Electromagnetism Basics

In Earlier times, Greeks and Etruscans found that a locally found rock, a lodestone, would always point in the same direction. This is the beginning of the study of magnetism. Only a few naturally occurring objects exhibit magnetic properties, but electrical currents are also affected by magnetic fields, which makes their study much more important.
The lodestone, now known as magnetite, is predominantly an iron alloy. Other elements such as nickel and cobalt can also be magnetized. Recently, many of the rare earth elements have been made into magnets using a high temperature ceramic process. The strongest of these are the ones made of neodymium.
Some magnets are permanent magnets since they do not lose their magnetic ability quickly, but rather stay magnetized for a long time, often for years. Other magnets are temporary magnets since they lose their magnetic ability very quickly. A refrigerator magnet would be a permanent, while an electromagnet would be a temporary one. Materials which are easy to magnetize (such as iron) also lose their ability soon, while those which are hard to magnetize (like nickel and cobalt) tend to hold on to it much longer.

Magnetic polarity

Magnetic materials possess a polarity; two distinct regions of opposite affect. When suspended, a magnetic object will rotate until one end will point toward the north pole of the earth.. If displaced, it will return to this position. This would be the north pole of the magnet and the opposite end would be the south pole.
Poles of a magnet are similar to opposite electrical charges in their interactions: like poles repel and unlike poles attract. Unlike electrical charges, magnetic poles are never found alone. They always exist in pairs.

Magnetic fields

Since magnetic forces are detected at a distance from the magnetic object their must be a magnetic field. Similar to electric fields in shape and effect, magnetic fields can be drawn to show the intensity and direction at any point around the source of the field.

Magnetic field lines start and end on either magnetic poles
or infinity, they do not cross, and their density indicates
the strength of the field. The direction of the field lines shows
the direction of the magnetic force on a north mono pole
(if one existed).

Electromagnetism

In the late 1700"s it was observed that magnetic compasses (small free floating magnets) placed near current carrying wires would be effected when the current was on, but returned to normal when the current was off. It wasn't until the mid 1800's that the connection between the two was formalized into the science of electromagnetism. It is now well understood that moving charges create magnetic fields and that changing magnetic fields cause forces on electrical charges

A moving charge creates a magnetic field which moves radially away from the path of the charge. As in electrical definitions, the direction of the magnetic field is defined by conventional current. The shape of the conductor has a great effect on the shape of the resulting magnetic field.
For a single straight wire, the resulting field lines for a current in the wire look like concentric rings around the wire. There are no poles in this field, since the lines are circles and do not touch the wire. Since the field gets weaker with distance from the wire, the lines should get farther apart.

The direction of the field can be determined by applying a right hand rule; the thumb of the right hand points in the direction of conventional current, the fingers curl around the wire in the direction of the magnetic field.
If the current conductor is wound into a coil, the individual magnetic fields of each of the loops of the coil superimpose to create a stronger, more direct field. Such a current carrying coil is called a solenoid. The field for a solenoid has polarity, one end of the coil acts a a north pole while the other end becomes a south pole. If a soft iron core is placed in the coil, the field becomes more concentrated and stronger in the region of the iron. Notice the similarity of the field for a solenoid and for that of a bar magnet.

The polarity of the coil can be determined through the application of a second right hand rule. If the fingers of the right hand curl around the coil in the same direction as conventional current, the thumb points to the north pole of the coil.

Magnetic Force

If a charged particle sits in a magnetic field, nothing special happens. If the particle is in motion, however, it gets interesting. When a charge moves through a magnetic field, as a single particle or as a current, there is a magnetic force on the particle perpendicular to its motion and also to the magnetic field.

The magnitude of the force depends on three factors; the size of the charge (q), the speed of the particle (v), and the strength of the magnetic field (B).

This equation can be used to experimentally define the strength of a magnetic field. If a one coulomb charge moves through a magnetic field at one m/s and experiences a force of one Newton, then the strength of the field is one Tesla. A Tesla is a large unit of field strength, with even the largest of laboratory magnets reaching a few tens of Tesla. Most everyday magnetic fields, like the earth's or a refrigerator magnet, are in the micro to nanoTesla range.

The direction of the magnetic force on a charged particle can be determined using another right hand rule; the fingers of the right hand point in the direction of the magnetic field, the thumb points in the direction of the moving charge (current), and the force on the particle is perpendicularly away from the palm.

Since the force is always perpendicular to the motion of the charge, a single charge will experience a centripetal force and travel in a circular path, or at least part of one. A device known as a mass spectrometer uses this principle to identify the size and charge of unknown particles.

If the moving charge is part of a current in a conductor then the concuctor will experience a force perpendicular to itself and the field. The rules for the strength of the force and its direction on the conductor are the same as for a single charge.

When two parallel current carrying conductors are close to one another, their respective magnetic fields will superimpose and create a force between the conductors. If the currents are in the same direction, the field between the conductors is weaker than the field outside which causes an unbalanced force pulling the two together (attraction). If the currents are in the opposite direction, then the inner field is stronger and the conductors will be pushed apart (repulsion).

For many applications this causes little concern, but if you are working with power lines, or electrical cords to appliances which use large currents the effect can be noticeable and must be considered.

Applications of electromagnetic

The use of electromagnetic factors in mass spectrometers and power lines has already been mentioned. A use closer to home would be their use in audio speakers.

The guts of a speaker includes a large permanent magnet with a voice coil wrapped around its core. The cone of the speaker (the part we see) is attached to the coil. As the current in the coil changes in response to the sound originally recorded, the force on the coil changes size and direction. This makes the coil move forward and backward in response to the force. As the coil moves, so does the cone, which pushes the air in accordance with the sound and electrical signals involved.

Induced current

You have already seen that a moving charge in a magnetic field can experience a force. The key elements are; a moving charge (current), a magnetic field, and a force. A current created the force.
Now, just turn that around: a wire forced through a magnetic field may have a current created in it. No source required, just move a wire through the magnetic field. Of course, as before, certain factors determine the amount and direction of the current.
The speed, direction, and orientation of the movement of the conductor through the field determines the aspects of the induced current. The maximum amount of current will be induced if the conductor moves perpendicular to the field, is moved at a faster speed, and if there is more conductor present (more wire).

The direction of the induced current is a little more complicated because once the charges are forced to move (current) by the magnetic field they possess a magnetic field of their own. These two fields (the one creating the force and the one of the current) will superimpose to create a separate, new force on the conductor.
If you think about it, the direction of this additional force can not be in the same direction as the force on the conductor in the first place. If it was then a small nudge of the wire would create the current, which would create a force on the wire in the same direction, which would make the wire move faster, which would create more current, more force, etc. It would be a run away situation. Somebody along time ago would have started a wire, and it would continue to move faster ever since.
This situation compels the new force to be in the opposite direction of the force which creates the motion. Think of it as induced electrical friction. The larger the induced current, the larger this opposing force. Many times it is referred to as back emf, or back current.
The formal statement of this phenomenon is Lenz's law; The magnetic field of the induced current opposes the change in the applied magnetic field. In other words, the direction of the induced current is opposite to the current which would create the motion in the first place.

We usually think of wires moving in a stationary magnetic field, but the wire could be stationary and the field move past it. The results would be the same, but sometimes the direction of the currents and fields can be a little challenging to determine.


Self inductance is a term used to describe the creation of a back current in a conductor when the current in a wire changes. If the current is increasing then the induced current tend to keep the current value from increasing as fast. If the current is decreasing, then the induced current would tend to keep it from decreasing as fast. Since solenoids have a larger magnetic effect due to the interaction of one section's magnetic field on another section, the self inductance of a solenoid is more pronounced than for a single, straight wire.

Applications of induction

There are many applications of induced currents and their magnetic fields. One of the more common ones is the recording of sound on magnetic tape (cassettes) or recording computer data on disks. Both of these systems are essentially the same.

A wire carrying a current which changes its magnitude and direction according to the information it carries is wrapped around a piece of iron. The changing current induces a changing magnetic field in the iron. The recording medium is a thin film of iron particles embedded in plastic. As they pass by the "head" of the iron magnet the regions of the iron particles align themselves with the magnetic field at that point in time. As the pattern of current changes, the pattern of the iron particles in the plastic changes accordingly. You can see why bringing a strong magnet close to these materials could ruin the data stored on them.

A device which utilizes the concept of induction is the transformer. A transformer consists of two separate coils of wire wrapped around the same piece of iron. Any changes in current in one of the coils creates a changing magnetic field in the iron. Since the other coil is also wrapped around the iron, any change in the iron's magnetic field causes a change in the current in that coil. Notice that a transformer only transfers electrical energy between the two coils when the current changes. If the current doesn't change in the first coil, there is no current in the second coil. Transformers are usually only used in alternating current circuits, where the current is continually oscillating back and forth. Transformers provide a way of isolating one electrical circuit from another since there is no electrical connection between the two coils, but a magnetic link which connects the two. The amount of energy (voltage and current) which is transferred depends on the sizes of each of the coils as well as their orientation on the iron core.

Generators

A generator is a device which converts mechanical energy into electrical energy. By moving a wire through a magnetic field, you have seen that a current can be produced. If this process can be done repeatedly the current can be maintained. In simplest terms a generator consists of a coil of wire which rotates in a magnetic field. Permanent magnets are usually used to create the magnetic field, and the coil is wrapped around an iron core to concentrate its field. In the diagram below only a single loop of the coil is shown for simplicity

As you can see from the diagram the magnitude and direction of the current changes as the loop turns through the magnetic field. The only parts of the loop which are involved in the generation of current are the sides which are perpendicular to the field (colored in yellow and blue). The other sides are parallel to the field and do not create current as they rotate.
In positions A and C the loop is creating the maximum current since the wires are moving perpendicular to the field. In positions B and D there is no current generated since the wires are moving parallel to the field. Notice that the current, or EMF, follow a sinusoidal curve. Since the current changes its magnitude and direction in a cyclic pattern it is referred to as an alternating current.

Alternating Current

An alternating current follows the same concepts as the waves you have studied in the past; they have an amplitude and frequency. The amplitude of an alternating current is referred to as its maximum value. This is not that useful of a number since the maximum value only exists for a fraction of each rotation.
A more useful value for EMF or current is the effective value. Consider a light bulb connected to both a battery and a generator. The bulb receives energy when the charges move through its filament, no matter which direction they are traveling. The power of the bulb at any point in time is the product of the current and voltage at that point in time (Watt's law).
Since the battery has a constant voltage and current, this calculation is simple to make and the result remains constant. For the generator, these values are changing so the calculation is not constant. This is where the effective value of AC comes in.
If you want both bulbs to have the same brightness (power) then the effective value of the current, or EMF, is 0.707 times the maximum value. If the maximum voltage of a generator is 100 V, then its effective voltage would be 70.0 V. This means that it would create the same brightness in a light bulb as one connected to a 70.7 V DC source.

It is the effective voltage that is usually expressed for AC. American household voltage is around 110 Vac. It is the effective voltage which is 110 V, the maximum voltage would be around 164 V, and the voltage swing during one cycle of the AC would be close to 328 V. That's one reason being shocked by household current is so dangerous.

Magnets

The easiest way to consider motors are that they are generators run in reverse. Since a generator converts mechanical energy into electrical energy, a motor changes electrical energy into mechanical energy. The parts of a motor can be identical to the parts of a generator, but instead of turning the loop to induce a current, current is sent through the loop to make it turn.

A motor involves a turning coil in a magnetic field. This was the condition that created current in the generator. That means that a motor also generates current while it is running. Remember, according to Lenz's law, this generated current is opposite to the direction which is making the motor turn. Because of friction and lack of involvement of the entire magnetic field, the back current is only a fraction of the motor's current.
When a motor first starts to turn there is almost no back current since the coil is barely turning. This means that the motor is drawing the maximum current from the source. As the motor increases the speed of the coil, the back current increases until it reaches a steady value. At this point the motor draws the least current from the source.
You may have noticed this when a large motor is started in your home. For a brief time as the motor is starting, the lights in the house dime, but come back to normal when the motor is running at full speed.
This situation is also why it is more economically responsible to leave a motor running rather than turn it on and off. Also, when a load is attached to the shaft of the motor, the speed of the coil is reduced, this reduction in speed reduces the back current, which increases the current in the coil, and gives the motor more power to maintain its speed as the load is applied. Pretty neat, huh?

Fermionic Condensate

Bose-Einstein and Fermionic Condensates

The fermionic condensate is a new phase of matter, first created on December 16, 2003 by physicist Deborah S. Jin. It is the sixth known phase of matter, and is related to the Bose-Einstein condensate, which was first created in 1995. The five other phases of matter are gases, solids, liquids, plasmas and Bose-Einstein condensates.

Bose-Einstein condensates are formed at low temperatures when a significant number of bosons collapse into the same quantum state. A similar condensate that uses fermions instead of bosons is known as a fermionic condensate. Such a condensate is far more difficult to achieve because the Pauli exclusion principle prohibits two or more fermions from occupying the same quantum state.

In 1957, John Bardeen, Leon Cooper and Robert Schrieffer proposed that electrons, a type of fermion, could pair-up to form what are now known as Cooper pairs - such pairs essentially act like bosons. If a similar sort of pairing were possible amongst fermionic atoms then the formation of a fermionic condensate might be possible. However until recently many believed that the temperature required for such a pairing would be too cold to achieve.

In 2001, physicist Murray Holland of the Joint Institute for Laboratory Astrophysics (JILA) speculated that fermionic atoms could be coaxed into pairing up at higher temperatures by subjecting them to a magnetic field. In 2003, Deborah Jin of the same institute (JILA) and Rudolf Grimm of the University of Innsbruck were able to coax fermionic atoms into bonding to form molecular bosons and thus able to form a Bose-Einstein condensate but not a fermionic condensate.

On December 16 2003, Deborah Jin observed the pairing and formation of a fermionic condensate among fermionic atoms for the very first time. The experiment involved 500,000 Potassium-40 atoms cooled to a temperature of 5 × 10-8 kelvin with a time-varying magnetic field applied to them. The findings were published in the online edition of Physical Review Letters on January 24, 2004.

The Helium Superfluids and the Condensates

Superfluidity is an unusual state characterised by the complete absence of viscosity. Superfluids exist only at low temperatures and possess a number of strange characteristics. For example, when placed in an open container a superfluid will gradually flow up and over the sides of the container due to film flow. All Bose-Einstein and fermionic condensates are superfluids however there are also two Helium-based superfluids. These Helium-based superfluids were created prior to Bose-Einstein and fermionic condensates and are noteworthy because of the similarities they share with Bose-Einstein and fermionic condensates.

In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered superfluidity amongst Helium-4 atoms (which are bosons) by cooling them to a temperature of less than 2.17 kelvin. Like a Bose-Einstein condensate the Helium-4 superfluid arose because of the collapse of a significant number of bosons into the same quantum state. However the Helium-4 superfluid was formed from liquid Helium-4 as opposed to a Bose gas and therefore could not be considered a Bose-Einstein condensate.

In 1971, Douglas Osheroff, David Lee and Robert Richardson discovered superfluidity amongst Helium-3 atoms (which are fermions) by cooling them to a temperature of less than 2.6 millikelvin. Like a fermionic condensate this Helium-3 superfluid arose because pairing between the fermions allowed them to collapse into a single quantum state. However the Helium-3 superfluid was formed from liquid Helium-3 as opposed to a fermi gas and therefore could not be considered a fermionic condensate.

The Condensates and Superconductivity

Superconductivity is a phenomenon characterized by the complete absence of electrical resistance. Currently superconductors exist only at low temperatures however an affordable room temperature superconductor would revolutionize the electrical and electronic industries allowing for more efficient electric power generation and transmission as well as more effective electromagnets.

Conventional superconductivity is currently best explained by BCS theory. This theory suggests that conventional superconductivity is dependent on the Cooper pairing of electrons. As mentioned before, a similar pairing mechanism is necessary in order to form fermionic condensates. However the pairing mechanism involved in Jin's fermionic condensate is much stronger than that in the Cooper pairing of electrons. In this context, Jin notes that “the strength of pairing in our fermionic condensate, adjusted for mass and density would correspond to a room temperature superconductor."

How Do Free Electrons Originate?

Scientists at Max Planck Institute of Plasma Physics (IPP) in Garching and Greifswald and Fritz Haber Institute in Berlin have discovered a new way in which high-energy radiation in water can release slow electrons. Their results have now been published in the journal,Nature Physics. Free electrons play a major role in chemical processes. In particular, they might be responsible for causing radiation damage in organic tissue.

When ionising radiation impinges on matter, large quantities of slow electrons are released. It was previously assumed that these electrons are ejected by the high-energy radiation from the electron sheath of the particle hit -- say, a water molecule. In their experiment the Berlin scientists bombarded water clusters in the form of tiny ice pellets with soft X-radiation from the BESSY storage ring for synchrotron radiation. As expected, they detected the slow electrons already known. In addition, however, they discovered a new process: Two adjacent water molecules work together and thus enhance the yield of slow electrons.

First the energy of the X-radiation is absorbed in the material: A water molecule is then ionised and releases an electron. But this electron does not absorb all of the energy of the impinging X-ray photon. A residue remains stored in the ion left behind and causes another electron to be released just very few femtoseconds later. (A femtosecond is a millionth of a billionth of a second. For example, the electrons in a chemical process take a few femtoseconds to get re-arranged.) This process is known as autoionisation, i. e. the molecule ionises itself.

The Max Planck scientists have now discovered that two adjacent water molecules can work together in such an autoionisation process. Working in conjunction, they achieve a state that is more favourable energy-wise when each of them releases an electron. What happens is that the molecular ion produced first transfers its excess energy to a second molecule, which then releases an electron of its own. This energy transfer even functions through empty space, no chemical bonding of the two molecules being necessary.

This discovery did not really come as a surprise. More than ten years ago theoreticians at the University of Heidelberg around Lorenz Cederbaum had predicted this "Intermolecular Coulombic Decay." It had already been observed in frozen rare gases. Identifying it beyond doubt now in water called for a sophisticated experimentation technique by which the two electrons produced are identified as a pair.

By demonstrating that the process is possible in water -- thus presumably in organic tissue as well -- the IPP scientists might now be able to help clarify the cause of radiation damage. "Slow electrons released in an organism may have fatal consequences for biologically relevant molecules," states Uwe Hergenhahn from the Berlin IPP group at BESSY: "It was just a few years ago that it was found that deposition of such electrons can cut organic molecules in two like a pair of scissors. Very little is known as yet about how this and other processes at the molecular level give rise to radiation damage. What is clear, though, is that this constitutes an important field of research." Intermolecular Coulomb decay is also important for other chemical processes: The paired action of a water molecule and a substance dissolved in the water could clarify how dissolving processes function at the molecular level.

The results of the IPP scientists were recently published in the journal, Nature Physics. The same issue also features a complementary experiment in which a research group at the University of Frankfurt observed intermolecular Coulombic decay in the tiniest possible water cluster conceivable, comprising just two water molecules.

Atoms and Molecules: Using Magnetic Toys as Inspiration, Researchers Tease out Structures of Self-Assembled Clusters

Rendering of various cluster shapes.

Scientists have long studied how atoms and molecules structure themselves into intricate clusters. Unlocking the design secrets of Nature offers lessons in engineering artificial systems that could self-assemble into any desired form.

In the January 29th issue of Science, a team from Harvard led by Vinothan Manoharan and Michael Brenner, presents additional clues to how and why groups of atoms and molecules may favor less symmetrical and more complex, flexible geometric patterns.

The answer relates to a familiar concept in physics -- entropy. The researchers literally first caught sight of the link by using magnetic "stick and ball" construction toys.

Manoharan, Associate Professor of Chemical Engineering and Physics in Harvard's School of Engineering and Applied Science (SEAS) and Department of Physics, and his colleagues used colloidal particles, a suspended chemical mixture seen in semi-solid foods like mayonnaise, to simulate the clustering behavior of atoms and molecules.

"To allow clusters to form, we put a few tiny polystyrene spheres in microscopic cylindrical wells filled with water. The particles act as 'sticky' hard spheres and naturally cluster together just like groups of nearby interacting atoms and molecules do," says Manoharan.

The researchers expected that simple, highly symmetric shapes would arise most often. Instead, two surprising, related, and scalable phenomena arose when the number of particles used in their experiments reached six or rose above nine.

Six particles can form into a symmetrical octahedron and into a more complex tri-tetrahedron shape. In terms of chemical structure, each shape results in 12 bonds, and hence, has the same amount of potential energy.

With the potential energy being equal, Manoharan and colleagues thought that both shapes would occur in equal proportion. They found, however, that the tri-tetrahedron occurs 20 times more often than the octahedron.

"The only possible explanation was entropy," says Manoharan. "Most people are familiar with entropy as a measure of 'disorder', but the most useful definition of entropy is simply the number of different ways a bunch of particles can arrange themselves."

Natalie Arkus, a former applied mathematics graduate student who worked with Brenner, Glover Professor of Applied Mathematics and Applied Physics, provided a hint to solving the puzzle, as she discovered a method to calculate all the possible structures that could be formed using geometric magnetic toys (composed of magnetic metal rods and silver ball bearings).

As there are more ways for the complicated tri-tetrahedron structure to form (something that can be directly seen by labeling the toy spheres and counting the ways they can be put together) the shape appears far more frequently than the octahedron. In general, among clusters with the same potential energy, highly symmetric structures are less likely to arise.

The researchers also found that when the number of particles reaches nine or higher, entropy plays another important role.

Because the number of possible structures with nine or more particles is vast, the team focused on what are called non-rigid, or flexible structures. Non-rigidity occurs when a cluster has half-octahedra that share at least one vertex, allowing the cluster to twist without breaking or forming another bond (something also easily seen with the toys).

"Because they can move flexibly, the non-rigid clusters have high vibrational entropy," explains Manohran. "In cases with nine or more particles, symmetric clusters do not appear as often due to rotational entropy. The ability to rotate is useful, as it allows clusters to have extra bonds."

As a general rule, the team found that for all clusters up to 8 particles and a select number of structures with up to 12, the most symmetric structures occurred the least often due to entropy.

"Our findings illustrate, in a tangible way, what the concept of entropy means," says Manohran.

Looking ahead, the researchers are interested in using their results to understand the emergence of bulk crystallization, or how particles come together in the early stages of forming a crystal.

Manoharan and Brenner's co-authors included Guangnan Meng, a research associate in the Department of Physics at Harvard University, and Natalie Arkus, a graduate of SEAS and now a postdoctoral fellow at The Rockefeller University. The authors acknowledge support from the National Science Foundation and the Defense Advanced Research Projects Agency.

Universe Is 30 Times More Run Down Than Thought, Astronomers Find

Telescope image of the Pleiades star cluster.

Cars run out of petrol, stars run out of fuel and galaxies collapse into black holes. As they do, the universe and everything in it is gradually running down. But how run down is it? Researchers from The Australian National University have found that the universe is 30 times more run down than previously thought.

PhD student Chas Egan and Dr Charley Lineweaver from the ANU Research School of Astronomy and Astrophysics have computed the entropy of the universe. Scientists compute entropy to find out how efficient an engine is or how much work can be extracted from a fuel or how run down and disordered a system is. Using new data on the number and size of black holes they found that the universe contains 30 times more entropy than earlier estimates.

"We considered all contributions to the entropy of the observable universe: stars, star light, the cosmic microwave background. We even made an estimate of the entropy of dark matter. But it's the entropy of super-massive black holes that dominates the entropy of the universe. When we used the new data on the number and size of super-massive black holes, we found that the entropy of the observable universe is about 30 times larger than previous calculations," said Mr Egan.

"Contrary to common opinion, the maintenance of all the complicated structures we see around us -- galaxies, stars, hurricanes and kangaroos -- have the net effect of increasing the disorder and entropy of the universe. But to be fair, their contributions are negligible compared to the entropy of super-massive black holes," added Dr Lineweaver.

The researchers' results have important implications for terrestrial and extraterrestrial life. "The universe started out in a low entropy state and, in accordance with the second law of thermodynamics, the entropy has been increasing ever since," Mr Egan said. "This is important because the amount of energy available to life in the universe, including terrestrial life, depends on the entropy of the universe. We'd like to know how much energy will be available to life forms anywhere in the universe, and where this energy is. The first step in this procedure is to determine the entropy of the universe. That is what we did."

Dr Lineweaver said that the next step in the research is to out how close we are to maximum entropy, how much entropy is being produced and how much time we have left before the universe and all life in it dies in the inevitable heat death.

Warping the cone on a Topological Insulator

(Left) In the simplest (“hydrogen-atom”) case, the energy-momentum relationship of the surface states in a topological insulator takes the form of a Dirac cone. The constant energy surfaces are then circles of different radii. (Right) Results of a first-principles computation of the spin texture of the surface states in Bi₂Te₃ that we have carried out, showing in quantitative detail how the Dirac cone is warped due to the effect of the crystal potential. The actual shape of the Fermi surface is determined by the natural chemical potential and can be a hexagon or a snowflake. As a result of deviations from the ideal Dirac cone dispersion, a nonzero out-of-the plane spin polarization develops and the conventional nesting channels (red arrows) open up possibilities for magnetic order on the surface.

Unlike common ordered states of matter such as the crystalline solids and magnets characterized by some broken symmetry, the topological states of quantum matter are characterized by boundary properties that are highly robust or stable in the presence of disorder, fluctuations, and perturbations [1, 2]. The first example of a two-dimensional topological insulator was provided by the quantum Hall effect in an electron gas [1], which has no distinct three-dimensional topological generalization [3,4, 5, 6]. Recent theoretical and experimental work indicates that strong spin-orbit coupling in insulators can produce a three-dimensional topological insulator, which is a completely new state of matter [3, 5, 6, 7,8, 9, 10, 11, 12, 13]. This state is characterized by a surface Berry’s phase, realized via an odd number of Dirac cones protected against disorder such as alloying [3, 4, 5, 6]. This is in sharp contrast to the case of graphene, which possesses an even number of Dirac cones.

The thermoelectric Bi-Sb alloys discovered last year provided the first example of a truly three-dimensional topological insulator [7, 8, 9]. The experimental discovery of another thermoelectric family followed—the Bi₂X₃ (X=Se,Te) series—harboring the same topological insulator states with a simpler surface spectrum [10, 11, 12, 13]. The large band gap topological insulator Bi₂Se₃ features a nearly ideal Dirac cone [10, 12], but in the smaller band gap material Bi₂Te₃, with trigonal crystal potential, the cone is warped [11, 12]. Even though the cone is deformed with hexagonal symmetry, the Berry phase quantifying the topological invariant remains unchanged, consistent with its topological order [12, 13]. Now, in an article appearing in the current issue of Physical Review Letters [14], Liang Fu of Harvard University in the US presents a perturbation calculation, which yields a hexagonal modulation of the surface states analogous to the existence of a nonlinear term in the spin-orbit interaction in the bulk crystal. (See Fig. 1 for an illustration using our results of a full-potential, self-consistent computation in Bi₂Te₃.) This leads to a number of remarkable predictions concerning the unusual physical properties induced by the warping of the surface states, which are distinct from the simpler “hydrogen-atom” version of the topological insulator discovered previously [10].

The spin and linear momentum of an electron on the surface of a topological insulator are locked-in one-to-one, as revealed in recent spin-resolved photoemission measurements [8, 9, 12, 13]: the electron is forbidden from moving backwards or taking a “U-turn”—an effect that hinders its localization [15]. This can be realized in a real material if the spins of electrons lying on the Dirac cone, but away from the apex (the Dirac node), rotate around in a circle at the Fermi level. Spins arranged in this way lead to a geometrical quantum phase known as Berry’s phase with a value of π [8, 12]. The work of Fu effectively shows that in the presence of a hexagonally deformed cone, the spins must acquire finite out-of-the-plane components to conserve the net value of Berry’s phase, thus preserving the bulk topological invariant. The resulting finite value of the out-of-the-plane component opens up new possibilities for observing spectacular quantum effects.

Fascinating classes of new particles such as a Majorana fermion, or a magnetic monopole image, or a massive spin-textured Dirac particle can live on the surface of a topological insulator if a finite gap can be induced in the surface band with the chemical potential near the Dirac node [3, 4, 5, 6, 16]. One approach is to create an interface between the topological insulator and a more conventional material such as a magnet or a superconductor, which can induce an energy gap in the Dirac bands. Alternatively, a magnetic field can be applied perpendicular to the sample surface. Until now it had been thought that a parallel magnetic field would neither open a gap nor lead to a surface Hall effect. However, when the surface states are warped significantly in a truly bulk insulating sample, as the work of Fu shows, a parallel magnetic field would open up a gap in the surface spectrum, enabling quantum Hall experiments to be carried out using relatively modest magnetic fields (less than 7 tesla) accessible in a typical laboratory setting. This is an exciting prospect since such experiments have so far not been possible because very high magnetic fields are needed for opening a gap.

In a truly bulk insulating sample, where the chemical potential is such that the hexagonal warping of the Fermi surface survives, one would expect to see a rare realization of the quantum Hall effect on the surface without the presence of Landau levels. In this way, one could observe a half-integer quantum of conductance per surface, which has not been possible so far in any known two-dimensional electron gas, including graphene, and thus allow experimental tests of currently debated scenarios for unusual antilocalization-to-localization transitions on a topological insulator [5]. Presently realized Bi₂Te₃ is quite bulk-metallic with small resistivity [11, 12], whereas spin-orbit insulators with high bulk resistivity values coupled with surface carrier control are needed to realize a functional topological insulator with which many more interesting experiments could be possible. Recent progress in making highly resistiveBi₂Se₃ [12, 17] coupled with surface carrier control [18] heralds the potential dawn of a topological revolution. Similar improvements in sample resistivity are also called for in Bi₂Te₃.

The hexagonal warping of the surface cone and the resulting deviations from a circular Fermi surface provide a natural explanation for the enhanced magnitude of surface electronic interference effects recently reported around crystalline defects in Bi₂Te₃ [19, 20]. For a circular Fermi surface (Fig. 1, left), the electronic states at the Fermi surface cannot backscatter since in a fully spin-polarized band [12, 13] a consequence of time-reversal symmetry is that there are no available states into which electrons can scatter elastically. As a result, interference effects around the defects are small. With hexagonal warping, however, new electronic scattering channels open up, connected by various simple nesting vectors on the Fermi surface (see Fig. 1), and consistent with recent scanning tunneling measurements, the scattering due to defects becomes enhanced [19, 20].

The ideal topological insulators displaying a perfect Dirac cone dispersion cannot support the formation of charge- or spin-density waves. However, in the presence of hexagonal or other forms of deformations of the surface band, such ubiquitous ordering instabilities are no longer forbidden, but rather present a competing ground state, which can circumvent the topological protection. Since electrons with opposite momenta possess opposite spins, a spin-density-wave state is favored if the Fermi surface has parallel segments that can be nested, as is seen to be the case for hexagonal deformation (see Fig 1, right). As one goes around the hexagonal Fermi surface, electrons will experience rapid changes in the out-of-the-plane spin component (σ_z) to maintain their overall Berry’s phase value at π [8, 12], making the resulting spin-density wave appear quite exotic. A stripelike one-dimensional order on the topological surface may also be energetically possible.

Even more complex spin-density-wave orderings could be supported by other topological insulators that presently lack a tractable theoretical model. For example, Bi_1-xSb_x alloys display a central hexagonally deformed Dirac cone and additional surface states of complex character [8]. Despite the complexity of its five surface bands and hexagonal warping, the Bi_1-xSb_x alloy topological insulator precisely exhibits π Berry’s phase and antilocalization just like the “hydrogen-atom” single-surface-Dirac-cone systems [8, 10, 12]. This is a remarkable manifestation and demonstration of a deep and profound principle of physics—topological invariance. Thus the power and beauty of the topological order and invariance in nature are rather firmly established by studying systems that have complex “geometries,” i.e., complex and warped surface states as in the Bi_1-xSb_x alloy series or Bi₂Te₃.

The most remarkable property of topological surface states is that they are expected to be perfect metals in the sense that they are robust and stable against most perturbations and instabilities towards conventional ordered states of matter. The warping of the surface states, even though it opens up interesting possibilities for studying this unusual state of matter in detail, ironically also opens up competing channels which provide a route for departure from the much sought after paradigm of perfect topological stability of the spin-textured metal [3, 4]. Most topological insulators lying hidden in nature are likely to harbor complex and warped surface states as in Bi_1-xSb_x and Bi₂Te₃.

Interestingly, even though the first three-dimensional topological insulators and their spin-textures carrying nontrivial Berry’s phases were discovered only recently, we already have new materials emerging with physical properties lying in a regime where quantum effects associated with competing orders such as the spin-density waves can be foreseen. Parallel progress in theoretical modeling, hand-in-hand with experimental advances in spin-sensitive measurement methods, will likely be essential for unraveling and eventually exploiting the exotic spin textures [21, 22] and other beautiful properties of topological quantum matter.

References

1. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
2. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
3. L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
4. J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 (2007).
5. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
6. X. L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).
7. D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 452, 970 (2008).
8. D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Science 323, 919 (2009).
9. A. Nishide, A. A. Taskin, Y. Takeichi, T. Okuda, A. Kakizaki, T. Hirahara, K. Nakatsuji, F. Komori, Y. Ando, and I. Matsuda,arXiv:0902.2251.
10. Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature Phys. 5, 398(2009).
11. Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science 325, 178 (2009).
12. D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin,A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 460, 1101 (2009).
13. D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Phys. Rev. Lett. 103, 146401 (2009).
14. L. Fu, Phys. Rev. Lett. 103, 266801 (2009).
15. P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Nature460, 1106 (2009).
16. L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
17. J. G. Checkelsky, Y. S. Hor, M.-H. Liu, D.-X. Qu, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. 103, 246601 (2009).
18. Y. Xia, D. Qian, D. Hsieh, R. Shankar, H. Lin, A. Bansil, A.V. Fedorov, D. Grauer, Y.S. Hor, R.J. Cava, and M.Z. Hasan,arXiv:0907.3089.
19. Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, arXiv:0908.0371.
20. T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, arXiv:0908.4136.
21. C. Day, Phys. Today 62, No. 4, 12 (2009).
22. H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nature Phys 5, 438 (2009).