Acoustic plasmons at the crossover between the collisionless and hydrodynamic regimes in two-dimensional electron liquids

TL;DR

This paper theoretically investigates acoustic plasmon modes in two-dimensional electron liquids, identifying signatures of the transition between collisionless and hydrodynamic regimes using near-field optical microscopy.

Contribution

It introduces a comprehensive non-local conductivity model that includes various collision effects and many-body interactions to analyze plasmon dispersion and damping.

Findings

Identification of key experimental signatures of the collisionless-hydrodynamic crossover.

Derivation of dispersion relations for acoustic plasmons including damping effects.

Analysis of coupling between plasmons and near-field probes in different regimes.

Abstract

Hydrodynamic flow in two-dimensional electron systems has so far been probed only by dc transport and scanning gate microscopy measurements. In this work we discuss theoretically signatures of the hydrodynamic regime in near-field optical microscopy. We analyze the dispersion of acoustic plasmon modes in two-dimensional electron liquids using a non-local conductivity that takes into account the effects of (momentum-conserving) electron-electron collisions, (momentum-relaxing) electron-phonon and electron-impurity collisions, and many-body interactions beyond the celebrated Random Phase Approximation. We derive the dispersion and, most importantly, the damping of acoustic plasmon modes and their coupling to a near-field probe, identifying key experimental signatures of the crossover between collisionless and hydrodynamic regimes.

Equations

$$-$$

$$\begin{array}{rl}& {\sigma }_{\mathrm{L}}(q,\omega )=\frac{i{\mathcal{D}}^{\ast }/\pi }{\omega +i\gamma +\frac{\omega +i\gamma +i{\gamma }_{\mathrm{ee}}}{2}\frac{{\mathcal{D}}^{\ast }}{\mathcal{D}}\frac{v_{\mathrm{F}}}{v_{\mathrm{F}}^{\ast }}\left[\sqrt{1-{\left(\frac{v_{\mathrm{F}}^{\ast }q}{\omega +i\gamma +i{\gamma }_{\mathrm{ee}}}\right)}^2}-1\right]-\frac{1}{2}\frac{{\mathcal{D}}^{\ast }}{\mathcal{D}}\frac{K}{K^{\ast }}\frac{v_{\mathrm{F}}^2{q}^2}{\omega }}.\end{array}$$

$${\sigma }_{\mathrm{L}}^{\mathrm{h}}(q,\omega )=\frac{i{\mathcal{D}}^{\ast }/\pi }{\omega +i\gamma +q^2\left(i{\nu }^{\ast }-\frac{{\mathcal{D}}^{\ast }}{\bar{n}m\mathcal{D}{K}^{\ast }\omega }\right)},$$

$${ϵ}_{\mathrm{L}}(q,\omega )=1-{\Lambda }^{-1}\frac{(-i)\pi q^2{v}_{\mathrm{F}}^2{\sigma }_{\mathrm{L}}(q,\omega )}{2\omega \mathcal{D}},$$

$$\Lambda =\frac{C}{e^2\mathcal{N}},$$

$$S_{\mathrm{c}}$$

$${\Gamma }_{\mathrm{c}}$$

$$S_{\mathrm{h}}$$

$${\Gamma }_{\mathrm{h}}$$

$${\eta }_z(\omega )\approx \frac{3\pi |Z|c^3[\mathrm{Re}(q_{\mathrm{p}}){]}^3}{\overline{ϵ}{\omega }^3}\times \{\begin{array}{l}d\mathrm{Re}(q_{\mathrm{p}})e^{-2\mathrm{Re}(q_{\mathrm{p}})z}z>0\\ [d\mathrm{Re}(q_{\mathrm{p}}){]}^{-1}0>z>-d,\end{array}$$

Full text

Introduction.—In electron systems a collective charge mode exists at frequency above the threshold for intra-band electron-hole excitations. This mode is called “plasmon” giuliani_vignale_book ; pines_nozieres_book and is particularly useful for technological applications in the case of two-dimensional (2D) electron systems. In this case indeed plasmons are gapless modes typically falling in the mid-infrared fei_nature_2012 ; chen_nature_2012 ; woessner_natmat_2015 or Terahertz (THz) ju_nat_nano_2011 ; alonso_gonzalez_natnano_2017 ; lundeberg_science_2017 frequency ranges.

In recent years plasmons in 2D materials grigorenko_natphot_2012 ; basov_science_2016 ; low_natmat_2017 such as graphene have attracted a great deal of attention because of their ability to confine light on length scales much shorter than the free-space wavelength lundeberg_science_2017 ; alcaraz_science_2018 , their long lifetimes woessner_natmat_2015 ; ni_nature_2018 , and their gate tunability yan_nat_nano_2012 ; fei_nature_2012 ; chen_nature_2012 ; woessner_natmat_2015 .

Due to the long-range nature of the bare electron-electron (e-e) interaction, plasmons in 2D materials on a dielectric substrate have a long-wavelength “unscreened” dispersion of the form giuliani_vignale_book ; pines_nozieres_book $\omega\propto\sqrt{q}$, where $\omega$ is the angular frequency and $q$ is the in-plane wave vector. Conversely, if the long-range part of the e-e interaction is screened by e.g. a nearby conducting gate, the plasmon dispersion is modified into an acoustic one (see e.g. Ref. santoro_prb_1988, ), $\omega\propto q$.

Acoustic plasmons (APs) alonso_gonzalez_natnano_2017 ; lundeberg_science_2017 ; santoro_prb_1988 ; principi_ssc_2011 ; principi_prb_2018 are particularly interesting because they can achieve larger mode confinement with respect to their unscreened counterpart. This happens for two reasons. First, an AP is more confined in the vertical direction due to the presence of the metallic gate alcaraz_science_2018 , with the largest part of the electromagnetic energy density being localized between the gate and the 2D material. Second, due to the screening of the long-range part of the Coulomb interaction, APs are softer (because the restoring force is reduced) and carry high values of $q$, for a given value of $\omega$. This allows the study of interesting quantum non-local effects lundeberg_science_2017 , which become important when the plasmon dispersion gets close to the boundary of the intra-band electron-hole continuum located at $\omega=v_{\rm F}^{*}q$, $v_{\rm F}^{*}$ being the quasiparticle velocity. With the term “quasiparticle” velocity we mean the Fermi velocity as dressed by electron-electron (e-e) interactions giuliani_vignale_book ; pines_nozieres_book ; kotov_rmp_2012 . The same jargon and notation will be used below for the Drude weight ${\cal D}^{*}$, the density-of-states at the Fermi energy ${\cal N}^{*}$, etc. The same quantities without the “$*$” symbol, e.g. $v_{\rm F}$, ${\cal D}$, ${\cal N}$, etc, will denote instead the non-interacting counterparts.

In 2D conducting materials of extremely high electronic quality, such as graphene encapsulated in hexagonal Boron Nitride wang_science_2013 , e-e interactions induce, in the intermediate-to-high-temperature regime, the so-called hydrodynamic transport regime. In this regime, e-e collisions are so frequent that they can establish a local thermal quasi-equilibrium. This happens when the e-e mean-free-path $\ell_{\rm ee}\equiv v_{\rm F}^{*}\tau_{\rm ee}$ (here $\tau_{\rm ee}$ is the e-e scattering time giuliani_prb_1982 ; quian_prb_2005 ; li_prb_2013 ; polini_normale_2016 ; principi_prb_2016 ) is much shorter than both the mean-free-path for momentum-relaxing collisions with phonons or impurities $\ell\equiv v_{\rm F}^{*}\tau$ and the characteristic wavelength torre_prb_2015 ; bandurin_science_2016 $1/q$ of external perturbations. In the ac regime, we should also require torre_prb_2015 ; bandurin_science_2016 ; pellegrino_prb_2017 the angular frequency of the perturbation $\omega$ to be much smaller than the e-e scattering rate $1/\tau_{\rm ee}$. Transport signatures of hydrodynamic behaviour have been found in different high-quality materials like single- and bi-layer graphene bandurin_science_2016 ; krishna_kumar_natphys_2017 ; crossno_science_2016 ; bandurin_natcomm_2018 , GaAs/AlGaAs heterostructures dejong_prb_1995 ; braem_arxiv_2018 , and ${\rm PdCoO}_{2}$ moll_science_2016 .

The rate $\gamma\equiv 1/\tau$ of momentum non-conserving collisions with impurities and phonons and the e-e scattering rate $\gamma_{\rm ee}\equiv 1/\tau_{\rm ee}$ define several regimes in the $q$-$\omega$ plane, which are sketched in Fig. 1.

In the hydrodynamic regime landaufluidmechanics and at the level of linear-response theory, the electron liquid can be described by the continuity equation $i\omega n(\bm{r},\omega)=\nabla\cdot\bm{J}(\bm{r},\omega)$, $n(\bm{r},\omega)$ being the deviation of the particle density from its equilibrium value $\bar{n}$ and $\bm{J}(\bm{r},\omega)$ the particle current, and the Navier-Stokes equation torre_prb_2015 ; bandurin_science_2016 ; pellegrino_prb_2017

[TABLE]

Here, ${\bm{E}}(\bm{r},\omega)$ is the electric field, $e$ is the elementary charge, $m\equiv\hbar k_{\rm F}/v_{\rm F}$ is the bare effective mass, $k_{\rm F}$ being the Fermi wave vector, $K^{*}=[\bar{n}\partial P/\partial\bar{n}]^{-1}$ is the compressibility pines_nozieres_book ; giuliani_vignale_book ; asgari_adp_2014 , $P=P(\bar{n})$ being the pressure, $\nu^{*}$ is the kinematic viscosity torre_prb_2015 ; bandurin_science_2016 ; pellegrino_prb_2017 ; landaufluidmechanics , $\mathcal{D}^{*}$ ($\mathcal{D}$) is the Drude weight of the interacting abedinpour_prb_2011 ; levitov_prb_2013 (non-interacting) electron system. A derivation of Eq. (1) is given in Sect. I of Ref. note_supplementary, .

In this work we identify signatures of the transition between the hydrodynamic ($\omega\ll\gamma_{\rm ee}$) and collisionless ($\omega\gg\gamma_{\rm ee}$) regimes in the dispersion and, most importantly, the damping of AP modes. In the case of single-layer graphene (SLG) at room temperature, for example, $\tau_{\rm ee}\approx 0.15~{}{\rm ps}$ at typical carrier densities principi_prb_2016 ($\bar{n}=1.0\times 10^{12}~{}{\rm cm}^{-2}$, say) and the crossover is expected to occur in the THz range. Our work is structured as follows. We first introduce the two main ingredients of our theory: the non-local longitudinal conductivity $\sigma_{\rm L}(q,\omega)$—Eq. (2)—and the interaction potential $v_{q,\omega}$, both calculated in the long-wavelength limit. We then find AP modes, which are described by an equation of the form $q_{\rm p}=q_{\rm p}(\omega)$ for every real frequency $\omega$. Here $q_{\rm p}$ is a complex wave vector $q_{\rm p}={\rm Re}(q_{\rm p})+i{\rm Im}(q_{\rm p})$, which gives access to both dispersion and damping. Finally, we analyze the coupling of these modes to a near-field probe and discuss our results.

The non-local conductivity from Landau kinetic theory.—The response of a 2D electron liquid to an external scalar potential can be calculated using Landau kinetic equation pines_nozieres_book ; giuliani_vignale_book for a normal Fermi liquid, which governs the response of the quasiparticle distribution function to slowly-varying electromagnetic fields conti_prb_1999 ; pellegrino_prb_2017 . Its use is justified when the excitation wavelength is sufficiently long compared to the inverse of the Fermi wave vector $k_{\rm F}$, and when the excitation energy $\hbar\omega$ is sufficiently small compared to the Fermi energy $E_{\rm F}$, and to the energy of the lowest inter-band excitation $E_{\rm g}$.

As detailed in Sects. I-II of Ref. note_supplementary, , the linearized kinetic equation can be solved by using a simple ansatz pellegrino_prb_2017 . After lengthy but straightforward algebra, we find the following expression for the longitudinal non-local conductivity note_conductivity , which controls the current response to an electric field parallel to ${\bm{q}}$:

[TABLE]

Here, $\mathcal{D}=\pi e^{2}\bar{n}/m$ ($K={\cal N}/\bar{n}^{2}$) is the Drude weight (compressibility) of the non-interacting system, ${\cal N}=N_{\rm f}m/(2\pi\hbar^{2})$ being the density-of-states at the Fermi energy and $N_{\rm f}$ the number of fermion flavors (e.g. $N_{\rm f}=4$ for graphene). In Landau theory of Fermi liquids pines_nozieres_book ; giuliani_vignale_book , $K/K^{*}=(v_{\rm F}^{*}/v_{\rm F})(1+F^{\rm s}_{0})$ and $\mathcal{D}^{*}/\mathcal{D}=(v_{\rm F}^{*}/v_{\rm F})(1+F_{1}^{\rm s})$, where $F^{\rm s}_{0(1)}$ is the spin-symmetric dimensionless Landau parameter in the $s$ ($p$) angular momentum channel pines_nozieres_book ; giuliani_vignale_book ; note_landau_parameters . The many-body corrections $v_{\rm F}^{*}/v_{\rm F}$, $K^{*}/K$, and $\mathcal{D}^{*}/\mathcal{D}$ can be calculated from approximated theories kotov_rmp_2012 ; asgari_adp_2014 ; abedinpour_prb_2011 ; lundeberg_science_2017 and are fundamental for a quantitative interpretation of experimental data since, for example, $v_{\rm F}^{*}/v_{\rm F}\approx 1.3$ kotov_rmp_2012 , $K^{*}/K\approx 0.8$ lundeberg_science_2017 , and $\mathcal{D}^{*}/\mathcal{D}\approx 1.5$ abedinpour_prb_2011 in SLG at densities on the order of $10^{12}~{}{\rm cm}^{-2}$.

In deriving Eq. (2) we made the following assumptions. i) The momentum-conserving and the momentum-relaxing collisions are described by one parameter each, i.e. differences between the relaxation times of the different angular components of the distribution function ledwith_arxiv_2017 and the difference between $\tau_{\rm ee}$ and the viscosity time $\tau_{\rm v}$ principi_prb_2016 are neglected. ii) Only the zeroth- and first-order, spin symmetric, Landau parameters $F^{\rm s}_{0(1)}$ are considered. Higher-angular-momentum Landau parameters $F^{\rm s}_{l}$ with $l\geq 2$ are typically smaller, unless the system is highly correlated. We used these assumptions to derive the simplest yet highly-non-trivial model for the non-local longitudinal conductivity. However, the technique we used in our derivation, based on analytical inversion of tridiagonal matrices lorentzen_book ; note_supplementary , easily allows the introduction of different scattering rates for the different harmonics of the distribution function ledwith_arxiv_2017 as well as higher-order Landau parameters.

Eq. (2) is the first important result of this work because, despite its simplicity, it i) embodies a wealth of physical effects, including many-body effects beyond the Random Phase Approximation (RPA), ii) allows us to span the whole frequency range, from the hydrodynamic to the collisionless regime, and iii) is valid with no assumptions on the relative values of the parameters, other than the ones mentioned previously for the applicability of Landau kinetic equation. In what follows we will anyway assume that $\gamma_{\rm ee}\gg\gamma$ because the hydrodynamic regime is relevant only in this case.

We now look at four special limits of Eq. (2). i) We first set $q=0$, i.e. we consider the local conductivity. In this case, Eq. (2) reduces to a Drude-like formula with a renormalized Drude weight $\mathcal{D}^{*}$ and a damping rate $\gamma$ induced solely by momentum-non-conserving collisions. The e-e collision rate $\gamma_{\rm ee}$ appears at order $q^{2}$. Note that e-e interactions fully disappear from $\sigma_{\rm L}(0,\omega)$ in a Galilean invariant electron system where ${\cal D}^{*}={\cal D}$ because in this case  pines_nozieres_book ; giuliani_vignale_book $v^{*}_{\rm F}/v_{\rm F}=1/(1+F^{\rm s}_{1})$. ii) Second, expanding to second order in $|v_{\rm F}^{*}q/(\omega+i\gamma+i\gamma_{\rm ee})|$ the square root in the denominator of Eq. (2) and taking the limit $\omega\ll\gamma_{\rm ee}$, we obtain the hydrodynamic non-local conductivity principi_prb_2016

[TABLE]

where $\nu^{*}\equiv\mathcal{D}^{*}v_{\rm F}^{*}v_{\rm F}/[4\mathcal{D}(\gamma_{\rm ee}+\gamma)]$. Ignoring many-body renormalizations, our result for $\nu^{*}$ reduces to the “classical” formula for the viscosity of an electron gas steinberg_pr_1958 ; pellegrino_prb_2017 , while for Galileian invariant systems it reduces to the expression given in Ref. conti_prb_1999 with $F_{2}^{\rm s}=0$. The quantity $\sigma_{\rm L}^{\rm h}(q,\omega)$ can be obtained directly by using Eq. (1) coupled to the continuity equation. iii) Third, if both many-body renormalizations and e-e collisions are neglected we recover the response function used in Ref. giuliani_prb_1984 to discuss the effect of diffusion (i.e. electron-impurity collisions) on 2D unscreened plasmons. iv) Finally, if the scattering rates $\gamma$ and $\gamma_{\rm ee}$ are both sent to zero, the long-wavelength ($q\ll k_{\rm F}$) limit of the collisionless conductivity of a 2D electron system stern_prl_1967 with parameters renormalized by e-e interactions is recovered.

The screened e-e interaction.—The dispersion of plasmons in a material depends also on the interaction potential $v_{q,\omega}$ between charges in the material itself. This quantity relates the Fourier transform of $n(\bm{q},\omega)$ to the Fourier transform of the induced (i.e. Hartree) scalar potential $V_{\rm ind}(\bm{q},\omega)$, i.e. $V_{\rm ind}(\bm{q},\omega)=v_{q,\omega}n(\bm{q},\omega)$. In 2D materials the interaction potential is strongly affected by the presence of nearby dielectrics or conductors. The interaction potential for generic layered structures can be easily calculated tomadin_prl_2015 ; alonso_gonzalez_natnano_2017 . For example, for a graphene sheet encapsulated between hBN slabs of different thickness and in the presence of a metallic gate has been calculated in Ref. alonso_gonzalez_natnano_2017, . For low frequencies (i.e. low compared to all, e.g. phonon, features in the dielectric functions of the nearby dielectrics) and long wavelengths (i.e. for $q$ much smaller than the inverse of the dielectric thickness) $v_{q,\omega}$ can be safely replaced by its limit $v_{q,\omega}\approx\lim_{q,\omega\to 0}v_{q,\omega}\equiv e^{2}/C$, $C$ being the capacitance per unit area of the structure (see Sect. III of Ref. note_supplementary, ). If we consider a structure made of a perfectly conducting gate parallel to the 2D electron system and separated along the $\hat{\bm{z}}$-direction by a dielectric spacer of thickness $d$ and dielectric tensor $\bar{\bm{\epsilon}}$, the capacitance per unit area is $C=\bar{\epsilon}_{zz}/(4\pi d)$, where ${\bar{\epsilon}}_{zz}$ denotes the tensor component along the $\hat{\bm{z}}$ direction. For all realistic experimental geometries alonso_gonzalez_natnano_2017 ; lundeberg_science_2017 using e.g. graphene encapsulated in hBN, the plasmon wavelength is much longer than the thickness of the whole device and, therefore, the replacement $v_{q,\omega}\to e^{2}/C$, i.e. the so-called local capacitance approximation (LCA), is fully justified in the THz regime where the hydrodynamic-ballistic crossover takes place. All results reported in Figs. 2 and 3 refer to SLG encapsulated in hBN.

AP velocity and damping.—Mathematically, plasmons are zeroes of the longitudinal dielectric function pines_nozieres_book ; giuliani_vignale_book $\epsilon_{\rm L}(q,\omega)$ of the 2D electron system, $\epsilon_{\rm L}(q,\omega)=1+iq^{2}v_{q,\omega}\sigma_{\rm L}(q,\omega)/(e^{2}\omega)$. Using the LCA the latter becomes

[TABLE]

where

[TABLE]

is a dimensionless parameter that characterizes how much the e-e interaction is screened by the nearby dielectric environment. Numerical values of this important parameter are given in Sect. IV of Ref. note_supplementary for graphene.

The plasmon equation $\epsilon_{\rm L}(q,\omega)=0$ with $\epsilon_{\rm L}(q,\omega)$ as in Eq. (4) can be solved for the plasmon wave vector $q_{\rm p}$. We find $q_{\rm p}(\omega)=(\omega/S_{\omega})\sqrt{1+2i\Gamma_{\omega}/\omega}$, where $S_{\omega}$ and $\Gamma_{\omega}$ are real functions of the frequency representing the velocity and the damping of the mode respectively. These two functions can be calculated analytically (see Sect. V of Ref. note_supplementary, ) and the result is shown in Fig. 2. We are now interested in the asymptotic behavior of $S_{\omega}$ and $\Gamma_{\omega}$ for $\omega\gg\gamma_{\rm ee}$ (collisionless limit) and $\omega\ll\gamma_{\rm ee}$ (hydrodynamic limit). In the former we find

[TABLE]

while in the latter we find

[TABLE]

Eqs. (6)-(9) are the second important result of this work. In particular, Eqs. (8)-(9) can be obtained by directly solving Eq. (4) with the conductivity given in Eq. (3) and ignoring terms of order higher than one in $\omega/\gamma_{\rm ee}$.

From these results one can easily understand why achieving high values of the screening parameter $\Lambda$ is of pivotal importance to observe the crossover from the collisionless to the hydrodynamic regime. Indeed, in the limit $\Lambda\to 0$ we have $S_{\rm h}=S_{\rm c}=v_{\rm F}\sqrt{\mathcal{D}^{*}/(2\mathcal{D}\Lambda)}$ and $\Gamma_{\rm h}=\Gamma_{\rm c}=\gamma/2$. Therefore, for small values of $\Lambda$ no crossover can be observed as $S_{\rm h}=S_{\rm c}$ and $\Gamma_{\rm h}=\Gamma_{\rm c}$, and the damping of the AP mode is completely controlled by momentum-relaxing collision, with $\gamma_{\rm ee}$ dropping out of the problem.

On the other hand, for $\Lambda\gg 1$ the velocities in the two regimes converge to distinct values. The velocity of the AP mode in the collisionless regime tends to a value which is close (ignoring here, for the sake of simplicity, many-body corrections) to the Fermi velocity, $S_{\rm c}\to v_{\rm F}$, while in the hydrodynamic regime it converges to the speed of sound in a neutral Fermi liquid phan_arxiv_2013 ; lucas_prb_2018 , i.e.  $S_{\rm h}\to v_{\rm F}\sqrt{(\mathcal{D}^{*}K)/(2\mathcal{D}K^{*})}\approx v_{\rm F}/\sqrt{2}$. The situation is even more dramatic for the damping $\Gamma_{\omega}$. In the hydrodynamic regime, and for $\Lambda\gg 1$, we have $\Gamma_{\rm h}\approx\gamma/2+\omega^{2}/[4(\gamma+\gamma_{\rm ee})]$, while $\Gamma_{\rm c}\approx\gamma+\gamma_{\rm ee}$, implying that the extrinsic dissipation controlled by $\gamma$ becomes twice more efficient with respect to the $\Lambda\ll 1$ case and a new damping mechanism controlled by $\gamma_{\rm ee}$ kicks in. In Fig. 2 we show the impact of $\Lambda$ on the real and imaginary parts of $q_{\rm p}$. When frequency increases, the damping starts to acquire a significant contribution from e-e collisions. This shows up as viscous dissipation in the hydrodynamic regime—see the second term in Eq. (9). In this regime, indeed, the contribution to the damping is proportional to $q^{2}$ and therefore to $\omega^{2}$, since we are probing the damping along the AP dispersion. When frequency is further increased above $\gamma_{\rm ee}$, the e-e contribution to the damping saturates to a finite value. Note that since in hydrodynamic electron liquids $\gamma_{\rm ee}\gg\gamma$, this contribution can be the dominant one even with moderate values of $\Lambda$ and lead to a significant increase of the imaginary part of $q$, as shown in Fig. 2(b).

Coupling efficiency to a near-field probe.—In order to design experiments that are able to probe the collisionless to hydrodynamic crossover with light, it is important also to consider the coupling strength of APs to an external field. We characterize the coupling to an external near-field probe using the quantity $\eta_{z}(\omega)$ defined by the ratio between the power fed into the AP mode by a dipole source of strength $p$ and frequency $\omega$, located at an height $z$, with its axis perpendicular to the 2D liquid, and the power radiated by the same source in vacuum, given by Larmor’s formula (see Sect. VI of Ref. note_supplementary ).

In Fig. 3 we show the numerically-calculated dependence of $\eta_{z}(\omega)$ on frequency for different vertical positions $z$ of the dipole for the aforementioned case of a 2D material separated from a perfect metal located at $z=-d$ by a dielectric spacer.

For long wavelengths, and assuming small dissipation we can approximate $\eta_{z}(\omega)$ as

[TABLE]

where $Z\equiv[{\rm Re}(q_{\rm p})\partial_{q}\epsilon_{\rm L}(q,\omega)|_{q=q_{\rm p}}]^{-1}=-\{2+q_{\rm p}\partial_{q}\log[\sigma_{\rm L}(q_{\rm p},\omega)]\}^{-1}\approx-1/2$. Since $q_{\rm p}d$ is a small number, we see that the AP modes are much more coupled to a dipole located between the material and the gate. This happens because the electric field of AP modes is mainly concentrated in the spacer region woessner_acsphot_2017 . This suggest that to couple efficiently to these modes, structures specially designed for launching plasmons should be put in the region where the field is concentrated.

In summary, we have studied the dispersion and damping of APs in a 2D electron liquid at the crossover between the hydrodynamic and collisionless regimes. We have found that, in the presence of strong screening by an external gate, both the velocity and the damping of AP modes are enhanced in the collisionless regime, with the enhancement being more dramatic for the damping. If the screening is strong enough, i.e. if $\Lambda>1$, well defined APs with a phase velocity smaller than the Fermi velocity $v_{\rm F}$ (but larger than the sound velocity $\approx v_{\rm F}/\sqrt{2}$) are allowed in the hydrodynamic regime.

Notice that some properties of plasmons in 2D Fermi liquids have been discussed in two recent publications, Refs. lucas_prb_2018, and svintsov_prb_2018, . However, the former mainly focusses on the difference between long-range and short-range interactions, and considers only the many-body compressibility renormalization. In the latter work, effects beyond RPA are neglected, and so are momentum non-conserving processes. We have, however, demonstrated that the latter processes are important to correctly describe the plasmon damping and introduce the possibility of having overdamped excitations at low frequencies and long wavelengths, as shown in Eqs. (7) and (9). The non-linear electromagnetic response of a Dirac electron fluid at the crossover between the collisionless and hydrodynamic regimes has been discussed in Ref. sun_pnas_2018, .

Acknowledgements.—This work has been sponsored by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 785219—“Graphene Core2” and via the European Research Council (ERC) grant agreement No. 786285. B.V.D. is supported by a post-doctoral fellowship of the Flemish Science Foundation (FWO-Vl). F.H.L.K. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0522), support by Fundacio Cellex Barcelona, Generalitat de Catalunya through the CERCA program, and the Mineco grant Plan Nacional (FIS2016-81044-P) and the Agency for Management of University and Research Grants (AGAUR) 2017 SGR 1656. We thank Niels Hesp and Hanan Hertzig Sheinfux for useful discussions.