^{1,a)}and M. J. Martín

^{1}

### Abstract

In this paper, the diffusivity in suspended monolayer graphene at low and high electric fields is investigated. The knowledge of this quantity and its dependence on the electric field is of primary importance not only for the investigation of the electronic transport properties of this material but also for the development of accurate drift-diffusion models. The results have been obtained by means of an ensemble Monte Carlo simulation. For the calculation of the diffusion coefficient, two different methods are considered, one based on the second central moment and the other one based on the Fourier analysis of velocity fluctuations, which are directly related to the noise behaviour at high frequencies. The diffusion coefficient is analyzed considering both parallel and transversal directions with regard to the applied field. Taking into account the importance of degeneracy in this material, the calculations are properly performed by considering an excess electron population obeying a linearized Boltzmann transport equation, which allows studying in an adequate fashion the diffusivity phenomena. The results show the importance of degeneracy effects at very low fields in which transport is mainly dominated by acoustic phonon scattering. Values of the diffusion coefficient larger than 40 000 cm^{2}/Vs are obtained for a carrier concentration equal to 10^{12} cm^{−2}. The correlation function of instantaneous velocity fluctuation is explained in terms of the wavevector distribution, and their power spectral density is evaluated in the THz range, showing an important dependence on the applied field and being strongly related to microscopic transport processes.

This work has been supported by research project SA188A11 from the Consejería de Educación de la Junta de Castilla y León.

I. INTRODUCTION

II. SIMULATION PROCEDURE

A. Monte Carlo model

B. Diffusion coefficient calculation

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Diffusion
- 39.0
- Graphene
- 39.0
- Electric fields
- 38.0
- Correlation functions
- 15.0
- Carrier density
- 10.0

## Figures

Diffusion coefficient for parallel (circles) and perpendicular (squares) directions for background (grey) and excess (black) carriers obtained by means of the second central moment. White symbols show the results obtained from velocity fluctuations just for some selected electric field values in order to provide an insight of the agreement of both methods and avoid excessive symbol overlapping in the figure. The inset shows an ampliation of the low-field region. The carrier density is equal to 10^{12} cm^{−2}.

Diffusion coefficient for parallel (circles) and perpendicular (squares) directions for background (grey) and excess (black) carriers obtained by means of the second central moment. White symbols show the results obtained from velocity fluctuations just for some selected electric field values in order to provide an insight of the agreement of both methods and avoid excessive symbol overlapping in the figure. The inset shows an ampliation of the low-field region. The carrier density is equal to 10^{12} cm^{−2}.

Wavevector distribution function for background carriers at E = 0.01 kV/cm (a), 0.1 kV/cm (b), and 1 kV/cm (c) and excess carriers distribution at E = 0.01 kV/cm (d), 0.1 kV/cm (e), and 1 kV/cm (f). The carrier density is equal to 10^{12} cm^{−2}. The circles indicate the equilibrium Fermi surface as a reference.

Wavevector distribution function for background carriers at E = 0.01 kV/cm (a), 0.1 kV/cm (b), and 1 kV/cm (c) and excess carriers distribution at E = 0.01 kV/cm (d), 0.1 kV/cm (e), and 1 kV/cm (f). The carrier density is equal to 10^{12} cm^{−2}. The circles indicate the equilibrium Fermi surface as a reference.

Percentage of scattering events as a function of the applied field for background carriers (grey symbols) and excess carriers (black symbols) fornS = 10^{12} cm^{–2} (a)–(c). Average electron energy as a function of the applied field for background (grey diamonds) and excess (black diamonds) carriers (d).

Percentage of scattering events as a function of the applied field for background carriers (grey symbols) and excess carriers (black symbols) fornS = 10^{12} cm^{–2} (a)–(c). Average electron energy as a function of the applied field for background (grey diamonds) and excess (black diamonds) carriers (d).

Average time between scatterings for background (grey circles) and excess carriers (black circles) for nS = 10^{12} cm^{–2} (a) and quasi-equilibrium diffusion coefficient as a function of the carrier density (circles: Monte Carlo results; triangles down: results from Eq. (3) ; triangles up: results from Eq. (4) ) for background electrons and diamonds: for excess electrons.

Average time between scatterings for background (grey circles) and excess carriers (black circles) for nS = 10^{12} cm^{–2} (a) and quasi-equilibrium diffusion coefficient as a function of the carrier density (circles: Monte Carlo results; triangles down: results from Eq. (3) ; triangles up: results from Eq. (4) ) for background electrons and diamonds: for excess electrons.

Average velocity distribution function in the parallel (a) and perpendicular (b) directions for several values of the applied field (excess carriers).

Average velocity distribution function in the parallel (a) and perpendicular (b) directions for several values of the applied field (excess carriers).

Instantaneous velocity for a given particle in the parallel (a) and perpendicular (b) direction, for an applied field equal to 1 kV/cm.

Instantaneous velocity for a given particle in the parallel (a) and perpendicular (b) direction, for an applied field equal to 1 kV/cm.

Autocorrelation function of parallel and perpendicular velocity fluctuations for excess carriers, for nS = 10^{12} cm^{−2} and several values of the applied electric field, 0.01 kV/cm (a), 0.1 kV/cm (b), 1 kV/cm (c), and 10 kV/cm (d). Figure (a) includes also the results for the background carrier population for comparison.

Autocorrelation function of parallel and perpendicular velocity fluctuations for excess carriers, for nS = 10^{12} cm^{−2} and several values of the applied electric field, 0.01 kV/cm (a), 0.1 kV/cm (b), 1 kV/cm (c), and 10 kV/cm (d). Figure (a) includes also the results for the background carrier population for comparison.

Parallel (a) and perpendicular (b) power spectral density of velocity fluctuations for nS = 10^{12} cm^{−2} and several values of the applied electric field.

Parallel (a) and perpendicular (b) power spectral density of velocity fluctuations for nS = 10^{12} cm^{−2} and several values of the applied electric field.

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