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# Time-dependent simulation of electron transport in graphene structures

ABSTRACT: This work consists in simulate using the software BITLLES, the electron transport through. double gate FET. The channel has been modeled by a material with a parabolic band structure (silicon) and with another linear band structure (Graphene). The dimensions of the channel are 10x1x8 nm. Semiclassical dynamics of electrons is considered and simulations are computed with the Monte Carlo technique. The linear band structure of graphene implies that the group velocity (Vg) of electrons WAVE PACK is limited to a maximum. Graphene with group velocities below 1E6 m / s, it means, 1E4 m / and 1e5 m / s will be studied to maximitze its effect on DC and transient features.

It is observed that for the same voltage Graphene-FET offers increased power (an order of magnitude higher). Also at slower group velocity, the slower the device to voltage changes (transient). Finally, for low group velocities graphene, the order of 1E4 m / s, there is a selflocking effect of carriers.

## The Boltzmann Transport Equation then reads:

where:

APPROACHES: Neglect the exact correlation between carriers, as they interact through the electric field. Analyze particles as Newtonian, considering the collisions as binary, instantaneous and space located.

## VELOCITY GROUP VS ENERGY

Have been considered lower graphene velocities to study how this novel characteristic can affect the DC and transient behavior of graphene FETs.

## MONTE CARLO TECHNIQUE.

The Monte Carlo (MC) technique, applied to the solution of the Boltzmann equation, simulates electronic devices providing an accurate solution of the Boltzmann equation of the dynamics of electrons by using trajectories. Then the I-V characteristic of the devices is obtained, as the local velocity distribution or the local electric Some techniques: Single particle Ensemble Monte Carlo of devices.

## We are using the MONTE CARLO OF DEVICES.

ADVANTAGES: Give an indirect instant solution of Boltzmann equation. Good performance of the random transport of carriers, that is propitious for studying the electronic noise in devices. So, it is a key tool for the analysis of the non stationary effects, although the computational limitations (cost).

LIMITATIONS: The classical approach of electron transport, considering them as Newtonian particles.

## Jordi El Mariachet Carreo NIU: 1139230 UAB-Master EMNE

BITLLES SIMULATOR
We are going to use only its semiclassical approach algorithms. It means that the equations to be implemented with the Fortran code are the shown lines above (See Semiclassical approach to the electron transport) and some effects must be neglected, as indicated also ut supra. Further considerations, as scattering or phonon interactions, impurities, etc. will be disabled for the simulation.

## Jordi El Mariachet Carreo NIU: 1139230 UAB-Master EMNE

BITLLES SIMULATOR
SETTINGS: DC or TRANSIENT external bias for D, S and Gates. Silicon (Parabolic Band Structure) or Graphene (Linear Band Structure) for the materials. Fermi Level. Total time of simulation, maximum time of particles life on the volume of simulation. Number of steps or iterations (resolution) Saving time for each step or iteration. Solving method. Boundary conditions for solving POISSON. Time step for solving Schrdinger Equation. Time steps for measurements Range of energy for computation of scattering rates. Type of injection of electrons: Discrete or continuous. MAIN DATA FILES: PARTICLES10*.TXT, CURRENT1OVERL10*.TXT, TRAJECTORY10*.TXT .

SILICON:

GRAPHENE (i/ii):

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## DC CHARACTERISTIC: Id=f(Vg, Vd)

GRAPHENE (ii/ii):

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## TRANSIENT CHARACTERISTIC: GRAPHENE Vs SILICON Vs VELOCITY GROUP (II/II)

The main conclusions of the TRANSIENT simulation are: 1.- For three simulations, the saturation values of the current for large times (when the transient is finished) fit well with the Dc discussion done previously.

2.- Both graphene SAMPLES have a faster transit time (tau) from 0 to 1 than the silicon SAMPLE.

3.- The group velocity (Vg) seems to control somehow the transit time (tau) of the graphene SAMPLES. As expected, the lower the group velocity, the slower the device.

## Jordi El Mariachet Carreo NIU: 1139230 UAB-Master EMNE

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CONCLUSIONS:

SELF LOCKING PHENOMENA:the presence of barriers generated by the low group velocity of graphene. The velocity is so small (when saturated) that even a very high electric field is unable to eliminate it. When electrons follow parabolic (Newtonian) dynamics, a the higher the electric field, the higher the velocity.

The transient behavior of the graphene has been analyzed and again the group velocity seems to have a relevant role: the lower the group velocity, the slower the device

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