Comment on "Negative Landau damping in bilayer graphene"

TL;DR

This paper critically examines previous claims of plasmon instabilities in bilayer graphene with electron drift, demonstrating that such instabilities are artifacts caused by calculation errors and emphasizing the importance of physical effects like broken Galilean invariance.

Contribution

It clarifies that the predicted plasmon instability in bilayer graphene is an artifact and highlights the significance of physical factors like broken Galilean invariance in plasmon behavior.

Findings

The supposed plasmon instability is due to calculation errors.

Broken Galilean invariance suppresses plasmon instabilities.

Spatial dispersion of conductivity is crucial for accurate predictions.

Abstract

In [Phys. Rev. Lett. vol. 119, p. 133901 (2017)] it was argued that two parallel graphene layers in the presence of electron drift support unstable plasmon modes. Here we show that the predicted plasmon instability is an artifact of errors upon evaluation of graphene polarizability in the presence if electron drift. Crucial role of broken Galilean invariance and spatial dispersion of conductivity for suppression of plasmon instabilities is highlighted.

Equations

$$\Pi (\mathbf{q},\omega )=4\int \frac{d^2\mathbf{p}}{(2\pi {)}^2}\frac{\mathbf{q}{\nabla }_{\mathbf{p}}{f}_{\mathbf{p}}}{\omega +i\delta -{\mathbf{qv}}_{\mathbf{p}}};$$

$${\Pi }_{\pm }=\frac{2}{\pi }\frac{{\epsilon }_F}{(1\mp s\beta {)}^2}\left(\sqrt{1-{\beta }^2}-\frac{s\pm \beta }{\sqrt{s^2-1}}\right);$$

$${\tilde{\Pi }}_{\pm }\approx -\frac{{\epsilon }_F}{\pi }{\left(\frac{q{v}_0}{\omega \pm q{u}_0}\right)}^2$$

Full text

Comment on ”Negative Landau damping in bilayer graphene”

Dmitry Svintsov

Laboratory of 2D Materials’ Optoelectronics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

  

Victor Ryzhii

Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577 Japan

In Ref. Morgado and Silveirinha, 2017 it was argued that two parallel graphene layers in the presence of electron drift support unstable plasmon modes. Here we show that the predicted instability is an artifact of two errors in the calculation of graphene polarizability $\Pi({\bf q},\omega)$ (here $\bf q$ is the wave vector and $\omega$ is the frequency) (1) long-wavelength expansion of $\Pi$ in the short-wavelength domain $\omega\lesssim qv_{0}$ ($v_{0}$ is the electron Fermi velocity) (2) application of Doppler shift $\omega\rightarrow\omega-{\bf qu}_{0}$ for the evaluation of polarizability in the presence of drift with velocity ${\bf u}_{0}$ (Eq. (1) of [Morgado and Silveirinha, 2017]). The latter is invalid in graphene Duppen et al. (2016); Svintsov et al. (2013) due to the absence of Galilean invariance Abedinpour et al. (2011); Levitov et al. (2013).

Upon renouncement of long-wavelength expansion, graphene polarizability acquires a square-root singularity at the threshold of single-particle excitations (SPEs) ${\rm Re}\Pi({\bf q},\omega)\propto[\omega^{2}-q^{2}v_{0}^{2}]^{-1/2}$ Hwang and Das Sarma (2007); Ryzhii et al. (2007). This singularity keeps plasmon phase velocity above Fermi velocity, $\omega/q>v_{0}$ (see recent experiment Lundeberg et al. (2017)), and protects the waves from Landau damping. Below we show that such singularity persists at finite drift velocity. Therefore, the frequency of graphene plasmons cannot be pushed into the domain of negative Landau damping $\omega<qu_{0}$ by current, contrary to the case of massive two-dimensional systems Krasheninnikov and Chaplik (1980).

The proper calculation of $\Pi(q,\omega)$ exploits Lindhard (RPA) formula with drifting electron distribution function $f_{\bf p}=[1+e^{(pv_{0}-{\bf pu}_{0}-\varepsilon_{F})/T}]^{-1}$. We set $\hbar=k_{B}=1$ and restrict ourselves to the classical limit ($\omega\ll\varepsilon_{F}$) where the polarizability reads:

[TABLE]

here ${\bf v}_{\bf p}=v_{0}{\bf p}/p$ is the carrier velocity. The integral is evaluated analytically using the residue theorem Kukhtaruk et al. (2016). We present the final result for waves propagating parallel ($\Pi_{+}$) and anti-parallel ($\Pi_{-}$) to the drift direction:

[TABLE]

we have denoted $s=(\omega+i\delta)/qv_{0}$ and $\beta=u_{0}/v_{0}$ 111At finite temperature, Fermi energy should be replaced according to $\varepsilon_{F}\rightarrow T\ln(1+e^{\varepsilon_{F}/T})$. This expression is radically different from Doppler-shifted polarizability in the $q\rightarrow 0$ limit used in [Morgado and Silveirinha, 2017],

[TABLE]

The persistence of square-root singularity at $\omega=qv_{0}$ and the lack of Galilean invariance are apparent from Eq. (2). The singularity persists in full-quantum Lindhard formalism Duppen et al. (2016) and upon different assumptions about the form of distribution function (e.g. shifted Fermi disc model Sabbaghi et al. (2015)).

We substitute the polarization (2) into the dispersion relation for plasmons in double graphene layer (Eq. (2) of [Morgado and Silveirinha, 2017]) and observe that it has no unstable roots (Fig. 1), in contrast to the dispersion with approximated polarization (3). The proper plasmon frequencies retain outside of SPE domain at finite $u_{0}$ and thus have zero imaginary part. Inclusion of interband polarizability and/or electron collisions makes these modes decaying. We have verified that instabilities are absent at all accessible drift velocities $u_{0}<v_{0}$, Fermi energies, and propagation angles. The instabilities might re-appear in hydrodynamic regime where the singularity at $\omega=qv_{0}$ is removed Svintsov et al. (2013). However, calculations in this regime require the methods essentially different from used both in Ref. Morgado and Silveirinha, 2017 and here.

In conclusion, the plasmon instability predicted in [Morgado and Silveirinha, 2017] is an artifact of unjustified approximations to graphene polarizability. Accurate treatment shows that the collisionless electron plasma in graphene double layers is stable at arbitrary drift velocity.

This work was supported by Grant No. 16-19-10557 of the Russian Science Foundation.