Experimental Review of Graphene

My first academic paper was officially published as of the beginning of this year.

Experimental Review of Graphene (pdf)

Experimental Review of Graphene
Daniel R. Cooper, Benjamin D’Anjou, Nageswara Ghattamaneni, Benjamin Harack, Michael Hilke, Alexandre Horth, Norberto Majlis, Mathieu Massicotte, Leron Vandsburger, Eric Whiteway, and Victor Yu
ISRN Condensed Matter Physics
Volume 2012 (2012), Article ID 501686, 56 pages, doi:10.5402/2012/501686

The full-text paper is available as html, and as a pdf by ISRN. A preprint (with chapter headings) is available on arxiv (full-text pdf).

The review is intended to provide an introduction to the study of graphene from an experimental perspective. The topics covered within are:

  1. Electronic structure
  2. Vibrational properties (phonons)
  3. Synthesis (fabrication)
  4. Characterization (measurement / detection techniques)
  5. Electronic transport and field effect (scattering, mobility, conductivity)
  6. Magnetoresistance and the quantum Hall effect
  7. Mechanical properties (micromechanical oscillators, actuators)
  8. Graphene transistors
  9. Optoelectronics (transparent conducting electrodes, photodetectors, light-emitting diodes, photovoltaics, quantum dots)
  10. Sensors (electrochemical and biosensors)

Gordon E. Moore: The Semiconductor Prophet

The insights in Gordon Moore’s world-famous paper, Cramming more components onto integrated circuits, have been validated again and again in the decades since its publication.

Upon reading the paper, many startlingly accurate statements are likely to jump out at you. Startlingly accurate that is - because the date of original publication was April 1965.

Here are some of the prophetic insights that leaped out at me:

He says that memory may be distributed throughout the machine rather than concentrated in a single unit. My primary experience with this phenomenon is in the construction of personal computers. Today’s PCs have hard disks, RAM, and CPU cache in order of increasing speed and decreasing size. Additionally, specialized devices such as video cards are increasingly being fitted with their own RAM and even sometimes flash memory. Memory accessibility has proven to be one of the salient difficulties of computer design. Spreading the memory around has made even faster operations possible.

He accurately predicted that semiconductor integrated circuits will come to dominate electronics. The rise of the PC age is a good indication of this domination. Today we are beginning to see semiconductor integrated circuits in pretty much anything that has electric power flowing through it.

His ‘day of reckoning’ thing sounds a lot like the frequency wall that we hit in the early 2000’s. Since the early 2000’s, clock frequencies in mainstream computers have not increased. Today, our top CPU manufacturers focus on improving performance per clock cycle and per watt of power.

He says that we may find it more economical to build larger systems out of smaller functions. Look at our multi-core personal computers, computer clusters, cell computers, and cloud computing. As a consequence of the frequency wall and economics, today’s supercomputers are dominated by multicore and multiprocessor systems. In the last few years we have also been watching the rise of the cloud computing system. Using the power of the Internet, staggeringly huge supercomputers are created out of smaller cells linked to each other through the network. We have only just scratched the surface of how cloud computing is going to change the face of our computing world.

Lastly, this is the piece in which Moore first described the economic relationship that would come to be known as Moore’s Law. His observations are often misquoted and misinterpreted in popular media. He identified a definite trend in the cost of production of integrated circuit components and the number of components per integrated circuit. This has been extrapolated by later thinkers into a plethora of versions of “Moore’s Law” that are claimed to be representative. The accuracy of the later versions is highly questionable. However, Moore’s actual prediction has been remarkably accurate for over four decades.

Scratching the surface of surface science

The primary resource for this material was a lecture by Peter Grütter.

Why has it been reseached?

The semiconductor industry is probably the primary reason why surface science has received so much attention. It is well established in industry and in science. The development of the tools and techniques has been driven primarily by the semiconductor industry.

Another major driver is catalysis. This is the area of study of how to catalyse reactions/processes so that they can happen more quickly and/or with lower energy requirements. Surfaces can be extremely important for catalysis. With regards to catalysis, the sites of interest on the surface are actually the kinks and defects rather than the flat surface itself.

Small features can be of primary importance in many of these condensed matter fields of study. Dr. Grütter calls attention to the fact that in the semiconductor industry, the doping atoms among the silicon of crucial to the operation of the devices.

Introduction to Surface Science

In this class, we will be talking primarily about solid-vacuum interfaces rather than solid-liquid interfaces. We are building on the knowledge we gained in the introductory sections on vacuum systems.

Surfaces are 3 dimensional. They are not merely two-dimensional planes. They are a layer of transition from bulk conditions to vacuum conditions.

The dipole layer is an interesting physical phenomenon that takes place at the surface of a material. Electron density does not drop off to zero once we are outside the surface atoms. It tapers off, becoming negligible some small distance away from the surface. This distance is on the order of one fermi wavelength, which would vary depending on the material.

So some negative charge ends up outside the surface. The only picture I can find of this effect online is here, even though it is given in terms of electrostatic potential rather than electron density. Rather than a smooth drop in electron density, we end up with a periodic (on the scale of fermi wavelengths) charge density as we look into the surface. Thus, just inside the surface we actually have a higher electron density than we do further into the bulk of the material. This interface between the high internal electron density and the low external electron density is called the dipole layer.

The dipole layer can stop atoms from diffusing out of the surface. As they diffuse towards the surface, they suddenly come up against a larger density of electrons, which push them away. In the image linked above, the diffusion would be taking place from right to left. The lower potential pushes back on the atom’s electrons, causing it to have more difficulty getting through the surface than it had moving throughout the bulk of the material.

A few Observations

As we already know, taking an electron out of the surface will take some energy. The amount of energy depends on several things such as strength of bond to ion core, interaction of electrons with each other, etc. This is known as the work function. There are two versions, one considers the energy needed to move the electron to just outside the solid surface, while the other considers the move of the electron to infinity).

The work function depends primarily on the dipole layer. Can be different work functions for different surfaces (faces) of crystals! Depends on the orientation of the atoms. Work function also depends on step density. What is a step? Consider a perfect planar surface of atoms. Now consider adding another layer to half the surface, so that there is a ‘step’ up to the second layer. There can be many such steps. As a heavy and long-time computer user, one of the first things I visualized was the fact that an angled line on a computer monitor is not smooth, it has ‘steps’ made of straight sections. Similarly with a surface viewed at the nano scale. The closer the steps are to each other, the higher the step density. Step density changes the work function because of the details of the dipole layer at each step.

Question that you must learn to ask yourself: You must ask yourself if what you are studying is affected by small defects in the system. In the history of science this has been overlooked many times. How big of an effect can these things have? Well, it turns out that a 5% difference in work function for Tungsten can be created by step density. Even more astounding, a 1 eV difference in work function can be measured depending on tungsten orientation! 1 eV at the nanometer scale indicates a huge difference in electric field. These hugely different electric fields can help explain why such small defects can often have a large effect on chemical reaction rates via catalysis.

Surface Energy

The simplest way to explain surface energy that I can find is from Wikipedia, where it is stated that surface energy can be defined as the excess energy at the surface of a material compared to the bulk of the material.

In class, the first thing we discuss about surface energy is the jelly model (jellium), which is quite similar to the plum pudding model. It feels almost heretical to be talking about this, since this class is in a building named after the man who proved that the plum pudding model was wrong (Ernest Rutherford).

We can calculate surface energy for jellium quite easily. This tends to agree with experiment at low densities, then eventually becomes very broken at higher densities. The more complicated (and accurate) models are quite difficult to calculate. Additionally, the surface energy is very hard to measure experimentally.

Surface energy is crucial to our understanding of many physical aspects of surfaces. For example, it helps us understand how we can grow materials on other materials. Will we get island growth or layer-by-layer growth?

One of the reasons this is difficult to model correctly is that the electron correlation effect between d-orbitals are difficult to calculate. This is why estimating the surface energy of elements such as gold, iron, etc involves very complicated calculations.

It turns out that finding the minima of surface energy will show us the shape of an equilibrium crystal. Real crystals may not completely agree because our physical crystal growth is not perfect. In closing, surface energy is important for studying crystal shapes as well as understanding what materials we can grow on what substrates and how they grow.

Surface Structure

There are three major ways in which the surface structure can be very different from the bulk structure.

Relaxation

The spacing between surface atoms and second layer is often not equal to the distance between the 2nd and third. Surface atoms tend to get pulled in little bit because they do not have a bond on one side. This is true for both covalent bonds and metals. This relaxation may be up to three layers deep (distances grow towards lattice standard as we go deeper).

Reconstruction

Where the surface structure is different from the bulk. For example, there might be more atoms on the surface layer than in a bulk layer. They may be connected to each other at different angles. Thus, the unit cell of the surface crystal can be very different from the bulk unit cell. We actually cannot calculate some of these structures because they are too complicated.

Related aside, Dr. Grütter began talking about silicon (111). He said, “This was the Guinea Pig or Drosphila of surface science for a number of years.” Apparently about 20 years of work went into understand silicon (111), which has what is called a “7x7 reconstruction” comprised of 64 atoms in 4 layers. The problem was eventually solved by a combination of scanning tunneling microscopy and diffraction studies.

Aside from the aside: This is not the industrially relevant silicon unit cell. That role is filled by silicon (001). Dr. Grütter says that it is very important that one can grow very smooth layers of oxide on silicon (001).

Aside3: Dr. Grütter says that silicon cannot be used as a photon emitting material very well because this would violate momentum conservation. However, gallium arsenide is capable of being a useful photon emitting material.

Composition

Most materials are an alloy, there are multiple constituent elements. Will the surface layer be the same composition as a bulk layer? It turns out that often surface layers are usually completely different than the bulk in terms of composition. Surface might be all of one element. Second layer might be a split of some kind. Third layer might be a different split.

This fact has huge implications for surface characteristics such as the ability to catalyze reactions, corrosion resistance, hardness, etc.

Surface Complexity

We tend to think of surfaces as atomically flat, but they are not. A decent flat surface might have truly flat areas that are 10nm in length. We might be able to get 100nm of nice flat area if we try really hard and employ a lot of tricks.

Some of the forms of imperfections in a surface are:

  1. kinks
  2. terraces
  3. vacancies
  4. adatoms
  5. monoatomic steps
  6. step-adatoms

Curious about what these are? Check out this Wikipedia page which includes some of their definitions.

A fair amount of research has been done on the subject of the effects of these imperfections in surfaces. For example, we have learned that electromigration is affected. Defects can backscatter electrons. This can become important when the surface atoms are a notable number of total atoms in the wire, which happens at the nano scale.

General Trends in the History of Circuits

The material for the post is based primarily on a lecture by Thomas Szkopek in the class Nanoelectronic Devices that I am taking at McGill.

Electrons are very light and have a definite (constant) charge. Charge to mass ratio is a primary reason why electrons are better than nucleons or mechanical devices for the creation of semiconductor electronics.

What is a transistor? It is ‘transferred resistance’. We can control the resistance of a lump of material. By controlling the resistance we can control to flow of current through a semiconductor. This is the primary basis for decision making at the circuit level.

We have built more transistors than anything else? (is this true or is it computer bits (like those in a hard drive)? not sure what he said).

In semiconductors, germanium was eventually replaced by silicon. Why?

Not because of cost or availability (initially). It was primarily a question of easier fabrication. The key facet was the quality of the oxide you can grow on the silicon rather than on the germanium.

There has been a lot of talk for years about how this or that material was going to replace silicon. None of them have yet done so because silicon is really well established and quite good at what it does. “Gallium Arsenide is the material of the future and it always will be.” - Szkopek.

Why smaller and smaller integrated circuits? By making the parts smaller and closer together, we can eliminate a lot of the resistances, capacitances and inductances as well as reducing our overall material usage. This should mean cheaper integrated circuits that require less materials to create and less power to run.

Gordon Moore

Gordon Moore was a chemist by training but was also one of the most successful electronic engineers of all time. What was Moore’s major contribution? He figured out how to grow high quality oxide on silicon.

As a computer scientist, I am well aware of some of the many different ways Moore’s Law has been mis-represented. So what is it actually? We read “Cramming More Components onto Integrated Circuits” the famous paper that Moore wrote in 1965, from which ‘Moore’s Law’ was extrapolated.

Moore's Law: Relative manufacturing cost per component and number of components per integrated circuit (diagram in paper, or on the Wikipedia article).

Cost increase as we move to the right is primarily because fewer chips are successfully made when we try to jam more components onto the wafer. There is a minimum cost for each technology level.

Atomic Scale

What happens when transistor dimensions approach 10nm? We are looking at atomic scale.

He then showed us some pictures taken with scanning-tunneling microscopy of a tiny surface with iron atoms on it. The atoms were physically arranged into a circle. As this symmetry is created you can see the creation of a symmetric pattern of standing waves of electron position in the center. This is an incredible (and graphic) demonstration of quantum mechanics in action. The pictures are from this paper on “Confinement of electrons to quantum corrals on a metal surface.”

One of Szkopek’s main points with regards to these photos is that atomic scale disorder is going to be present when we are working at such ridiculously small scales. Some of this disorder can be dealt with through more careful manufacturing and usage techniques, but we are definitely getting into the realm where we are starting to touch upon the omnipresent low-level disorder of the universe.

Szkopek says that Intel is currently using a 1.2nm oxide layer. That is 4 layers of atomic oxide. We are at the atomic scale, and will have to be considering the facts that govern it.

In closing, Szkopek talked a bit about how the nice formulas we tend to see in the theoretical sections of courses devolve into complicated, ugly looking things when we try to do real problems. There is a tendency to term this “things getting crazy”. Szkopek makes his point clear when he closed the class with: “Things don't get crazy, they get physical!”