The Physics of the Graphene Revolution
The Graphene lattice. Image from: https://singularityhub.com/2018/08/05/beyond-graphene-the-promise-of-2d-materials/

It is a poorly kept secret that graphene is a material with the potential to revolutionise the world of electronics. Alongside its optical transparency, its flexibility and its unparalleled physical strength, it is among the most efficient conductors of electricity known to science.

In some ways, it is remarkable that graphene conducts electricity at all. It is, after all, a lattice of carbon, which occupies the “non-metal” bin of high-school chemistry, so we might expect that it would be a poor conductor of electricity like, say, crystalline sulfur or phosphorus. And indeed diamond, a crystal of carbon, is an extremely good insulator. So why is graphene any different?

The electrons around any atom occupy distinct regions of space known as orbitals. These can come in a wide range of different shapes and sizes, but for carbon there are only two types of orbital present: two spherical s-orbitals and one dumbbell shaped p-orbital. Each of these contain two of the carbon atom’s six electrons.

Examples of s and p atomic orbitals. The p orbital shown is in the z direction, which is the orbital left unhybridised in the carbon atoms of the graphene lattice. Image from the Encylcopaedia Brittanica https://www.britannica.com/science/orbital/images-videos

More exciting is what happens when atoms bond. To do this, orbitals combine in a process known as hybridisation, and this can happen in a few different ways. In diamond, the outer s-orbital and the p-orbital hybridise to form 4 sp³ hybrid orbitals, each containing one electron which can then be shared with a neighbouring carbon atom to form an atomic bond. This process utilises all of carbon’s 4 valence electrons, meaning that there are no electrons free to move to neighbouring carbon atoms and carry a current through the material.

In graphene, and its layered counterpart graphite, the hybridisation is different. Instead of forming 4 sp³ orbitals, the orbitals hybridise into 3 sp² hybrid orbitals, leaving one electron alone in an unhybridised p-orbital. It is the interactions between these orbitals of atoms in adjacent layers which hold crystals of graphene together. More importantly for our purposes, these orbitals are not involved in atomic bonding, so electrons can freely move between the unhybridised orbitals of neighbouring atoms, allowing a current to be carried through the crystal.

Ok, so graphene can conduct electricity. But why is it so good at it? The best measure of how conductive a material is, unsurprisingly, its conductivity. This is the ratio between the current generated by a voltage applied to a material, and the magnitude of that voltage, with some factors related to the size of the sample to allow comparison between materials. So, if a small voltage is applied to a material, and it generates a large current, then the material has a large conductivity and is a good conductor of electricity.

In the simplest terms, a voltage corresponds to extra energy given to electrons within a material. If that extra energy isn’t large enough to break electrons free of their atoms, then no current can flow at all, and we have an insulator. Conversely, if it is easy to break an electron free of its atom, most of the extra energy goes to its kinetic energy — the energy associated with its movement through the material — and the electron can move quickly. Current is the rate of flow of charge, so electrons moving quickly through a material corresponds to a large current.

The electrical conductivity of some common metals. Notice that group 11 elements occupy the top 3 most conductive metals. Image from http://wiki.robotz.com/index.php?title=File:Conductivitymetalchart0.jpg

For example, in the most conductive metals, group 11 elements such as gold, silver and copper, the outermost valence electron is on its own in a higher energy subshell than the rest of the atom’s electrons, meaning it takes comparatively little energy for it to break free from its atom. This means that when that electron receives energy from a voltage, most goes to kinetic energy meaning that it can move quickly through the metal. Combined with these outermost electrons of neighbouring sites all moving similarly fast, the current generated by the voltage is large, and these metals have extremely high conductivities.

So what about graphene? Graphene is a honeycomb lattice: its atoms form hexagons such as those one might find in the local beehive. We find that the dispersion relation, which describes how the energy of an electron varies with its momentum, has two bands: an upper band known as the conduction band, and a lower band known as the valence band. The gradient of the dispersion relation gives the electron group velocity, which can be broadly thought of as the velocity of the electrons as they move through the lattice.

The dispersion relation for a honeycomb lattice such as graphene. Here, momentum is normalised against the magnitude of the momentum at the K and K’ points, so the two bands meet at k=1 and k=-1.

Crucially, the bands touch at two momenta, which we refer to as the K and K’ points. Near these points, the dispersion relation is linear, which means that the momentum of the electrons increases at a constant rate as energy of those electrons increases. This is the behaviour of a massless particle!

When you increase the energy of a photon, for example, its momentum increases at a constant rate, namely that of the speed of light c. Just like photons can’t travel at any speed other than the speed of light, a conduction electron in graphene cannot travel at a speed other than a velocity of around 1 million m/s. So, even for extremely small energies, the electrons move at very high speeds. This means that these small voltages generate a large current, which mean graphene has a large conductivity.

The dispersion relation of a honeycomb lattice close to the K point, which has been defined as q=0. Colour has been used to differentiate the conduction band (red) and the valence band (blue). Notice that the dispersion relation is linear, such as would be expected for a massless particle.

However, for fast electrons to be generating a large current, they must, for the most part, be moving in the same direction through the material. Ideally, the voltage ensures this by setting up an electric field in the material, attracting the negatively charged electrons to the positive terminal, but the universe is not all sunshine and rainbows. Electrons can be deflected, or scattered, by numerous impediments. Indeed, a wire is an ever-changing assault course of impurities, vibrating atoms and other electrons, all of which can disrupt the progress of any one electron moving under the influence of the voltage.

We thus have another factor which affects conductivity: how often electrons are scattered as they move through a material. If the electrons are scattered often, then most will be prevented from moving in the direction of the electric field associated with the applied voltage, and the current associated with that voltage will therefore be small. This leads to low, poor conductivity.

A diagram of the generic honeycomb lattice, indicating the inequivalent A and B sites, as well as the primitive lattice vectors and nearest-neighbour vectors. Image from Gilles Montambaux. Artificial graphenes: Dirac matter beyond condensed matter. Comptes Rendus Physique, Elsevier Masson, 2018, 19 (5), pp.285–305. ff10.1016/j.crhy.2018.10.010f

Does this crush our dreams for graphene? Thankfully not. On a honeycomb lattice such as graphene, there are two “types” of lattice sites, which we call the A and B sublattice. Any electron state can be thought of a superposition of being located “on” the A sites and the B sites, and we can treat this behaviour, known as a pseudospin, in the same way that we do the spin of the electron. For example, an electron on an A site can be thought of as being spin up and an electron on a B site can be thought of as being spin down.

Importantly, for the electrons which carry the current in graphene, this pseudospin depends on the direction of the group velocity. Near the K point, an electron with a positive group velocity is spin up, while an electron with a negative group velocity is spin down. This coupling of spin and velocity is known as chirality, and we can think of it as electrons moving in a positive direction on the A sites and in a negative direction on the B sites.

The dispersion relation near the K point of a honeycomb lattice with the pseudospin in each band, here represented as left and right as opposed to up and down, superimposed. Note that any backscattering, represented by a horizontal translation on the graph, requires a flip of pseudospin direction.

So, what happens when a chiral electron encounters a scatterer? For it to scatter backwards, it would require a flip of this pseudospin since the group velocity changes sign. In other words, to backscatter would require a move from the A sublattice to the B sublattice, which is not possible in most cases. We therefore see complete suppression of backscattering, and an overall suppression of all scattering.

As such, electrons in graphene can travel for up to micrometre distances without scattering at all. This behaviour is known as ballistic transport, and this reduced scattering ensures that the full power of the speed of the conduction electrons in graphene can be brought to bear.

The final factor which affects the conductivity of a material is simply the density of conduction electrons within a material, known as charge carrier density. Electrons are what carry the charge and so the more of them that move, the greater the associated current. In graphene, we can control this quantity by doping. This is where atoms such as phosphorus or nitrogen, which can donate two electrons for conduction compared to carbon’s one, replace some carbon atoms in the lattice to increase the number of available electrons for carrying a current.

Even while the carrier density of graphene could never be as high as in most metals, its conductivity is still remarkable. The theoretical limit of its conductivity is up to a million times better than that of silver, the best known metallic conductor. Indeed, the conductivity of graphene samples is actually often limited by interactions with the material on which they are mounted, rather than the properties of the graphene itself.

Even so, for a material as light as graphene to exhibit such impressive conductivity is extremely exciting. For example, one of the major hurdles in the world of electric transport is the sheer weight of lithium-ion batteries which dramatically limits the range of such vehicles. Current graphene-based cells are an order of magnitude more efficient, an energy-to-mass ratio of 1000 Wh/kg, compared to 180 Wh/kg for Li-ion batteries.

Energy efficiency is as important a goal as any as our global society hurtles headlong into an increasingly inevitable climate crisis. More efficient conductors such as graphene, and potentially even the long elusive room-temperature superconductors, have the power to reduce energy losses in just transporting energy around our power grids. This could provide one of the many necessary spears in the phalanx of efforts to combat the encroaching climate disaster.
['Jason Segall']
['Electronics', 'Technology', 'Physics', 'Graphene', 'Science']