physics notes

FUNDAMENTALS OF SUPERCONDUCTORS

The theoretical understanding of superconductivity is extremely complicated and involved. It is far beyond the scope of this video booklet to attempt to discuss the quantum mechanics of superconductors. However, in this section fundamental terms and phenomena of superconductors will be discussed.

Superconductors have the ability to conduct electricity without the loss of energy. When current flows in an ordinary conductor, for example copper wire, some energy is lost. In a light bulb or electric heater, the electrical resistance creates light and heat. In metals such as copper and aluminum, electricity is conducted as outer energy level electrons migrate as individuals from one atom to another. These atoms form a vibrating lattice within the metal conductor; the warmer the metal the more it vibrates. As the electrons begin moving through the maze, they collide with tiny impurities or imperfections in the lattice. When the electrons bump into these obstacles they fly off in all directions and lose energy in the form of heat. Figure (3) is a drawing that shows atoms arranged in a crystalline lattice and moving electrons bouncing off the atoms that are in their way.

Inside a superconductor the behavior of electrons is vastly different. The impurities and lattice are still there, but the movement of the superconducting electrons through the obstacle course is quite different. As the superconducting electrons travel through the conductor they pass unobstructed through the complex lattice. Because they bump into nothing and create no friction they can transmit electricity with no appreciable loss in the current and no loss of energy.

The ability of electrons to pass through superconducting material unobstructed has puzzled scientists for many years. The warmer a substance is the more it vibrates. Conversely, the colder a substance is the less it vibrates. Early researchers suggested that fewer atomic vibrations would permit electrons to pass more easily.However this predicts a slow decrease of resistivity with temperature. It soon became apparent that these simple ideas could not explain superconductivity. It is much more complicated than that.

The understanding of superconductivity was advanced in 1957 by three American physicists-John Bardeen, Leon Cooper, and John Schrieffer, through their Theories of Superconductivity, know as the BCS Theory. The BCS theory explains superconductivity at temperatures close to absolute zero. Cooper realized that atomic lattice vibrations were directly responsible for unifying the entire current. They forced the electrons to pair up into teams that could pass all of the obstacles which caused resistance in the conductor. These teams of electrons are known as Cooper pairs.Cooper and his colleagues knew that electrons which normally repel one another must feel an overwhelming attraction in superconductors. The answer to this problem was found to be in phonons, packets of sound waves present in the lattice as it vibrates. Although this lattice vibration cannot be heard, its role as a moderator is indispensable.

According to the theory, as one negatively charged electron passes by positively charged ions in the lattice of the superconductor, the lattice distorts. This in turn causes phonons to be emitted which forms a trough of positive charges around the electron. Figure (4) illustrates a wave of lattice distortion due to attraction to a moving electron. Before the electron passes by and before the lattice springs back to its normal position, a second electron is drawn into the trough. It is through this process that two electrons, which should repel one another, link up. The forces exerted by the phonons overcome the electrons' natural repulsion. The electron pairs are coherent with one another as they pass through the conductor in unison. The electrons are screened by the phonons and are separated by some distance. When one of the electrons that make up a Cooper pair and passes close to an ion in the crystal lattice, the attraction between the negative electron and the positive ion cause a vibration to pass from ion to ion until the other electron of the pair absorbs the vibration. The net effect is that the electron has emitted a phonon and the other electron has absorbed the phonon. It is this exchange that keeps the Cooper pairs together. It is important to understand, however, that the pairs are constantly breaking and reforming. Because electrons are indistinguishable particles, it is easier to think of them as permanently paired. Figure (5) illustrates how two electrons, called Cooper pairs, become locked together.

By pairing off two by two the electrons pass through the superconductor more smoothly. The electron may be thought of as a car racing down a highway. As it speeds along, the car cleaves the air in front of it. Trailing behind the car is a vacuum, a vacancy in the atmosphere quickly filled by inrushing air. A tailgating car would be drawn along with the returning air into this vacuum. The rear car is, effectively, attracted to the one in front. As the negatively charged electrons pass through the crystal lattice of a material they draw the surrounding positive ion cores toward them. As the distorted lattice returns to its normal state another electron passing nearby will be attracted to the positive lattice in much the same way that a tailgater is drawn forward by the leading car.

The electrons in the superconducting state are like an array of rapidly moving vehicles. Vacuum regions between cars locks them all into an ordered array as does the condensation of electrons into a macroscopic, quantum ground state. Random gusts of wind across the road can be envisioned to induce collisions, as thermally excited phonons break pairs. With each collision one or two lanes are closed to traffic flow, as a number of single-particle quantum states are eliminated from the macroscopic, many-particle ground state.

The BCS theory successfully shows that electrons can be attracted to one another through interactions with the crystalline lattice. This occurs despite the fact that electrons have the same charge. When the atoms of the lattice oscillate as positive and negative regions, the electron pair is alternatively pulled together and pushed apart without a collision. The electron pairing is favorable because it has the effect of putting the material into a lower energy state. When electrons are linked together in pairs, they move through the superconductor in an orderly fashion.

As long as the superconductor is cooled to very low temperatures, the Cooper pairs stay intact, due to the reduced molecular motion. As the superconductor gains heat energy the vibrations in the lattice become more violent and break the pairs. As they break, superconductivity diminishes. Superconducting metals and alloys have characteristic transition temperatures from normal conductors to superconductors called Critical Temperature (T). Below the superconducting transition temperature, the resistivity of a material is exactly zero. Superconductors made from different materials have different T values. Among ceramic superconductors, YBaCuO T, is about 90 K while for HBaCaCu0 it is up to 133 K. Figure (6) is a graph of resistance versus temperature for YBaCuO.
Since there is no loss in electrical energy when superconductors carry electrical current, relatively narrow wires made of superconducting materials can be used to carry huge currents. However, there is a certain maximum current that these materials can be made to carry, above which they stop being superconductors. If too much current is pushed through a superconductor, it will revert to the normal state even though it may be below its transition temperature. The value of Critical Current Density (J) is a function of temperature; i.e., the colder you keep the superconductor the more current it can carry. Figure (7) is a graph of voltage versus current for a superconductive wire.

For practical applications, J values in excess of 1000 amperes per square millimeter (A/mm), are preferred.

An electrical current in a wire creates a magnetic field around a wire. The strength of the magnetic field increases as the current in the wire increases. Because superconductors are able to carry large currents without loss of energy, they are well suited for making strong electromagnets. When a superconductor is cooled below its transition temperature (T) and a magnetic field is increased around it ,the magnetic field remains around the superconductor. Physicists use the capital letter H as the symbol for Magnetic Field. If the magnetic field is increased to a given point the superconductor will go to the normal resistive state.

The maximum value for the magnetic field at a given temperature is known as the critical magnetic field and is given the symbol H. For all superconductors there exist a region of temperatures and magnetic fields within which the material is superconducting. Outside this region the material is normal. Figure (8) demonstrates the relationship between temperature and magnetic fields.

Figure (9) demonstrates what occurs as a superconductor is placed into a magnetic field. When the temperature is lowered to below the critical temperature,(T), the superconductor will "push" the field out of itself. It does this by creating surface currents in itself which produces a magnetic field exactly countering the external field, producing a "magnetic mirror". The superconductor becomes perfectly diamagnetic, canceling all magnetic flux in its interior. This perfect diamagnetic property of superconductors is perhaps the most fundamental macroscopic property of a superconductor. Flux exclusion due to what is referred to as the Meissner Effect, can be easily demonstrated in the classroom by lowering the temperature of the superconductor to below its T and placing a small magnet over it. The magnet will begin to float above the superconductor. In most cases the initial magnetic field from the magnet resting on the superconductor will be strong enough that some of the field will penetrate the material, resulting in a nonsuperconducting region. The magnet, therefore, will not levitate as high as one introduced after the superconductive state has been obtained.

There are two types of superconductors, Type I and Type II. Very pure samples of lead, mercury, and tin are examples of Type I superconductors. High temperature ceramic superconductors such as YBaCuO (YBCO) and BiCaSrCuO are examples of Type II superconductors. Figure (10) is a graph of induced magnetic field of a Type I superconductor versus applied field. Figure (10) shows that when an external magnetic field (horizontal abscissa) is applied to a Type I superconductor the induced magnetic field (vertical ordinate) exactly cancels that applied field until there is an abrupt change from the superconducting state to the normal state. Type I superconductors are very pure metals that typically have critical fields too low for use in superconducting magnets. Magnetic field strength is measured in units of gauss. The earths magnetic field is approximately 0.5 gauss. The field strength at the surface of a neodymium-iron-boron magnet is approximately 16 kilogauss. The strongest type-I superconductor, pure lead has a critical field of about 800 gauss. The unit of a gauss is a very small unit. A much larger unit of field strength is the tesla (T). Ten kilogauss (1 x 10 gauss) is equal to 1 tesla. Figure (11) is a graph of induced magnetic field of a Type II superconductor versus applied field. Figure (11) shows a Type II superconductor in an increasing magnetic field. You will notice that this graph has an Hc1 and Hc2. Below Hc1 the superconductor excludes all magnetic field lines. At field strengths between Hc1 and Hc2 the field begins to intrude into the material. When this occurs the material is said to be in the mixed state, with some of the material in the normal state and part still superconducting. Type I superconductors have Hc too low to be very useful. However, Type II superconductors have much larger Hc2 values. YBCO superconductors have upper critical field values as high as 100 tesla.

Type II behavior also helps to explain the Meissner effect. When levitating a magnet with a Type I superconductor, a bowl shape must be used to prevent the magnet from scooting off the superconductor. The magnet is in a state of balanced forces while floating on the surface of expelled field lines. Because the field at the surface of a samarium-cobalt magnet is about 600 G, and the Hc1 for the YBCO superconductor is less that 200 G the pellet is in the mixed state while you are performing the Meissner demonstration. Some of the field lines of the magnet have penetrated the sample and are trapped in defects and grain boundaries in the crystals. This is known as flux pinning. This "locks" the magnet to a region above the pellet.

The superconducting state is defined by three very important factors: critical temperature (Tc), critical field (Hc), and critical current density (Jc). Each of these parameters is very dependant on the other two properties present. Maintaining the superconducting state requires that both the magnetic field and the current density, as well as the temperature, remain below the critical values, all of which depend on the material. The phase diagram in Figure (12) demonstrates relationship between Tc, Hc, and Jc. The highest values for Hc and Jv occur at 0 K, while the highest value for Tc occurs when H and J are zero. When considering all three parameters, the plot represents a critical surface. From this surface, and moving toward the origin, the material is superconducting.Regions outside this surface the material is normal or in a lossy mixed state. When electrons form Cooper pairs, they can share the same quantum wave-function or energy state. This results in a lower energy state for the superconductor. Tc and Hc are values where it becomes favorable for the electron pairs to break apart. The current density larger than the critical value is forced to flow through normal material.This flow through normal material of the mixed state is connected with motion of magnetic field lines past pinning sites. For most practical applications, superconductors must be able to carry high currents and withstand high magnetic field without reverting to its normal state.

Higher Hc and Jc values depend upon two important parameters which influence energy minimization, penetration depth and coherence length. Penetration depth is the characteristic length of the fall off of a magnetic field due to surface currents.Coherence length is a measure of the shortest distance over which superconductivity may be established. The ratio of penetration depth to coherence length is known as the Ginzburg-Landau parameter. If this value is greater than 0.7, complete flux exclusion is no longer favorable and flux is allowed to penetrate the superconductor through cores known as vortices. Currents swirling around the normal cores generate magnetic fields parallel to the applied field. These tiny magnetic moments repel each other and move to arrange themselves in an orderly array known as a fluxon lattice.This mixed phase helps to preserve superconductivity between Hc1 to Hc2. It is very important that these vortices do not move in response to magnetic fields if superconductors are to carry large currents. Vortex movement results in resistivity. Vortex movement can be effectively pinned at sites of atomic defects, such as inclusions, impurities, and grain boundaries. Pinning sites can be intentionally introduced into superconducting material by the addition of impurities or through radiation damage.

Up to this point those properties of superconductors which are commonly referred to as macroscopic properties, such as the Meissner effect and zero resistance have been discussed. We will now focus on those properties which are often referred to as quantum mechanical or microscopic properties. An example of microscopic properties is the phenomenon of electron tunneling in superconductors. Tunneling is a process arising from the wave nature of the electron. It occurs because of the transport of electrons through spaces that are forbidden by classical physics because of a potential barrier. The tunneling of a pair of electrons between superconductors separated by an insulating barrier was first discovered by Brian Josephson in 1962. Josephson discovered that if two superconducting metals were separated by a thin insulating barrier such as an oxide layer 10 to 20 angstroms thick, it is possible for electron pairs to pass through the barrier without resistance. This is known as the dc Josephson Effect, and is contrary to what happens in ordinary materials, where a potential difference must exist for a current to flow. The current that flows in through a d.c. Josephson junction has a critical current density which is characteristic of junction material and geometry. A Josephson junction consists of two superconductors separated by a thin insulating barrier. Pairs of superconducting electrons will tunnel through the barrier. As long as the current is below the critical current for the junction, there will be zero resistance and no voltage drop across the junction. If it is placed next to a wire with a current running through it, the magnetic field generated by the wire lowers the critical current of the junction. The actual current passing through the junction does not change, but has become greater than the critical current which was lowered. The junction then develops some resistance which causes the current to branch off. Figures (13) and (14) demonstrate the Josephson effect and a Josephson junction. Figure (13) illustrates the Josephson effect. Figure (14) is a graph of the current-voltage relation for a Josephson junction.

The Josephson junction is a superfast switching devise. Josephson junctions can perform switching functions such as switching voltages approximately ten times faster than ordinary semiconducting circuits. This is a distinct advantage in a computer, which depends on short, on-off electrical pulses. Since computer speed is dependent on the time required to transmit signal pulses the junction devices' exceptional switching speed make them ideal for use in super fast and much smaller computers.