1 Introduction

Graphene is a gapless semiconductor made of a single layer of carbon atoms arranged into a honeycomb hexagonal lattice [1]. In view of applications in graphene-based electron devices, it is crucial understanding the basic transport properties of this material.

An important step in the analysis of the electrical features of graphene is the determination of the mobility curves that can then be inserted in the simulation CAD tools already available for several semiconductor materials, e.g. Silicon and GaAs. The mobilities are functions of the electric field and depend also on factors like Fermi level, temperature and presence of impurities. The direct way to determine the mobility curves is by experiments. However the measurements are rather delicate and very sensible to the specific specimen one is dealing with. In particular the determination of the low field mobility is subject to a rather wide uncertainty, for example see [2] for Silicon.

An indirect theoretical way is instead based on the solutions of the transport equation; in fact once the distribution of electrons has been obtained, one can evaluate the current as a suitable average quantity.

Usually the numerical solutions of the transport equation are obtained by DSMC approaches with the unavoidable stochastic noise due to the statistical fluctuations. Here, in the case of monolayer graphene, we derive the mobility curves numerically solving the electron semiclassical Boltzmann equation with a deterministic DG method [3, 4]. At last the numerically obtained mobility is fitted with some analytical formulas which are widely used for other semiconductors such as Si or ZnO.

A remarkable point is that the low field mobility is obtained without the intrinsic huge noise in DSMC simulation at very low electric fields.

The plan of the paper is as follows. In Section 2 the transport equation for charge carriers in graphene is presented along with the derivation of the mobility expressions from the electron distribution functions. In Section 3 we illustrate the DG method for solving the Boltzmann equation and in the last section the numerical results of the mobilities are shown and fitted with analytical expressions.

2 Semiclassical transport equation for graphene and mobilities

The electron energy in graphene depends on a two dimensional wave-vector k belonging to a bi-dimensional Brillouin zone \(\mathcal{B}\) which has a hexagonal shape.

Most of the electrons are in the valleys around the vertices of the Brillouin zone, called Dirac points or K and \(K'\) points. Usually the K and \(K'\) valleys are treated as a single equivalent one.

In a semiclassical kinetic setting, the charge transport in graphene is described by four Boltzmann equations, one for electrons in the valence band (π) and one for electrons in the conduction band (\(\pi^{*}\)), that in turn can belong to the K or \(K'\) valley,

$$ \frac{\partial f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k}) }{\partial \mathrm {t}} + \mathbf{v}_{\ell,s} \cdot\nabla_{{\mathbf {x}}} f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k}) - \frac{e}{\hbar} \mathbf{E} \cdot\nabla _{\mathbf {k}} f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k}) = \frac{df_{\ell,s}}{d \mathrm {t}}(\mathrm {t}, \mathbf {x},\mathbf {k}) \bigg| _{e-ph} , $$
(1)

where \(f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k})\) represents the distribution function of charge carriers, in the band π or \(\pi^{*}\) (\(s = -1\) or \(s=1\)) and valley ℓ (K or \(K'\)), at position x, time t, and with wave-vector k. We denote by \(\nabla_{\mathbf {x}}\) and \(\nabla_{\mathbf {k}}\) the gradients with respect to the position and the wave-vector, respectively. The group velocity \(\mathbf{v}_{\ell,s}\) is related to the band energy \(\varepsilon _{\ell,s}\) by

$$\mathbf{v}_{\ell,s} = \frac{1}{\hbar} \nabla_{\mathbf {k}} \varepsilon _{\ell ,s} . $$

With a very good approximation [1] a linear dispersion relation holds for the band energies \(\varepsilon _{\ell,s}\) around the equivalent Dirac points; so that \(\varepsilon _{\ell,s} = s \hbar v_{F} \vert \mathbf {k}-\mathbf {k}_{\ell} \vert \), where \(v_{F}\) is the (constant) Fermi velocity, ħ the Planck constant divided by 2π, and \(\mathbf {k}_{\ell}\) is the position of the Dirac point ℓ. The elementary (positive) charge is denoted by e, and E is the electric field, here assumed as external. The right hand sides of Eqs. (1) are the collision terms representing the interactions of electrons with acoustic, optical (with wave-vector close to the Γ point of the first Brillouin zone) and K phonons (with wave-vector close to the zone edge of the first Brillouin zone). Acoustic phonon scattering is intra-valley and intra-band. Optical phonon scattering is intra-valley and can be longitudinal optical (LO) and transversal optical (TO); it can be intra-band, leaving the electrons in the same band, or inter-band, pushing the electrons from the initial band toward another one. Scattering with optical phonons of K type pushes electrons from a valley to a nearby one (inter-valley scattering). We assume that phonons are a bath at thermal equilibrium. Hence, the general form of the collision term can be written as (see [1, 5, 6] for more details)

$$\begin{aligned}[b] \frac{d f_{\ell,s}}{d \mathrm {t}}(\mathrm {t},\mathbf {x},\mathbf {k})\Big|_{e-ph} ={}& \sum _{\ell', s'} \biggl[ \int_{\mathcal{B}} S_{\ell', s', \ell, s}\bigl(\mathbf {k}', \mathbf {k}\bigr) f_{\ell',s'}\bigl(\mathrm {t},\mathbf {x},\mathbf {k}'\bigr) \bigl( 1 - f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k}) \bigr)\,d\mathbf {k}'\\ &{} - \int_{\mathcal{B}} S_{\ell, s, \ell', s'}\bigl(\mathbf {k}, \mathbf {k}'\bigr) f_{\ell,s}(\mathrm {t},\mathbf {x},\mathbf {k}) \bigl( 1 - f_{\ell',s'}\bigl(\mathrm {t},\mathbf {x}, \mathbf {k}'\bigr) \bigr)\,d\mathbf {k}' \biggr], \end{aligned} $$

where the total transition rate \(S_{\ell', s', \ell, s}(\mathbf {k}', \mathbf {k}) \) is given by the sum of the contributions of the several types of scatterings described above

$$\begin{aligned} S_{\ell', s', \ell, s}\bigl(\mathbf {k}', \mathbf {k}\bigr) =& \sum _{\nu} \bigl\vert G^{(\nu)}_{\ell', s', \ell, s}\bigl( \mathbf {k}', \mathbf {k}\bigr) \bigr\vert ^{2} \bigl[ \bigl( n^{(\nu)}_{\mathbf{q}} + 1 \bigr) \delta \bigl( \varepsilon _{\ell,s}(\mathbf {k}) - \varepsilon _{\ell',s'}\bigl(\mathbf {k}'\bigr) + \hbar \omega^{(\nu)}_{\mathbf{q}} \bigr) \\ &{} + n^{(\nu)}_{\mathbf{q}} \delta \bigl( \varepsilon _{\ell,s}( \mathbf {k}) - \varepsilon _{\ell',s'}\bigl(\mathbf {k}'\bigr) - \hbar \omega^{(\nu)}_{\mathbf{q}} \bigr) \bigr] . \end{aligned}$$
(2)

The index ν labels the νth phonon mode. The \(\vert G^{(\nu )}_{\ell', s', \ell, s}(\mathbf {k}', \mathbf {k})\vert ^{2} \)’s are the electron-phonon coupling matrix elements, which describe the interaction mechanism of an electron with a ν-phonon, from the state of wave-vector \(\mathbf {k}'\) belonging to the valley \(\ell'\) and band \(s'\) to the state k belonging to the valley ℓ and band s. The symbol δ denotes the Dirac distribution, \(\omega^{(\nu )}_{\mathbf{q}}\) is the νth phonon frequency, \(n^{(\nu)}_{\mathbf{q}}\) is the Bose-Einstein distribution for the ν-type phonons

$$\begin{aligned} n^{(\nu)}_{\mathbf{q}} = \frac{1}{e^{\hbar \omega^{(\nu )}_{\mathbf{q}} /k_{B} T} - 1} , \end{aligned}$$
(3)

where \(k_{B}\) is the Boltzmann constant and T the graphene lattice temperature. If, for a \(\nu_{*}\)-type phonon, \(\hbar \omega^{(\nu _{*})}_{\mathbf{q}} \ll k_{B} T\), then the corresponding scattering can be assumed elastic. In this case, we eliminate in Eq. (2) the term \(\hbar \omega^{(\nu_{*})}_{\mathbf{q}}\) inside the delta distribution and we use the Laurent approximation \(n^{(\nu _{*})}_{\mathbf{q}}\approx{k_{B} T/\hbar \omega^{(\nu _{*})}_{\mathbf{q}}} - \frac{1}{2} + O(\hbar \omega^{(\nu _{*})}_{\mathbf{q}} / k_{B} T)\).

It is preferable to treat the electrons in the valence band as holes for insuring the integrability of the corresponding distribution function. However, in this paper we consider the case of a high value of the Fermi energy, which is equivalent for conventional semiconductors to a n-type doping. Under such a condition, electrons belonging to the conduction band do not move to the valence band and vice versa. Therefore the hole dynamics is neglected. A reference frame centered in the K-point will be used and in order to simplify the notation the indices s and ℓ will be omitted.

Under the above hypotheses the scattering rates read as follows.

For acoustic phonons, we consider the elastic approximation according to which

$$ \bigl(2 n^{(ac)}_{\mathbf{q}} + 1 \bigr) \bigl\vert G^{(ac)} \bigl(\mathbf {k}', \mathbf {k}\bigr) \bigr\vert ^{2} = \frac{1}{(2 \pi )^{2}} \frac{\pi D_{ac}^{2} k_{B} T}{2 \hbar \sigma_{m} v_{p}^{2}} ( 1 + \cos\vartheta_{\mathbf {k},\mathbf {k}'} ) , $$
(4)

where \(D_{ac}\) is the acoustic phonon coupling constant, \(v_{p}\) is the sound speed in graphene, \(\sigma_{m}\) is the graphene areal density, and \(\vartheta_{\mathbf {k},\mathbf {k}'}\) is the convex angle between k and \({\mathbf {k}'}\).

The electron-phonon matrix elements related to the longitudinal optical (LO), the transversal optical (TO) and the K phonons are (see for example [5])

$$\begin{aligned}& \bigl\vert G^{(LO)}\bigl(\mathbf {k}', \mathbf {k}\bigr) \bigr\vert ^{2} = \frac{1}{(2 \pi)^{2}} \frac{\pi D_{O}^{2}}{\sigma_{m} \omega_{O}} \bigl( 1 - \cos( \vartheta_{\mathbf {k},\mathbf {k}' - \mathbf {k}} + \vartheta_{\mathbf {k}' ,\mathbf {k}' - \mathbf {k}} ) \bigr), \end{aligned}$$
(5)
$$\begin{aligned}& \bigl\vert G^{(TO)}\bigl(\mathbf {k}', \mathbf {k}\bigr) \bigr\vert ^{2} = \frac{1}{(2 \pi)^{2}} \frac{\pi D_{O}^{2}}{\sigma_{m} \omega_{O}} \bigl( 1 + \cos( \vartheta_{\mathbf {k},\mathbf {k}' - \mathbf {k}} + \vartheta_{\mathbf {k}' ,\mathbf {k}' - \mathbf {k}} ) \bigr), \end{aligned}$$
(6)
$$\begin{aligned}& \bigl\vert G^{(K)}\bigl(\mathbf {k}', \mathbf {k}\bigr) \bigr\vert ^{2} = \frac{1}{(2 \pi)^{2}} \frac{2 \pi D_{K}^{2}}{\sigma_{m} \omega_{K}} ( 1 - \cos \vartheta_{\mathbf {k},\mathbf {k}'} ) , \end{aligned}$$
(7)

where \(D_{O}\) is the optical phonon coupling constant, \(\omega_{O}\) the optical phonon frequency, \(D_{K}\) is the K phonon coupling constant and \(\omega_{K}\) the K phonon frequency. The angles \(\vartheta_{\mathbf {k},\mathbf {k}' - \mathbf {k}}\) and \(\vartheta_{\mathbf {k}' ,\mathbf {k}' - \mathbf {k}}\) denote the convex angles between k and \(\mathbf {k}' - \mathbf {k}\) and between \(\mathbf {k}'\) and \(\mathbf {k}' - \mathbf {k}\), respectively.

In the literature there are several values for the coupling constants entering into the collision terms. For example for the acoustic deformation potential one can find values ranging from 2.6 eV to 29 eV. A similar degree of uncertainty is found for the optical and K phonon coupling constants as well. In our numerical simulations of monolayer graphene, the parameters proposed in [7, 8] have been adopted. They are reported in Table 1.

Table 1 Physical parameters for the scattering rates
Full size table

From the semiclassical transport equations, using a procedure developed for other semiconductors (see for example [9–14]), one can formulate macroscopic models that are more suited for CAD purposes because they avoid the numerical solutions of the Boltzmann equations, even if introduce some approximations for the needed closure relations, see e.g. [9, 15, 16].

Macroscopic quantities can be defined as moments of the distribution functions with respect to some suitable weight functions \(\{\psi(\mathbf{k}) \}\), assuming a sufficient regularity for the existence of the involved integrals. Of particular relevance are the following two moments

$$\begin{aligned}& \mbox{average density}\quad \rho_{i} = \frac {4}{(2 \pi)^{2}} \int_{\mathbb {R}^{2}} f_{i} (\mathrm {t},\mathbf {x},\mathbf {k})\,d\mathbf{k}, \end{aligned}$$
(8)
$$\begin{aligned}& \mbox{average velocity}\quad \rho_{i} \mathbf{V}_{i} = \frac{4}{(2 \pi)^{2}} \int_{\mathbb {R}^{2}} f_{i} (\mathrm {t},\mathbf {x},\mathbf {k}) \mathbf{v}\,d\mathbf{k} \end{aligned}$$
(9)

(i = electrons, holes), where the factor 4 arises from taking into account both the spin states and the two equivalent valleys. The term \(1/(2 \pi)^{2}\) is the standard normalization of the elementary cell in the phase space for a two-dimensional crystal lattice [17].

By integrating the Boltzmann equations with respect to k, one has the following balance equation involving the above-defined macroscopic quantities

$$\begin{aligned} \frac{\partial}{\partial t} \rho_{i} + \nabla_{\mathbf{x}} \cdot ( \rho_{i} \mathbf{V}_{i} ) = \rho_{i} C_{i}, \end{aligned}$$
(10)

where the terms at the right hand sides are the density productions (the reader is referred to [16] for details).

If we introduce the current densities

$$\begin{aligned} \mathbf{J}_{i} = \mathsf {e}_{i} \rho_{i} \mathbf {V}_{i},\quad i = e, h, \end{aligned}$$
(11)

with \(\mathsf {e}_{i} \) equal to −e for electrons and e for holes, from Eqs. (10), one gets the celebrated drift-diffusion equations

$$\begin{aligned} \frac{\partial}{\partial t} (\mathsf {e}_{i} \rho_{i} ) + \nabla_{\mathbf{x}} \cdot \mathbf{J}_{i} = \mathsf {e}_{i} (G -R ),\quad i = e, h, \end{aligned}$$
(12)

where the collision term \(\mathsf {e}_{i} \rho_{i} C_{i}\) has been splitted as the sum of the generation and recombination terms G and R.

In Eq. (12) closure relations for the current and the recombination-generation terms must be prescribed. The classical approach models the currents as the sum of the diffusive and the drift part

$$\begin{aligned}& \mathbf{J}_{e} = D_{e} \nabla_{\mathbf{x}} \rho_{e} + e \rho_{e} \mu_{n} \mathbf{E}, \end{aligned}$$
(13)
$$\begin{aligned}& \mathbf{J}_{h} = - D_{h} \nabla_{\mathbf{x}} \rho_{h} + e \rho_{h} \mu_{p} \mathbf{E}, \end{aligned}$$
(14)

with the introduction of the diffusion coefficients \(D_{i}\) and the mobilities \(\mu_{i}\), \(i = e, h\). If one considers as valid the Einstein relations, then \(D_{i} = \mu_{i} k_{B} T\). The mobilities \(\mu_{i}\) are assumed to be functions of the electric field and can parametrically depend also on other physical quantities. We will investigate their dependence on the Fermi level \(\varepsilon _{F}\) or equivalently on the electron density. Moreover we will assume that the mobility is isotropic that is it depends only on the modulus of the electric field E. Therefore we will take as mobility functional dependence

$$\begin{aligned} \mu_{i} = \tilde{\mu}_{i} (E, \rho_{i} ). \end{aligned}$$
(15)

More sophisticated models could take into account also a dependence on the lattice temperature and its gradient or the electron temperature.

In the sequel, as said, we will limit our analysis to the case of positive Fermi energies and therefore only the electrons contribute significantly to the current but it is straightforward to extend the analysis to holes as well.

In order to obtain the expression of the electron mobility, it is enough to consider the homogeneous case where

$$\begin{aligned} \mathbf{J}_{e} = e \rho_{e} \mu_{n} \mathbf{E}. \end{aligned}$$
(16)

In the space-homogeneous case the Boltzmann equation relative to the K valley reduces to (we drop the valley and band indices for the sake of simplifying the notation)

$$\begin{aligned} \frac{\partial f(\mathrm {t},\mathbf {k})}{\partial \mathrm {t}} - \frac{e}{\hbar} \mathbf{E} \cdot\nabla_{\mathbf {k}} f(\mathrm {t},\mathbf {k}) =& \int S\bigl(\mathbf {k}', \mathbf {k}\bigr) f\bigl(\mathrm {t},\mathbf {k}' \bigr) \bigl( 1 - f(\mathrm {t},\mathbf {k}) \bigr)\,d\mathbf {k}' \\ &{} - \int S\bigl(\mathbf {k}, \mathbf {k}'\bigr) f(\mathrm {t},\mathbf {k}) \bigl( 1 - f\bigl( \mathrm {t},\mathbf {k}'\bigr) \bigr)\,d\mathbf {k}' . \end{aligned}$$
(17)

A similar equation holds for the \(K'\) valley. As initial condition we take the Fermi-Dirac distribution

$$f(0, \mathbf {k}) = \frac{1}{1 + \exp ( \frac{\varepsilon (\mathbf {k}) - \varepsilon _{F}}{k_{B} T} )} , $$

where \(\varepsilon _{F}\) is the Fermi energy, which is related to the initial charge density by

$$ \rho(0) = \frac{4}{(2 \pi)^{2}} \int f(0, \mathbf {k})\,d\mathbf {k}. $$
(18)

As said, in (18) the factor 4 arises because we are considering both the two states of spin and the degeneracy (equal to 2) of the valley. As alternative one can consider the population of a single valley, and put equal to one the valley degeneracy and take in (18) \(\rho(0)/2\) for the electron density. Note that in the unipolar case ρ remains constant, \(\rho(\mathrm {t}) = \rho(0)\), as a consequence of the charge conservation.

If one fixes the electric field, the Fermi energy and the lattice temperature, as \(\mathrm {t}\mapsto+\infty\) the solution of (17) gives the stationary distribution function which, inserted in relationship (9), allows us to evaluate \(\mathbf{J}_{e}\). Therefore, if we are able to solve numerically the semiclassical Boltzmann equation, it is possible to get in a rather simple way the numerical values of the mobility as function of the electric field once the lattice temperature and Fermi energy have been assigned.

3 Application of the DG method to the electron transport equation in graphene

Lately several efficient numerical schemes have been applied for getting deterministic solutions of the Boltzmann equation for charge transport in semiconductors. Several works based on weighted essentially non oscillatory (WENO) schemes can be found in the literature about simulation of Silicon and Gallium Arsenide electron devices [18, 19] and recently also for suspended monolayer graphene [5]. Here we adopt the DG method for discretizing Eq. (17).

We choose a bounded domain \(\Omega \subseteq \mathbb {R}^{2}\) such that \(f(\mathrm {t},\mathbf {k}) \approx0\) for every \(\mathbf {k}\notin \Omega \) and \(\mathrm {t}> 0\), and introduce a finite decomposition \(\lbrace C_{\alpha} \rbrace_{\alpha= 1}^{N}\) of Ω, with \(C_{\alpha}\) appropriate open sets having a regular boundary, such that

$$C_{\alpha} \cap C_{\beta} = \emptyset \quad\mbox{if } \alpha\ne \beta, \mbox{ and } \bigcup_{\alpha= 1}^{N} \overline{C_{\alpha}} = \Omega. $$

The distribution function is assumed to be constant in each cell \(C_{\alpha}\). If we denote by \(\chi_{\alpha}(\mathbf {k})\) the characteristic function over the cell \(C_{\alpha}\), then the approximation of the distribution function f is given by

$$f(\mathrm {t},\mathbf {k}) \approx f^{\alpha}(\mathrm {t})\quad \forall \mathbf {k}\in C_{\alpha}\quad \Longleftrightarrow\quad f(\mathrm {t},\mathbf {k}) \approx\sum_{\alpha= 1}^{N} f^{\alpha}(\mathrm {t}) \chi_{\alpha}(\mathbf {k}) \quad \forall \mathbf {k}\in\bigcup _{\alpha= 1}^{N} C_{\alpha} . $$

This assumption replaces the unknown f, which depends on the two variables t and k, with a set of N unknowns \(f^{\alpha}\), which depend only on time t. In order to obtain a set of N equations for the new unknowns \(f^{\alpha}\), we integrate Eq. (17) with respect to k over every cell \(C_{\alpha}\) and replace f with its approximation. The derivative of f with respect to the time is treated easily. We have

$$\int_{C_{\alpha}} \frac{\partial f(\mathrm {t},\mathbf {k})}{\partial \mathrm {t}}\,d\mathbf {k}\approx M_{\alpha} \frac{d f^{\alpha}(\mathrm {t})}{d \mathrm {t}}, $$

where \(M_{\alpha}\) is the measure of the cell \(C_{\alpha}\). It is clear that the numerical method yields a system of ordinary differential equations by discretizing the collision operator and the drift term as discussed below. The final set of ordinary differential equations in time can be integrated by using a total variation diminishing (TVD) Runge-Kutta scheme [20].

3.1 Discretization of the collision operator

Since for each \(\mathbf {k}\in C_{\alpha}\)

$$\begin{aligned} &\int S\bigl(\mathbf {k}', \mathbf {k}\bigr) f\bigl(\mathrm {t},\mathbf {k}' \bigr) \bigl( 1 - f(\mathrm {t},\mathbf {k}) \bigr)\,d\mathbf {k}' - \int S\bigl(\mathbf {k}, \mathbf {k}'\bigr) f(\mathrm {t},\mathbf {k}) \bigl( 1 - f\bigl( \mathrm {t},\mathbf {k}'\bigr) \bigr)\,d\mathbf {k}' \\ &\quad\approx \sum_{\beta= 1}^{N} \biggl[ \int_{C_{\beta}} S\bigl(\mathbf {k}', \mathbf {k}\bigr) f^{\beta}( \mathrm {t}) \bigl( 1 - f^{\alpha}(\mathrm {t}) \bigr)\,d\mathbf {k}' - \int_{C_{\beta}} S\bigl(\mathbf {k}, \mathbf {k}'\bigr) f^{\alpha}( \mathrm {t}) \bigl( 1 - f^{\beta}(\mathrm {t}) \bigr)\,d\mathbf {k}' \biggr] \\ &\quad= \sum_{\beta= 1}^{N} \biggl[ f^{\beta}(\mathrm {t}) \bigl( 1 - f^{\alpha}(\mathrm {t}) \bigr) \int_{C_{\beta }} S\bigl(\mathbf {k}', \mathbf {k}\bigr)\,d\mathbf {k}' - f^{\alpha}(\mathrm {t}) \bigl( 1 - f^{\beta}(\mathrm {t}) \bigr) \int_{C_{\beta }} S\bigl(\mathbf {k}, \mathbf {k}'\bigr)\,d\mathbf {k}' \biggr], \end{aligned}$$

if we define

$$ A^{\alpha, \beta} = \int_{C_{\alpha}} \biggl[ \int_{C_{\beta}} S\bigl(\mathbf {k}, \mathbf {k}'\bigr)\,d\mathbf {k}' \biggr]\,d\mathbf {k}, $$
(19)

we obtain

$$\begin{aligned} &\int_{C_{\alpha}} \biggl[ \int S\bigl(\mathbf {k}', \mathbf {k}\bigr) f\bigl(\mathrm {t},\mathbf {k}' \bigr) \bigl( 1 - f(\mathrm {t},\mathbf {k}) \bigr)\,d\mathbf {k}' - \int S\bigl(\mathbf {k}, \mathbf {k}'\bigr) f(\mathrm {t},\mathbf {k}) \bigl( 1 - f\bigl( \mathrm {t},\mathbf {k}'\bigr) \bigr)\,d\mathbf {k}' \biggr]\,d\mathbf {k}\\ &\quad \approx \sum_{\beta= 1}^{N} \bigl[ A^{\beta, \alpha} \bigl( 1 - f^{\alpha}(\mathrm {t}) \bigr) f^{\beta }(\mathrm {t}) - A^{\alpha, \beta} f^{\alpha}(\mathrm {t}) \bigl( 1 - f^{\beta}(\mathrm {t}) \bigr) \bigr] . \end{aligned}$$

In doing so, the integral collision operator is replaced by quadratic polynomials. We note that the numerical coefficients \(A^{\alpha, \beta}\) depend only on the scattering terms and the domain decomposition and can be evaluated analytically. The interested reader is referred to [21] for the detailed calculations.

3.2 Discretization of the drift term

We must approximate the term

$$- \frac{e}{\hbar} \mathbf{E} \cdot \int_{C_{\alpha}} \nabla_{\mathbf {k}} f(\mathrm {t},\mathbf {k})\,d\mathbf {k}= - \frac{e}{\hbar} \mathbf{E} \cdot \int_{\partial C_{\alpha}} f(\mathrm {t}, \mathbf {k}) \mathbf{n}\,d\sigma, $$

where n is the external unit normal to the boundary \(\partial C_{\alpha}\) of the cell \(C_{\alpha}\). Since, due to the Galerkin method, the approximation of f is not defined on the boundary of the cells, we must introduce a numerical flux, that furnishes reasonable values of f on every \(\partial C_{\alpha}\), depending on the values of the approximation of f in the nearest neighborhoods of the cell \(C_{\alpha}\) and on the sign of \(\mathbf{E} \cdot \mathbf{n}\). The simplest numerical flux is given by the upwind rule, that uses only the nearest adjacent cells. For the sake of shortness we refer the interested reader to [21, 22] for the details.

4 Electron mobility in monolayer graphene

By using the numerical scheme outlined in the previous section, we have numerically solved Eq. (17) and evaluated the electron mobility in monolayer graphene. The robustness and accuracy of the scheme has been investigated in [22], where a cross comparison with DSMC solutions has clearly validated both the approaches for a wide range of electric fields and Fermi energies.

The case of room lattice temperature (300 K) has been considered, for electron densities \(\rho_{e}=0.5, 1, 2, 5, 8, 10\times10^{12}\mbox{/cm}^{2}\). First of all, by extrapolation we have evaluated the low field mobility \(\mu_{0}\) defined as

$$\mu_{0} = \lim_{E \mapsto0^{+}} \mu (E, \rho ). $$

The results, reported in Figure 1, show that the low field mobility decreases with the rise of the carrier density as happens for graphene on SiO2 [23]. The data are well fitted by the following empirical function, already used for Si device mobility [23],

$$\begin{aligned} \mu_{0} = \mu_{0} (\rho) = \frac{\tilde{\mu}}{1 + (\rho/\rho _{ref} )^{\alpha}}, \end{aligned}$$
(20)

with \(\rho_{ref} = 0.5 \times10^{12}~\mbox{cm}^{-2}\), \(\alpha= 1.2916\) and \(\tilde{\mu} = 2 \mu_{0} (\rho_{ref}) = 373.4306 \times10^{4}~\mbox{cm}^{2}\mbox{/V s}\).

Figure 1
figure 1

Low field mobility. Low field mobility versus the electron density. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

Full size image

The low field mobility is then used in a formula, suitably adapted from that of Arora et al. [24], to fit the dependence of the mobility on the electric field at the considered values of the Fermi energy

$$\begin{aligned} \mu(E) = \frac{\mu_{0} + \frac{v_{F}}{E} (\frac{E}{E_{ref}} )^{\beta_{1}}}{1 + (\frac{E}{E_{ref}} )^{\beta_{2}} + \gamma (\frac{E}{E_{ref}} )^{\beta_{3}}}, \end{aligned}$$
(21)

where \(E_{ref} = 1 \mbox{k V/cm}\) and \(\mu _{0}\) is given by (20).

The values of the fitting coefficients are reported in Table 2. The data are well approximated for each value of ρ as can be seen from Figures 2-7.

Figure 2
figure 2

\(\pmb{\rho_{e}=0.5\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Figure 3
figure 3

\(\pmb{\rho_{e}=1\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Figure 4
figure 4

\(\pmb{\rho_{e}=2\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Figure 5
figure 5

\(\pmb{\rho_{e}=5\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Figure 6
figure 6

\(\pmb{\rho_{e}=8\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Figure 7
figure 7

\(\pmb{\rho_{e}=10\times10^{12}\mbox{/cm}^{2}}\) . Mobility versus the electric field. The circles represent the results obtained by the DG scheme, the crossed line the fitting function.

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Table 2 Values of the fitting coefficients
Full size table