Tuesday, May 22, 2012

Quasiparticles

Physicists can now begin to peer into and manipulate the environment in which atoms live.  Within many different materials they are finding out that atoms and molecules in close proximity exhibit properties and effects that don't occur in the individual atoms themselves. The physics that explores the mechanisms behind these various physical material properties is called condensed matter physics. Especially hot right now is the study of how electrical and heat conductivity arise within various materials. Although it is a complex and challenging field, for those of us who are curious about matter and energy, it is worth a closer inspection. This is where physicists are discovering that atoms in solids and liquids exhibit some very peculiar behaviours. It is here where the enigmatic quasiparticle lives.

If we take a look at the particles listed in the Standard Model of physics we will not find the quasiparticle. It is neither a real nor a theoretical particle. As the name implies, a quasiparticle is a sort-of-particle, and it comes in many different kinds. When energy is applied to the closely interacting subatomic particles inside a solid or liquid, new behaviours emerge. These behaviours can act like particles. We can better define what a quasiparticle is by first defining what a particle is in physics. As you will see, this definition has undergone some major renovation, especially over the last 100 years.

Particle Evolution

The notion that matter is made up of tiny elementary particles dates back over 2000 years.  John Dalton, in the 1800's, took up this notion and named the smallest indivisible particle of nature the atom. The Greek word "atomos" means indivisible. Several decades later, scientists began to realize that atoms weren't the smallest unit of matter after all. Electrons and nuclei could be distinguished within them. This was just the beginning. Collider experiments in the 1950's and 1960's unleashed a whole zoo of subatomic particles. The Standard Model was formulated in the 1970's to "house" and organize all of them.

Meanwhile, as new particles were continuously being discovered, their nature was also under investigation. The concept of the atom evolved (http://www.nobeliefs.com/atom.htm) from Dalton's nondescript billiard ball atom into J.J. Thomson's "plum pudding" model, and then into Ernest Rutherford's planetary model, Bohr's atomic orbital atom, and finally into our current, and complex, quantum mechanical model. 

The Bohr model is still used today, mostly for simpler atoms. For larger complex atoms we have to turn to the quantum mechanical model and, although this model is incredibly useful, it is not intuitively easy to use.

Particles of matter simultaneously act like waves. This means that it is impossible to know the exact position and momentum of any particle at the same time. This particle-wave duality is evidence of Heisenberg's uncertainty principle, which is incorporated into the quantum mechanical atom model. The electrons inside atoms, rather than being solid little particles orbiting the nucleus, are instead clouds of probability that are assigned specific orbital shapes. A series of numbers, called quantum numbers, are used to describe each electron inside an atom. They include the principle quantum number, angular momentum number, magnetic number and quantum spin number. The principle number describes the average distance from the electron's orbital to the nucleus. The angular momentum number describes the shape of the electron orbital. The magnetic quantum number describes how the electron orbital is oriented in space. And the spin quantum number defines which of two possible directions the electron is spinning within a magnetic field. Yes, the atom has grown complicated, thanks to the contributions of many scientists over the past 100 or so years. It is hard for most of us, including me, to imagine that the everyday materials around us ultimately break down not into tiny dust-like particles but into tiny clouds of probability. And hold on, we're not done. The subatomic landscape is about to get even more surreal.

The Modern Particle

Particles are constituents of matter or energy (they are equivalent according to Einstein), and according to quantum mechanics, they are excitations of quantum fields. Physicists now believe that quantum fields underlie all physical phenomena. These fields tend to be associated with the four fundamental forces in nature: electromagnetism, the weak force, the strong force and gravity. Let's see how this "excitation of quantum fields" business works:

We will take electromagnetism, the fundamental force behind the propagation of light, magnetism and all of chemistry and biology, as our example. Electromagnetism can be described as something called a quantum electrodynamic field. Physicists can assign numerical values to every possible space-time point within such a field, and in this way they can fully describe the electromagnetic force at any point in three spatial dimensions as well as in time. Physicists can also describe various matter particles within this field as well as the force particles that mediate the field itself, which in this case are photons. Physicists do this by assigning values of various operators to each particle. These operators include position, charge, momentum, angular momentum and energy. It is no accident that they are reminiscent of the quantum numbers that describe the electron. Operators are mathematical constructs that are more complex than simple numerical values. When physicists work with quantum mechanics, they have to incorporate Schrodinger's equation* which means they can't get exact values of these particle properties. The properties themselves are all wave functions, so to best describe the nature of each property you start with a one-dimensional differential equation. Differential equations solve for a dependent variable over one or more independent variables. If you have taken some calculus, you have a feeling for them. You use these equations when you have to describe something that is continuously varying. These calculations alone can become very complex and you're not done yet. You then have to solve that equation for three spatial dimensions. All this makes a particle a complicated beast indeed!

*A little aside for interest's sake: Around 1925, both Heisenberg and Schrodinger developed mathematical theories to describe Bohr's wave-function electron orbits. Heisenberg's uncertainty principle interprets the electron as a particle with quantum behaviour. Schrodinger interprets the electron as an energy wave. These contemporaries didn't like each other very much and both thought the other's theory was suspect. Heisenberg, in fact, is quoted as saying that Schrodinger's theory is "disgusting." Nevertheless, in 1926, Schrodinger was one of a handful of people that proved that matrix mechanics (Heisenberg) and wave mechanics (Schrodinger) are mathematically identical.




Heisenberg is on right (credit: Deutsches Bundesarchiv (German Federal Archive), Bild183-R57262, Wikipedia).

To sum up then, a particle is an energy excitation in a field. It possesses a set of values of various operators. A particle such as an electron in an atom, a photon of light, or a gluon mediating the strong force can all be described this way. It is the fullest, and most modern, way of describing a particle.

From Particle To Quasiparticle

Now, if you take all these descriptors, and generalize "field" to "system," you can describe any arrangement of atoms, molecules or even rubber balls! And that is just what people studying condensed matter physics do. Condensed matter physics is the science that describes the behaviour of systems in which atoms are in close enough proximity to significantly interact with each other. This includes solids as well as some liquid systems. Physicists can use the idea of quasiparticles to help them describe complicated microscopic behaviours of various solids and liquids.

Remember, quasiparticles are not real or even theoretical particles. They are phenomena that "act" like particles. More specifically, they are an example of what is called emergent phenomena and they occur when a complex system of atoms or molecules behaves as if it contains fictitious particles that behave as if they are moving through empty space (this out-there statement will hopefully make more sense as we continue). Quasiparticles can be assigned all the quantum mechanical values that are associated with real particles.

Quasiparticle Phenomena Can Be Strange

Some of these phenomena are pretty weird, a ghost-in-the-machine kind of weird. Here are two examples:

Example 1: When an electron moves through a semiconductor it interacts with all the other electrons as well as the atomic nuclei in that conductor. These interactions can be very complex. They disturb the electron's motion in such a way that it begins to look like an electron with a different mass altogether moving through empty space.

Example 2: Even more strangely, the general movement of electrons in the valence band of a semiconductor (this is the region where electrons are bound to individual atoms in contrast with mobile conduction electrons elsewhere in the material) resembles what would happen if the semiconductor contained positively charged particles instead of electrons. Physicists call these positively charged quasiparticles holes.

Quasiparticles, like real particles, can be fermions with half integer spins (like holes). These emergent phenomena act like particles. Or, quasiparticles can be bosons with whole integer spins (phonons are examples of these and I will describe them shortly). Bosonic quasiparticles are often called collective excitations. These phenomena tend to act more like fields than particles.

Quasiparticles Tackle the Many-Body Problem

Physicists can make good use of emergent quasiparticle phenomena. For example, imagine trying to describe every single subatomic particle in a macroscopic system. Physicists call this the many-body problem in physics. Consider a grain of sand, a very small macroscopic system. According to Ask A Scientist, each grain contains about 2 x 1019 atoms. Inside this grain, each atom interacts with the other atoms via their outer shell electrons according to Coulomb's law. Each atom has many choices in terms of with, who and how it interacts. To describe these interactions, and all the choices involved with each one, you would need a differential equation that accommodates all these possibilities. Physicists call these possibilities degrees of freedom and there would be an almost infinite number of them even in a tiny sand grain. Then you would need to solve it in 3-dimensional space (this adds even more degrees of freedom). Even with powerful computers, it isn't possible to fully and directly describe exactly what's going on inside a single grain of sand! But physicists have an option. They can sometimes use quasiparticle dynamics to simplify such complex systems of atoms, enough so that they can understand and predict some of the physical properties of that system.

To do this, the system of particles is treated as a whole, like a single quantum system in other words. Like any quantum system (a quantum particle is a quantum system), it has a ground state and various excited states. Theoretically, it should have an infinite number of such states, each associated with higher and higher energy. However, only low excited states close to ground state are likely to occur. Boltzmann distribution assigns very low probabilities to highly excited states at any given temperature, and that helps to keep things simple (-ish). I'll try to illustrate this idea by using a theoretical perfect crystal as an example.

An Introduction To the Phonon

A perfect crystal at absolute zero is in the ground state. This means that the atoms in the crystal are arranged in an orderly lattice with lowest possible energy. Its entropy is exactly zero. This means that every particle is in its proper place and there is no rotational or vibrational energy in the lattice (This, however, doesn't mean that atoms within the lattice are not vibrating even at absolute zero). If the temperature (energy) is increased, the number of possible lattice arrangements increases, and it is now in a low excited state. This means that the lattice will vibrate very slightly at a particular frequency. Physicists call this a collective excitation associated with a single phonon. A phonon is a type of quasiparticle. It is a vibration that acts and can be described as a single quantum state. Increasing the energy of this lattice will introduce new phonons to the system.

Few low-lying excited states are as simple as a perfect crystal. Most solid and liquid materials contain several phonons as well as other quasiparticles and collective excitations, each of which are treated as if they act simultaneously and independently of each other, which is never true. However, this treatment is often close enough to reality to be a useful approximation. If you add another phonon to a real crystalline solid, such as a metal for example, it won't have exactly twice the excitation energy of one phonon because the resulting vibration will tend to be anharmonic to some (usually very small) degree. These small aberrations can be corrected using formulas that describe various interactions between collective excitations. Even so, in strongly correlated materials, like many transition metal oxides used in superconductors, collective excitations are so far from being independent from each other that you can't use the quasiparticle approach even as a starting point to describe their microscopic behaviour.

To sum up then, the beauty of the quasiparticle approach is that rather than trying to deal with a huge number of possible particle interactions, each with several variables attached to it, you can use the formulas for a few fairly independent elementary excitations instead. It's not an exact picture of the microscopic behaviour of a system, and for some systems it won't work, but under certain circumstances it can be a very good approximation.

Phonons Are One Kind of Collective Excitation

We can move on now to a more in-depth definition of the phonon. The phonon in the crystal I mentioned can be defined as a collective excitation in a periodic elastic arrangement of atoms (or molecules) in condensed matter such as a solid or liquid. This periodic elastic arrangement of atoms or molecules is usually some kind of lattice. Crystal structures are lattices. Lattice models, and there are many kinds of them, are very useful in theoretical physics, especially solid-state physics. Many solids and some liquids have crystalline atomic or molecular arrangements. Snowflakes, diamonds and table salt are all crystals. Table salt is an especially interesting crystal. Each of the two atoms, sodium and chloride, forms a separate cubic lattice, with the two lattices interpenetrating each other, as shown to the right:



Above left is the mineral form of this salt, called rock salt (credit: Rocker1984; Wikipedia). You can see its macroscopic crystalline organization.

At the opposite end of the spectrum are amorphous solids without any periodic order. Glass (SiO2) is an example. Its atomic structure is shown here:


Even these materials have phonons but they tend to be strongly scattered. The liquid crystal in electronic displays is an example of a liquid that flows but its molecules are arranged in a crystal-like way. These rod-shaped molecules tend to line up along a common axis, giving the material a structure that falls in between that of a solid and a liquid, an unusual physical state called a mesophase. Soap and many detergents are also liquid crystals.

These crystal arrangements have some give to them; they can vibrate and they can recover after being deformed. For this reason they are called elastic. Engineers refer to the elastic limits of various structural components when they design structures such as office towers and bridges, which come under periodic stress.

A phonon represents an excited state in a microscopic structure, which is expressed as a vibration of a particular frequency. Phonons play an important role in describing the physical properties of a solid, such as its thermal and electrical conductivity. For example, the Debye model treats the specific heat of a material as "phonons in a box." The phonons move as free particles but they are confined to the material - the box. Both heat and electrical conductivity arise from this free movement of quasiparticle electrons. A material would have infinite heat and electrical conductivity if its lattice didn't vibrate and scatter these mobile quasiparticle electrons. For metals, which tend to have good lattice structures, these two properties correlate directly with each other according to the Wiedermann-Franz law. Phonons in nonmetals, however, tend to become more significant as heat carriers than as carriers of electrical current, where the relationship between molecular or atomic arrangement to heat becomes more complex and the energy of the real electrons themselves becomes a factor.

How Collective Excitation Works

Let's take a closer look at the nature of the vibrational modes themselves. In classical mechanics, vibrational modes move like waves. Examples are sound waves and water waves. In quantum mechanics, however, vibrational modes are a little different. They have both particle and wave properties, or particle-wave duality.

Let's look at a simplified lattice wave from a classical mechanics viewpoint and than from a quantum mechanical viewpoint. We can think of these viewpoints as two different building block sets.

Classic Mechanical Lattice Wave

Imagine a very thin slice of a solid material that displays a classical 2-dimensional lattice of atoms, a simple arrangement that looks like this:


This is the model we want to make. The forces acting between the atoms in this lattice are those in nature; they can be any combination of Van der Waals forces, covalent bonds, ionic bonds, etc., all of which are ultimately due to the electric force. They keep each atom in its equilibrium position. The black lines can be bouncy little springs in our construction set. To keep things simple, we will keep magnetic forces and gravitational forces out of our set. With no external magnetic field and so few atoms, those forces are negligible. The force between each pair of atoms has potential energy, just like a spring does. The little lattice we're making has a total potential energy which is a function of the distances between all of its atoms (blue circles). Mathematically, the potential energies can be treated as harmonic potentials. They have natural harmonic frequencies just as springs do. The displacement of one or more atoms from this lattice will set vibration waves propagating through the lattice, and the lattice will experience a restoring force, bringing it back to its original arrangement. This set is a classical mechanic set, so this lattice is treated like a system of point particles where electrons and nuclei move in step with each other and the forces between them act like elastic springs. Each atom is treated like an independent harmonic oscillator, and all the atoms vibrate with the same frequency. Think of beads woven into an elastic mesh. You tug on one bead and what happens? In this set, vibration is akin to sound waves moving through it. You'll see what I mean shortly.

Quantum Mechanical Lattice Wave

A quantum mechanical lattice wave model looks a bit different. The springs are still there but each atom, each "bead in the "mesh," is much more complex. It is defined by mass, as well as by position and momentum operators. When quantum mechanical equations such as Schrodinger's equation, are incorporated into the classical formulas, a value for the wave number of the phonon emerges that will take on quantized values. You get only discrete specific allowable vibrational modes and none in between. This means you need a minimum amount of energy to move the system to its next excitation level. If you tug on an atom in this model, it won't vibrate unless you tug on it with sufficient energy. The phonon (as well as other quasiparticles) emerges as a quantum of vibrational energy in this quantum mechanical lattice. When you do get a vibration set up within it, you might be surprised by how it behaves . . .

Quasiparticles are behind some very peculiar phenomena in some materials. An especially interesting example is that of graphene. Graphene is a layer of graphite just one carbon atom thick, and it can exhibit electron mobility (electrical conductivity) ten times greater than the silicon used today in computer chips. Here, quasiparticle electrons (researchers call them Dirac fermions) move through the material is if they were massless particles traveling through a vacuum at light speed. Ordinary electrons have a small but definite mass as they move through materials. They interact with other particles and that results on a drag on their momentum. They can't travel anywhere near light speed inside a material (or in a vacuum either thanks to their mass). Dirac fermions experience no such problem, and this suggests the very enticing possibility of a future carbon-based super-fast computer.

Phonons Have Two Modes of Vibration

Sound waves are basically long wavelength longitudinal wave phonons. Phonons, like any wave disturbance, can take the form of longitudinal or transverse waves.



In longitudinal waves, vibration motion is in the same direction as the direction the wave is moving. In transverse waves, vibration motion is perpendicular to the direction the wave is moving. Sound is transmitted through gases, plasma and liquids as longitudinal waves. But through solids, sound can be transmitted as both longitudinal and as transverse waves. Solids with more than one kind of atom in them, each one with a different bonding strength or mass for example, can exhibit both kinds of phonons. Transverse wave phonons in solids are sometimes called optical phonons because they act like photons, which is where the name "phonon" comes from.

Sound phonons in solids are coherent movements of atoms within a lattice. In the classic lattice sample above, the vibration would be best described as a sound phonon. A classical lattice system cannot account for the unusual quasiparticle phenomena that some materials exhibit, but it can approximately describe the flow of heat through a material as mechanical energy transported through a classical perturbation of the lattice. Unlike sound phonons, optical phonons are out-of-phase movements of atoms within a lattice. These movements occur in lattices containing two or more different atoms or molecules. Optical phonons are most apparent in ionic crystals, such as the sodium chloride crystals described earlier. When this lattice is excited by infrared radiation, negative Cl- ions move in one direction in the field and positive Na+ ions move in the other direction. In reality, both types of phonon are present in all lattice systems. Every atom in a 3-dimensional lattice has 3 degrees of freedom - it can vibrate along one or more of three spatial coordinates. This translates into 2 possible transverse modes of vibration as well one possible longitudinal mode. All of this might seem a bit confusing. Try using this JAVA applet to help you get more of a feel for the concept.  Play with the mass ratio, atom separation and phonon wavelength to see what the resulting sound and optical phonons look like.

You might be wondering - if phonons are like photons, why don't they move right through glass, my example of an amorphous solid, rather than scatter internally as I suggested? Visible light photons move right through glass; that's why we see it as transparent. Photons of visible light don't have enough energy to interact with the electrons in glass atoms (these atoms happen to have extra big band gaps, gaps between electron energy levels and that makes the "jump" more difficult), and so electrons remain in ground state as photons move right through the glass. UV photons have more energy - they usually can't get through ordinary window glass as a result. If your eyes could see only in the UV range, most windows would be opaque to you. So, if this is the case, don't phonons move right through glass too, rather than being scattered around by glass's irregular atomic soup? The answer, and it's a complicated one as I understand it, depends, like photons, on the energy of the phonons involved.

How A Material Vibrates Determines Many of Its Physical Properties

Normal vibrational modes give wavelengths and frequencies specific to a particular material, based on the vibrational modes of its atoms or molecules. All atoms in every molecule are in constant motion while the molecule itself experiences translational and rotational motion. A molecule made of only two atoms has only a single possible vibrational motion, but molecules made of more than two atoms have more than one kind of possible vibration, each of which is called a normal vibrational mode. Here are 3 of 6 possible vibrational modes in a three-atom molecule:


These normal modes define that material's physical properties. Specific vibrational frequencies inside insulators, for example, determine their capacity to store heat. These vibrational frequencies can be in the form of phonons and/or other quasiparticles such as excitons and plasmons, with each making a separate contribution to the heat capacity of the material.

This kind of approach also helps scientists predict earthquake propagation through various materials in Earth's crust and mantle. Structural engineers use these equations to predict the strength and elastic properties of various construction materials.

Remember that quasiparticles can be fermions (like electron holes) or bosons (the phonon and others). Hopefully the phonon discussion above gives you an idea of how a qausiparticle as a collective excitation works. Now let's focus on quasiparticles with fermion-like behaviours, where they act more like matter particles than microscopic vibrations within a material.

Quasiparticles Act Like "Particles" With Amazing Results

Recently, physicists managed to split electrons into quasiparticles, marking a new era in condensed matter research. Electrons are believed to be fundamental particles. Like the quarks in atomic nuclei, they can't be split into smaller particles. However, in the 1980's physicists predicted that an electron could be split into three quasiparticles: a holon carrying the electron's charge, a spinon carrying its spin and an orbiton carrying its orbital location. These quasiparticles, each carrying one of the electron's features, can move in different directions and at different speeds. This seems to contradict the notion that electrons are indivisible particles, but it doesn't. Here's how: When they are confined inside atoms, electrons behave like waves so their wave function dominates. When excited, these waves can be broken into multiple waves, each carrying off a different characteristic of the electron, with the restriction that they can't exist independently outside the material. Remember they are not actually particles; they are vibrations.

These researchers (Schlappa et al., 2012) repeatedly hit an electron with X-ray photons, causing it to gain energy and move up into higher oribitals, When it did so, it split into either a holon and spinon or into an orbiton and spinon. That means, in the first case for example, that the electron's charge goes off one direction and its spin moves off in another. That's beyond weird isn't it? And yet it's now experimentally verified. The next step will be to try to produce all three quasiparticles at the same time. Spinons, and especially orbitons, might someday be very useful in constructing a quantum computer. If information can be encoded in the orbital transitions of electrons for example, they can be used to perform ultrafast calculations, overcoming a major stumbling block in quantum computing, which is that quantum effects are usually destroyed before calculations can be performed. It also has huge implications for superconductor technology, especially high-temperature superconductivity. Most materials become superconductors only when they are just a few degrees above absolute zero and that makes them impractical to use. Now, using methods similar to that described above, they might be able to move electricity around in bulk inside the material without losing much of it as waste heat when the real electrons themselves scatter against atoms.

Now for my personal favourite quasiparticle story, which I've saved for last: Quasiparticles are beginning to hint at how the Higgs field might work.

The behaviour of electrons and other particles depends on their environment. Researchers (Gomes et al., 2012) created a special environment for electrons by arranging carbon monoxide molecules into a 2-dimensional hexagonal pattern like that of graphene, except that here they can change the spacing of individual molecules. When they changed the spacing of the molecules they could change the masses of quasiparticles associated with the electrons. They could also get the quasiparticles to act as if they were interacting with electric and magnetic fields when no fields were present. Finally, they could get them to act like relativistic particles with a "speed of light" they could adjust (according to special relativity, particles with mass have a built-in speed limit. If they are in a vacuum it is the speed of light. If they are inside a material, the speed limit is a property of the electronic structure of the system. In this way they could get them to act like the Dirac fermion quasiparticles I mentioned earlier. These are massless excitations that travel at the speed of light. They found that they can even make the quasiparticles transition between massless and massive states by distorting the lattice, a process that many researchers believe is analogous to how the Higgs field might work in particle physics (with the admission that we are comparing a lattice of atoms with a force field). Researchers believe the Higgs field gives particles mass. Some particles moving through the Higgs field are "sticky." They tug at and distort the field as they travel through it and that distortion registers as mass. Other particles, such as photons, which are massless, are not "sticky." They don't distort the field and therefore they register as massless. I think this experiment beautifully helps to bring out the full weird potential of the quantum mechanical nature of matter, and begs for more research to be done.

The idea of the quasiparticle originated in 1958 by physicist Lev Landau as a theory for Fermi liquids. This theory describes how fermions interact in metals and why some of the properties of such a system are very similar to those of a Fermi gas, in which fermions don't interact. As well as metals, it describes the how liquid He-3 behaves at low temperatures (but not low enough to be in a superfluid phase). He started with a system at ground state and then added energy to it. At ground state, all the fermions in the system occupy lower momentum states with the upper ones being empty. As energy is applied, the spin, charge and momentum of the fermions corresponding to the occupied momentum states remains the same, but their properties such as mass, magnetic moment, and so on are renormalized (an adjustment allowable in quantum mechanics) to new values. As a result, the particle excitations of a Fermi gas directly correspond to those in a Fermi liquid system, to which metals belong. In metals these excitations are the quasiparticles we explored here.

Over just a few decades, our understanding of how atoms and molecules interact with each other has grown enormously. Continuing research into strange and exotic quasiparticle phenomena is likely to continue to unearth new properties within materials that will form the basis for exciting new electronic technologies. This is certainly a story you will want to stay tuned to . . .

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