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Graphene is both strong and lightweight, providing many potential applications of graphene-based electrical and structural devices. Thanks to COMSOL Multiphysics®, with its state-of-the-art solvers, optimized by Intel® to run on their processors, simulating such applications is now straightforward.

 

 

What is graphene?

 

In short, graphene is a special type of material made up of a single layer of carbon atoms arranged in a hexagonal lattice. It was discovered in its stable form at the University of Manchester in 2003, leading to Nobel Prizes in 2010 for the two researchers who discovered the material. Since 2004, when graphene was originally isolated in its planar form using adhesive tape, the race has been on to mass-produce the material utilizing economical fabrication techniques. These days, a more popular method involves performing epitaxial growth on silicon carbide by heating it to high temperatures at very low pressures. There are plenty of other techniques out there as well, all accompanied by their own advantages and limitations. As I mentioned, graphene is both strong and lightweight – to be more specific, it is about 200 times stronger than steel and weighs less than 1 milligram per square meter. Furthermore, compared to copper, graphene has a higher electrical and thermal conductivity. This can be attributed to the unusually high room temperature electron mobility of 15,000 cm2/(V-s).

 

 

Introducing a Band Gap Into Graphene

 

Researchers have attempted to use graphene as a replacement for silicon in semiconductor devices. The problem with graphene in its unaltered form is that it does not have a band gap. This means that, if used in a fast electrical switch (like a MOSFET), there is no way to turn the current off. The basic idea is to chemically modify the graphene so that it retains its unusually high mobility, but introduces a band gap to make it behave like a semiconductor.

 

In simple terms, the idea of a MOSFET is to apply a gate voltage to control the drain-to-source resistance and thus the drain current, as such:

 

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At a certain gate-to-source voltage (VGS), and at low drain-to-source voltages (VDS), the drain current is almost linearly dependent on VDS. When VDS increases, the drain current saturates. The saturation level is dependent on the gate-to-source voltage, while the switching time is dependent on the semiconductor's mobility. In other words, the higher the mobility of the semiconducting material, the faster the current can be switched on and off.

 

The fact that the room temperature electron mobility of graphene is an order of magnitude greater than silicon means that lower drain-source voltages can be applied to generate the same electrical current. This results in lower electric fields for the same electrical current, hence less power dissipation (since the power dissipation is the dot product of the current density and electric field). From a thermal management and maintenance cost perspective, this is very attractive. Another advantage of the high electron mobility is that the on/off switch time will be an order of magnitude faster than in the pure silicon case, which is significant in the power electronics industry.

 

Since the physics of semiconductors is rather complicated, the modeling approach most commonly used is to solve a set of drift diffusion equations coupled to Poisson’s equation:

 

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Where n = number density of electrons, p = number density of holes, V = electrostatic potential, Rn = electron recombination rate, Rp = hole recombination rate, Jn = electron current, and Jp = hole current.

 

If you solve this set of equations, you can construct the voltage-current characteristics, step response, and efficiency of semiconducting devices. Device engineers who are working on the applications of graphene-based semiconductors can already make "what-if" decisions for how well a specific device will work using appropriate semiconductor device simulation software.

 

 

Manufacturing and Large-Scale Production Issues

 

There are many competing manufacturing methods employed to attempt mass-producing graphene – including exfoliation, epitaxial growth on a suitable substrate, reduction of graphene oxide, pyrolisis, or growth from metal-carbon melts. Another method is the use of thermal decomposition at high vacuum. Engineers might look to simulation in order to design and optimize high-vacuum systems that are suitable for producing graphene. However, there are few modeling tools out there capable of aiding in this, particularly if the system is nonisothermal. As the mean free path of gas molecules become comparable to the length scale of flow, kinetic effects become important. Therefore, conventional fluid dynamics tools cannot be used for modeling gases at low pressures. The pressure on surfaces primarily depends on the line of sight with respect to molecular sources and sinks in the vacuum system.

 

As an example, take the nonisothermal vacuum system described in this thesis on growth of graphene films on Pt(111). The system shown there would inevitably be costly to build, but if design optimizations can be made prior to construction, it would save a lot of money later on. The thesis outlines some estimations on the fluxes onto surfaces and deposition rates, but these do not take the geometry of the system; the notion that different surfaces have different temperatures; or the location of the pumps into consideration.

 

If we were to accurately model nonisothermal molecular flows and deposition rates onto substrates on arbitrarily complicated geometries, we would need to take a sophisticated modeling approach. The flow of gas is determined by collisions with the surfaces in the system because gas molecules interact with surfaces more frequently than they interact with one another. Therefore, we would need to solve a complicated integral equation in order to compute the molecular flux, pressure, heat flux, and number density in the system. The molecular flux can further be used together with a suitable differential equation to determine the deposition rate and the deposited film thickness. This allows different process configurations and concepts to be tested before building the physical process chamber.

 

 

Simulation and Experiments Go Hand-in-Hand

 

The potential improvements in a wide range of today's devices along with the potential to create new devices, makes graphene too powerful of a substance to ignore. Even though mass-producing graphene in a monetarily and environmentally efficient way is an ongoing process, researchers are already simulating and prototyping graphene-based devices. Due to its unusual properties, any simulation tool should be chosen carefully to fully take into account all the different physical processes that need to be taken into consideration. The characteristics of prototypical graphene-based devices need to be carefully measured, quantified, and compared with the expected performance. In this sense, the simulation and experiments go hand-in-hand – the simulations can provide guidance as to what experiments to perform; the experiments can provide data to validate the simulation, giving confidence that further simulations will reproduce the physical characteristics of the actual device.

 

If you are interested in reading more about graphene, check out the graphene series on the COMSOL Blog.

 

 

COMSOL® is a rapidly growing, high-tech engineering software company, providing solutions for physics-based modeling and simulation. With COMSOL Multiphysics® and the suite of add-on modules, engineers and scientists can simulate any physics-based system. Learn more at http://www.comsol.com/.

 

Daniel Smith received his Masters degree in applied mathematics from the University of St. Andrews in 2002 and a Masters in Numerical computing from the University of Manchester in 2003. Following this, Daniel worked as a scientist at MKS Instruments in the Corporate Advanced Technology group.

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