Intrinsic plasmons in two-dimensional Dirac materials

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Intrinsic plasmons in two-dimensional Dirac materials

S. Das Sarma and Qiuzi Li

Phys. Rev. B 87, 235418 – Published 17 June 2013

Abstract

We consider theoretically, using the random phase approximation (RPA), low-energy intrinsic plasmons for two-dimensional (2D) systems obeying Dirac-like linear chiral dispersion with the chemical potential set precisely at the charge neutral Dirac point. The “intrinsic Dirac plasmon” energy has the characteristic $\sqrt{q}$ dispersion in the 2D wave vector $q$ , but vanishes as $\sqrt{T}$ in temperature for both monolayer and bilayer graphene. The intrinsic plasmon becomes overdamped for a fixed $q$ as $T \to 0$ since the level broadening (i.e., the decay of the plasmon into electron-hole pairs due to Landau damping) increases as $1 / \sqrt{T}$ as temperature decreases, however, the plasmon mode remains well defined at any fixed $T$ (no matter how small) as $q \to 0$ . We find the intrinsic plasmon to be well defined as long as $q < \frac{k_{B} T}{e^{2}}$ . We give analytical results for low and high temperatures, and numerical RPA results for arbitrary temperatures, and consider both single- and double-layer intrinsic Dirac plasmons. We provide extensive comparison and contrast between intrinsic and extrinsic graphene plasmons, and critically discuss the prospects for experimentally observing intrinsic Dirac point graphene plasmons.

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Received 3 May 2013

DOI:https://doi.org/10.1103/PhysRevB.87.235418

Authors & Affiliations

S. Das Sarma and Qiuzi Li

Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

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Vol. 87, Iss. 23 — 15 June 2013

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Figure 1
Plasmon dispersion of intrinsic MLG. (a) Presents the plasmon dispersion in the high-temperature regime ( $k_{B} T ≫ ℏ v_{F} q$ ). The dashed and solid lines correspond to the analytical results [given in Eq. (7)] and the numerical results, respectively. (b) Numerical results for the plasmon dispersion as a function of $ℏ v_{F} q / (k_{B} T)$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 2
Plasmon damping rate of intrinsic MLG. (a) Presents the plasmon damping rate in the high-temperature regime ( $k_{B} T ≫ ℏ v_{F} q$ ). The dashed and solid lines correspond to the analytical results [given in Eq. (8)] and the numerical results, respectively. (b) Numerical results for $ℏ γ / (k_{B} T)$ as a function of $ℏ v_{F} q / (k_{B} T)$ . (c) $γ_{p} / ω$ as a function of $ℏ v_{F} q / (k_{B} T)$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 3
Results for extrinsic MLG with $q / k_{F} = 0.01$ . (a) Presents the plasmon dispersion in the low-temperature regime. The dashed and solid lines correspond to the analytical results [given in Eq. (11)] and the numerical results, respectively. (b) Numerical results for $ω_{p} / E_{F}$ as a function of $T / T_{F}$ . (c) Numerical results for $γ / E_{F}$ as a function of $T / T_{F}$ . (d) Numerical results for $γ / ω_{p}$ as a function of $T / T_{F}$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 4
Results for extrinsic MLG with $q / k_{F} = 0.5$ . (a) Numerical results for $ω_{p} / E_{F}$ as a function of $T / T_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $T / T_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $T / T_{F}$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 5
Results for extrinsic MLG with $q / k_{F} = 1.0$ . (a) Numerical results for $ω_{p} / E_{F}$ as a function of $T / T_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $T / T_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $T / T_{F}$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 6
Plasmon dispersion as a function of $q / k_{F}$ for extrinsic MLG. (a) and (b) present $ω_{p} / E_{F}$ in the long-wavelength limit. The dashed and solid lines correspond to the analytical results [given in Eq. (12)] and the numerical results, respectively. (a) is for $T = 0$ and (b) is for $T / T_{F} = 0.1$ . (c) and (d) present $ω_{p} / E_{F}$ for different temperatures. (c) is for $κ = 1$ and (d) is for $κ = 5$ .Reuse & Permissions
Figure 7
Plasmon damping rate as a function of $q / k_{F}$ for extrinsic MLG. (a), (c) $γ / E_{F}$ as a function of $q / k_{F}$ for different temperatures. (b), (d) Present $γ / ω_{p}$ as a function of $q / k_{F}$ . (a), (b) For $κ = 1$ ; (c), (d) for $κ = 5$ . Note that the legend applies to all subfigures.Reuse & Permissions
Figure 8
Results for extrinsic MLG with $T / T_{F} = 10.0$ . (a) Presents plasmon dispersion in the long-wavelength limit. The dashed and solid lines correspond to the analytical results [given in Eq. (17)] and the numerical results, respectively. (b) Plasmon damping rate versus $q / k_{F}$ . The dashed and solid lines correspond to the analytical results [given in Eq. (18)] and the numerical results, respectively. (c) $γ / ω_{p}$ versus $q / k_{F}$ . The dashed and solid lines correspond to the analytical results [given in Eq. (17)] and the numerical results, respectively.Reuse & Permissions
Figure 9
Results for extrinsic MLG. (a), (c) Present plasmon dispersion versus $T / T_{F}$ . The dashed and solid lines correspond to the analytical results [given in Eq. (17)] and the numerical results, respectively. (b), (d) Present plasmon damping rate versus $T / T_{F}$ . (a), (b) For $q / k_{F} = 0.1$ ; (c), (d) for $q / k_{F} = 0.5$ . The dashed and solid lines correspond to the analytical results [given in Eq. (18)] and the numerical results, respectively.Reuse & Permissions
Figure 10
Results for extrinsic MLG with $q / k_{F} = 0.5$ over a wide $T / T_{F} = 0 - 10$ range. (a) Numerical results for $ω_{p} / E_{F}$ as a function of $T / T_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $T / T_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $T / T_{F}$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 11
Results for extrinsic MLG with $q / k_{F} = 1.0$ over a wide $T / T_{F} = 0 - 10$ range. (a) Numerical results for $ω_{p} / E_{F}$ as a function of $T / T_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $T / T_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $T / T_{F}$ . The dashed and solid lines correspond to $κ = 1$ and 5, respectively.Reuse & Permissions
Figure 12
Results for extrinsic MLG with $κ = 5$ . (a) Numerical results for $ω_{p} / E_{F}$ as a function of $q / k_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $q / k_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $q / k_{F}$ . The solid, dashed, dotted, and dotted-dashed lines correspond to $T / T_{F} = 0, 1, 5$ , and 10, respectively.Reuse & Permissions
Figure 13
Results for extrinsic MLG with $κ = 1$ . (a) Numerical results for $ω_{p} / E_{F}$ as a function of $q / k_{F}$ . (b) Numerical results for $γ / E_{F}$ as a function of $q / k_{F}$ . (c) Numerical results for $γ / ω_{p}$ as a function of $q / k_{F}$ . The solid, dashed, dotted, and dotted-dashed lines correspond to $T / T_{F} = 0, 1, 5$ , and 10, respectively.Reuse & Permissions
Figure 14
Results for an intrinsic double-layer graphene system in the small- $q$ regime with $κ = 1$ , $d = 300$ Å, and $T = 100$ K. (a) Plasmon dispersion versus $ℏ v_{F} q / (k_{B} T)$ . (b) Plasmon damping rate versus $ℏ v_{F} q / (k_{B} T)$ . The dashed and solid lines correspond to the analytical results [given in Eqs. (22, 23, 24, 25)] and the numerical results, respectively.Reuse & Permissions
Figure 15
Numerical results for an intrinsic double-layer graphene system. (a), (b) Plasmon dispersion versus $ℏ v_{F} q / (k_{B} T)$ ; (c), (d) plasmon damping rate versus $ℏ v_{F} q / (k_{B} T)$ ; (e), (f) $γ / ω_{p}$ versus $ℏ v_{F} q / (k_{B} T)$ . (a), (c), and (e) are for the optical plasmon mode. (b), (d), and (f) are for the acoustic plasmon mode. For $κ = 5$ , $d = 40$ Å, and $T = 20$ K, the acoustic plasmon mode $ω_{a p}$ is degenerate with the boundary of the single-particle excitation region. Note that the legend applies to all subfigures.Reuse & Permissions
Figure 16
Results for a mixed double-layer intrinsic-extrinsic graphene system with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.0$ , $d = 300$ Å, and $κ = 1$ . (a) $ω_{p} / E_{F}$ versus $q / k_{F}$ for $T / T_{F} = 0.1$ . (b) $γ / E_{F}$ versus $q / k_{F}$ for $T / T_{F} = 0.1$ . (c) $ω_{p} / E_{F}$ versus $T / T_{F}$ for $q / k_{F} = 0.01$ . (d) $γ / E_{F}$ versus $T / T_{F}$ for $q / k_{F} = 0.01$ . The dashed and solid lines correspond to the analytical results [given in Eqs. (30, 31, 32, 33)] and the numerical results, respectively.Reuse & Permissions
Figure 17
Numerical results for a mixed double-layer intrinsic-extrinsic graphene system over a wide $q / k_{F} = 0 - 1$ range with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.0$ , $d = 300$ Å, and $κ = 1$ . (a) $ω_{p} / E_{F}$ versus $q / k_{F}$ for different temperatures. (b) $γ / E_{F}$ versus $q / k_{F}$ for different temperatures. (c) $γ / ω_{p}$ versus $q / k_{F}$ for different temperatures. For $T / T_{F} = 0$ , the acoustic plasmon mode $ω_{a p}$ is degenerate with the boundary of the single-particle excitation region. Note that the legend applies to all subfigures.Reuse & Permissions
Figure 18
Numerical results for a mixed double-layer intrinsic-extrinsic graphene system over a wide $T / T_{F} = 0 - 1$ range with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.0$ , $d = 300$ Å, and $κ = 1$ . (a) $ω_{p} / E_{F}$ versus $T / T_{F}$ for different $q / k_{F}$ . (b) $γ / E_{F}$ versus $T / T_{F}$ for different $q / k_{F}$ . (c) $γ / ω_{p}$ versus $T / T_{F}$ for different $q / k_{F}$ . Note that the legend applies to all subfigures.Reuse & Permissions
Figure 19
Numerical results for a mixed double-layer intrinsic-extrinsic graphene system over a wide $q / k_{F} = 0 - 1$ range with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.0$ , $d = 40$ Å, and $κ = 5$ . (a) $ω_{p} / E_{F}$ versus $q / k_{F}$ for different temperatures. (b) $γ / E_{F}$ versus $q / k_{F}$ for different temperatures. (c) $γ / ω_{p}$ versus $q / k_{F}$ for different temperatures. For $T / T_{F} = 0$ , the acoustic plasmon mode $ω_{a p}$ is degenerate with the boundary of the single-particle excitation region. Note that the legend applies to all subfigures.Reuse & Permissions
Figure 20
Numerical results for a mixed double-layer intrinsic-extrinsic graphene system over a wide $T / T_{F} = 0 - 1$ range with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.0$ , $d = 40$ Å, and $κ = 5$ . (a) $ω_{p} / E_{F}$ versus $T / T_{F}$ for different $q / k_{F}$ . (b) $γ / E_{F}$ versus $T / T_{F}$ for different $q / k_{F}$ . (c) $γ / ω_{p}$ versus $T / T_{F}$ for different $q / k_{F}$ . Note that the legend applies to all subfigures.Reuse & Permissions
Figure 21
Results for a double-layer extrinsic graphene system with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.25$ , $d = 300$ Å. (a), (b) $ω_{p} / E_{F 1}$ versus $q / k_{F 1}$ . (c), (d) $ω_{p} / E_{F 1}$ versus $T / T_{F 1}$ . The dashed and solid lines correspond to the analytical results [given in Eqs. (34) and (35)] and the numerical results, respectively.Reuse & Permissions
Figure 22
Results for a double-layer extrinsic graphene system with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.25$ , $d = 300$ Å, and $κ = 1$ . (a) $ω_{p} / E_{F 1}$ versus $q / k_{F 1}$ for different temperatures. (b) $γ / E_{F 1}$ versus $q / k_{F 1}$ for different temperatures. (c) $γ / ω_{p}$ versus $q / k_{F 1}$ for different temperatures. Note that the legend applies to all subfigures.Reuse & Permissions
Figure 23
Results for a double-layer extrinsic graphene system with $n_{1} = 10^{12}$ cm $^{- 2}$ , $n_{2} / n_{1} = 0.25$ , $d = 300$ Å, and $κ = 1$ . (a) $ω_{p} / E_{F 1}$ versus $T / T_{F 1}$ for different $q / k_{F 1}$ . (b) $γ / E_{F 1}$ versus $T / T_{F 1}$ for different $q / k_{F 1}$ . (c) $γ / ω_{p}$ versus $T / T_{F 1}$ for different $q / k_{F 1}$ . Note that the legend applies to all subfigures.Reuse & Permissions
Figure 24
The temperature scale for the crossover of the collective mode from being intrinsic to being extrinsic. The dashed curve shows $T_{1}^{*} (n) = \sqrt{π n} ℏ v_{F} / k_{B}$ . The solid curve shows $T_{2}^{*} (n) = \sqrt{π n} \frac{ℏ v_{F}}{k_{B}} (1 + \frac{r_{s}}{8} \ln \frac{n_{c}}{n})$ including Fermi velocity renormalization due to electron-electron interactions (see Ref. 120 for details), where $r_{s} = 0.4$ ( $κ = 5$ ) and $n_{c} = 10^{15}$ cm $^{- 2}$ is the high-density cutoff.Reuse & Permissions

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