Understanding the Electrical Conductivity of Graphene
Properties, Band Structure, and Applications
By now, you’ve probably heard of graphene. It’s the only 2D material, just single layer of graphite. Astonishingly simple. It’s a hexagonal carbon lattice structure (sheet), but it’s the strongest material, clocking in at 200 times stronger than steel. Oh, it’s also the thinnest and most conductive material known. 🤩
It’s flexible, transparent, light, nonpermeable (waterproof), and more. This is due to many phenomena that occur within graphene due to it’s unique band structure. These things give it so many applications in so many different fields! For example, it can be used in:
- biomedical engineering (sensors, tissue engineering🏥)
- in space tech (space elevator🚀)
- energy storage (lithium ion batteries🔋)
- electronics (flexible electronics📱)
- environmental remediation (water filtration💧)
So it’s basically the dream material. It has consistently been referred to as the wonder material. Well, have you ever wondered why graphene is so amazing? What makes it special? Let’s dive into the material science behind it.
Hey! I also made a video about this! You can check it out here :) 🤩
Background
Before we get into the technicalities of all this, you’re gonna need some background info. Let’s start with some definitions.
💃Conductivity — how well a material can transport thermal or electrical energy. This is the ability of valence electrons in the material to freely move.
🤧Impurity — something in the material that’s different from the rest of it, the material is no longer pure, it has more than one substance in it.
👌Defect — imperfection in the structure.
➕Doping — adding an impurity(which would add more positive charges or more negative ones) to a semiconductor to tailor the conductivity.
🌊Brillouin zone — used to describe the behavior of waves in a periodic medium, for example a crystal lattice. It’s kind of like a graph, or a 3D model. We’ll talk about it more later
📊Fermi energy — the energy difference between the highest and lowest occupied single particle states in a quantum system of non-interacting fermions at absolute zero temperature.
🔝Fermi level — the highest energy level an electron can occupy at absolute zero temperature.
➗Fermi surface — the Fermi energy divides allowed occupied states from those that cannot be occupied, and and abstract 3D “momentum space” is created to illustrate this. The surface of this momentum space is known as the Fermi surface and it separates occupied electron states within the momentum space from the unoccupied ones below it.
⚖️Fermi temperature — the temperature at which the thermal energy of the particles is similar to their Fermi energy.
🧠Quasiparticles — a way to simplify descriptions of complex systems because they treat it as though they were made up of individual particles when the reality is much more complex.
↗️ Mean free path — the mean distance a particle can travel through a material before it collides with another impurity/particle; how long it can travel before being scattered.
🎯Antilocalization — affects the electronic properties of materials significantly. Weak localization causes the electrons to scatter off impurities in the material, which affect’s the electron’s wave function and how it moves through the material. Antilocalization is when the wave function of the electron is enhanced, for example the electron moves in a curved path and interferes with itself, but also happens to enhance it’s delocalization. In short, it can help to reduce scattering.
🫂Adatoms — atoms that are adsorbed onto the surface of a material. It’s where atoms, ions or molecules of a gas liquid or dissolved solid adhere to the surface of a solid/liquid material.
✌️Quantized — apply quantum theory to, especially form into quanta, in particular restrict the number of possible values or states a system can have so that certain variables can assume only certain discrete values.
🤝Lattice — ordered array of points describing the arrangement of particles that form a crystal.
📝Other terms: (don’t worry I’ll explain later) quantum hall effect, massless Dirac fermions, massless Dirac dispersion, Berry phase, half integer quantum hall effect, Boltzmann equation, effects of Zitterbewegung.
Synthesis Practices
So just in case you’re completely new to graphene, I’ll give a quick rundown of the different ways it’s formed. For starters, synthesizing graphene is an incredibly challenging process, preventing it from taking over the world. Plus, only graphene of pristine quality exhibits the properties we want.
Synthesizing graphene is also expensive, and getting it to be pristine quality is challenging. This is mostly due to the fact that it is on such a microscopic scale. Since it is only one layer, every atom needs to be in the exact perfect position.
The majority of these methods use exfoliation, from a larger block of graphene.
- Mechanical exfoliation- peeling off layers of graphite with tape or something similar until you are left with one atom thick. This practice is not very effective, as the quality is low and its super time consuming. Quick fact: this is how it was discovered. Out of pencil lead! ✏️
- CVD- combining gas molecules with heated underlayer of the reaction chamber they are in. A reaction occurs that creates a material film on the surface, and this film is made of graphene. This practice is pretty good, with higher quality, and it is obviously much easier than mechanical exfoliation. 👍
- Chemical exfoliation — This first reduces the inter layer Van der Waals forces (weak intermolecular forces), and then exfoliates the graphene to single layers using rapid heating or sonication (with sound). Basically, fights the attractive forces in between the layers, than exfoliates the graphene down to single layers. This also works quite well, producing decent quality graphene. Overcoming the Van der Waals forces between sheets of graphene is called Liquid phase exfoliation. Another method of chemical exfoliation is reducing GO (graphene oxide) to graphene through electrochemical reduction. 🧪
- Epitaxial growth — deposition of a single crystalline film on a single crystalline substrate produces epitaxial film. Basically, you deposit single carbon atoms onto a single crystalline substance to form the graphene. This practice uses thermal decomposition, which is chemical decomposition that is caused by heat. It can produce large amounts of pure graphene.👌
A bit about the band structure of graphene
Let’s start with this: what is a band structure?
In solid state physics (doesn’t that sound so fancy???🧠), the band structure describes the arrangement of energy levels that are available to electrons in the material. These energy levels are also known as bands, hence the name.
Due to the regular arrangement of atoms in a crystalline material (like graphene), a repeating pattern of positive and negative charges in space are formed. This interacts with the energy levels in the crystal lattice (the symmetrical three-dimensional arrangement of atoms inside a crystal) and causes them to be quantized.
When something is quantized, it can only exist in discrete energy levels. (on or off, 1 or 0, spin up or spin down — if you know anything about quantum computing 🤝).
Let me give you a better explanation, because this is pretty important for you to understand. Picture a mermaid. 🧜♀️(bear with me here) — a mermaid has a tail in water, and legs on land, right? It can’t have both. It is discrete, only able to exist in specific states. In this case, the electrons are only allowed to be in certain energy levels, or energy bands.
So, the allowed energy levels are grouped into the energy bands we talked about. And these bands are separated by something called band gaps. (shocker)🤯
Band structures have a very particular shape, which depends on the crystal structure of the material, the atomic orbitals of the atoms of the material, and more. The band structure also determines many of the properties of the material, such as its electrical conductivity, and also the optical properties of the material, as well as its response to magnetic and electric fields.
Band gaps are actually quite important in conductivity, they are what determines if something is a conductor, semiconductor, or insulator. As you can see in the picture, a metal has an overlap, a semiconductor a small band gap, and an insulator a large band gap.
The valence band (the red) contains the energy levels that are filled by electrons in the material at equilibrium, and the conduction band (blue) contains the energy levels that are empty and available for electrons to jump into when they get excited.
🤔When you think about it, it makes sense! If there is an overlap with these two bands, excitation is going to be much easier, therefore making something more conductive. When excitation occurs, the electrons are able to “jump” into the conduction band. And we know that metals are good conductors. At the Fermi level (the dotted line on the picture, I’ll explain more in detail later), the band is only partially filled in conductors, which allows for easy movement of electrons through the material, creating good conductivity.
On the other hand, having a large gap makes it hard for the electrons to become excited and move bands, hence why that is an insulator’s band structure. An insulator has a completely filled valence band, which means there are no available states for electrons to move into, making conductivity low. The electrons are also bound tightly to the atoms and require a large amount of energy to be excited into the conduction band.
Semiconductors have a small band gap, which allows for some movement of electrons. They can conduct energy under certain conductions, such as when they are doped with impurities.
Graphene is a semiconductor, but its band structure is quite unique. The valence and conduction bands touch, not overlap (despite the high conductivity). Sometimes there can be a small band gap though.
📝To summarize: metals have high conductivity due to overlapping valence and conduction bands, semiconductors can conduct electricity under certain conditions due to a small band gap, and insulators have a large band gap and are not very good at conducting electricity.
Don’t get scared, I promise it isn’t that bad. It looks super cool so I wanted to show you. 😊
Graphene has a really interesting band structure. It is a semiconductor, but has a band gap of zero. So the valence and conduction bands are not overlapping, but they are touching. Sometimes graphene can also have a very small band gap.
The valence and conduction bands meet at the 6 vertices of the hexagonal Brillouin zone and form linearly dispersing Dirac cones. Wow, does that ever sound complex🤣. A picture is worth a thousand words, so here you go:
Do you see all those Ks? Those are called Dirac points. It’s where the conduction and valence bands meet. You also might have notices that there are 6 of them, and they’re at the top of a cone like structure. That’s why they’re called Dirac cones. The Dirac points are known as K and K’.
The special thing about this is that electrons near the Dirac points behave as massless Dirac fermions, which have a high mobility. (yep I’m going to explain later 😊)
Ok so now you have a little background on band structure, which will be useful as we dive into the depth of it. 🚀
What is conductivity?
Let’s start off with what conductivity is. In short, it’s how well a material can conduct (transport) either electrical or thermal energy.⚡This is due to the ability of the valence electrons to move freely. The more they are able to move, the higher the conductivity. I’m going to be focusing on the electrical conductivity of graphene in this section.💡
I’ll try to answer the questions that might arise, in an attempt to make this easier to understand. :)
❓Question #1: What stops these electrons from moving quickly?
In short, scattering. This is when the electrons collide with another particle or impurity in the material, causing them to change direction.
❓Question #2: How can we fix this?
1️⃣To start, have only one layer. All the best properties of graphene come when there is only one layer, including maximal conductivity.
SLG (single layer graphene) has only 2 free electron channels (where electrons can move) and is thus the most conductive. When there are interlayer channels, the electrons are affected by phonon and electronic scattering.
Ok, so phonon and electronic scattering. What on 🌍 is this? Well, phonon scattering is where the lattice vibrations📳, which are called phonons, interact with electrons or other phonons. This causes them to scatter and lose energy.
Lattice vibrations are the movement back and forth of atoms in a solid around their original, equilibrium position.
Electronic scattering is the electrons in a material interacting with each other, impurities, defects, or lattice vibrations and scattering. This can be due to forces such as the Coulomb scattering, which causes electrons to scatter due to the repulsive forces between them (remember opposites attract🥰, and like charges repel each other). Phonon scattering is more important for thermal conductivity, and electronic scattering for electrical conductivity.
Do you remember how graphene is just a single layer of graphite? Well, this is why graphene is so much more conductive than graphite.🫶
2️⃣Also, defectless graphene is best. This has no defects for the electrons to scatter off of. This maximizes the conductivity. 🥳
This is a fantastic time to introduce another term: mean free paths. This is the distance an electron can travel before being scattered in another direction. When the MFP (mean free path) is high, electrons can move a longer distance without being scattered, leading to higher conductivity.
When the MFP distance is lower, electrons aren’t able to travel as far due to the higher concentration of impurities, defects, and lattice vibrations. This leads to lower conductivity.
Also, something called backscattering can also occur in more graphene/substances with a higher defect concentration. Backscattering is caused by an electron bumping into or entering another substance or different density and changing direction. And when the electrons are consistently changing direction, they are not moving efficiency, resulting in lowered conductivity. 😭
Doping is also quite interesting, and there are 2 types — p-type doping and n-type doping. Doping can be used to alter the charges and charge carriers in a material by adding intentional impurities.
Doping involves adding different atoms to your material, and that atom will either accept or donate more electrons into your material. In this example, silicon was doped with Arsenic and there’s an extra electron. Silicon was also doped with aluminum, and now there’s a hole.
P-type doping has a high hole concentration, and comes from adding acceptor impurity atoms (they will accept electrons.) This increases the number of holes (positive charge carriers) in the material. This would be adding aluminum to the silicon, in our example.
N-type doping is increasing the number of free electrons (negative charge carriers) in the material. This is adding Arsenic to our silicon.
Sometimes unintentional doping can occur, and it also affects the conductivity. The larger amount of unintentional doping that is found the more likely the electrons are to backscatter. And you can probably guess why; the electrons would scatter off of the impurities.
In addition, defects can change the band structure. Basically what happens is that a defect in sublattice B can introduce a localized state, which can cause changes in the band structure. These localized states can trap charge carriers, causing changes in the material’s conductivity.
The ability of defectless graphene to conduct electricity is significantly reduced when a certain amount of positive or negative charges are added to the surface rendering it neutral, creating a NCP (neutrality compensation point). 📉So, even if there are no defects, conductivity can still be lowered.
This is because of the charge carriers.✍️ When a material is doped, you are changing the amount of charge carriers. Increasing the holes or increasing the free electrons.
The NCP is a point in a doped semiconductor where the concentration of ionized dopants exactly balances out the concentration of free charge carriers. At this point, the semiconductor is electrically neutral. 😐
This point also has zero net electrical conductivity😭. One reason for this lack of conductivity at the NCP may be non-interacting quasiparticles.
❓Question #3: What the heck are quasiparticles?
Quasiparticles are a way to simplify descriptions of complex systems because they treat it as though they are made up of individual particles, when the reality is much more complex. Some examples would be photons (carry heat + light) and excitons (carry energy in semiconductors).
To further explain, imagine your muscle. We just call it a muscle, but a doctor might think of it as epimysium, nerves, myosin, actin, among other components. We are simplifying the complex system that is a muscle. 💪
Non-interacting quasiparticles is where the behavior of each particle is described as independent of the other particles, despite the fact that they are constantly interacting.
A simple example of this, going back to our muscle analogy, would be each muscle fiber acting as if it was working independently of the others to lift something, but they are actually all working together. 🫂
Non-interacting quasiparticles have been established in Fermi liquids, but we are not sure if their impacts can be found when applied directly to graphene. 💭
Fermi liquids is a type of material where the particles, usually electrons, interact strongly with each other but still behave as non-interacting particles near the Fermi surface, which is the boundary between occupied and unoccupied energy levels in a material.
When an electron moves through graphene, it can behave as if it has no mass, and its motion is described by the behavior of these massless quasiparticles. The energy levels of these quasiparticles are determined by the electronic structure of graphene, which creates a gap between the energy levels of the electrons that can move through it.
When the material is doped away from the NCP, the charge carriers can be separated by an electric field, and electrons and holes move in the same direction in momentum space. They move in opposite directions in free space. When the doping occurs close to the NCP, the neural particle-hole pairs which are strongly coupled by Coulomb’s forces behave like a collection of neutral “atoms”. This shows that the conductivity of graphene is determined by Coulomb’s interactions.
❓Question #4: Wait but what if it’s engineered defects?
Ah, yes, look who’s been paying attention! 👏I mentioned that we can control conductivity with doping, so that leads to the logical conclusion that we can control the conductivity of the material in general.
Yes, you can tailor the conductivity by adding intentional defects🤩. I know at the start, I said that you want perfect, defectless graphene, and yes, you do . . . for the most part.
BUT. There’s a crazy interesting phenomenon that happens when the defect concentration is at just the right amount. 😼
Contrary to expectations, the conductivity of graphene increases with increasing concentration of vacancy defects. Defects decrease the mobility of the electrons (remember scattering) but they might increase the number of carriers (electrons + holes). This can lead to much higher conductivity, especially in graphene because there are no carriers at Fermi energy.
Fermi energy is the energy difference between the highest and lowest occupied single particle states in a quantum system of non-interacting subatomic particles at absolute zero temperature.
Let’s break up graphene into 2 sublattices. A lattice is an “ordered array of points describing the arrangement of particles that form a crystal.” Graphene and it’s honeycomb lattice is typically broken up into 2 sublattices, A and B. You can see that here:
If there is a defect in sublattice A, the Dirac point remains. However, in sublattice B, this may cause a localized state to form, which can act as a mid gap energy level. So, the effect a vacancy (empty space) in sublattice A is much more noticeable in sublattice B.
Mid gap states are electronic states that can be found within the energy gap of a material, which is the range of energies that an electron cannot occupy in the material. These states are usually associated with defects or impurities in the material, and are found in the middle of the energy gap.
The energy gap between the highest occupied electronic state (valence band) and the lowest unoccupied state (conduction band) should be a well-defined boundary. With impurities or defects however, states can form within the gaps, and they trap electrons, preventing them from contributing to electrical conductivity.
This new mid gap state indicates that the B sublattice becomes metallic in the presence of a vacancy in sublattice A. This indicates that graphene becomes metallic in the neighborhood around the di-vacancy (when 2 adjacent atoms are missing).
📝TL;DR: vacancies induce a metallic component in the electronic structure, and since the metallic structure extends to several lattice constraints, it should enhance the conductivity.
❓Question #5: Metallic component…?
The metallic component should be viewed as a transition from a system where the effects of Zitterbewegung are more prominent with limited conductivity, to a system of highly conductive graphene.
The effects of Zitterbewegung are a rapid back and forth movement of a particles position and momentum due to the relativistic design of the particle. It is more pronounced in particles that travel at relativistic speeds, for example electrons in graphene. 😼This can cause a shift in the particle’s average position which leads to the Quantum Hall effect, and can also affect the electron’s scattering and transport properties, affecting the conductivity.
The key thing here is that since metal is more conductive (due to the overlapping valence and conduction bands, red and blue), this metallic component can enhance conductivity.
This is driven by the defects in the graphene, through the mid gap states they create. They produce the metallic character with the help of the small shift of the Fermi levels away from the Dirac point.
Defects also donate or accept charge, which can lead to a shift in the Fermi energy (difference between highest and lowest occupied states at absolute zero.) A study compared 2 graphene samples, one with a 0.1% defect concentration, and the other with a 1% defect concentration. The only difference was the Fermi energy by 0.016 Ry. The second sample had greatly enhanced conductivity. 🤩
Once this point has been reached (where the metallic component is formed), the addition of more defects produces scattering which reduces the conductivity of the material and produces a more conventional behavior.
At higher defect concentrations, the scattering due to impurities becomes relevant, and balancing these two effects allows the resistivity to grow with larger defect concentrations.⚖️
📝TL;DR: this increase in conductivity is only for certain amounts of defect concentration, the scattering due to impurities and this metallic component effect must be balanced.
❓Question #6: What are some other effects of impurities?
In the Brillouin zone, as quality decreases, the Dirac points move away from Vg = 0 (Vg representing a gate bias voltage, which is used to control the flow of current between the source and the drain electrode).
Soooo. What does that mean? Well first, what’s a Brillouin zone?
A Brillouin zone is used to describe the tendency to occur at regular intervals of the crystal lattice structure of a material. The energy bands of a crystal are determined by the wave functions of the electrons in the Brillouin zone. The Brillouin zone also determines the allowed electronic transitions in a crystal, which are important for understanding the optical and electrical properties of materials.
Do you remember at the start when I was talking about Dirac cones and conduction bands and stuff? I didn’t actually explain what the Brillouin zone is, and why graphene’s is so special. 💀
So let’s start off with clearly establishing that graphene has a very unique Brillouin zone, and it has cone shapes in it.
When the Brillouin zone is conelike, it means that the energy of the electrons (or other particles) in the material vary linearly with the wave vector in a particular direction. The wave vector🌊 is a way of describing the size and direction of waves, which are a disturbance that travels through a medium or space, carrying energy without the transport of matter. These can be light waves, sound waves, water waves, or electromagnetic waves.
If the band was flat, the energy of the electrons would remain constant. This conelike shape is often referred to as a “Dirac cone” and it is observed in materials that have a specific type of band structure where the dispersion of energy is governed by the linear dispersion relation. This means the change in frequency is directly proportional to a change in wave vector (size and direction of waves).
A Dirac cone in the Brillouin zone of a material can have important impacts on it’s electronic and optical properties. 🏂
The Dirac points are the points in the electronic band structure where the valence and conduction bands meet, which forms a linear dispersion relation. They are also at the top of the Dirac cones.🤯
In the Brillouin zone of graphene, there are 2 cone like shapes in the electron dispersion, which is known as Massless Dirac Dispersion. This indicates that the energy of the electrons/other particles in the material vary linearly with with the wave vector in a particular direction. These cones create things such as massless charge carriers, long spin lifetimes, and strong electron photon coupling. The points of these cones are known as K and K’.
❓Question #7: Ok, so what?
Electrons near the Dirac points behave as massless Dirac fermions! These enhance conductivity. 🤩
When the particles have a linear dispersion relation, the energy varies linearly with the momentum instead of the quadratic relationship seen in the dispersion relation of a free particle. The Dirac equation was developed to describe the behavior of electrons in the context of relativistic quantum mechanics.
When the mass term in this equation is set to zero, the solution describes a massless Dirac fermion which has lots of unique properties. To start, their energy varies linearly with their momentum. The presence of massless Dirac fermions in graphene can lead to unusual electronic properties, such as:
- high electron mobility (linear dispersion relation means they behave as massless particles)
- lack of a bandgap (material will be able to conduct electricity over a wide energy range)
- non-trivial topological properties (due to graphene’s unique band structure which we already talked about)
Massless Dirac fermions enhance conductivity because they exhibit particular electronic properties that lead to things like high electron mobility and unusual quantum transport phenomena. In materials that have massless Dirac fermions, the electronic band structure has a pair of linearly dispersing bands that cross at a single point in the Brillouin zone, known as the Dirac point.
(Full circle moment 😍🔥)
The Dirac point is, as a reminder, where valence and conduction bands touch each other. The electrons behave as if they have no effective mass, like relativistic particles. This leads to high electron mobility.📈
Linear dispersion means that electrons can be easily excited to higher states with a small energy input. 🔥
This also leads to the Quantum Hall Effect, which quantizes the conductivity of the material in discrete steps. The Hall conductance is proportional to the number of occupied electronic states at the Dirac point.
Hall conductance is a measure of the strength of the Hall effect, which is the generation of a voltage across a semiconductor or conductor. The voltage is transverse to the direction of the electric current when the electric current is exposed to a magnetic field perpendicular to the current.
❓Question #8: So, what’s the Quantum Hall effect?
The first thing you need to know is that when things get small, their properties change. Think nanomaterials/nanotech. For example, when gold particles get down to really small, nanometer size, they look red, not gold! 🟥
And that’s the same in this case too! When materials get really small, in graphene’s case, one atom thick, their properties can change. That’s one of the reasons graphene is so special overall, because its a 2D material.
But anyways, back to the quantum Hall effect. When the material is only 2 dimensions, some really interesting quantum phenomena can occur. One of these such phenomena is the quantum Hall effect, or QHE. Graphene also has a very unusual QHE, its a half-integer QHE. 👀
The electrons undergo some strange behavior under this effect.
So, what does that mean?
First let’s talk about some basics. You remember quantized states right? Only able to be specific values? Well, that’s the whole thing behind the QHE! The energy levels of electrons can only be specific values, rather than a continuous increase. Here, like the one on the left, not the one on the right. See the difference?
The plateaus represent the allowed energy values; if there was a consistent increase there wouldn’t be quantized values. The fact that graphene has a half-integer QHE means that these discrete values are shifted by a half integer when compared to the QHE in other systems.
Some theories as to why the QHE is shifted by a half integer in graphene is because of the Landau levels, combined with the particle hole symmetry of graphene.
❓Question #9: Landau levels????😭
Landau levels are formed when a 2D system, such as graphene, is exposed to a magnetic field normal to the graphene plane. This quantizes the in-plane motion of charge carriers (electrons + holes) into LL (or Landau levels.) :]
Landau levels also have degeneracy, specifically gs = 4. This includes both spin degeneracy and sublattice degeneracy, for a total degeneracy of 4 fold. Degeneracy means that the LLs are losing some of their defining characteristics, losing a property that is usually present.
When exposed to a stronger magnetic field, the degeneracies of the four fold n=+/- LLs are partially resolved, with a degeneracy of two fold in each of those. (don’t worry about the equation, it only indicates the hole and electron (+/-) Landau levels, respectively).
This is important for the conductivity of graphene because it can allow you to get specific conductivity levels, as the QHE is extremely stable. In the case of graphene and its half integer quantum hall effect, it can be used to create highly accurate and precise electrical components like resistors, capacitors and inductors. This may also allow development of new electronic devices. 🤩
The Quantum Hall effect also leads to a lot of the fascinating properties in graphene. For example, massless Dirac fermions lead to the very high conductivity because they are able to move super quickly. 📈
Now let’s talk about how and where this occurs. It occurs when electrons are confined to only 2 dimensions. For example, a 2D material. 🤯
When these 2D systems are exposed to strong magnetic fields and low temperatures, the QHE can be found. The QHE also can be found at higher temperatures too, although it is generally associated with lower ones. 💭
Remember the massless quasiparticles? There are gaps between the energy levels of the quasiparticles. And these gaps are so big that even when a strong magnetic field is applied, it cannot affect the particles in the way it normally would. This gap is called a cyclotron gap, and it’s the reason why the QHE can survive if the magnetic field isn’t as strong, or the temperature isn’t as low. 🏖️
The gaps are important because they determine how the quasiparticles respond to external influences, such as a magnetic field. In the case of the Quantum Hall Effect, this gap is large enough that it creates a stable plateau in the conductivity of the material, (this is literally the definition of the QHE🤩) which allows the QHE to survive even in the presence of imperfections or disorder in the material. So it all comes down to the energy level gaps in the quasiparticles. ;)
Now let’s look at Berry’s phase. :D
❓Question #10: What’s Berry’s phase and why does it matter?
A fundamental concept in studying the QHE, along with many other phenomena in physics, is Berry’s phase. This is describing how a particle’s wave function changes as it is undergoing a cyclic evolution, or an orbit.
When a particle is exposed to changes in its environment, the wave function describing the particle’s state changes. 🌊Berry’s phase is the additional phase shift that builds up in the wave function during cyclic evolution. This can create interference patterns of affect the behavior of particles in a magnetic field.
This affects the transport properties of the material, and leads to the anomalous (not normal) QHE.
Berry’s phase is found in systems with non-trivial (special)✨ arrangement of their energy bands. In graphene, this non-trivial arrangement is due to the 2 inequivalent valleys in the Brillouin Zone exhibited by the electronic band structure. This causes the band structure to exhibit a non-zero Berry phase, which is also a special type of Berry’s phase.
Berry’s phase affects the conductivity of graphene through the quantum Hall effect.🏂 Do you remember how the QHE was half-integer? That’s in part because of Berry’s phase — that indicates that there is a non-trivial Berry’s phase.
TL; DR:
That was a LOT. Vielen Dank fürs Lesen🫶. Let me recap the science behind the conductivity of graphene, and then we’ll take a look at some of the applications.
- 🩹band structures — graphene’s valence and conduction bands touch at the Dirac points. When the valence and conduction bands overlap, its a metal with higher conductivity, and the further away these are, the lower the conductivity
- 1️⃣one layer of graphene allows you to minimize scattering, which helps to increase the conductivity
- 👌defectless graphene reduces scattering as well, except when it’s intentional defects, like doping. Another possibility is a specific amount of defect concentration so that the metallic component will form, enhancing conductivity
- ⛷️graphene has a linear dispersion relation, meaning the change in frequency is proportional to the change in wave size and length
- 📈at the Dirac points (K and K`), there are massless Dirac fermions, which leads to high electron mobility because these particles have no effective mass
- 💫The quantum Hall effect is caused by the unique band structure of graphene and Berry’s phase, which creates a shift in the cyclic evolution of a particle, changing how it behaves under the influence of a magnetic field
- 🪜The quantum Hall effect means that the conductivity increases in discrete steps, rather than a continuous increase. Graphene’s is shifted by a half integer due to the Landau levels and Berry’s phase. The quantum Hall effect and it’s stable conductivity values could make the amount of conductivity more precise
Applications
Let’s take a look at some of the super cool applications of graphene. 🤩
Lithium Ion Batteries
To start, Lithium ion batteries! These are a revolutionary type of battery, and graphene has the potential to level them up even more.
In this case, graphene is used for the cathode, which the negatively charged side in a battery. 🔋Top level up the battery, we’re looking for a material with low apparent density (density including pores and water of the material) and very high conductivity to coat the LiFePO4 found in the battery’s cathode (negative terminal).
Well, graphene fits that criteria quite well, it is incredibly light and also very conductive.
Graphene nanosheets (GN) work so well for this because they are the most exfoliated form of graphene, which gives them lower apparent density and more contact surface, which reduces the amount of additives needed.
GNs spread out well throughout the cathode, which is important for the conductivity. If all the graphene agglomerates (groups together), the conductivity will be severely limited. GNs tend not to do this at lower concentrations.
The results of a study looking at how GNs in lithium ion batteries would affect the performance found some interesting stuff. I’ll give you a rundown here. :)
A Fe/GN (iron graphene nanosheet) cathode with 2 wt% (weight percentage) compared to a Fe/SP cathode with 20 wt%; the first performs better. SP is known as “super P” and is a type of conductive carbon black.
The discharge plateau performance (whether the battery’s performance plateaus and drops off; this can indicate greater stability of the output voltage) and the specific capacity (measure of energy per unit of mass/volume of active material in the battery) were both much better in the GN battery. 📈
In addition, the cycling performance🚲 (the capacity of the battery to maintain its performance over the course of many cycles of charging and discharging) of the GN battery remained mostly above 150 mAg^(-1) and wasn’t affected by the cycle number.
However, in the SP battery, there was a lower specific capacity and slightly worse cycling performance, which is likely due to the unstable charge/discharge process which is caused by SP agglomeration and side reactions.
So, to conclude, the Fe/GN cathode with a weight percentage of 2% performed much better than the typical Fe/SP cathode with a weight percentage of 20%. It has a better discharge plateau performance, specific capacity, and cycling performance. 🚲
When the fraction is lower (0.5–1%) a stable discharge platform is not formed, which may be because the content is not high enough to create a proper conducting network. When the fraction of GN is higher, it tends to aggregate(clump), and displays some electrochemical behaviors similar to graphite. 😭The charge discharge behavior also becomes worse.
GNs also bridge the active LiFePO4 particles by a plane to point method, which is more efficient than the typical SP point to point bridging method.
Biomedical — Tissue engineering
More research is needed to determine if graphene is in fact biocompatible, but seeing as it is made of carbon which is present in our daily lives, it looks as though graphene will be fine to use.
Graphene has high potential due to its excellent aqueous processability, amphiphilicity, surface functionalizability, surface enhanced Raman scattering (SERS) property, and fluorescence quenching ability.😳
Yeah that’s a lot of massive words. 😭Basically what they are trying to say is that graphene can be processed in water or a water-like solution, it has a surface that doesn’t like water and one that does, has a surface that can be modified to change its physical, chemical, or biological characteristics.
Surface enhanced Raman scattering is a technique that makes it easier to detect and analyze molecules by increasing the intensity of their Raman scattering. 🧠And finally, fluorescence quenching ability refers to any process that decreases the fluorescence intensity of a sample, which is how much light (photons) are emitted by a substance that has absorbed light or other electromagnetic radiation. This allows scientists to detect + analyze substances.
These are all really key features for something to be used in biomedical engineering.
Graphene may be used as a scaffold for cell culturing, (growing cells under specific conditions, like in a lab).
Another way graphene can be used for tissue engineering is called suspending graphene in culture media. This is dispersing graphene particles in a liquid medium that is used to grow and support cells. The cells can interact with the graphene particles and then can be used for various tissue engineering applications. For example, creating scaffolds for cell growth or studying the effects of graphene on cell behavior.
Again, it’s important to note here that more research is needed here before graphene can be applied for tissue engineering. 📝
Some other potential applications are 📱flexible electronics (due to the fact that graphene is highly conductive but also flexible), a 🌌space elevator (because graphene is super strong and light), and 🧬biosensors (although more research is needed to determine if graphene is biocompatible or not).
TL;DR
- graphene has a very unique band structure, which leads to super cool properties. There are Dirac cones and a linear dispersion relation
- one layer is the most conductive
- defectless is best — unless it’s engineered and a specific concentration
- massless Dirac fermions exist due to the band structure and have very high mobility
- quantum hall effect and Berry’s phase also contribute to the high conductivity, they are caused because electrons are confined to 2D
- lithium ion batteries — graphene can increase the performance a lot
- biomedical — more research is needed but it looks promising
Thank you so so so much for reading, I hope you now know a whole bunch about graphene. I wish you the best, and see you in the next one! 🥳
— Sasha
Citations
Also, if you’re interested, I summarized a bunch of these research papers here, so feel free to check it out!
Biomedical Applications of Graphene and Graphene Oxide | Accounts of Chemical Research (acs.org)
Xiao_2010.pdf;jsessionid=A631E1D376F6F2A8F940056A3CFF86C0 (utexas.edu)
Conductivity engineering of graphene by defect formation — IOPscience
https://arxiv.org/ftp/cond-mat/papers/0509/0509355.pdf
Phys. Rev. Lett. 95, 226801 (2005) — Quantum Spin Hall Effect in Graphene (aps.org)
Phys. Rev. B 78, 085415 (2008) — Conductivity of defectless graphene (aps.org)
