Charge transport and mobility in monolayer graphene
- Armando Majorana^{1},
- Giovanni Mascali^{2} and
- Vittorio Romano^{1}Email author
DOI: 10.1186/s13362-016-0027-3
© Majorana et al. 2016
Received: 30 December 2015
Accepted: 18 July 2016
Published: 2 August 2016
Abstract
For electron devices that make use of innovative materials, a basic step in the development of models and simulation computer aided design (CAD) tools is the determination of the mobility curves for the charge carriers. These can be obtained from experimental data or by directly solving the electron semiclassical Boltzmann equation. Usually the numerical solutions of the transport equation are obtained by Direct Simulation Monte Carlo (DSMC) approaches with the unavoidable stochastic noise due to the statistical fluctuations. Here we derive the mobility curves numerically solving the electron semiclassical Boltzmann equation with a deterministic method based on a discontinuous Galerkin (DG) scheme in the case of monolayer graphene. Comparisons with analytical mobility formulas are presented.
Keywords
graphene mobility Boltzmann equation discontinuous Galerkin methodMSC
82D37 82C70 82C801 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.
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.
Physical parameters for the scattering rates
\(\sigma_{m}\) | \(7.6 \times10^{-8}~\mbox{g/cm}^{2}\) |
\(v_{F}\) | \(10^{6}~\mbox{m/s}\) |
\(v_{p}\) | \(2 \times10^{4}~\mbox{m/s}\) |
\(D_{ac}\) | 6.8 eV |
\(\hbar \omega_{O}\) | 164.6 meV |
\(D_{O}\) | \(10^{9}~\mbox{eV/cm}\) |
\(\hbar\omega_{K}\) | 124 meV |
\(D_{K}\) | \(3.5 \times10^{8}~\mbox{eV/cm}\) |
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].
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.
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).
3.1 Discretization of the collision operator
3.2 Discretization of the drift term
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.
Values of the fitting coefficients
ρ ( \(\boldsymbol {10^{12}}\textbf{cm}\boldsymbol {^{-2}}\) ) | \(\boldsymbol {\beta_{1}}\) | \(\boldsymbol {\beta_{2}}\) | \(\boldsymbol {\beta_{3}}\) | γ |
---|---|---|---|---|
0.5 | 2.5203 | 3.5728 | 1.0382 | 32.4631 |
1 | 2.7357 | 3.4656 | 1.1659 | 22.2200 |
2 | 3.1256 | 3.6552 | 1.1625 | 12.6928 |
5 | 2.0462 | 1.2446 | 1.2395 | 7.9534 |
8 | 1.7225 | 1.2159 | 1.2154 | 7.5381 |
10 | 1.7846 | 1.6393 | 1.2186 | 7.5336 |
Declarations
Acknowledgements
The authors (AM and VR) acknowledge the financial support by the project FIR 2014 Charge Transport in Graphene and Low dimensional Structures: modeling and simulation, University of Catania, Italy. The author (GM) acknowledges the financial support from GNFM Progetto giovani 2015.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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