Figure 3 shows a plot of current vs source-drain bias for varying gate voltages in a back-gated device. We focus here on the curves taken at densities far from the charge neutrality point, such as the curve singled out in Figure 4. Here we see that at low bias the slope of the curve is constant, and at high bias the curve turns down, approaching a linear behavior with a reduced slope.

To describe this saturating decrease in conductivity, we propose a model based on an electric field-dependent carrier velocity *v*
_{
d
}(*E*) of the form:

{v}_{d}\left(E\right)={\left(\frac{1}{{\mu}_{0}E}+\frac{1}{{v}_{\mathit{\text{sat}}}}\right)}^{-1}

(1)

where *μ*
_{0} is the low field mobility and *v*
_{
sat
} is a phenomenologically introduced saturation velocity. The total current through the device is given by

I=\mathit{\text{jW}}=-{\mathit{\text{nev}}}_{d}W

(2)

We assume that the capacitance to the back gate dominates in determining the charge density in the channel, so that

\mathit{\text{ne}}={C}_{g}\left(V\left(x\right)-({V}_{g}-{V}_{\mathit{\text{CNP}}})\right)={C}_{g}(V-{V}_{0})

(3)

where *V* = *V*(*x*) is the potential at position x along the channel, and we have defined *V*
_{0}≡*V*
_{
g
}−*V*
_{
CNP
}. Using the relation *E*=*d* *V*/*d* *x*, we have

{I}^{-1}=-\frac{1}{{\mathit{\text{WC}}}_{g}(V-{V}_{0})}\left(\frac{1}{{\mu}_{0}\mathit{\text{dV}}/\mathit{\text{dx}}}+\frac{1}{{v}_{\mathit{\text{sat}}}}\right)

(4)

Rearranging terms and integrating gives the current

I={\mathit{\text{WC}}}_{g}({V}_{0}-{V}_{b}/2)\frac{{\mu}_{0}{V}_{b}/L}{1+{\mu}_{0}{V}_{b}/{v}_{\mathit{\text{sat}}}L}

(5)

In its limiting forms, Equation 5 for the current qualitatively gives the behavior seen in Figure 4. At low *V*
_{
b
}, current is linear in *V*
_{
b
} with a conductivity *WC*
_{
g
}
*V*
_{0}
*μ*
_{0}/*L*, determined by the low field mobility, as expected. At high *V*
_{
b
}, current is again linear in *V*
_{
b
}, but now with a conductivity of *WC*
_{
g
}
*v*
_{
sat
}/2 and an offset determined by the gate voltage. At low fields, the variation in carrier density is small and the linear *I–V* results from the linear form of *v*
_{
d
}(*E*)≈*μ*
_{0}
*E* in this regime. At high fields, *v*
_{
d
} approaches a constant value *v*
_{
sat
}, and the linear dependence of the carrier concentration on *V*
_{
b
} is responsible for an *I–V* characteristic approaching linear behavior. Note this is in contrast to the case of carbon nanotubes, where there are a set number of conducting channels, so that the current saturates with the drift velocity.

The expression in Equation 5 for *I*=*I*(*V*
_{
b
}) was fit to the *I–V* characteristics in Figure 3; the result is shown in Figure 5. For ribbon devices, the geometry is not well approximated by a parallel plate capacitor, so the gate capacitance was calculated numerically.For the device in Figure 5, the capacitance was calculated to be 47.5 nF/cm^{2} using a numerical calculation based on the finite element method. The model fits well for curves taken at densities far at high carrier densities, and begins to break down for curves measured near the charge neutrality point, as seen in Figure 5 for *V*
_{
g
}=−10 V. This fit has two free parameters, *v*
_{
sat
} and *μ*
_{0}. For this dataset, this model gives *μ*
_{0} values between 400 and 600 cm^{2}/Vs, compared to the value of 700 cm^{2}/Vs from low bias sweeps of *G–V*
_{
g
}.

The values of *v*
_{
sat
} obtained from this fit are plotted against *V*
_{
g
} in Figure 6(a). In Figure 6(b), we plot *v*
_{
sat
} against the inverse of the Fermi energy

{E}_{F}=\hslash {v}_{F}\sqrt{\pi {C}_{g}({V}_{g}-{V}_{\mathit{\text{CNP}}})}

(6)

Converting *V*
_{
g
} to *E*
_{
F
} involves the value of *V*
_{
CNP
}, which commonly drifts throughout measurement due to changes in adsorbed molecules and positions of trapped charges. Here the black circles correspond to conversion of *V*
_{
g
} to *E*
_{
F
} using *V*
_{
CNP
} = −15 V, the same value used in Equation 5 for the original fit. Red triangles represent a conversion to *E*
_{
F
} using *V*
_{
CNP
} = −8 V so that the a linear fit of *v*
_{
sat
} vs.{E}_{F}^{-1}intersects the origin.

In order to understand the inverse relationship between *v*
_{
sat
} and *E*
_{
F
}, we seek a physical understanding of the electric field dependent carrier velocity, or drift velocity, in Equation 1. This expression corresponds to scattering by optical phonons, which would produce an electric field dependent mean free path. By Matthiessen’s rule, mean free paths add as

\frac{1}{l}=\frac{1}{{l}_{\mathit{\text{sc}}}}+\frac{1}{{l}_{\mathit{\text{op}}}}

(7)

where *l* is the total mean free path and *l*
_{
sc
} is the mean free path for elastic impurity scattering and quasi-elastic acoustic scattering, and *l*
_{
op
} is the mean free path for optical phonon emission. If electrons are immediately scattered upon reaching the optical phonon energy, so that

{l}_{\mathit{\text{op}}}=\frac{\hslash \Omega}{\mathit{\text{eE}}}

(8)

where *E* is the electric field and *Ω* is the relevant optical phonon frequency, then the mobility *μ* is given by

\frac{1}{\mu}=\frac{1}{{\mu}_{0}}+\frac{E}{{v}_{\mathit{\text{sat}}}}

(9)

This form of the mobility results in the expression for the drift velocity *v*
_{
d
} = *μ* *E* given in Equation 1. For electrons and holes in graphene, which have a constant carrier velocity of *v*
_{
F
}, drift velocity can be understood as the time averaged velocity of carriers when scattering is taken into account.

From the above calculation we see that our phenomenological velocity saturation model can be understood in terms of a picture where electrons scatter by optical phonon emission upon reaching the phonon energy\hslash \Omegaunder the influence of the applied electric field. With this in mind, we derive an expression for current density using a different approach, in order to gain insight into our measured values for the saturation velocity. Current density is given by

\overrightarrow{j}=-e\int d\overrightarrow{k}{D}_{k}\overrightarrow{v}\left(\overrightarrow{k}\right)g\left(\overrightarrow{k}\right)

(10)

where *D*
_{
k
} = 2/(2*π*)^{2} is the density of electronic states in k-space,\overrightarrow{v}\left(\overrightarrow{k}\right)={v}_{F}is the electron velocity, andg\left(\overrightarrow{k}\right)is the distribution function. In the relaxation time approximation, we have

g\left(\overrightarrow{k}\right)={g}^{0}\left(\overrightarrow{k}\right)-e\overrightarrow{E}\xb7\overrightarrow{v}\left(\overrightarrow{k}\right)\tau \left(\epsilon \right(\overrightarrow{k}\left)\right)\left(-\frac{\partial f}{\partial \epsilon}\right)

(11)

where{g}^{0}\left(\overrightarrow{k}\right)is the equilibrium distribution function, *τ* is the relaxation time, and *f* is the Fermi-Dirac distribution function. For a device with its length in the *x* direction, we seek\overrightarrow{j}=j\widehat{x}, so we consider only\overrightarrow{E}=E\widehat{x}, and

\overrightarrow{E}\xb7\overrightarrow{v}\left(\overrightarrow{k}\right)={\mathit{\text{Ev}}}_{F}cos\theta

(12)

where *θ* is the angle betweend\overrightarrow{k}\overrightarrow{E}. We assume that electrons are immediately scattered upon reaching the energy threshold for phonon emission, giving

\tau =\frac{\hslash \Omega}{{\mathit{\text{eEv}}}_{F}}

(13)

So that for Equation 10 we have

j=e\int \frac{\mathit{\text{dk}}}{{\pi}^{2}}{v}_{F}\stackrel{2}{cos}\theta \hslash \Omega \left(-\frac{\partial f}{\partial \epsilon}\right){|}_{\epsilon =\hslash {v}_{F}k}

(14)

In polar coordinates

\begin{array}{cc}j& =e{\int}_{0}^{\infty}\frac{\mathit{\text{dk}}}{{\pi}^{2}}{\int}_{0}^{2\pi}d\theta {v}_{F}\stackrel{2}{cos}\theta \hslash \Omega \delta (\hslash {v}_{F}k-{E}_{F})\\ =\frac{e}{\pi}\Omega \frac{{E}_{F}}{\hslash {v}_{F}}\end{array}

(15)

At high fields, we assume *j*=*nev*
_{
sat
} and use{E}_{F}=\hslash {v}_{F}\sqrt{\pi n}to obtain

\frac{{v}_{\mathit{\text{sat}}}}{{v}_{F}}=\frac{\hslash \Omega}{{E}_{F}}

(16)

Using this expression with *v*
_{
F
}=10^{8} cm/s [2, 3], we obtain a value of\hslash \Omega =62.0meV from the linear fit (dashed line) in Figure 6(b). This is well below the value of the longitudinal zone-boundary phonon for graphene, which has\hslash \Omega =200meV [28]. We suggest that our measured phonon energy corresponds to the SiO_{2} surface phonon energy\hslash \Omega =55meV [29–31], although we note that values measured in other ribbon devices of different geometries vary widely (from ≈22 meV to ≈120 meV), possibly due to discrepancies in determining the relevant device geometry, the corresponding capacitance, and the position of the charge neutrality point.