer and a top oxide layer of thickness $t_{\mathrm{ox}}$.

Figure 2

Dispersion relation of coupled interfacial plasmon-phonon system with $n={10}^{12}{\mathrm{cm}}^{-2}$ for a substrate of SiO${}_2$ and 20 nm of HfO${}_2$ top dielectric. The five IPP branches are drawn in thick lines, while the SPP branches are shown in dotted lines. The graphene plasmon dispersion (${\omega }_p$) is plotted with a dot-dashed line. The cutoff $Q=Q_c$ is determined from the crossing of $\omega =v_{F}Q$ and the plasma branch ${\omega }_p\left(Q\right)$. In the limits $Q\to 0$ and $Q\to \infty$, the IPP branches converge to the pure phonon and plasmon branches. In between, they are a mix of the pure branches.

Figure 3

(a) Plot of $M_Q^{\left(l\right)}{(\Omega /A)}^{1/2}$ for $n={10}^{12}{\mathrm{cm}}^{-2}$ in SiO${}_2$-supported graphene with 20 nm HfO${}_2$ top dielectric. (b) The same plot but at small $Q$. $M_Q^{\left(\mathrm{SPP}1\right)}$ (solid square) and $M_Q^{\left(\mathrm{IPP}2\right)}$ (open diamond) are highlighted. As $Q\to 0$, the two converge. Similarly, $M_Q^{\left(\mathrm{SPP}3\right)}$ (solid circle) and $M_Q^{\left(\mathrm{IPP}4\right)}$ (open triangle) converge as $Q\to 0$.

Figure 4

Surface polar phonon dispersion in SiO${}_2$-supported SLG with (a) 2 nm of and (b) 200 nm of HfO${}_2$ top dielectric at $n={10}^{12}$ cm${}^{-2}$. The SPP modes (${\omega }_Q^{\left(\mathrm{SPP}1\right)}$, ${\omega }_Q^{\left(\mathrm{SPP}2\right)}$, ${\omega }_Q^{\left(\mathrm{SPP}3\right)}$, and ${\omega }_Q^{\left(\mathrm{SPP}4\right)}$) are drawn with solid lines. The dispersion of SiO${}_2$-supported and HfO${}_2$-supported SLG with no top dielectric are plotted in dashed lines. $K_F$ is the Fermi wave vector.

Figure 5

Electron-SPP coupling coefficients in SiO${}_2$-supported SLG with (a) 200 nm, (b) 20 nm, and (c) 2 nm of HfO${}_2$ top dielectric at $n={10}^{12}$ cm${}^{-2}$. The uncoupled electron-SPP coupling coefficients ($M_Q^{\left(\mathrm{SPP}1\right)}$, $M_Q^{\left(\mathrm{SPP}2\right)}$, $M_Q^{\left(\mathrm{SPP}3\right)}$, and $M_Q^{\left(\mathrm{SPP}4\right)}$) are drawn with dashed lines and solid symbols. The dispersion of SiO${}_2$-supported and HfO${}_2$-supported SLG with no top dielectric are plotted in solid lines and open symbols. $K_F$ is the Fermi wave vector.

Figure 6

Plasmon dispersion at (a) $n={10}^{11}$ cm${}^{-2}$, (b) ${10}^{12}$ cm${}^{-2}$, and (c) ${10}^{13}$ cm${}^{-2}$ for $t_{\mathrm{ox}}=2$, 5, 10, 20, and 200 nm plotted in solid lines. In each subfigure, the lowest (highest) branch corresponds to $t_{\mathrm{ox}}=2$ nm ($t_{\mathrm{ox}}=2$ nm). The electron energy dispersion is drawn with dotted lines. The dashed lines show the Landau cutoff ($Q_c$) at the intersection of the different plasmon dispersion curves and the electron energy dispersion. The leftmost (rightmost) line corresponds to $t_{\mathrm{ox}}=2$ nm ($t_{\mathrm{ox}}=200$ nm).

Figure 7

Plot of (a) electron-SPP and (b) electron-IPP coupling coefficients at $n={10}^{11}$ cm${}^{-2}$ for $t_{\mathrm{ox}}=2$ nm. The corresponding electron-SPP and electron-IPP coupling coefficients plots for $t_{\mathrm{ox}}=200$ nm are shown in (c) and (d), respectively. The coupling coefficients without Landau damping are plotted in dashed lines in (b) and (d). The dotted line in (b) and (d) indicates the position of the cutoff wave vector $Q_c$.

Figure 8

Plot of (a) electron-SPP and (b) electron-IPP coupling coefficients at $n={10}^{12}$ cm${}^{-2}$ for $t_{\mathrm{ox}}=2$ nm. Only a small fraction of the IPP modes are not Landau damped. The corresponding electron-SPP and electron-IPP coupling coefficients plots for $t_{\mathrm{ox}}=200$ nm are shown in (c) and (d), respectively. The coupling coefficients without Landau damping are plotted in dashed lines in (b) and (d). The dotted line in (b) and (d) indicates the position of the cutoff wave vector $Q_c$. In (d), a large fraction of the IPP modes are coupled to the plasmons because the onset of Landau is significantly delayed.

Figure 9

Plot of (a) electron-SPP and (b) electron-IPP coupling coefficients at $n={10}^{13}$ cm${}^{-2}$ for $t_{\mathrm{ox}}=2$ nm. The corresponding electron-SPP and electron-IPP coupling coefficients plots for $t_{\mathrm{ox}}=200$ nm are shown in (c) and (d), respectively. The coupling coefficients without Landau damping are plotted in dashed lines in (b) and (d). The dotted line in (b) and (d) indicates the position of the cutoff wave vector $Q_c$.

Figure 10

Plot of (a) ${\mu }_{\mathrm{SPP}}$ and (b) ${\mu }_{\mathrm{IPP}}$ at $n={10}^{11}$, ${10}^{12}$, and ${10}^{13}$ cm${}^{-2}$ for different $t_{\mathrm{ox}}$ values.

Figure 11

Plot of the (a) conductance and (b) mobility for the SPP and IPP remote phonon models. For $t_{\mathrm{ox}}=30$ nm, ${\sigma }_{\mathrm{IPP}}$ and ${\mu }_{\mathrm{IPP}}$ (${\sigma }_{\mathrm{SPP}}$ and ${\mu }_{\mathrm{SPP}}$) are drawn in solid (dashed) lines. For $t_{\mathrm{ox}}=5$ nm, ${\sigma }_{\mathrm{IPP}}$ and ${\mu }_{\mathrm{IPP}}$ (${\sigma }_{\mathrm{SPP}}$ and ${\mu }_{\mathrm{SPP}}$) are drawn in dash-dot (dotted) lines. The inset in (b) shows a zoomed-in plot of the mobility (${\mu }_{\mathrm{SPP}}$ and ${\mu }_{\mathrm{IPP}}$) for $t_{\mathrm{ox}}=5$ and 30 nm between $n=0$ to $2\times {10}^{12}$ cm${}^{-2}$.

Figure 12

Plot of ${\mu }_{\mathrm{IPP}}$ from $T=80$ to 400 K for $n={10}^{12}$ to ${10}^{13}$ cm${}^{-2}$ at intervals of $\Delta T=20$ K and $\Delta n={10}^{12}$ cm${}^{-2}$. ${\mu }_{\mathrm{IPP}}$ scales approximately as ${\mu }_{\mathrm{IPP}}\propto T^{-1.8}$ at $n={10}^{13}$ cm${}^{-2}$.

Figure 13

(a) Plot of the IPP-limited mobility (${\mu }_{\mathrm{IPP}}$) with (solid) and without (dot-dash) interband transitions for $t_{\mathrm{ox}}=30$ nm. Without interband transition, ${\mu }_{\mathrm{IPP}}$ is higher. (b) Plot of the increase in ${\mu }_{\mathrm{IPP}}$ ($\Delta {\mu }_{\mathrm{IPP}}$) with respect to the electron density $n$. The increase is maximum at around $n={10}^{12}$ cm${}^{-2}$.