Back in Track: Cuprates Superconducting Phase

After a short detour into the basic physics of normal (or low temperature) superconductors, we are back on the road towards open issues on high temperature cuprate superconductors. One of the things those of you who are curious enough might be wondering, after telling you that cuprates become, on doping,

the best conductors in the universe at temperatures considerably higher than those for normal superconductors, is why then they have not been exploited technologically more than normal superconductors. The reason is a law of life: nothing in life is for free. And nothing is physics is for free, particularly in the case of superconductivity. The price to pay for having a superconductor survive at high temperatures is that their superfluid density is low. Superfluid density is a measure of how strong the supercurrents are. The higher the superfluid density, the stronger the currents the superconductor can sustain. An it is ultimately these supercurrents what are of technological value.

There are plenty of mysteries and head twisters associated with High superconductors. Let’s start with one of the features that is somewhat less of a puzzle. Why is the superfluid density so low in cuprates? The obvious answer is that there are not too many charge carriers, as superconductivity starts when the materials have been doped with about only 7% or 8% carriers from the undoped insulating states. Also the bands are thin and thus unable to provide too much superconducting charge carriers. While these two conditions would lead to smaller superfluid densities, the superfluid densities are quite smaller than what one would expect even if one takes into account these two facts.

Remember that high temperature superconductors are very strongly correlated materials. We have seen this in previous posts. There is a large repulsion between valence electrons, which imposes a penalty cost for high local electronic densities. Repulsion also increases phase fluctuations of the superconducting order parameter [as an aside note for non-experts, you can consider the order parameter as a quantity which represents what is ordered in the phase of matter. For example, in the case of a normal, low temperature superconductor it would be the amplitude of Cooper pairs]. The increased phase fluctuations lead to a suppression of the superfluid density. This happens independent of the mechanism causing the superconductivity. (We will explain exactly how this reduction of superfluid density due to increased phase fluctuations caused by repulsion happens in a future post.) Another aspect that can contribute to the low superfluid density is that there are many different phases competing with superconductivity in the underdoped region. Interestingly enough, these competing phases also arise from strong correlations.

Another theme for debate is whether electrons are Cooper paired in cuprates as they are in normal, low temperature superconductors. Many people think they are, for good reasons I will state in a second. Some other people disagree and work out theories to explain superconductivity in cuprates based on spin-charge separation, among other types of theories based on more exotic fractionalization mechanisms. Those who believe electrons are Cooper paired in cuprates base their belief in the outcomes of several experiments. For example, in Angle Resolved Photoemission Spectroscopy (ARPES) experiments, electrons are shot into a piece of material. Those electrons scatter off the particles of the material, exchanging energy and momentum with them. Because of this, if you measure the energy and momentum of the electrons that bounced off after scattering, that will reflect the energy and momentum of the particles. That is how ARPES people can determine the energy spectrum of particles in a material. In the case of cuprates, ARPES measures an energy dispersion for the particles which is consistent with the dispersion of Cooper paired electrons. Even more dramatic, experiments have been done to study Josephson tunneling between a normal, low temperature superconductor and a cuprate superconductor. We already learned about Josephson tunneling in the previous post. Once a piece of normal, low temperature superconductor is brought into contact with a piece of cuprate superconductor, a Josephson current is established across the junction. The only way known to date in which this can happen is if both materials are Cooper paired. In that case a Josephson current will flow as both materials try to equalize the phase of their pairs. Once the phases are equal, such current stops. The fact that tunneling occurs between a normal superconductor and the cuprates suggests that cuprates are Cooper paired.

On another aspect, the order parameter in high temperature superconductors has a symmetry associated with it. Experiments seem to point toward a d-wave symmetry. What does this mean? It means that in the case where electrons are paired, they are paired so that their momentum is in a d-wave state. That is, the electrons that are paired are electrons in orbitals d of the atoms. [As a point of reference, in a normal, low temperature superconductor the electrons in the Cooper pairs have, most commonly, an s-wave orbital (or momentum) symmetry. That is, the electrons paired are those in orbitals s of the atoms.]

There are also several puzzling features in the underdoped or pseudogap region of the phase diagram of cuprates. For example, some experiments in the underdoped region seem to point towards the existence of three different energy scales in the materials. Two of those are what physicists call energy gaps. These represent the amount of energy you have to input in the material to create different types of excitations.

If electrons are paired, then there are basically two ways to lose superconductivity. One of them is by breaking the pairs. The temperature at which you start breaking pairs represents one of the energy scales measured in the system, the so called “depairing gap”. There exists also an energy scale characterizing a temperature where the phase of the wavefunction of the pairs starts fluctuating so much that all pairs cease to have the same phase. This is called loss of phase coherence. As it was precisely the rigidity associated with the phase coherence what made possible for the material to superconduct, a dephased superconductor does not superconduct. This is then the second way in which superconductivity can be lost. The energy scale associated with loss of phase coherence is at a lower temperature than the depairing gap. What is the third energy scale measured in the underdoped region? This is a second gap and it is not associated with the superconductivity. No one knows its exact nature. It could arise from one of the orders competing with superconductivity in this region. It could very well be associated to an electronic, charge order, or may be even the antiferromagnetic parent state itself. But this is something that stills need to be uncovered.

Another puzzling experimental result in the pseudogap phase is found in ARPES. When ARPES measures the energy spectrum of excitations (i.e. quasiparticles, as excitations are commonly also called) in the pseudogap phase, it finds a node in such spectrum which survives till the point where the antiferromagnets just starts arising. (The node is nothing more than a region or point where an energy gap goes to zero.) Moreover, when ARPES measures a little bit inside the antiferromagnetic phase, an anisotropy is seen in the momentum distribution. This indicates that the antiferromagnetic energy gap is lower at those positions where there were nodes before.

A minute ago, I told you that the symmetry of the order parameter in cuprate superconductors is d-wave. A d-wave symmetry has areas where it is a maximum (called antinodes) and areas where it is zero (called nodes). For example, it can behave like , which can vanish if . Because a d-wave symmetric gap would have nodes, and the superconducting order parameter has d-wave symmetry, some physicists think that the result of ARPES experiments might suggest a coexistence of superconductivity with the antiferromagnet.

Now let us supposed that it is established that electrons in cuprates are paired. In that case, what could be providing the “binding interaction” among themselves? Well, the first guess would be phonons, for this is what pairs electrons in a normal superconductor. The drawback is that phonons in cuprates, although they are quite strong, do not seem to be strong enough to provide a binding as robust as the one in cuprates. Is there another mode in the system capable of providing the binding? Yes, magnons are waves formed as the spin of particles in a material changes direction from one point to another. Some physicists have started to try to work out theories where magnons would mediate an attractive, binding interaction between electrons. Nonetheless, magnons are not Coulombic in nature but rather magnetic. Magnetic interactions are a lot weaker than Coulombic ones. Hence, magnons should pretty weak, making this mechanism be probably less likely than the phonons themselves. Still some other people think another Coulombic or electronic type of interaction might bind the electrons.

One more mechanism being considered by physicists as a possible explanation for the pairing derives from the Hubbard repulsion in the system. Suppose you a theory which includes a Hubbard repulsive interaction, together with the imposition of electrons being paired in the same way they are in low temperature superconductors. You will find that there is an energy cost for having electrons do this because the Hubbard term is repulsive (it is positive). It seems like we cannot have pairing of electrons due to , which you might be thinking is quite obvious. After all, how is it possible to get binding from repulsion? But if we want to be more careful in doing the calculations, we need to include, together with , the kinetic energy of the electrons. It is in fact due to this that spins align antiparallel in an antiferromagnet (and remember the parent state of superconductors is an antiferromagnet). There can be a bit of tunneling among the wavefunctions of two neighboring electrons. This happens because the electrons lower they energy by spreading their wavefunctions a bit. But this would mean there can be a small overlap of both wavefunctions. The only way to allow such overlap is to have spins aligned antiparallel to each other, for otherwise we cannot beat the Pauli exclusion principle. So now, if we include all these effects together with , and we calculate the effective antiferromagnetic interaction between electrons, we see that a pairing of electrons with d-wave symmetry can occur. The effective interaction is attractive for a “d-wave pairing”. The relevant energy scale involved in this mechanism is , where represents the kinetic energy. When is large, as it is in cuprates, this scale is lower than . Nonetheless, it is very close to the characteristic energy scale of superconductivity in the materials. So it looks like we can have a pairing of electrons, in the same way it occurs in low temperature superconductors, with a d-wave symmetry. The exception is that in this case the pairing happened via an indirect interaction rather than by exchanging some mode like it happens in the case of exchange of phonons.

Ultimately, what is the truth? It is still our task to find out. It is like a hide and seek game. The truth is there hiding somewhere, waiting for us to find it. But it turns out it is doing a pretty good job of hiding!

Now let us go back around our confusion circle, to where we started, and close the circle. Why is the superfluid density so low? Can it be that the same mechanism responsible to provide such a strong attraction between the electrons in the high temperature superconductor be the same one responsible, by virtue of its own strength, of keeping the superfluid density low?

I have probably already risen a storm of questions, confusion and doubts about cuprates. Yes, it seems to be all covered with a huge cloud that needs to be cleared. This same storm has been tormenting the mind of the best physicists of the world for over 20 years. So if by the end of this post, it is in you too, feel assured you are in very good company. Puzzling your mind, Dr. Damian Rudelberg

Detour Into Normal Superconductors

Last Thursday night, I witnessed by accident a pretty interesting conversation. Two young men, called Peter and John, were having dinner together at a Diner. I was quiet in the table behind them, listening to the conversation. While enjoying a quite delicious cookies ‘n cream shake and a succulent bacon cheeseburger, they started chatting about a great new blog they found on the internet, called Quantum Matters. John just read some very good posts on superconductivity on the blog, and decided to make a few questions to Peter, who happened to be a physicist and had worked for a while on high .

John:

So Peter, as I was telling you, I was reading two posts they have on superconductivity. They started talking about the parent state of cuprate superconductors, and moved from there to talk about the pseudogap region. Nonetheless, the posts stopped there. Right when they were reaching about 8% doping, they let us readers hanging, saying we should stay tuned. I will definitely stay tuned, but I am quite anxious to know the end of the story! So, how does the game continues? What is the deal after I dope the material past the pseudogap phase?

Peter:

Well John, the “magic” is that close to 8% doping, these materials, which in their parent state are insulators (and even in the pseudogap region their conductivity is nothing to brag about!) become the best conductors in the universe! The temperature at which they lose superconductivity varies with doping. It begins increasing at about 8% doping, reaches a maximum close to 20% doping, and then decreases to zero again, forming a dome.

John:	How does that happen?
Peter:	Ok, let’s change things a bit here and better than talking about the particular case of cuprates, let me explain you what happens in normal superconductors, for cuprates are a little bit special. At very low temperatures, electrons in a normal superconductor like to bind in pairs. It turns out that they lower their energy when they are paired. And in physics my friend, everything is governed by energetics. So an electron with momentum and spin up () pairs with an electron with momentum and spin down (). The pairs are called Cooper pairs.

John:	Wo, wo ,wo! Wait a second! Excuse my ignorance, but according to one of the first things I learned about physics in high school, electrons repel each other. How can they bind in pairs then?
Peter:	I see you are awake and alert! Well, there are two things in here. Electrons interact among themselves via an electric force, or Coulomb interaction, which is indeed repulsive. On the other hand, they live in a material made of atoms. And the atoms form a lattice. The ions in such lattice can vibrate, and such vibrations behave as quantum particles called phonons. This is analogous to light being a wave which also behaves as a quantum particle called photon. Electrons can feel such vibrations, and in fact, they can “talk” to each other by exchanging phonons. In such a way, they can feel an attractive interaction among themselves. See how it works John? They can attract each other via the mediation of a phonon. That’s how they bind in pairs.
John:	Ok, so we have these Cooper pairs. Now, how do they lead to superconductivity?
Peter:	Ahh, good question! But before I tell you, would you order a basket of onion rings? You know, knowledge costs something!
John:	What a briber!
Peter:	Ok, it turns out that these Cooper pairs, like any other particle in quantum mechanics, have a wavefunction associated with them. And this wavefunction naturally has a phase. In a superconductor, not only electrons are paired, but all pairs’ wavefunctions have the exact same phase. And it costs a whole lot of energy to change that phase. That is what gives the rigidity to the ground state (or fundamental, i.e. lowest energy, state), which in turns allows it to superconduct. This is because once a current is set in the material, changing that current implies changing the phase of the Cooper pairs, which is simply energetically too costly. Hence, the current stays there forever, unaltered. See, the same principle is what is responsible for a superconductor being able to levitate a magnet. If you try to put a magnetic field through a piece of superconductor, the superconductor hates it, because it means changing their phase. It prefers to set up a current along its edges, which in turns produces a magnetic field opposing that of the magnet, and expel the field of the magnet. And voilà! The magnet levitates! Impressive isn’t it?

Photo from Lawrence Berkeley Laboratory. Click here for photo source

Well my readers, there you had it. Quite an interesting conversation! Nonetheless, there is another important and quite cool phenomena exhibited by superconductors, which Peter forgot to tell John about. But that is why I’m here. You guys have Damian to tell you.

It turns out that if you take two pieces of superconductors, different superconductors, with different , and put them in contact, so that you form a junction out of the two pieces, there is current flowing from one side of the junction to the other, even if you apply no voltage at all! Surprised? Well, you should be! The reason there is current flowing even with no voltage is that the Cooper pairs in each piece of superconductor have the same phase within that piece, but such a phase is different from the phase in the other piece of superconductor. When you put both pieces together, because they like to have the same phase all over, they try to equalize the phase across the junction. This causes the current. The effect is called Josephson tunneling, for the pairs in one side tunnel to the other side through the junction. I should mention briefly that the Cooper pairs can only tunnel if their spins are paired in the same way inside the pairs (that is, an up spin with a down spin for example — called a singlet –, or two ups, or two downs — called a triplet –, etc). This was an important experimental tool which allowed physicists not only to determine the type of pairing in different superconductors (knowing the pairing of one of the superconductors, you can tell the pairing of the second if Josephson tunneling occurs), but to make intelligent guesses on whether cuprates are really Cooper paired or not. But this is a subject that requires a bit of explanation. I’ll tell you more about it in my next post.

Till then,

Dr. Damiam Rudelberg