# High-Frequency Limits of Graphene Field-Effect Transistors with Velocity Saturation
## 4 Simple Model of GFETs  
### Substrate Phonon Scattering Model  
The scattering by optical phonons has been identified as an important phenomenon of high-bias transport in high mobility materials like carbon nanotubes [23]. However, the influence of the substrate cannot be neglected, and the impact of surface polar phonons (SPP) on transport properties has been invoked in several studies to explain the current saturation [8; 12; 14; 15; 24]. Those studies have been conducted with graphene on oxide samples, and to our knowledge, the effect of HPhPs has only recently been demonstrated using high-mobility exfoliated graphene on h-BN transistors [11]. It was found that velocity saturation is provided by the lower \(\Omega_{1}\)-HPhP band and hot electron cooling by the upper \(\Omega_{1I}\)-HPhP band. The out-of-plane optical phonons of the \(\Omega_{1}\) band lift the sublattice degeneracy of graphene leading to an (unscreened) electron-phonon coupling reminiscent of the mass disorder mechanism reported in Ref. [25]. In Refs. [11; 16] exfoliated graphene was used and the mobility was high enough (\(\mu>3\) m\({}^{2}\)V\({}^{-1}\)s\({}^{-1}\)) to access interband transport at very high bias, the so-called Zener-Klein regime, which is not reached in the present study due to limited mobility. The heuristic model used to describe HPhP scattering is inspired by the optical phonon-electron scattering model used by [PERSON] et al. for carbon nanotubes [23]. One considers that electrons are accelerated by an electric field \(\mathcal{E}=V_{ds}^{0}/L\) up to the HPhP energy (here \(\hbar\Omega_{I}\simeq 0.1\) eV) to emit a phonon. In this notation, \(V_{ds}^{0}\) is the voltage across the channel without potential drop at the contacts. This interaction can be expressed as a scattering length for the electron-HPhP interaction \(l_{HPhP}=\frac{\hbar\Omega_{I}}{e\mathcal{E}}\), which is density-independent as represented by the dashed lines in Figure 3a for different electric fields. Our samples are not totally ballistic, and we need to account for a finite impurity scattering length \(l_{imp}=\mu\hbar\sqrt{\tau n}/e\), shown as a black solid line (the effective mobility is \(\mu=1\) m\({}^{2}\)/V/s). The resulting scattering length is given by the Matthiessen rule \(l_{scatt}^{-1}=l_{imp}^{-1}+l_{HPhP}^{-1}\), which is shown at \(\mathcal{E}=1\) V \(\upmu\)m\({}^{-1}\) in Figure 3a as a green solid line. As seen in the figure, at high bias, the transport is indeed dominated by HPhP scattering. The total scattering length \(l_{scatt}\) is translated into conductance \(G^{0}=\frac{4e^{2}}{\hbar}\frac{k_{F}W}{\pi}\frac{l_{scatt}}{L+l_{scatt}}\). It can be more conveniently expressed as a resistance:  
\[R=\frac{h\pi}{4e^{2}k_{F}W}+\frac{L}{Wne\mu}+\frac{h\pi}{4e^{2}k_{F}W}\frac{ eV_{ds}^{0}}{\hbar\Omega_{I}} \tag{1}\]  
The first term of Equation (1) is the ballistic resistance, the second is the well-known diffusive term, and the third corresponds to the HPhP scattering contribution; it is responsible for saturation and proportional to \(V_{ds}^{0}\). Note that the linear \(V_{ds}\) dependence of the DC resistance (negative quadratic deviation in the current) depends only on the carrier density and the HPhP energy. That is how we can identify the \(\hbar\Omega_{I}\simeq 0.1\pm 0.01\) eV energy of h-BN HPhPs as the main source of scattering and rule out the scattering by intrinsic graphene optical phonons or the \(\Omega_{II}\)-HPhP band, as \(\hbar\Omega_{OP}\)\(\sim\)\(\hbar\Omega_{II}\)\(\sim\)0.2 eV. Note that the role of intrinsic OPs or \(\Omega_{II}\)-HPhPs in energy relaxation can be identified using noise thermometry respectively in suspended [26] and h-BN supported graphene [11; 16].