On High Temperature Superconductivity: Pseudogap Region

In the previous post we reviewed a bit of the nature of the parent state of the cuprates, i.e. the Mott antiferromagnet. And we said we were ready to move and start looking for the action in these cuprate guys. It is time to pick up our “cachybachies” and move to the action then!

The action starts when atoms between the Copper-Oxide planes are interchanged by atoms that take electrons from the planes. In this way, the planes are hole doped. Somewhat similar physics occur when the material is electron doped, but the effects are considerably larger for hole doping so we concentrate on this case. Also, historically hole doped materials were discovered first and are easier to bake. (Most of these materials are literally grown in ovens. The art of growing relatively good large crystal is a black art perfected by a secretive order of Japanese monks.)

Well, when hole doped, the first change in the High Copper-Oxides shows up at about 3% doping: the local moments and the antiferromagnetic order vanishes into oblivion! A tiny bit of doping induces a phase transition. Now the material goes into a state that can be described as exhibiting some “dissociative identity disorder (DID)”. This phase has no magnetic behavior, but there is evidence that at the lowest temperatures it is a spin glass. This might be a red herring as the glassy behaviour could well be due to material imperfections.

Cuprates phase diagram. The left hand side corresponds to electron doping, while the right hand side corresponds to hole doping.

The “DID phase” beyond 3% doping exhibits extremely puzzling behavior. First there are experimental hints of different types of both charge and spin order that are not fully developed as they do exhibit long range order, but they are fluctuating dynamically. They have dynamical signals. The evidence points to stripe charge and spin order that, if it were to develop, would probably be a charge and/or spin density wave order (CDW or SDW). There is evidence of possibly Wigner crystal order, where the charge order is fully two dimensional rather than anisotropic as in 1D stripes or CDW-SDW’s. The periodicities of these orders depend on the doping and tend to be commensurate with the lattice.

In this region beyond 3% doping, all of this phase competition arises because of the large repulsion and the strong correlations it creates. Whenever one have such strong correlation, the notion of well defined electron-like or hole-like quasiparticle should break down in the sense that if a hole or an electron is excited, it should not propagate coherently. This picture is certainly borne out by experiments in the sense that they either see no quasiparticle peak but only very broad spectra, or they see a very broad peak with a width comparable or larger than the energy. This behavior is seen over broad range of temperatures and dopings. It becomes more pronounced as one goes to smaller dopings where correlations are stronger.

This strongly correlated behavior has lead to a plethora of theories to describe the physics, which is extremely interesting and include all sorts of exotica. Most of those theories include the electrons or holes breaking up into pieces that carry charge and no spin, and spin and no charge. It purports the elementary excitations to be fractionalized, i.e. to carry fractions of the quantum numbers of the electron or hole.

They are mediated by gauge interactions. While these theories are very beautiful and constitute state of the art field theories, none of them have been successful. They have failed in explaining the superconductivity.

These theories do explain some of the incoherent spectra of the cuprates, but they do not explain this in a unique way in the sense that most of them give incoherent spectra, yes. But proper calculation in a region with large gives the incoherent spectra with no need to invoke exotica! The proponents of this theories make fuss about the incoherent spectra, but it is a fact that in theories in which the electron has no stability and decays into pieces, the electronic spectra will necessarily be incoherent. Hence success accounting for this spectra is not strong evidence in favor of this type of physics.

To be explicit, fractionalization and gauge physics implies incoherent electronic spectra, but the converse need not be true. In fact, fairly often it is not true. While it is true that this exotic physics can occur for strongly correlated systems, as evidenced by the fractional Quantum Hall effect and some one dimensional systems, in the cuprates they have not provided an explanation of the most important effects and in particular of the superconductivity.

Ok, enough of the pseudogap region for the moment. Shall we move into the superconducting phase? Are you ready to rumble?

Well, better hold your horses till the next post.

Stay tuned for even more!

Dr. Damian Rudelberg

On High Temperature Superconductivity: The Parent State

How about giving our second step in the race through the “quantum world”? We propose it to be into the amusements of High Temperature (high ) Superconductors. For those not familiar with the topic, do not worry. We’ll start, of course, by providing you with a review of the basic properties of high temperature superconductors, so you can feel comfortable following us in future posts where we will be commenting actively on experiments, theory related to experiments, and so on.

Imagine a having a material where you can set up a current, and such current will never decay! No resistance, no energy dissipation, wait for as many years as your life allows you, and the current stays there, unaltered. Imagine the same material levitating a magnet!

Magnet levitating on top of a piece of superconductor. Photo from Lawrence Berkeley Laboratory. Click here for photo source

Sounds like fantasy, like dreams, doesn’t it? Well, it is not. Nature has provided us with this wonder. These materials are known as superconductors.

And now you might wonder, what are exactly high temperature superconductors? How do they look like? Well, they are a family of materials which when undoped are insulating salts.

Piece of high temperature superconductor Bi-2223.Photo from Wikimedia Commons. Click here for photo source

Piece of high temperature superconductor YBCO. Photo from UBC Materials Research. Click here for source

They consist of Copper-Oxygen planes separated by a bunch of other elements in the structure between the planes.

Crystal Structure of Bi-2212. You can see here the Copper-Oxide layer, where Copper is shown in purple and Oxygen in brown. Stuff in between are Bismuth in green, Calcium in pink and Strontium in orange. Photo from Lawrence Berkeley Laboratory. Click here for photo source

The different families differ in the structure between the planes and also in the number of Copper-Oxide planes in the unit cell. There are materials with two consecutive Copper-Oxygen planes (these are denominated “bilayer superconductors”), and some with more consecutive planes (multilayer materials) in the unit cell. Because of the presence of the Copper-Oxygen planes, and their importance, as all the cool action happens in the planes, these materials are colloquially known as “cuprates”

The cuprates are interesting even when undoped. When undoped, the Copper-Oxygen planes have an odd number of electrons. According to standard band theory (which works for most materials), it should have an unfilled band and thus be a metal. The cuprates are not only not a metal, but are insulators with a very healthy gap! This is because the on-site repulsion is very large.

The large per Copper site prevents the electrons from tunneling from site to site, effectively leaving one valence electron per Copper site. Now, hybridization of the Copper atoms with the oxygen leads to a splitting of the valence band. The valence band now accommodates two electrons per Copper and is half filled. Having two electrons in a site costs energy and thus the insulator has a gap proportional to . The material is insulating due to strong correlation effects.

Materials like the cuprates that should be metallic, but are insulating are called Mott insulators in honor of Sir Nevill Mott who first recognized that electronic repulsion and its consequent strong correlation effects could turn metals into insulators. In fact, strong correlations among particles are the cradle of plenty of exotic, cutting edge materials physics. One could arguably state that the major part of modern day condensed matter physics is the study of strong correlations and the nontrivial physics that arises from them.

Now, even though there is no real tunneling in these materials, there is virtual tunneling driven by the lowering of kinetic energy of the on site localized electron by spreading somewhat to neighboring sites. This virtual tunneling leads to an effective antiferromagnetic interaction between the spins of the neighboring localized spins, which prefer to line up antiparallel to each other. The reason that the interaction is antiferromagnetic is that if the spins are aligned parallel to each other, the Pauli exclusion principle prohibits tunneling, be it virtual or real. This prohibition vitiates the possible lowering of kinetic energy due to localization. If the spins are antiferromagnetically aligned in neighboring sites, then tunneling fluctuations and their concomitant energy reduction can and does occur. This is the reason why the half filled, or undoped, cuprates are antiferromagnetically ordered Mott insulators.

We are ready to move now into where the action really is in this materials. But this will be the subject of another post.

Stay tuned for more!

Dr. Damian Rudelberg

On Quantum Measurement

Every important race starts with a first step. With some time, we hope to put you in the vanguard of condensed matter physics principally, but physics in general. But before we start running, we think it is better to start with a simple walk.

For the benefit of non-specialists and students, who might not have such an extensive knowledge of physics, we will start reviewing some basic, yet extremely important topics. The first question that might come to your mind could be: Why condensed matter physics? Why should I care? Well, look around you. Almost everything you see is governed, in its most fundamental principles, by the physics of materials. Starting from the patterns you see if oil is mixed with water, to the color of the sky, the melting of water, the functioning of resistors and ultimately your computer, to the existence of more exotic phenomena like superconductivity, superfluidity, exotic phases of matter, or the dreams of sophisticated technology advances like functional quantum computers. You name it! The physics of materials pervade our world and daily lives from head to toes. This is what condensed matter physics is all about. And since most of condensed matter physics is concerned with the quantum mechanics of many body systems, we will be discussing some aspects of quantum mechanics in general, and quantum field theory of materials.

Let us start our walk by taking some time to share a few thougts on quantum measurement. This is one of the most mysterious aspects of quantum mechanics. Quantum Measurement is important for abstract questions of how one goes from quantum to classical behavior in an intrinsically quantum world. It must be better understood and controlled as it is one of the important puzzles to be cracked if the promising and exciting dreams of quantum computation are to be realized.

Many physicists consider the problem of quantum measurement unsolved. We will certainly not solve it in this post, but with the simple and cliched example of the double slit experiment, we will present what we know about decoherence from interaction with a measurement apparatus and present the two main
interpretations of measurement: Copehagen (wave function collapse in this context) or Many worlds interpretations.

First, we plagiarize the textbooks presentation of the double slit experiment and start with a double slitted wall. Far to the left is a particle gun and far to the right is a screen where the particles become stuck and are collected. The gun shoots evenly in all directions.

A) The experiment with classical particles

• If the experiment is run with the upper slit open and the lower slit closed, the distribution of bullets at the screen is .

• If the experiment is run with the lower slit open and the upper slit closed, the distribution of bullets at the screen is .

• If the experiment is run with both slits open, the distribution of bullets at the screen is .

B) The experiment with quantum particles

The two most important difference between quantum and classical mechanics are

1. particles or aggregates of particles (that is, matter) behave like waves
2. quantum mechanics is a statistical theory in the sense that projections of the square of the quantum system’s wave function onto a state , , represents the probability of measuring the system in state .

So while quantum evolution is perfectly deterministic, something not completely straightforward happens at “the moment of” measurement. Nonetheless, we can predict accurately the statistical behavior of the system, that is, the probability distribution of the outcomes.

• If the experiment is run with the upper slit open and the lower slit closed, the upper slit acts as a source of waves for the particle which has wavefunction with space dependece . This gives a distribution of particles at the screen .

• If the experiment is run with the lower slit open and the upper slit closed, the lower slit acts as a source of waves for the particle which has wavefunction with space dependece . This gives a distribution of particles at the screen .

• If the experiment is run with both slits open, both slits act as sources of waves for the particle which has wavefunction with space dependece . This gives a distribution of particles at the screen .

Here we see one of the most important differences between quantum and classical mechanics: the wave aspect of matter. This aspect is illustrated by the the wave interference term , which is absent when the experiment is performed with classical particles.

If instead of shooting the particles all at once, they are shot one by one, and after a large number has accumulated at the screen one looks at the distribution, the results do not change. There is still interference. Each particle “interferes” with itself. Actually the wavefunction corresponding to each particle interferes with itself.

You can have a bit of fun with two nice applets found here:

http://www.colorado.edu/physics/2000/applets/twoslitsa.html

http://www.colorado.edu/physics/2000/applets/twoslitsb.html