\section{Introduction}

Surface acoustic waves (SAWs) \cite{Landau1986} have been used for decades as a valuable scientific and technological tool.
In the context of electronics,
they are often excited in piezoelectric materials
\cite{Hutson1962,Morgan2010,Auld1990,Royer2000,Royer2000a,Newnham2004},
where the mechanical and electrical fields are coupled. In particular, they have been applied as experimental probes of the quantum Hall effects in
two-dimensional electron gases (2DEG) \cite{Wixforth1986}.

On the other hand, there is an increased interest in 2DEG since the isolation of graphene in 2004 and the production of other two-dimensional (2D) materials which followed it.
Due to their unusual character, the properties of graphene electrons have been intensively studied during the last decade
\cite{CastroNeto2009}. Although graphene on a substrate has received considerable attention, relatively few studies have been devoted to the case of graphene in contact with a piezoelectric material.
These include the propagation of surface acoustic waves on graphene \cite{Thalmeier2010}, some acoustoelectric effects \cite{Miseikis2012} \cite{Bandhu2013}, and the relaxation induced by the
surface acoustic wave quanta on graphene electrons \cite{Zhang2013}, among others.
Recently, it has been proposed that surface acoustic waves (SAW) can provide a
diffraction grating for the conversion of light into graphene plasmons \cite{Schiefele2013}.

The coupling of the piezoelectric SAW to the electrons in
a 2DEG or in graphene has been computed, within certain simplifying assumptions and for definite substrate crystal structures, in Ref. \cite{Simon1996,Knabchen1996}. The derivation of the electron-surface phonon interaction for a piezoelectric material has been performed only within a purely elastic Rayleigh wave approximation \cite{Knabchen1996} or for definite propagating directions \cite{Simon1996}. But these methods fail in stronger piezoelectric materials and for other crystal symmetries. For instance, the isotropic Rayleigh wave approximation in the case of lithium niobate leads to surface acoustic wave velocities about 15 \% too low and lacking the correct angular dependence \cite{Morgan2010,Farnell1970}, and this material is not the one with the largest electromechanical couplings at all. Moreover, the obtained vertex are expressed in terms of a matching constant whose physical interpretation is rather obscure, allowing for just order of magnitude estimates.

In the present work we calculate a general electron-phonon interaction [see Eq. (\ref{eqn:PAVertex})], which is expressed solely in terms of physical quantities characterizing the response of the substrate surface. We emphasize that all the quantities appearing in the vertex are both computable from linear piezo-elasticity theory and experimentally measurable. One of them, the {\it electromechanical coupling coefficient}, $K_R$, will turn out to be central to all computations, serving as a natural dimensionless parameter which provides the scale for the effect of the substrate piezoelectricity on the 2D electron system. Moreover, from very general considerations explained in Appendix \ref{app:PiezoElectricVertex}, we are able to provide bounds on its size: $0\le K_R<1$ [see Eq. (\ref{eqn:interactionsRatio})]. It is important to note that the vertex written here is derived within the framework of linear piezo-elastic theory, which means that its validity should be restricted to low amplitude, low frequency and long wavelength phenomena. Bulk modes are also left aside in this work. On the other hand, our study is not restricted to any approximation based on the symmetry or the piezoelectric softness of the substrate.

Equipped with the effective electron-electron interaction which results from taking into account the exchange of these acoustic phonons between the electrons in graphene (or other 2D materials), the question can be raised of whether these interactions might be attractive and, depending on some material parameters and the tunable electronic density of graphene, perhaps strong enough to generate electron Cooper pairing and superconductivity \cite{Mahan2013}. We can further ask whether such a superconductivity could be observed at temperatures attainable in a laboratory without the recurring to huge non gate-achievable doping levels, as predicted for intrinsic graphene phonons \cite{Einenkel2011} or for Kohn-Luttinger or electronic superconductivity in other graphene heterostructures with repulsive interactions \cite{Guinea2012}. From the interaction vertex derived in the present work, it can be shown that the relative size of the static phonon-mediated electron-electron interaction with respect to the original Coulomb repulsion turns out to be exactly $K_R^2<1$, because of the aforementioned general inequality. However, by applying the Eliashberg formalism to graphene \cite{Einenkel2011}, we are able to assess, in terms of $K_R$, the influence that these low-frequency and long-wavelength phonons have on possible BCS type instabilities. The conclusion is that  present piezoelectric materials are not able to either induce $s-$wave pairing by themselves or affect in a significant way any pairing instability which could be already present in graphene. We note, however, that this conclusion could be substantially changed in case new hard piezoelectric materials would be found.

\section{Electron-phonon interaction}
\begin{figure}
	\centering
	\includegraphics[width=0.9\linewidth]{./PiezoSAWSchema}
	\caption{A flat piezoelectric substrate with an interface to air or vacuum, with $\mathbf{b}=\left[\cos(\theta),\sin(\theta)\right]$ the propagation direction of the piezoelectric SAW.}
	\label{fig:PiezoSAWSchema}
\end{figure}

The system to be considered is depicted in Fig.~\ref{fig:PiezoSAWSchema}. The $z=0$ interfacial surface is supposed to be free of tension and, when acting as a substrate to a deposited graphene sheet (or any other 2D charged system), free of electrodes as well. However, by introducing a surface charge, one can express the 2D response of the piezoelectric substrate, the {\it piezoelectric surface permittivity}, as the ratio of an electric displacement to an electric field. To be precise, let us allow for a surface charge with a harmonic dependence along $\textbf{q}=q[\cos \theta,\sin\theta]$ (here $\textbf{R}=(x,y)$)
\begin{align}
	\sigma(\mathbf{R},t)=\sigma(\mathbf{q},\omega) e^{i (\textbf{q}\cdot\textbf{R}-\omega t)},
\label{eqn:surfaceChargeDef}
\end{align}
and, from linearity, all quantities evaluated at the surface have the same 2D space-time dependence. We use SI units throughout this work. Because the medium at $z>0$ has a dielectric constant $\varepsilon_\text{vac}$, then, from Poisson's equation it follows that
\begin{align}
	\varepsilon_{\text{vac}}= \frac{D_3^+(\mathbf{q},\omega)}{q \varphi(\mathbf{q},\omega)},
\label{eqn:vaccumPermittivity}
\end{align}
where $D_3^+$ is the normal to the surface Fourier component of the electric displacement field over the surface (on the $z \rightarrow 0^+$ side) and $\varphi$ is the electric potential, which is continuous because we do not allow for anything more singular than surface charges. The same ratio taken below the surface ($z \rightarrow 0^-$ ) allows us to introduce the (relative) {\it piezoelectric surface permittivity},
\begin{align}
	\tilde{\varepsilon}(\mathbf{q},\omega) := -\frac{1}{\varepsilon_{\text{vac}}}\frac{D_3^-(\mathbf{q},\omega)}{q \varphi(\mathbf{q},\omega)} \, ,
\label{eqn:piezoSurfacePermittivityDef}
\end{align}
which can also be straightforwardly expressed in terms of the surface impedance tensor \cite{Ingebrigsten1969}.
From Poisson's equation and Eqs. (\ref{eqn:surfaceChargeDef}-\ref{eqn:piezoSurfacePermittivityDef}) it follows
\begin{equation}\label{eqn:surfaceChargePotentialRelation}
	\sigma(\mathbf{q},\omega)=D_3^+(\mathbf{q},\omega)-D_3^-(\mathbf{q},\omega)=q \varphi(\mathbf{q},\omega) [1+\tilde{\varepsilon}(\textbf{q},\omega)]\varepsilon_\text{vac}.
\end{equation}

Further analysis summarized in Appendix \ref{app:PiezoElectricVertex} shows that $\tilde{\varepsilon}(\mathbf{q},\omega)$ has a dependence of the form
$\tilde{\varepsilon}(\mathbf{q},\omega)=\tilde{\varepsilon}(\mathbf{q}/\omega)$. An immediate conclusion from Eq. (\ref{eqn:surfaceChargePotentialRelation}) is that a purely piezoelectric wave (i.e., without sources, $\sigma(\mathbf{q},\omega)=0$), can propagate without damping if and only if $\tilde{\varepsilon}(\mathbf{q}/\omega)+1=0$. Thus, if the phase velocity is $v_s(\theta)$, then $\tilde{\varepsilon}(\mathbf{b}/v_s(\theta))+1=0$ and the dispersion relation of the obtained, called piezoelectric Rayleigh waves (here referred to as SAW) is $\omega=v_s(\theta) q$.

For the high-frequency limit we introduce $\tilde{\varepsilon}_{\text{HF}}(\theta):=\tilde{\varepsilon}(\mathbf{q}/\omega), \omega\rightarrow\infty$. This should be the anisotropic dielectric function valid all the way up to the optical region, and should take into account all screening processes in the substrate except for the slow piezoelectric ones, which are estimated below for the substrate of the 2D electronic material.

A central quantity in the evaluation of devices which use piezoelectric Rayleigh waves is the {\it SAW electromechanical coupling coefficient} $K_R(\theta)$, introduced through the relation at $1+\tilde{\varepsilon}(\textbf{b}/v_s(\theta))=0$:
\begin{equation}
	\frac{K_R^2(\theta)/2}{\tilde{\varepsilon}_{\text{HF}}(\theta)+1} = \left[v_s(\theta)\frac{\partial \tilde{\varepsilon}(\textbf{b}/v)}{\partial v}\Bigr|_{v=v_s(\theta)} \right]^{-1} ~.
\label{eqn:emCouplingCoefficient}
\end{equation}

In Appendix \ref{app:PiezoElectricVertex} we show that very general considerations require
\begin{equation} \label{eqn:central-inequality}
0 \le K_R(\theta)<1\, , 
\end{equation}
which is one of the central results of this work. The inequality $(\ref{eqn:central-inequality})$  is crucial because, as we will show, it implies that at small frequencies piezoelectric phonons cannot provide the sufficient screening to overcome the bare Coulomb repulsion.

In Ref. \cite{Royer2000} it is shown that there is a relation between the amplitude of electric potential at the surface, $\varphi_0=\varphi(\mathbf{q},\omega)$ and the total energy $H_{\text{harm}}$, see Eq. (\ref{eqn:HamiltonianHarmonicPSAW}). Hence, standard quantization procedure (see the Appendix, subsection \ref{app:HamiltonianInterVertex}) shows that the interaction between the 2D electronic material sheet and the spontaneous piezoelectric Rayleigh waves can be written as
\begin{align}
	H_{\text{e-ph}}^{\text{PA}} &= \frac{1}{\sqrt{A}} {\sum_{\mathbf{k,q},\sigma}}\gamma_{\mathbf{q}}^{\text{PA}} \, a_{\mathbf{k+q},\sigma}^{\dagger}a_{\mathbf{k},\sigma}b_{\mathbf{q}}+\text{H.c.}~,
	\nonumber \\
	\gamma^{\text{PA}}_\mathbf{q} & = \frac{K_R(\theta)}{2} \sqrt{\frac{ \hbar e^2 v_s(\theta)}{\overline{\varepsilon}_{\text{HF}}(\theta)\varepsilon_\text{vac}}}=K_R(\theta)\sqrt{\frac{\pi\alpha_\text{fs}\hbar^2 v_F v_s(\theta)}{\overline{\varepsilon}_\text{HF}(\theta)}},
\label{eqn:PAVertex}
\end{align}
where $A$ is the area of the sample, $a_{\mathbf{k},\sigma},a^\dagger_{\mathbf{k},\sigma}$ the electron operators with $\sigma=\pm$ the electron spin, $b_{\mathbf{k}},b^\dagger_{\mathbf{k}}$ are the piezoelectric phonon operators, $\alpha_\text{fs}:=e^2/4\pi\varepsilon_\text{vac}\hbar v_F$ and  $\overline{\varepsilon}_{\text{HF}}(\theta) := (\tilde{\varepsilon}_{\text{HF}}(\theta)+1)/2$. The validity of the shown interaction hamiltonian Eq. (\ref{eqn:PAVertex}) requires two further assumptions: first, the 2DEG or multilayered graphene sample should be thin enough so that in Eq. (\ref{eqn:PSAWWaveVacuum}), $kd\ll 1$, where $z\equiv d$ is the width of the sample and $k$ is the maximum allowed phonon momentum. And second, this maximum allowed momentum should be sufficiently small for the classical piezo-elasticity theory, as shown in Eqs. (\ref{eqn:elasticNewton}-\ref{eqn:piezoConstitutive}), to remain valid. We assume that a maximum momentum on the order of $k_F \sim 10^6-10^7 \, \text{cm}^{-1}$ does not violate this last restriction.


The resulting total Hamiltonian for the combined system of 2D electron gas and piezoelectric Rayleigh phonon is
\begin{eqnarray}
	H= \sum_{\mathbf{k},\sigma}E_{\mathbf{k}} a_{\mathbf{k},\sigma}^{\dagger} a_{\mathbf{k},\sigma} + \hbar \sum_{\mathbf{q}}\omega_{\mathbf{q}}\, b_{\mathbf{q}}^{\dagger} b_{\mathbf{q}} \nonumber \\
	+  H_{\text{e-ph}}^{\text{PA}} + \frac{1}{2A} \sum_{\bf{q}} v_{\mathbf{q}}^{(0)} \rho(\mathbf{q}) \rho(\mathbf{-q} )~,
\label{eqn:PAHamiltonian}
\end{eqnarray}
where $E_\mathbf{k}$ is the electron energy for a 2D wave-vector $\mathbf{k}$,
$\omega_\mathbf{q}=v_s(\theta) q$ is the dispersion relation for the acoustic piezoelectric SAW phonon of 2D wave-vector $\mathbf{q}$ and $v_s(\theta)$ is the SAW propagation velocity, and
$\rho(\mathbf{q}) = \sum_{\mathbf{k},\sigma} a_{\mathbf{k+q},\sigma}^{\dagger} a_{\mathbf{k},\sigma}$ is the Fourier transform of the electron density.


We use the bare Coulomb electron-electron interaction as
\begin{align}
	v_{\mathbf{q}}^{(0)} &= \frac{ e^{2}}{2\overline{\varepsilon}_{\text{HF}}(\theta) \varepsilon_\text{vac} q},
\end{align}
which contains all high-frequency screening processes except for piezoelectric ones.

The bare electron-electron interaction mediated by phonons is \cite{Hwang2010,Mahan2013}
\begin{equation}
	V_{\text{ph}}^{\text{PA}}(\mathbf{q},\omega) = |\gamma_{\mathbf{q}}^{\text{PA}}|^{2} G_{0}^{\text{PA}}(\mathbf{q},\omega) \, ,
	\label{eqn:bareElectronPhononElectronInteraction}
\end{equation}
where
\begin{equation}
G_{0}^{\text{PA}}(\mathbf{q},\omega) = \frac{2\omega_\mathbf{q}}{\hbar\left(\omega^2-\omega_\mathbf{q}^2+i\eta\right)}.
\end{equation}
is the bare piezoelectric acoustic phonon propagator ($\eta \rightarrow 0^+$).  The resulting RPA-type approximation to the dielectric function and effective interaction are:
\begin{align}
	V_{\text{eff}}(\mathbf{q},\omega) &=\frac{ e^2}{2\varepsilon(\mathbf{q},\omega)\varepsilon_\text{vac}q} \nonumber \\
	 &=\frac{v_{\mathbf{q}}^{(0)}+V_{\text{ph}}^{\text{PA}}(\mathbf{q},\omega)}{1-\left[v_{\mathbf{q}}^{(0)}+V_{\text{ph}}^{\text{PA}}(\mathbf{q},\omega)\right]\,\Pi_{0}(\mathbf{q},\omega)}
	\label{eqn:V_eff}
\end{align}
which can also be written as
\begin{equation} \label{eqn:V-eff-RPA}
V_{\text{eff}}(\mathbf{q},\omega)
= \frac{v_{\mathbf{q}}^{(0)}}{\varepsilon_{\text{RPA}}(\mathbf{q},\omega)} + \left|\frac{\gamma_{\mathbf{q}}^{\text{PA}}}{\varepsilon_{\text{RPA}}(\mathbf{q},\omega)} \right|^{2} \tilde{G}^{\text{PA}}(\mathbf{q},\omega) ~.
\end{equation}
where $\varepsilon_{\text{RPA}}(\mathbf{q},\omega) \equiv 1-v_{\mathbf{q}}^{(0)}\Pi_{0}$, with  $\Pi_{0}$ the irreducible polarization function,
\begin{equation}
\tilde{G}^{\text{PA}}(\mathbf{q},\omega) =
\frac{G_{0}^{\text{PA}}(\mathbf{q},\omega)}
{1-\frac{|\gamma_{\mathbf{q}}^{\text{PA}}|^{2}G_{0}^{\text{PA}}(\mathbf{q},\omega)\Pi_{0}(\mathbf{q},\omega)}{\varepsilon_{\text{RPA}}(\mathbf{q},\omega)}}\, .
\end{equation}
In the low frequency limit, $\Pi_{0}(\mathbf{q},\omega\simeq 0)\simeq -D(E_F)=-2k_F/\pi \hbar v_F$, for monolayer graphene \cite{Wunsch2006}, or $\Pi_{0}(\mathbf{q},\omega\simeq 0)\simeq -D(E_F)=-m/2\pi \hbar^2$ for a 2DEG with effective mass $m$ \cite{Giuliani2005}. These two last static limits are exact for $q<2k_F$. In Eq. (\ref{eqn:V-eff-RPA}), the total interaction has been rewritten as the sum of a purely electronically screened Coulomb repulsion and a phonon-induced effective part in which the vertex and phonon-propagator are also screened by just the conducting electrons of the 2DEG or graphene \cite{Mahan2013,Mattuck2012}.
For frequencies small in the scale of the acoustic phonons (or the Bloch-Gr\"uneisen temperature $k_B T_{\rm BG}:=2\hbar v_s k_F $), the bare electron-phonon-electron interaction contributes to the long-range part of the total interaction with a ${\bf q}$-dependence similar to that of the Coulomb repulsion:
\begin{equation}
	V_{\text{ph}}^{\text{PA}}(\mathbf{q},\omega \simeq 0)=|\gamma_{\mathbf{q}}^{\text{PA}}|^{2}G_{0}^{\text{PA}}(\mathbf{q},\omega \simeq 0) = - \frac{2|\gamma_{\mathbf{q}}^{\text{PA}}|^{2}}{\hbar v_{s}q}~,
\label{eqn:Vstatic}
\end{equation}
 Note that it is the acoustic phonon propagator $G_{0}^{\text{PA}}(\mathbf{q},\omega \simeq 0)$ that introduces the coulombic long-range dependence in $q$ via the dispersion of the modes. In the next subsection we shall see that a similar final $q$ dependence has a different origin.

In the limit of low frequencies, $\omega \simeq 0$,  there can be no effective attraction for electrons close to the Fermi-surface because, as shown in Eq. (\ref{eqn:kSquareBound}), the following inequality is satisfied
\begin{equation}
	\frac{-V_{\text{ph}}^{\text{PA}}(\mathbf{q},\omega \simeq 0)}{v_{\mathbf{q}}^{(0)}}=K_R^2(\theta)<1 \label{eqn:interactionsRatio} \,
\end{equation}
which is, in conjunction with the interaction vertex given by Eq. (\ref{eqn:PAVertex}), a central result of this paper.

\subsection{Comparison with optical phonons}

For simplicity, we focus a single branch of the longitudinal optical (LO) for which we assume a constant frequency $\omega_0$. The total Hamiltonian reads as in Eq. (\ref{eqn:PAHamiltonian}) except for the replacements:
\begin{align}
\omega_{\mathbf{q}} & \rightarrow \omega_0 \\
v^{(0)}_{\mathbf{q}} & \rightarrow v^{(\infty)}_{\mathbf{q}} := \frac{e^2}{2\overline{\varepsilon}_\infty \varepsilon_\text{vac} q} \\
\overline{\varepsilon}_\infty & := \frac{\varepsilon_\infty+1}{2} \\
\gamma^{\text{PA}}_\mathbf{q} & \rightarrow \gamma^{\text{OP}}_\mathbf{q} := \sqrt{ g \frac{e^ 2 \hbar \omega_0}{2\varepsilon_\text{vac}q}} \\
g & := \left(\frac{1}{\varepsilon_\infty+1} - \frac{1}{\varepsilon_0+1}\right) >0\, ,
\end{align}
where standard notation for dielectrics is used: $\varepsilon_\infty$ the dielectric constant coming from very high frequency interband electronic transitions and $\varepsilon_0$ would be static dielectric constant in the absence of the piezoelectric phonons at frequencies much smaller than $\omega_0$. 
	
Again, as shown in the discussion around Eq. (\ref{eqn:Vstatic}), for small frequencies ($\omega \ll \omega_0$), the bare phonon-mediated electron-electron interaction contributes to the long-range part of the total interaction like the Coulomb repulsion:
\begin{equation}
V_{\text{ph}}^{\text{OP}}(\mathbf{q},\omega \simeq 0)=|\gamma_{\mathbf{q}}^{\text{OP}}|^{2}G_{0}^{\text{OP}}(\mathbf{q},\omega \simeq 0) = - g \frac{e^2}{\varepsilon_\text{vac} q}~. \label{eqn:VOPstatic}
\end{equation}
However, in contrast to the piezoelectric case, here it is the vertex that introduces the coulombic dependence in $q$.

At small frequencies, $\omega \ll \omega_0$, a single optical phonon is not enough to provide over-screening, because
\begin{equation}
	\frac{-V_{\text{ph}}^{\text{OP}}(\mathbf{q},\omega \simeq 0)}{v_{\mathbf{q}}^{(\infty)}}= \frac{\varepsilon_0-\varepsilon_\infty}{\varepsilon_0+1}<1~.
	\label{eqn:optInteractionsRatio}
\end{equation}	





\section{Effect of piezoelectric phonons on superconducting instabilities}

From (\ref{eqn:interactionsRatio}) and (\ref{eqn:V_eff}), we see that, in the static limit ($\omega \simeq 0$), and for $q<2k_F$,
$V_{\text{eff}}$ can be written in the form
\begin{equation}
V_{\text{eff}} (\mathbf{q},0) = \frac{[1-K_R^2(\theta)]v_{\mathbf{q}}^{(0)}}{1+\left[1-K_R^2(\theta)\right]v_{\mathbf{q}}^{(0)}D(E_F)}~, 
\label{eqn:Pairing2dpiezo}
\end{equation}
where we note that we have not assumed $q\ll k_F$, as discussed in the paragraph following (\ref{eqn:PAVertex}). From the inequality in (\ref{eqn:interactionsRatio}), we are led to conclude 
that over-screening of the Coulomb repulsion by the phonon-mediated attraction is not possible.
Moreover, and following standard textbook reasoning (see for example \cite{Ashcroft2011}), we conclude that BCS-type instabilities must also be ruled out. 
A similar result holds for a single branch of optical phonons, as can be seen from Eq. (\ref{eqn:optInteractionsRatio}) (see however \cite{Gorkov2015} for the effect of multiple optical phonon branches from the substrate on superconducting instabilities). 

Moreover, in case such over-screening occurred, the static dielectric constant from Eq. (\ref{eqn:V_eff}) would predict unphysical features such as unstable phononic modes with $\tilde{\omega}(q_{c}) = 0$ for some $q_{c} \neq 0$ and even imaginary frequencies for $q<q_{c}$. No matter how small the absolute difference $|1-K_R^2(\theta)|$ happened to be, there would always exist a pole for the static (\ref{eqn:V_eff}) at small enough $q$ (what cannot occur in standard BCS metals), signaling a different type of instability, possibly a charge density wave.

On the other hand, the result (\ref{eqn:interactionsRatio}) for the vertex could still lead to higher angular momentum pairing instabilities (as in the Kohn-Luttinger mechanism \cite{kohn1965new}) provided that $K_R^2(\theta)$ is sufficiently large and anisotropic, a case not considered by us.

\subsection{Eliashberg formalism \cite{Einenkel2011}}

The previous reasoning about the absence of superconducting instabilities, is incomplete and somewhat oversimplified. Three reasons support this claim: (i) Long-wave piezoelectric phonon excitations (as considered in the present work) can never be the only source of effective electron-electron interactions; in particular, we have not taken into account the short range electric fluctuations of the substrate. (ii) There is definitely some dynamic over-screening at high frequencies [see Eq. (\ref{eqn:bareElectronPhononElectronInteraction})]. And (iii) Coulomb interaction has to be properly renormalized by taking into account collisions with high momentum transfer, which diminishes the Coulomb repulsion and thus comparatively strengthens the other attraction mechanisms.

Leaving aside the first objection momentarily, we can use the Eliashberg formalism, as applied to graphene in Ref. \cite{Einenkel2011}, to deal with the other two objections. The effective interaction could cause superconducting instabilities if a dimensionless electron-phonon coupling $\lambda^\text{PA}$ happened to be greater than an also dimensionless Coulomb pseudopotential $\mu^*$ coming from high-energy renormalizations \cite{Morel1962,Einenkel2011}. The coupling constant $\lambda^\text{PA}$ in the Eliashberg formalism is the same appearing in (the real part of the) self-energy calculations to renormalize the Fermi velocity \cite{Gonzalez2015} and is given by:
\begin{align}
&\alpha_\text{PA}^2F(\omega)=\frac{|\gamma^\text{PA}|^{2}}{2 \pi^2 \hbar^2 v_s v_F}\frac{\sqrt{1-(\frac{\omega/v_s}{2k_F})^2}}{(1+\frac{k_\text{TF}}{\omega/v_s})^2}~, \nonumber \\
&\lambda^\text{PA}=2\int\limits_{0}^{\infty}\frac{\alpha_\text{PA}^2F(\omega)}{\omega}d\omega = \frac{r_s}{\pi} K_R^2\, F(2r_s)~, \label{eqn:lambda}\\
& F(x) = \int\limits_{0}^{1}\frac{t\sqrt{1-t^2}\,dt}{(t+x)^2}=-2+x\pi + \frac{(1-2x^2)\, \text{acosh}(x^{-1})}{\sqrt{1-x^2}}~, \nonumber
\end{align}
where $r_s(\theta):=\alpha_{\text{fs}}/\varepsilon(\theta)$, and the symbols $r_s, v_s, \gamma^{\text{PA}}, K_R^2$ stand for the Fermi surface angle-averaged quantities of the same name. The constant $\mu^*$ equals
\begin{equation}
\mu^* = \frac{\frac{1}{4}D(E_F)V}{1+\frac{1}{4}D(E_F)V \log(\frac{E_F}{\hbar \omega_c})}~,
\label{eqn:PseudoPotential}
\end{equation}
where $\omega_c$ is some energy cutoff which should satisfy $\omega_\text{Debye} \ll \omega_c \ll E_F/\hbar $ \cite{Einenkel2011} and $V$ comes from the Fermi surface average of the Thomas-Fermi renormalized Coulomb repulsion $v_\mathbf{q}^{(0)}/(1+\frac{k_\text{TF}}{q})$. We have
\begin{align}
&\frac{1}{4}D(E_F)V=\frac{r_s}{\pi}G(2r_s), \\
& G(x)=\int\limits_{0}^{1}\frac{\sqrt{1-t^2}\,dt}{t+x}=-1+\frac{\pi x}{2}+\sqrt{1-x^2}\,\text{acosh}(x^{-1})~, \nonumber
\end{align}
and therefore, provided that one takes $\hbar\omega_c \simeq k_B T_\text{BG}$, so that $\log\left(\frac{E_F}{\hbar \omega_c}\right) \simeq \log(\frac{v_F}{2v_s}) \simeq 5$. Thus, an estimate of the effective pseudo-potential is
\begin{equation}
\mu^*\simeq\frac{\frac{r_s}{\pi}G(2r_s)}{1+\frac{5r_s}{\pi}G(2r_s)} ~.
\end{equation}
There could be intravalley \footnote{In this analysis, we are taking into account only long-wavelength piezoelectric phonons, hence no intervalley pairing instability could occur. We address this question in the next paragraph.} superconducting instabilities provided that
\begin{equation}
1<\frac{\lambda^\text{PA}}{\mu^*}=K_R^2 \frac{F(2r_s)}{G(2r_s)}\left[1+\frac{5r_s}{\pi}G(2r_s)\right]~,
\end{equation}
which imposes a constraint on the value of $K_R^2$ from the piezoelectric substrate with respect to quantities depending on $r_s$. The coupling $K_R$ should be very large and actually greater than 1 for this choice of $\omega_c$, although there could exist superconductivity in this idealized case of a system consisting just of the graphene electrons and long wavelength piezoelectric phonons, provided that $E_F/\omega_c$ is larger and $K_R^2$ close to 1.

In order to amend the first objection, we have to consider proper phonons of the electronic system (here we go on considering graphene), in conjunction with the short range of the piezoelectric ones. Then, pairing instabilities due to intervalley scattering have to be considered as well, because intravalley scattering terms contribute also to the intervalley pairing gap.
With the notation in Ref. \cite{Einenkel2011}, an estimate on the critical temperature for the intravalley pairing is \cite{Einenkel2011} $T_c^\text{intra} = 1.13\omega_\text{Debye}\,\exp\left(-\frac{1+\lambda}{\lambda_{11}-\mu_{11}^*}\right)$, and a very similar is obtained for the intervalley transition
$T_c^{\text{inter}} = 1.13\omega_\text{Debye}\,\exp\left(-\frac{1+\lambda}{\lambda-\mu_{12}^*}\right)$, with $\lambda = \lambda_{11}+\lambda_{12}$ and the previously computed $\lambda^{\text{PA}}$ included into the intravalley term $\lambda_{11}$ ($\lambda_{12}$ denotes the contribution from all intervalley terms). Here the pseudo-potential $\mu_{12}^*$ is only slightly larger than $\mu_{11}$, and both are given by similar formulas as in Eq. (\ref{eqn:PseudoPotential}), but with $\omega_c\rightarrow \omega_\text{Debye}$. 

The upshot of this discussion is that the long-wavelength piezoelectric phonons work in favor of pairing instabilities, as shown in Fig. \ref{fig:tcPlot}. We emphasize, however, that we are not claiming that a piezoelectric substrate {\it per se} necessarily increases the critical temperature, since it could be the case that other piezoelectric fluctuations not considered in the present study (e.g. shorter wavelength modes) could work against pairing instabilities.

\begin{figure}
\raggedleft
\includegraphics[width=0.92\linewidth]{./tcPlot}
\includegraphics[width=0.9\linewidth]{./deltatcPlot}
\caption{(Color online) Critical temperature and variation from the \textquotedblleft bare\textquotedblright~one in \cite{Einenkel2011} for graphene on a piezoelectric substrate, as a function of the conduction band density. Three pairs of plots are given for three different values of $K_R$ and the two values of the constant $C = 3.5$ and 5 in Eq. (5.1) for $\lambda$ in the previous reference \cite{Einenkel2011}. }
\label{fig:tcPlot}
\end{figure}


\section{Conclusions}
In conclusion, we have derived a general expression for the two-dimensional electron-phonon piezoelectric interaction valid for any piezoelectric substrate covered by a two-dimensional electron system, as in the classical 2D Fr\"ohlich hamiltonian for the optical phonons, and characterized the magnitude of the interaction. Our results show that electron overscreening cannot be achieved just with the strongest piezoelectric phonons within our assumptions because $K_R^2<1$ is always satisfied. Nevertheless, these phonons could help further in other contexts where the 2D superconductivity is known to exist, for example in bulk few-layer $\text{MoS}_{2}$ with most of the carriers confined to the first layer \cite{Ye2012,Roldan2013}; or postulated to exist but not yet observed due to experimental difficulties (e.g. very heavily doped graphene \cite{Einenkel2011}). Other example is the recent high-temperature superconductor system of 2D FeSe on top of the ferroelectric $\text{SrTiO}_{3}$, whose optical phonons have been analyzed arriving at conclusions similar to ours \cite{Gorkov2015}, and where the strong piezoelectric phonons could play a role as well.
\begin{acknowledgments}
We want to thank Fernando Calle and J\"urgen Schiefele for valuable discussions. DGG acknowledges financial support from Campus de Excelencia Internacional (Campus Moncloa UCM-UPM). This work has been supported by the Spanish Ministry of Economy (MINECO) through Grants No. FIS2011-23713, FIS2013-41716-P; the European Research Council Advanced Grant (contract 290846), and the European Commission under the Graphene Flagship, contract CNECTICT- 604391.

\end{acknowledgments}


