A brief overview of the theories of superconductivity and the problems of high-temperature superconductivity are analyzed. School encyclopedia What is the phenomenon of superconductivity
Superconductivity - the property of some materials to have strictly zero electrical resistance when they reach a temperature below a certain value (critical temperature). Several dozen pure elements, alloys and ceramics are known that transform into a superconducting state. Superconductivity is a quantum phenomenon. It is also characterized by the Meissner effect, which consists in the complete displacement of the magnetic field from the volume of the superconductor. The existence of this effect shows that superconductivity cannot be described simply as ideal conductivity in the classical sense.
Opening in 1986-1993. a number of high-temperature superconductors (HTSC) has pushed back the temperature limit of superconductivity far and has made it possible to practically use superconducting materials not only at the temperature of liquid helium (4.2 K), but also at the boiling point of liquid nitrogen (77 K), a much cheaper cryogenic liquid.
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History of discovery
The basis for the discovery of the phenomenon of superconductivity was the development of technologies for cooling materials to ultra-low temperatures. In 1877, French engineer Louis Cayette and Swiss physicist Raoul Pictet independently cooled oxygen to a liquid state. In 1883, Zygmunt Wróblewski and Karol Olszewski liquefied nitrogen. In 1898, James Dewar managed to obtain liquid hydrogen.
In 1893, the Dutch physicist Heike Kamerlingh Onnes began to study the problem of ultra-low temperatures. He managed to create the best cryogenic laboratory in the world, in which he obtained liquid helium on July 10, 1908. Later he managed to bring its temperature to 1 degree Kelvin. Kamerlingh Onnes used liquid helium to study the properties of metals, in particular to measure the dependence of their electrical resistance on temperature. According to the classical theories that existed at that time, the resistance should gradually fall with decreasing temperature, but there was also an opinion that at too low temperatures the electrons would practically stop and stop conducting current altogether. Experiments conducted by Kamerlingh Onnes with his assistants Cornelis Dorsman and Gilles Holst initially confirmed the conclusion about a smooth decrease in resistance. However, on April 8, 1911, he unexpectedly discovered that at 3 degrees Kelvin (about −270 °C), the electrical resistance of mercury is practically zero. The next experiment, carried out on May 11, showed that a sharp jump in resistance to zero occurs at a temperature of about 4.2 K (later, more accurate measurements showed that this temperature is 4.15 K). This effect was completely unexpected and could not be explained by the then existing theories.
In 1912, two more metals were discovered that go into a superconducting state at low temperatures: lead and tin. In January 1914, it was shown that superconductivity is destroyed by a strong magnetic field. In 1919, it was discovered that thallium and uranium are also superconductors.
Zero resistance is not the only distinguishing feature of superconductivity. One of the main differences between superconductors and ideal conductors is the Meissner effect, discovered by Walter Meissner and Robert Ochsenfeld in 1933.
The first theoretical explanation of superconductivity was given in 1935 by Fritz and Heinz London. A more general theory was constructed in 1950 by L. D. Landau and V. L. Ginzburg. It has become widespread and is known as the Ginzburg-Landau theory. However, these theories were phenomenological in nature and did not reveal the detailed mechanisms of superconductivity. Superconductivity was first explained at the microscopic level in 1957 in the work of American physicists John Bardeen, Leon Cooper and John Schrieffer. The central element of their theory, called the BCS theory, is the so-called Cooper pairs of electrons.
It was later discovered that superconductors are divided into two large families: type I superconductors (which, in particular, include mercury) and type II (which are usually alloys of different metals). The work of L.V. Shubnikov in the 1930s and A.A. Abrikosov in the 1950s played a significant role in the discovery of type II superconductivity.
Of great importance for practical applications in high-power electromagnets was the discovery in the 1950s of superconductors that could withstand strong magnetic fields and carry high current densities. Thus, in 1960, under the leadership of J. Künzler, the Nb3Sn material was discovered, a wire from which is capable of passing a current with a density of up to 100 kA/cm² at a temperature of 4.2 K, being in a magnetic field of 8.8 T.
In 1962, the English physicist Brian Josephson discovered the effect that received his name.
In 1986, Karl Müller and Georg Bednorz discovered a new type of superconductors, called high-temperature superconductors. In early 1987, it was shown that compounds of lanthanum, strontium, copper and oxygen (La-Sr-Cu-O) experience a jump in conductivity to almost zero at a temperature of 36 K. In early March 1987, a superconductor was obtained for the first time at temperatures above boiling of liquid nitrogen (77.4 K): it was discovered that the compound of yttrium, barium, copper and oxygen (Y-Ba-Cu-O) has this property. As of January 1, 2006, the record belongs to the ceramic compound Hg-Ba-Ca-Cu-O(F), discovered in 2003, the critical temperature for which is 138 K. Moreover, at a pressure of 400 kbar, the same compound is a superconductor at temperatures up to 166 K.
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Phase transition to the superconducting state
The temperature range of transition to the superconducting state for pure samples does not exceed thousandths of a Kelvin and therefore a certain value of Tc - the temperature of transition to the superconducting state - makes sense. This value is called the critical transition temperature. The width of the transition interval depends on the heterogeneity of the metal, primarily on the presence of impurities and internal stresses. The currently known temperatures Tc vary from 0.0005 K for magnesium (Mg) to 23.2 K for the intermetallic compound of niobium and germanium (Nb3Ge, in film) and 39 K for magnesium diboride (MgB2) for low-temperature superconductors (Tc below 77 K , boiling point of liquid nitrogen), to approximately 135 K for mercury-containing high-temperature superconductors. Currently, the HgBa2Ca2Cu3O8+d (Hg−1223) phase has the highest known value of the critical temperature - 135 K, and at an external pressure of 350 thousand atmospheres the transition temperature increases to 164 K, which is only 19 K lower than the minimum temperature recorded under natural conditions at surface of the Earth. Thus, superconductors in their development have gone from metallic mercury (4.15 K) to mercury-containing high-temperature superconductors (164 K).
The transition of a substance to the superconducting state is accompanied by a change in its thermal properties. However, this change depends on the type of superconductors in question. Thus, for type I superconductors in the absence of a magnetic field at the transition temperature Tc, the heat of transition (absorption or release) goes to zero, and therefore suffers a jump in heat capacity, which is characteristic of a type II phase transition. This temperature dependence of the heat capacity of the electronic subsystem of a superconductor indicates the presence of an energy gap in the distribution of electrons between the ground state of the superconductor and the level of elementary excitations. When the transition from the superconducting state to the normal state is carried out by changing the applied magnetic field, then heat must be absorbed (for example, if the sample is thermally insulated, then its temperature decreases). And this corresponds to a phase transition of the 1st order. For type II superconductors, the transition from the superconducting to the normal state under any conditions will be a phase transition of type II.
Meissner effect
An even more important property of a superconductor than zero electrical resistance is the so-called Meissner effect, which consists in the superconductor pushing out a magnetic flux rotB = 0. From this experimental observation, it is concluded that there are continuous currents inside the superconductor, which create an internal magnetic field that is opposite to the external applied magnetic field and compensates for it.
A sufficiently strong magnetic field at a given temperature destroys the superconducting state of the substance. A magnetic field with intensity Hc, which at a given temperature causes a transition of a substance from a superconducting state to a normal state, is called a critical field. As the temperature of the superconductor decreases, the value of Hc increases. The dependence of the critical field on temperature is described with good accuracy by the expression
where Hc0 is the critical field at zero temperature. Superconductivity also disappears when an electric current with a density greater than the critical one is passed through the superconductor, since it creates a magnetic field greater than the critical one.
London moment
The rotating superconductor generates a magnetic field precisely aligned with the axis of rotation, the resulting magnetic moment is called the “London moment.” It was used, in particular, in the Gravity Probe B scientific satellite, where the magnetic fields of four superconducting gyroscopes were measured to determine their axes of rotation. Since the rotors of gyroscopes were almost perfectly smooth spheres, using the London moment was one of the few ways to determine their axis of rotation.
Applications of Superconductivity
The phenomenon of superconductivity is used to produce strong magnetic fields, since there is no heat loss when strong currents pass through a superconductor, creating strong magnetic fields. However, due to the fact that the magnetic field destroys the state of superconductivity, so-called so-called magnetic fields are used to obtain strong magnetic fields. Type II superconductors, in which the coexistence of superconductivity and a magnetic field is possible. In such superconductors, a magnetic field causes the appearance of thin threads of normal metal that penetrate the sample, each of which carries a magnetic flux quantum. The substance between the threads remains superconducting. Since there is no full Meissner effect in a type II superconductor, superconductivity exists up to much higher values of the magnetic field Hc2.
There are photon detectors based on superconductors. Some use the presence of a critical current, they also use the Josephson effect, Andreev reflection, etc. Thus, there are superconducting single-photon detectors (SSPD) for recording single photons in the IR range, which have a number of advantages over detectors of a similar range (PMTs, etc.) using other registration methods.
Vortexes in type II superconductors can be used as memory cells. Some magnetic solitons have already found similar applications. There are also more complex two- and three-dimensional magnetic solitons, reminiscent of vortices in liquids, only the role of current lines in them is played by the lines along which elementary magnets (domains) are lined up.
Electrons in metals
The discovery of the isotope effect meant that superconductivity was likely caused by interactions between conduction electrons and atoms in the crystal lattice. To figure out how this leads to superconductivity, we need to look at the structure of the metal. Like all crystalline solids, metals consist of positively charged atoms arranged in space in a strict order. The order in which the atoms are placed can be compared to a repeating pattern on wallpaper, but the pattern must repeat in three dimensions. Conduction electrons move among the atoms of the crystal at speeds ranging from 0.01 to 0.001 the speed of light; their movement is electric current.
The content of the article
SUPERCONDUCTIVITY, a state into which some solid electrically conductive substances transform at low temperatures. Superconductivity has been discovered in many metals and alloys and in a growing number of semiconductor and ceramic materials. Two of the most surprising phenomena observed in the superconducting state of matter are the disappearance of electrical resistance in the superconductor and the expulsion of magnetic flux ( cm. below) from its volume. The first effect was interpreted by early researchers as evidence of infinitely large electrical conductivity, hence the name superconductivity.
The disappearance of electrical resistance can be demonstrated by exciting an electric current in a ring of superconducting material. If the ring is cooled to the required temperature, then the current in the ring will exist indefinitely even after the current source that caused it is removed. Magnetic flux is a set of magnetic lines of force that form a magnetic field. While the field strength is below a certain critical value, the flux is pushed out of the superconductor, which is shown schematically in Fig. 1.
A solid that conducts electricity is a crystal lattice in which electrons can move. The lattice is formed by atoms arranged in a geometrically correct order, and the moving electrons are electrons from the outer shells of the atoms. Since the flow of electrons is an electric current, these electrons are called conduction electrons. If the conductor is in a normal (non-superconducting) state, then each electron moves independently of the others. The ability of any electron to move and therefore maintain an electric current is limited by its collisions with the lattice as well as with impurity atoms in the solid. For an electron current to exist in a conductor, a voltage must be applied to it; this means that the conductor has electrical resistance. If the conductor is in a superconducting state, then the conduction electrons combine into a single macroscopically ordered state, in which they behave as a “collective”; The entire “team” also reacts to external influences. Collisions between electrons and the lattice become impossible, and the current, once generated, will exist in the absence of an external current source (voltage). The superconducting state occurs abruptly at a temperature called the transition temperature. Above this temperature, the metal or semiconductor is in a normal state, and below it - in a superconducting state. The transition temperature of a given substance is determined by the relationship between two “opposite forces”: one tends to order the electrons, and the other tends to destroy this order. For example, the tendency towards ordering in metals such as copper, gold and silver is so small that these elements do not become superconductors even at temperatures only a few millionths of a kelvin above absolute zero. Absolute zero (0 K, –273.16° C) is the lower temperature limit at which a substance loses all its heat. Other metals and alloys have transition temperatures ranging from 0.000325 to 23.2 K ( see table). In 1986, superconductors were created from ceramic materials with unusually high transition temperatures. Thus, for ceramic samples YBa 2 Cu 3 O 7 the transition temperature exceeds 90 K.
Physicists call the superconducting state a macroscopic quantum mechanical state. Quantum mechanics, which is usually used to describe the behavior of matter on a microscopic scale, is here applied on a macroscopic scale. It is precisely the fact that quantum mechanics here allows us to explain the macroscopic properties of matter that makes superconductivity such an interesting phenomenon.
Opening.
A lot of information about a metal comes from the relationship between external voltage and the current it causes. Generally speaking, this relation has the form of equality V/I = R, Where V- voltage, I– current, and R- electrical resistance. According to this law (Ohm's law), electric current is proportional to voltage at any value R, which is the proportionality coefficient.
Resistance is usually independent of current, but is dependent on temperature. Having obtained liquid helium in 1908, G. Kamerlingh-Onnes from the University of Leiden (Netherlands) began to measure the resistance of pure mercury immersed in liquid helium and discovered (1911) that at liquid helium temperatures the resistance of mercury drops to zero. It was later discovered that many other metals and alloys also become superconducting at low temperatures.
The next important discovery was made in 1933 by the German physicist W. Meissner and his collaborator R. Ochsenfeld. They discovered that if a cylindrical sample is placed in a longitudinal magnetic field and cooled below the transition temperature, it completely expels the magnetic flux. The Meissner effect, as this phenomenon was called, was an important discovery because it made it clear to physicists that superconductivity is a quantum mechanical phenomenon. If superconductivity consisted only in the disappearance of electrical resistance, then it could be explained by the laws of classical physics.
PROPERTIES OF SUPERCONDUCTORS
In the physical literature, substances or materials that, under different conditions, can be in a superconducting or non-superconducting state are often called superconductors. The same simple (consisting of identical atoms) metal, alloy or semiconductor can be superconducting in some temperature ranges or external magnetic fields; at temperatures or fields of higher critical values, it is an ordinary (usually called normal) conductor.
After the discovery of the Meissner effect, a large number of experiments were performed with superconductors. Among the properties studied were:
1) Critical magnetic field - the field value above which the superconductor is in a normal state. Critical fields usually range from several tens of gauss to several hundred thousand gauss, depending on the superconductor and its metallophysical state. The critical field of a given superconductor varies with temperature, decreasing as it increases. At the transition temperature, the critical field is zero, and at absolute zero it is maximum (Fig. 2).
2) Critical current - the maximum direct current that a superconductor can withstand without losing the superconducting state. Like the critical magnetic field, the critical current strongly depends on temperature, decreasing as it increases.
3) Penetration depth - the distance to which the magnetic flux penetrates into the superconductor. The penetration depth turns out to be a function of temperature and varies in different materials: from 3H 10 –6 to 2H 10 –5 cm. The magnetic flux is pushed out of the superconductor by currents circulating in the surface layer, the thickness of which is approximately equal to the penetration depth.
To understand why the magnetic flux is pushed out, i.e. what causes the Meissner effect, we need to remember that all physical systems tend to a state with minimal energy. A magnetic field has some energy. A superconductor's energy increases in a magnetic field. But it decreases again due to the fact that currents arise in the surface layer of the superconductor. These currents create a magnetic field that compensates for the field applied from outside. The energy of a superconductor is higher than in the absence of an external magnetic field, but lower than in the case when the field penetrates inside it.
Complete expulsion of magnetic flux is not energetically beneficial for all superconductors. In some materials, a state of minimum energy in a magnetic field is achieved if some of the magnetic flux lines partially penetrate the material, forming a mosaic of superconducting regions where there is no magnetic field and normal regions where there is one.
4) Coherence length - the distance over which electrons interact with each other, creating a superconducting state. Electrons within the coherence length move in concert - coherently (as if “in step”). The coherence length for different superconductors varies from 5×10–7 to 10–4 cm. The existence of large coherence lengths (much larger than the atomic dimensions of the order of 10–8 cm) is associated with the unusual properties of superconductors.
5) Specific heat capacity - the amount of heat required to increase the temperature of 1 g of a substance by 1 K. The specific heat capacity of a superconductor increases sharply near the temperature of transition to the superconducting state, and decreases quite quickly with decreasing temperature. Thus, in the transition region, to increase the temperature of a substance in the superconducting state, more heat is required than in the normal state, and at very low temperatures, the opposite is true. Since specific heat capacity is determined primarily by conduction electrons, this phenomenon indicates that the state of the electrons is changing.
THEORIES OF SUPERCONDUCTIVITY
Before 1957, most attempts to explain experimental data were phenomenological in nature: they were based on artificial assumptions or loose modifications of existing theories and aimed at achieving agreement with experiment. An example of attempts of the first type is the two-fluid model, which postulates that at the transition temperature, some of the conduction electrons acquire the ability to move without experiencing resistance. This model explains the temperature dependence of the critical field, critical current and penetration depth, but does not provide anything for a physical understanding of the phenomenon itself, because does not explain such partial superconductivity.
Progress was made in 1935, when theoretical physicists, brothers F. and G. London, proposed to consider superconductivity as a macroscopic quantum effect. (Previously, only quantum effects were known that were observed on atomic scales - on the order of 10 -8 cm.) The Londons modified the classical equations of electromagnetism in such a way that they resulted in the Meissner effect, infinite conductivity and limited penetration depth. In the early 1950s, A. Pippard from the University of Cambridge showed that such a quantum state is in fact macroscopic, covering distances up to 10 –4 cm, i.e. 10,000 times the atomic radius.
While these efforts were important, they did not get to the heart of the fundamental interaction that drives superconductivity. Some indications of the nature of this interaction appeared in the early 1950s, when it was discovered that the temperature of the superconducting transition of metals made from different isotopes of the same element is not the same. It turned out that the higher the atomic mass, the lower the transition temperature. (Isotopes of the same element have the same number of electrons, but different nuclear masses.) The isotope effect indicated that the transition temperature depends on the mass of the atoms of the crystal lattice and, therefore, superconductivity is not a purely electronic effect.
Electrons in metals.
The discovery of the isotope effect meant that superconductivity was likely caused by interactions between conduction electrons and atoms in the crystal lattice. To figure out how this leads to superconductivity, we need to look at the structure of the metal. Like all crystalline solids, metals consist of positively charged atoms arranged in space in a strict order. The order in which the atoms are placed can be compared to a repeating pattern on wallpaper, but the pattern must repeat in three dimensions. Conduction electrons move among the atoms of the crystal at speeds ranging from 0.01 to 0.001 the speed of light; their movement is electric current.
Bardeen–Cooper–Schrieffer (BCS) theory.
In 1956 L. Cooper from the University of St. Illinois showed that if electrons are attracted to each other, then, no matter how weak the attraction, they must “condense” into a bound state. It can be assumed that this bound state is the sought-after superconducting state. As Cooper imagined, such attraction is possible between two electrons and should lead to the formation of bound pairs (called Cooper pairs) moving in the crystal lattice.
But back in 1950, G. Froelich suggested that electrons can be attracted to each other due to interaction with lattice atoms. This attraction mechanism is called electron-phonon interaction; it is as follows. An electron moving in a crystal lattice seems to distort it. This is due to the interaction between negatively charged electrons and positively charged lattice atoms. An electron moving through the lattice “brings together” its atoms. The second electron is then drawn into the "constricted region" under the increased influence of the positive charge. The energy of the first electron, expended on “lattice deformation,” is transferred without loss to the second member of the Cooper pair. Such a pair moves along the lattice, exchanging energy through the atoms of the lattice, but without losing its energy as a whole (Fig. 3).
This interaction is somewhat similar to the behavior of two heavy balls on a rubber membrane. When one ball rolls, it bends the membrane so that the second ball follows in its wake. Electrons, being similarly charged, unlike balls, repel each other. However, this mutual repulsion is strong only when the electrons are very close to each other, and quickly decreases as they move away. In interaction involving a lattice, or electron-phonon interaction, the electrons are quite distant from each other (at a distance of the order of 5×10 –7 –10 –4 cm). At such distances, electron repulsion is small compared to electron-phonon interaction, resulting in electrons being effectively attracted to each other. (A phonon is a quantum of vibrational energy of a crystal lattice.)
Until now, we have considered only one Cooper pair, whereas in reality there are approximately 10 20 Cooper pairs in 1 cm 3 of matter. It is easy to imagine that the lattice distortion created by one Cooper pair could disrupt the attraction in other pairs. In 1957, J. Bardeen, L. Cooper and J. Schrieffer proposed the so-called BCS (Bardeen – Cooper – Schrieffer) theory, for which they were awarded the 1972 Nobel Prize in Physics. According to this theory, pairs form a coherent state in which they all have the same momentum. These coherent electrons are said to be in a single quantum state; they form a so-called quantum, or superfluid, liquid. This coherence of electrons on a large scale is a remarkable macroscopic demonstration of quantum principles.
The BCS theory explains many of the properties of superconductors that we have already discussed. Electrons in a superconductor go into a collective state in such a way that their potential energy becomes minimal. Moving together, electrons are attracted to each other through the electron-phonon interaction mechanism, and the potential energy of the system turns out to be less than in the case of two electrons that do not attract each other. A superconductor in such a collective state is able to counteract the energy-increasing effects of a current or magnetic field; This implies the temperature dependence of the critical current and field. Above the transition temperature, the electrons have too much thermal energy and become "excited", i.e. transition from a lower-energy superconducting state to a normal, higher-energy state.
The isotopic effect is explained by the fact that in lighter isotopes the lattice is “perturbed” with less energy. The lattice of heavier isotopes is more difficult to deform, and therefore the transition to superconductivity occurs at lower temperatures. The BCS theory also explains why good conductors such as copper and gold are not superconductors. Conduction electrons in these substances easily pass through the atomic lattice, almost without interacting with it. This makes such materials good electrical conductors because they lose little energy due to lattice scattering. To achieve a superconducting state, a strong interaction between lattice atoms and electrons is necessary. For this reason, very good conductors of electricity are usually not superconductors.
Superconductors of the 1st and 2nd kind.
Based on their behavior in magnetic fields, superconductors are divided into type 1 and type 2 superconductors. Type 1 superconductors exhibit those ideal properties that have already been discussed. In the presence of a magnetic field, currents arise in the surface layer of the superconductor, which completely compensate for the external field in the thickness of the sample. If the superconductor has the shape of a long cylinder and is in a field parallel to its axis, then the penetration depth can be of the order of 3×10–6 cm. When the critical field is reached, superconductivity disappears and the field completely penetrates into the material. Critical fields for type 1 superconductors usually range from 100 to 800 Gas. Although type 1 superconductors have a shallow penetration depth, they have a large coherence length - on the order of 10 -4 cm.
Type 2 superconductors are characterized by a large penetration depth (about 2×10–5 cm) and a short coherence length (5×10–7 cm). In the presence of a weak magnetic field (less than 500 Gauss), all magnetic flux is pushed out of the type 2 superconductor. But higher N s 1 – the first critical field – the magnetic flux penetrates the sample, although to a lesser extent than in the normal state. This partial penetration persists until the second critical field - N s 2, which can exceed 100 kGs. With fields large N s 2, the flow penetrates completely and the substance becomes normal. The characteristics of various superconductors are presented in the table.
| CRITICAL TEMPERATURES AND FIELDS | |||
| Materials | Critical temperature, K | Critical fields (at 0 K), G | |
| Type 1 superconductors | |||
| Rhodium | 0,000325 | 0,049 | |
| Titanium | 0,39 | 60 | |
| Cadmium | 0,52 | 28 | |
| Zinc | 0,85 | 55 | |
| Gallium | 1,08 | 59 | |
| Thallium | 2,37 | 180 | |
| Indium | 3,41 | 280 | |
| Tin | 3,72 | 305 | |
| Mercury | 4,15 | 411 | |
| Lead | 7,19 | 803 | |
| Type 2 superconductors | Hc 1 | Hc 2 | |
| Niobium | 9,25 | 1735 | 4040 |
| Nb3Sn | 18,1 | – | 220 000 |
| Nb3Ge | 23,2 | – | 400 000 |
| Pb 1 Mo 5.1 S 6 | 14,4 | – | 600 000 |
| Yba 2 Cu 3 O 7 | 90–100 | 1000* | 1 000 000* |
| * Extrapolated to absolute zero. | |||
Josephson effect.
In 1962, B. Josephson, a graduate student at the University of Cambridge, thinking about what would happen if two superconductors were brought closer to a distance of several angstroms, suggested that Cooper pairs should, due to the “tunneling” effect, move from one superconductor to another at zero voltage.
Two remarkable effects were predicted. Firstly, a superconducting (non-dissipative) current can flow through a tunnel superconducting contact (a junction consisting of two superconductors separated by a dielectric layer). The critical value of this current depends on the external magnetic field. Secondly, if the current through the contact exceeds the critical junction current, then the contact becomes a source of high-frequency electromagnetic radiation. The first of these effects is called the stationary Josephson effect, the second - non-stationary. Both effects are clearly observed experimentally. In particular, oscillations of the maximum superconducting current through the junction were observed with increasing magnetic field. If the current specified by an external source exceeds a critical value, then a voltage appears at the junction V, periodically depending on time. The frequency of voltage oscillations depends on how much the current through the contact exceeds its critical value.
Of course, it is impossible to bring two superconductors closer to a distance of several angstroms. Therefore, in the experiments, a thin layer of superconducting material, such as aluminum, was sputtered onto the substrate, then it was oxidized from the surface to a depth of several angstroms, and another layer of aluminum was sputtered on top. Recall that aluminum oxide is a dielectric. Such a “sandwich” is equivalent to two superconductors located at a distance of several angstroms from each other.
The Josephson effect is caused by the phase relationships between electrons in the superconducting state. It was said above that the essence of the superconducting state is the coherent movement of Cooper pairs through the atomic lattice. The coherence of Cooper pairs in a superconductor is determined by the fact that pairs of electrons move “in phase.” Cooper pairs of two different superconductors move “out of phase.” Thus, each soldier of a marching company keeps pace with every other soldier in his company, but not in step with the soldiers of the other company. If two superconductors are brought closely together, Cooper pairs can tunnel through the gap between them. During tunneling, the phase of the Cooper pair changes. If the change is such that the Cooper pair begins to keep pace with the pairs in the second superconductor, then tunneling is possible. This is what happens in the stationary Josephson effect. The magnitude of the magnetic field determines the phase shift acquired by the tunneling pairs.
The transient Josephson effect occurs when the current through the junction exceeds the critical value for the steady-state Josephson effect. A voltage develops between the two superconductors, which causes the phases in the two superconductors to change over time. This in turn causes the tunnel current to oscillate (with a change in its direction) in accordance with changes in the phase difference in the two superconductors.
APPLICATIONS
From 1911 to 1986, many superconducting metals and alloys were investigated, but the highest measured transition temperature was 23.2 K. Cooling to this temperature required expensive liquid helium (4 He). Therefore, the most successful applications of superconductivity have remained at the level of laboratory experiments, which do not require large quantities of liquid helium.
At the end of 1986, K. Müller (Switzerland) and J. Bednorz (Germany), working at the IBM research laboratory in Zurich, discovered that a ceramic conductor built from lanthanum, barium, copper and oxygen atoms has a transition temperature to the superconducting state equal to 35 K. Soon, research groups around the world produced ceramic materials with a transition temperature of 90 to 100 K, which are capable of remaining superconductors (type 2, cm. higher) in magnetic fields up to 200 kG.
Ceramic superconductors are very promising for large-scale applications, mainly because they can be studied and used when cooled with relatively inexpensive liquid nitrogen.
Laboratory applications.
The first industrial application of superconductivity was the creation of superconducting magnets with high critical fields. Affordable superconducting magnets made it possible to obtain magnetic fields above 100 kG by the mid-1960s, even in small laboratories. Previously, creating such fields using conventional electromagnets required very large amounts of electricity to maintain electric current in the windings and huge amounts of water to cool them.
The next practical application of superconductivity relates to the technology of sensitive electronic devices. Experimental samples of devices with a Josephson contact can detect voltages of the order of 10–15 W. Magnetometers capable of detecting magnetic fields of the order of 10–9 Gauss are used in the study of magnetic materials, as well as in medical magnetocardiographs. Extremely sensitive detectors of gravity variations can be used in various fields of geophysics.
Superconductivity techniques and especially Josephson contacts are having an increasing impact on metrology. Using Josephson contacts, the 1 V standard was created. A primary thermometer was also developed for the cryogenic region, in which sharp transitions in certain substances are used to obtain reference (constant) temperature points. The new technique is used in current comparators, RF power and absorption coefficient measurements, and frequency measurements. It is also used in fundamental research, such as measuring the fractional charges of atomic particles and testing the theory of relativity.
Superconductivity will be widely used in computer technology. Here, superconducting elements can provide very fast switching times, negligible power losses when using thin-film elements, and high volumetric circuit packing densities. Prototypes of thin-film Josephson contacts are being developed in circuits containing hundreds of logic and memory elements.
Industrial applications.
The most interesting potential industrial applications of superconductivity involve the generation, transmission and use of electrical energy. For example, a superconducting cable a few inches in diameter can carry the same amount of electricity as a huge power transmission line network, with very little or no loss. The cost of insulating and cooling the cryoconductors must be offset by the efficiency of energy transfer. With the advent of ceramic superconductors cooled by liquid nitrogen, power transmission using superconductors becomes economically very attractive.
Another possible application of superconductors is in powerful current generators and small electric motors. Windings of superconducting materials could create enormous magnetic fields in generators and electric motors, making them significantly more powerful than conventional machines. Prototypes have long been created, and ceramic superconductors could make such machines quite economical. The possibilities of using superconducting magnets for storing electricity, in magnetohydrodynamics and for producing thermonuclear energy are also being considered.
Engineers have long wondered how the enormous magnetic fields created by superconductors could be used to maglev trains (magnetic levitation). Due to the mutual repulsion forces between the moving magnet and the current induced in the guide conductor, the train would move smoothly, without noise or friction, and would be capable of reaching very high speeds. Experimental maglev trains in Japan and Germany have reached speeds close to 300 km/h.
To do this, we may have to remember a few dates and start with 1911, when the Dutch physicist Kamerlingh-Onkes discovered the new phenomenon of superconductivity at the Leiden Laboratory. Then he was the first to achieve ultra-low temperatures and turn helium into liquid at minus 269 degrees. Finally, it became possible to cool substances in liquid helium and study their properties in a completely new, now accessible temperature range.
At that time, many believed (Onnes also shared the same opinion) that as one approaches -273 degrees, the electrical resistance of anyone should drop to zero. How tempting it was to finally check it out! But confirmation did not work. Maybe impurities are to blame? Onnes found mercury to be a suitable metal that could be examined in a very pure state. And indeed - as predicted by the electronic theory of metals - the resistance of mercury naturally decreased with decreasing temperature. Everything went fine until four degrees, when suddenly the resistance completely disappeared. It disappeared suddenly, all at once - abruptly.
However, Omnes took this quite calmly. He took this as confirmation of his theory of electrical resistance and called the new state of mercury he found “superconducting.” But it soon became clear that the paradoxical jump in resistance to zero could not be explained by any theory and that Onnes discovered something completely different from what he had expected.
What could have changed in the metal, why at a certain temperature (Onnes called it critical) nothing prevents the electrons from moving, why do they stop interacting with the atoms of the crystal lattice, or, as physicists say, stop being scattered by lattice vibrations?
Or maybe the resistance of the substance still remains, it just becomes so small that it cannot even be measured? Both Onnes himself and many experimenters tried to “catch” this residual resistance. They used the most sensitive methods to estimate the value of resistance from the attenuation of electric current in a superconducting ring. These experiments continued until very recently and culminated in the famous Collins experiment, where a superconducting lead ring with an electric current was preserved in liquid helium for about three years.
The most sensitive methods did not detect a decrease in current. This means not just good electrical conductivity, but superconductivity. There was no need to continue the experiment: it showed that the “resistance” of a superconductor is at least a billion times less than that of pure copper.
22 years passed before a second, no less astonishing discovery was made. It turned out that superconductivity is not only “ideal conductivity”, but also “ideal diamagnetism”. Let us recall that diamagnetic substances are substances that are “at odds” with a magnetic field. Placed in a magnetic field, they tend to displace it from themselves and occupy a position in space where the field strength is minimal. As an ideal diamagnetic, a superconductor does not tolerate the slightest magnetic field within itself. So, back in 1933, it became clear that zero resistance and zero magnetic field are two properties of the superconducting state.
Gradually, work on superconductivity began to unfold in all major centers of Europe and America. In the largest ones - because only the most powerful scientific institutions could afford to maintain expensive refrigeration facilities and helium liquefaction plants.
But neither the high cost nor the shortage of liquid helium prevented physicists from accumulating a large amount of factual material over the years - discovering hundreds of new superconductors and discovering a whole series of completely unexpected effects. We already know about a thousand superconducting substances - elements, compounds, alloys. Among them are over twenty elements of Mendeleev’s periodic table, up to technetium, a metal that does not exist on Earth under natural conditions (it is produced artificially in nuclear reactors). It turned out that superconductivity is possessed by metal alloys and inorganic compounds consisting of superconducting elements and - what is most surprising - not containing them. For a long time, the championship in the highest critical temperature was held by niobium nitride (-259 degrees), then superconductivity was discovered at -256 degrees in vanadium silicide, and in 1954 a record high critical temperature was recorded: -254.8 degrees in niobium stannide (alloy niobium with tin).
Based on some properties, mainly magnetic, superconducting substances began to be divided into superconductors of the first and second kind. All substances with high critical temperatures turned out to be type II superconductors. They also showed other important properties: high values of the critical magnetic field and critical current density. What does it mean? It was known: superconductivity can be “destroyed” not only by increasing the temperature above the critical temperature, but also by applying a magnetic field. So, samples of these compounds remained superconducting, even if currents with a density of up to a million amperes per square centimeter of cross-section were passed through them in an ultra-strong magnetic field.
During those same years, superconductivity was under intense attack from another direction. There were no complaints here about the lack of helium, or about the complexity and high cost of the equipment. The theorists faced other difficulties - mathematical ones. Who hasn't taken on the solution to the mystery of superconductivity? It was not until 1957 that the barriers were finally overcome.
Discovery of superconductivity
So, the general theory of superconductivity has emerged. Its main idea is this. Particles of the same sign must - according to Coulomb's law - repel each other. This law, of course, is also observed in superconductors. But in addition to this interaction, it turns out that there can be something else in a metal - a weak attraction that arises between electrons through an intermediate medium. This medium is the metal lattice itself, or, more precisely, its vibrations. And so, if conditions arise when this attraction becomes greater than the repulsive forces, superconductivity occurs.
Now no one doubts that the theory, basically, correctly explains the nature of superconductivity. But does this mean that all problems have been solved? Ask theorists: “Why does tin have a critical temperature of 3.7 degrees, and niobium 9.2?” Alas, theory still succumbs to such important questions...
The usual path in physics: the phenomenon was discovered - explained - learned to use. Most often, the development of theory and the development of methods of application go in parallel. Of course, in such an unusual area, far from everyday life, as superconductivity, the word “application” must be understood somewhat differently than usual - these are not tractors or washing machines. To apply means to use unique effects and make them “work”. Let at first only in the laboratory, even without noisy successes and sensations.
What if we try to make a superconducting magnet? - this question arose back in the twenties of the last century. It is known that the strongest magnetic fields are created with the help of electromagnets. Fields with a strength of up to 20 thousand oersteds can be obtained quite successfully using this method using relatively inexpensive installations. And if you need stronger fields - one hundred or more thousand oersteds? The power of the magnets increases to millions of watts. They need to be powered through special substations, and water cooling of the magnet requires the consumption of thousands of liters of water per minute.
Magnetic field - electric current - resistance are connected in a single chain. How tempting it would be, instead of these bulky, complex and expensive devices, to make a miniature coil of superconducting wire, place it in liquid helium and, powering it from a simple battery, obtain super-strong magnetic fields. This idea was realized much later - only when new materials with high critical fields and currents were discovered: first niobium, then an alloy of niobium with zirconium and titanium. And finally, niobium - tin. In many laboratories around the world, portable superconducting magnets are already in use, producing fields of about 100 thousand oersteds. And despite the high cost of liquid helium, such magnets are much more profitable than conventional ones.
Applications of Superconductivity
Strong magnetic fields are just one of many areas of possible and partly realized use of superconductivity. The most precise instruments of physical experiment - superconducting galvanometers and radiation detectors, resonators with superconducting coating for microwave technology and for linear accelerators of heavy particles, magnetic lenses for electronic devices, electric motors on superconducting bearings without friction, transformers and lossless transmission lines, magnetic screens, energy batteries, finally, miniature and high-speed “memory cells” of computers - this is a greatly reduced list of problems of today's applied superconductivity.
They are already saying that all classical electrical engineering can be “reinvented” if it is built not on ordinary conductors of electric current, but on superconducting materials.
Well, what if you dream a little? After all, in space there are ideal conditions for the operation of superconducting devices, ideal conditions for superconductivity. In the vacuum of outer space, a body can be heated from the outside only due to radiation (from the Sun, for example). If so, then any opaque screen is sufficient, and any object in space is completely thermally insulated. And since the elements of our imaginary machine themselves are superconducting and current flows through them without resistance, heat is not generated in them. There will be almost no liquid helium, which means the device will be able to operate indefinitely. Remember the experience of Collins, whose lead steering wheel retained current for almost three years.
Can you imagine that somewhere in orbit around the Moon there is a kind of cryogenic computer rotating, one serving entire sectors of the earth’s economy, science and transport? And what about superconducting magnets - maybe they will be the ones that will hold the plasma in the thermonuclear reactors of the future? Or cooled electrical cables, through which electrical energy can be transmitted over tens of thousands of kilometers without any loss?
Is this fantasy? Everything that has been said here is possible in principle. So it will be done. But when?
This is an excellent area both for imagination and for deep theoretical and experimental work.
In the meantime, the niobium-tin alloy remains the only substance with a maximum critical temperature of minus 254.8 degrees, and no one can understand for what advantages nature has distinguished it from thousands of other inorganic substances. No additions of other elements, no changes in the internal structure of this alloy could increase its critical temperature. The search for other, similar, double and triple alloys also turned out to be unsuccessful - no one has ever managed to rise above this enchanted number - minus 254.8 degrees. They began to say that, apparently, this temperature was not accidental; it was probably a limit that could not be crossed. All that remains is to find a theoretical justification for this fact, to find the reason why superconductivity cannot exist in metal systems at higher temperatures.
1. The phenomenon of superconductivity
2. Properties of superconductors
3. Application of superconductors
Bibliography
1. The phenomenon of superconductivity
Superconductors represent a special group of materials with high electrical conductivity. At low temperatures (currently at least below 18° K) certain metals and alloys acquire the ability to conduct current without any noticeable resistance; such solids are called superconductors.
This phenomenon has been known for a century; it was discovered in 1911 by Kamerlingh Onnes, who observed such a state in mercury at the temperature of liquid helium. Table 1 shows a list of some currently known superconductors and their transition temperatures to the superconducting state Tk. The transition usually occurs very abruptly: the resistance drops from its normal value to zero in the range of about 0.05 ° K.
Figure 1 - Change in electrical resistance in metals (M) and superconductors (M sv) in the low temperature range
With decreasing temperature, the electrical resistance of all metals decreases monotonically (Figure 1). However, there are metals and alloys in which the electrical resistance sharply drops to zero at a critical temperature - the material becomes a superconductor.
Superconductivity has been discovered in 30 elements and about 1000 alloys. Superconducting properties are exhibited by many alloys with the structure of ordered solid solutions and intermediate phases (o-phase, Laves phase, etc.). At ordinary temperatures these substances do not have high conductivity.
Table 1 – Superconductors and their transition temperatures to the superconducting state (ºK)
2. Properties of superconductors
The most general property of superconductors is the existence of a critical superconductivity temperature Tc, below which the electrical resistance of the substance becomes vanishingly small. According to recent estimates, the upper limit of the electrical resistance of a substance in the superconducting state (i.e. at a temperature below T k) is 10 -26 Ohm m.
Some elements can undergo allotropic transformations under the influence of high pressures (on the order of tens of thousands of atmospheres). The resulting crystallographic modifications (the so-called high-pressure phases) transform into a superconducting state when cooled, although at ordinary pressures these elements are not superconductors. For example, a superconductor is the TeII modification, formed at a pressure of 56,000 atmospheres, BiII (25 thousand atmospheres, T k= 3.9 K), BiIII (27 thousand atmospheres, T k=7.2 K). The high-pressure phases GaII and SbII remain superconductors even after the high pressure is removed, and at atmospheric pressure, the critical temperatures of the superconducting transition of these phases are 7.2 and 2.6 K, respectively. In the normal state, Be and Ga are not superconductors, but they become so upon deposition on substrates in the form of thin films. The appearance of superconductivity during film deposition from the vapor phase was also observed in Ce, Pr, Nd, Eu, and Yb.
It is characteristic that metals of subgroups IA, IB and IIA, which are good conductors of electricity at room temperature, are not superconductors (with the exception of beryllium in the thin-film state). Ferro- and antiferromagnetic elements are also not superconductors.
The superconducting properties of many elements, especially Mo, Ir and W, are very sensitive to the purity of the metal, which suggests that with the development of metal refining techniques, superconducting properties will be discovered in some other elements.
The transition from a normal state (with non-zero electrical resistance) to a superconducting state is observed not only in pure elements, but also in alloys and intermetallic compounds. Currently, more than a thousand superconductors are known. B. Matthias formulated rules connecting the existence of superconductivity with the valence Z.
1. Superconductivity exists only at 2< Z < 8.
2. In transition metals, their alloys and compounds at Z = 3, 5 or 7 the maximum temperatures of transition to the superconducting state are observed (see Figure 2).
3. For each given value Z certain crystal lattices are preferred (to obtain maximum T j) and T k increases rapidly with the atomic volume of the superconductor and decreases with increasing atomic mass.
Figure 2 - Presence of superconductivity and T to transition and simple metals
The most promising from the point of view of technical application are superconductors with a high critical temperature. Alloys and compounds of the transition metals niobium and vanadium have the highest Tc. These superconducting materials are divided into three groups: 1) alloys (solid solutions) with a body-centered cubic lattice - Nb-Ti, Nb-Zr. TK ~ 10 K and above; 2) compounds with a rock salt lattice, for example NbN and Nb (C, N), Tc ~ 18K; 3) compounds of niobium and vanadium with elements of the aluminum and silicon subgroups, having a crystal lattice of the β-W type and the stoichiometric formula A 3 B, where A -Nb or V, B is an element of the ShB or IVB subgroup, for example V 3 Si, Nb 3 Sn , Nb 3 (Al, Ge), T K ~ 21 K and higher.
The critical temperature of transition to the superconducting state and other superconducting characteristics, which will be discussed below, of A 3 B compounds are very sensitive to small deviations from stoichiometry, to the structural state of the sample (the presence of dispersed particles of other phases), defects in the crystal structure, and the degree of long-range order. Apparently, this explains the increase in Tc of the compounds Nb 8 Al, Nb 3 Ga, Nb 8 (Al, Ge) by several degrees after quenching at high temperatures and subsequent annealing. In particular, Tk of the Nb 3 Ge compound as a result of sharp quenching was increased from 11 to 17 K. On thin-film Nb 3 Ge samples obtained by sputtering, the following values were achieved: T k= 22 K and 23 K. Superconducting materials based on solid solutions have certain advantages over compounds of the A 3 B type due to their greater plasticity.
Substances in a superconducting state have specific magnetic properties. This is primarily manifested in the dependence of the critical temperature of superconductivity on the strength of the external magnetic field. The critical temperature is maximum in the absence of an external magnetic field and decreases with increasing field strength. At a certain external field strength N km, called critical Tk = 0. In other words, in fields equal to or greater than N km, the superconducting state does not arise in the substance at any temperature. This behavior of superconductors is illustrated by the H to (T) curve (Figure 3). Each point of this curve gives the value of the critical external field Hc at a given temperature T< Т к, вызывающего потерю сверхпроводимости. Эта кривая является кривой фазового перехода: сверхпроводящая фаза →нормальная фаза. В отсутствие магнитного поля этот переход является фазовым переходом второго рода. В присутствии внешнего магнитного поля - это переход первого рода.
Figure 3 - Dependence of the critical field of a superconductor on temperature
Another important magnetic property of superconductors is their diamagnetism. Inside a superconductor placed in a magnetic field, the induction is zero. If a superconductor is placed in a magnetic field at a temperature above the critical temperature, then when cooled below T k the magnetic field is “pushed out” of the superconductor and its induction in this case is also zero.
The destruction of superconductivity by an external magnetic field and the ideal diamagnetism of superconductors are associated with the fact that in order to maintain the superconducting state, the total momentum (kinetic energy) of electrons must be less than a certain value. Because of this, there is a certain limiting (critical) current density j c above which superconductivity breaks down and finite electrical resistance appears. The ideal diamagnetism of a superconductor is explained by the fact that an applied magnetic field induces currents on the surface of the superconductor that do not experience resistance. These currents circulate in such a way that the magnetic flux inside the superconductor is destroyed. Thus, the external magnetic field penetrates into the superconductor only to a very small depth (the so-called penetration depth) of the order of 10 -8 -10 -9 m. As the external magnetic field increases, the screening currents must increase in order to maintain the diamagnetism of the superconductor. If the external field is strong enough, the currents will reach a critical value and the substance will return to its normal state. The shielding currents disappear and the magnetic field penetrates the substance. The penetration depth of the magnetic field (at a constant field) increases with temperature and tends to infinity at T→ T k, which corresponds to the transition to the normal state.
Superconductors with a shallow penetration depth (sharp attenuation of the magnetic field near the surface) are called soft superconductors, or type I superconductors. There are also hard superconductors, or type II superconductors. Type II superconductors are characterized by higher values of critical fields and a larger width of the temperature region of transition to the superconducting state. For soft superconductors (tin, mercury, zinc, lead), the temperature range of transition to the superconducting state is about 0.05 K, while for hard superconductors (niobium, rhenium, compounds with the β-W structure), the temperature range of the superconducting transition is about 0. 5 K.
Introduction
Chapter 1 Discovery of the phenomenon of superconductivity
1.2 Superconducting substances
1.3 Meissner effect
1.4 Isotopic effect
Chapter 2 Theory of superconductivity
2.1 BCS theory
2.4 Formation of electron pairs
2.5 Effective interaction between electrons due to phonons
2.6 Canonical Bogolyubov transformation
2.7 Intermediate state
2.8 Type II superconductors
2.9 Thermodynamics of superconductivity
2.10 Tunnel contact and Josephson effect
2.11 Magnetic flux quantization (macroscopic effect)
2.12 Knight shift
2.13 High temperature superconductivity
Chapter 3. Application of superconductivity in science and technology
3.1 Superconducting magnets
3.2 Superconducting electronics
3.3 Superconductivity and energetics
3.4 Magnetic suspensions and bearings
Conclusion
Bibliography
Introduction
For most metals and alloys, at a temperature of about a few degrees Kelvin, the resistance abruptly goes to zero. This phenomenon, called superconductivity, was first discovered in 1911 by Kamerlingh Onnes. Substances with this phenomenon are called superconductors. In 1957, J. Bardeen, L. Cooper, J. Schrieffer developed a microscopic theory of superconductivity, which made it possible to fundamentally understand this phenomenon. The BCS theory explained the basic facts in the field of superconductivity (the absence of resistance, the dependence of Tc on the mass of the isotope, infinite conductivity (E = 0), the Meissner effect (B = 0), the exponential dependence of the electronic heat capacity near T = 0, etc.). A number of theoretical conclusions show good quantitative agreement with experiment. Many issues still need to be developed (distribution of superconducting metals in the periodic system, dependence of Tc on the composition and structure of superconducting compounds, the possibility of obtaining superconductors with the highest possible transition temperature, etc.). The successes of experimental and theoretical research have provided a real opportunity to begin work on mastering this physical phenomenon. For almost 100 years, developments have been going on in this area, new superconducting materials are being discovered, and the search for high-temperature superconductors is underway. In recent years, especially after the creation of the theory of superconductivity, technical superconductivity has been intensively developing.
Relevance. Today, superconductivity is one of the most studied areas of physics, a phenomenon that opens up serious prospects for engineering practice. Devices based on the phenomenon of superconductivity have become widespread; neither modern electronics, nor medicine, nor astronautics can do without them.
Target. Consider in more detail the phenomenon of superconductivity, its properties, practical application, study the BCS theory, and also find out the prospects for the development of this field of physics.
1) Find out what superconductivity is, the reasons for its occurrence and the conditions for the possible transition of a substance from a normal state to a superconducting state.
2) Explain the reasons influencing the destruction of the superconducting state.
3) Reveal the properties and applications of superconductors.
An object. The object of this course work is the phenomenon of superconductivity, superconductors.
Item. The subject is the properties of superconductors and their applications.
Practical use. The phenomenon of superconductivity is used to produce strong magnetic fields; superconductors are used in the creation of computers, for the construction of modulators, rectifiers, switches, persistors and persistrons, and measuring instruments.
Research methods. Analysis of scientific literature.
Chapter 1. Discovery of the phenomenon of superconductivity
1.1 First experimental facts
In 1911, in Leiden, the Dutch physicist H. Kamerlingh Onnes first observed the phenomenon of superconductivity. This problem was studied earlier; experiments showed that with decreasing temperature, the resistance of metals decreased. One of his first studies in the field of low temperatures was the study of the dependence of electrical resistance on temperature during an experiment with a mercury circuit. Mercury was then considered the purest metal that could be obtained by distillation. Studying the temperature variation of the electrical resistance of Hg, he discovered that at temperatures below 4.2 0 K, mercury practically loses its resistance. For this experiment, he used an apparatus (Fig. 1), which consisted of seven U-shaped vessels with a cross-section of 0.005 mm 2, connected inverted. This form of vessels was needed for free compression and expansion of mercury without breaking the continuity of the mercury thread. At points 1 and 2, current was supplied through tubes 3 and 4; at points 5 and 6, the voltage drop in sections of the mercury circuit was measured.
Figure 2 shows the results of his experiments with mercury. It should be noted that the temperature range in which the resistance decreased to zero is extremely narrow.
Rice. 2. Dependence of the resistance of platinum and mercury on temperature.
The graph shows that at a temperature of 4.2 0 K the electrical resistance of mercury suddenly disappeared. This state of a conductor in which its electrical resistance is zero is called superconductivity, and substances in this state are called superconductors. The transition of a substance to the superconducting state occurs in a very narrow temperature range (hundredths of a degree) and therefore it is believed that the transition occurs at a certain temperature Tc, called the critical temperature of the transition of the substance to the superconducting state.
Superconductivity can be observed experimentally in two ways:
1) by including a superconductor link in the general electrical circuit through which current flows. At the moment of transition to the superconducting state, the potential difference at the ends of this link becomes zero;
2) by placing a ring of superconductor in a magnetic field perpendicular to it. Having then cooled the ring below Tc, turn off the field. As a result, a continuous electric current is induced in the ring. The current circulates in such a ring indefinitely.
Kamerling - Onnes demonstrated this by transporting a superconducting ring with current flowing through it from Leiden to Cambridge. In a number of experiments, the absence of current attenuation in the superconducting ring was observed for about a year. In 1959, Collins reported that he observed no decrease in current for two and a half years. .
Experiments have shown that if a current is created in a closed loop from superconductors, then this current continues to circulate without an EMF source. Foucault currents in superconductors persist for a very long time and do not fade due to the lack of Joule heat (currents up to 300A continue to flow for many hours in a row). A study of the passage of current through a number of different conductors showed that the resistance of the contacts between superconductors is also zero. A distinctive property of superconductivity is the absence of the Hall phenomenon. While in ordinary conductors, under the influence of a magnetic field, the current in the metal is shifted, in superconductors this phenomenon is absent. The current in a superconductor is, as it were, fixed in its place.
Superconductivity disappears under the influence of the following factors:
1) increase in temperature;
As the temperature rises to a certain Tk, noticeable ohmic resistance almost suddenly appears. The transition from superconductivity to conductivity is steeper and more noticeable the more homogeneous the sample is (the steepest transition is observed in single crystals).
2) the action of a sufficiently strong magnetic field;
The transition from the superconducting state to the normal state can be accomplished by increasing the magnetic field at a temperature below the critical Tc. The minimum field Bc in which superconductivity is destroyed is called the critical magnetic field. The dependence of the critical field on temperature is described by the empirical formula:
where B 0 is the critical field extrapolated to absolute zero temperature. For some substances there appears to be a dependence on T to the first degree. If we begin to increase the external field strength, then at its critical value, superconductivity will collapse. The closer we get to the critical temperature point, the lower the external magnetic field strength must be to destroy the effect of superconductivity, and vice versa, at a temperature equal to absolute zero, the strength must be maximum in relation to other cases to achieve the same effect. This relationship is illustrated by the following graph (Fig. 3).
If we begin to increase the external field strength, then at its critical value, superconductivity will collapse. The closer we get to the critical temperature point, the lower the external magnetic field strength must be to destroy the effect of superconductivity, and vice versa, at a temperature equal to absolute zero, the strength must be maximum in relation to other cases to achieve the same effect. When a magnetic field acts on a superconductor, a special type of hysteresis is observed, namely if, by increasing the magnetic field, superconductivity is destroyed at (H - field strength, H to - increased field strength):
then, with a decrease in the field intensity, superconductivity will reappear under the field, varies from sample to sample and is usually 10% Hc.
3) a sufficiently high current density in the sample;
An increase in current strength also leads to the disappearance of superconductivity, that is, Tk decreases. The lower the temperature, the higher the maximum current strength ik at which superconductivity gives way to ordinary conductivity.
4) change in external pressure;
A change in external pressure p causes a shift in Tk and a change in the magnetic field strength, which destroys superconductivity.
1.2 Superconducting substances
Later it was found that not only mercury, but also other metals and alloys, the electrical resistance becomes zero when sufficiently cooled.
Niobium (9.22 0 K) has the highest critical temperature among pure substances, and iridium has the lowest (0.14 0 K). The critical temperature depends not only on the chemical composition of the substance, but also on the structure of the crystal itself. For example, gray tin is a semiconductor, and white tin is a metal that passes into the superconducting state at a temperature of 3.72 0 K. Two crystalline modifications of lanthanum (b-La and b-La) have different critical temperatures of transition to the superconducting state (for b -La T k =4.8 0 K, c-La T k =5.95 0 K). Therefore, superconductivity is not a property of individual atoms, but a collective effect associated with the structure of the entire sample.
Good conductors (silver, gold and copper) do not have this property, but many other substances that are very poor conductors under normal conditions do, on the contrary, do. It came as a complete surprise to the researchers and further complicated the explanation of this phenomenon. The bulk of superconductors are not pure substances, but their alloys and compounds. Moreover, an alloy of two non-superconducting substances can have superconducting properties. There are type I and type II superconductors.
Type I superconductors are pure metals; there are more than 20 of them in total. Among them there are no metals that are good conductors at room temperature, but, on the contrary, metals that have relatively poor conductivity at room temperature (mercury, lead, titanium, etc.).
Superconductors of the second type are chemical compounds and alloys, and these do not necessarily have to be compounds or alloys of metals, which in their pure form are superconductors of the first type. For example, the compounds MoN, WC, CuS are type II superconductors, although Mo, W, Cu and especially N, C and S are not superconductors. The number of type II superconductors is several hundred and continues to increase. .
For a long time, the superconducting state of various metals and compounds could be obtained only at very low temperatures, achievable with the help of liquid helium. By the beginning of 1986, the maximum observed value of the critical temperature was already 23 0 K.
1.3 Meissner effect
In 1933, Meissner and Ochsenfeld established that behind the phenomenon of superconductivity lies something more than ideal conductivity, that is, zero resistivity. They discovered that a magnetic field is pushed out of the superconductor regardless of whether the field is created by an external source or by a current flowing through the superconductor itself (Fig. 4). It turned out that the magnetic field does not penetrate into the thickness of the superconducting sample.
Figure 4. Pushing out a flux of magnetic induction from a superconductor.
At temperatures higher than the critical temperature of transition to the superconducting state, in a sample placed in an external magnetic field, as in any metal, the magnetic field induction inside is different from zero. If, without turning off the external magnetic field, the temperature is gradually reduced, then at the moment of transition to the superconducting state, the magnetic field will be pushed out of the sample and the magnetic field induction inside will become zero (B = 0). This effect was called the Meissner effect.
As is known, metals, with the exception of ferromagnets, have zero magnetic induction in the absence of an external magnetic field. This is due to the fact that the magnetic fields of elementary currents, which are always present in matter, are mutually compensated due to the complete randomness of their location.
Placed in an external magnetic field, they become magnetized, i.e. a magnetic field is “induced” inside. The total magnetic field of a substance introduced into an external magnetic field is characterized by a magnetic induction equal to the vector sum of the induction of the external and the induction of the internal magnetic fields, i.e. . In this case, the total magnetic field can be either greater or less than the magnetic field.
In order to determine the degree of participation of a substance in the creation of a magnetic field by induction, the ratio of induction values is found. The coefficient µ is called the magnetic permeability of a substance. Substances in which, when an external magnetic field is applied, the resulting internal field is added to the external one (µ > 1) are called paramagnets. At a coefficient >1, the external field in the sample decreases.
In diamagnetic substances (<1) наблюдается ослабление приложенного поля. В сверхпроводниках В=0, что соответствует нулевой магнитной проницаемости. В поверхностном слое металла возникает стационарный электрический ток, собственное магнитное поле которого противоположно приложенному полю и компенсирует его, что в результате и приводит к нулевому значению индукции в толще образца.
The existence of stationary superconducting currents is revealed in the following experiment: if a superconducting sphere is placed above a metal superconducting ring, then a continuous superconducting current is induced on its surface. Its occurrence leads to a diamagnetic effect and the emergence of repulsive forces between the ring and the sphere, as a result of which the sphere will float above the ring. The depth of field penetration into the sample is one of the main characteristics of a superconductor. Typically the penetration depth is approximately 100...400E. With increasing temperature, the depth of penetration of the magnetic field increases according to the law:
The simplest estimate of the depth of penetration of a magnetic field into a superconductor was given by the brothers Fritz and Hans London. Let us present this estimate. We will assume that we are dealing with fields that slowly change over time. Since superconductors are not ferromagnetic, we can neglect the difference between and and write the fundamental equations of electrodynamics in the form
Moreover, we will also neglect the difference between the partial and total derivatives with respect to time. Assuming that currents are created by the movement of only superconducting electrons, we will further write where is the concentration of such electrons. After differentiation with respect to time we get: The acceleration of the electron can be found from the equation if the effect of the magnetic field is neglected. Then
where the designation is introduced
Having differentiated the first equation (4) with respect to, excluding the quantities and from equations (4) and (5), we obtain
This equation is satisfied, but such a solution is not consistent with the Meissner effect, since there must be inside the superconductor. The extra solution was obtained because during the derivation the operation of differentiation with respect to time was used twice. To automatically eliminate this solution, the Londons introduced the hypothesis that in the last equation the derivative should be replaced by the vector itself. This gives
To determine the depth of penetration of the magnetic field into the superconductor, let us assume that the latter is limited by a plane on one side of it. Let's direct the axis inside the superconductor normal to its boundary. Let the magnetic field be parallel to the axis, so. Then
And equation (8) gives
The solution to this equation, which vanishes at, has the form
The integration constant gives the field on the surface of the superconductor. Over the length, the magnetic field decreases by a factor. The value is taken as a measure of the depth of penetration of the field into the metal.
To obtain a numerical estimate, we assume that for each metal atom there is one superconducting electron, assuming cm -3. then using formula (6) we find cm, which in order of magnitude coincides with the values obtained by direct measurements.
The surface layer of a superconductor has special properties associated with the nonzero magnetic field strength in it. These properties have a very significant impact on the production of superconductors with high critical fields.
A situation arises when surface currents, often called shielding currents, prevent the applied field from penetrating the magnetic flux into the sample. If the magnetic flux inside a substance in an external field is zero, then it is said to exhibit ideal diamagnetism. When the applied field density decreases to zero, the sample remains in its non-magnetized state. In another case, when a magnetic field is applied to the sample above the transition temperature, the final picture will change noticeably. For most metals (except ferromagnets), the relative magnetic permeability is close to unity. Therefore, the magnetic flux density inside the sample is almost equal to the flux density of the applied field. The disappearance of electrical resistance after cooling does not affect magnetization, and the distribution of magnetic flux does not change. If we now reduce the applied field to zero, then the magnetic flux density inside the superconductor cannot change; undamped currents appear on the surface of the sample, maintaining the magnetic flux inside. As a result, the sample remains magnetized all the time. Thus, the magnetization of an ideal conductor depends on the sequence of changes in external conditions.
The effect of pushing a magnetic field out of a superconductor can be explained on the basis of ideas about magnetization. If screening currents, which completely compensate the external magnetic field, impart a magnetic moment m to the sample, then the magnetization M is expressed by the relation:
where V is the volume of the sample. We can say that shielding currents lead to the appearance of magnetization corresponding to the magnetization of an ideal ferromagnet with a magnetic susceptibility equal to minus one.
The Meissner effect and the phenomenon of superconductivity are closely related and are a consequence of a general pattern, which was established by the theory of superconductivity, created more than half a century after the discovery of the phenomenon.
1.4 Isotopic effect
In 1950, E. Maxwell and C. Reynolds discovered the isotope effect, which was of great importance for the creation of the modern theory of superconductivity. A study of several superconducting isotopes of mercury showed that there is a relationship between the critical temperature of transition to the superconducting state and the mass of the isotopes. When the mass M of the isotope changed from 199.5 to 203.4, the critical temperature changed from 4.185 to 4.14 K. For this superconducting chemical element, a formula was established that is justified with sufficient accuracy:
where const has a specific value for each element.
The mass of an isotope is a characteristic of the crystal lattice, since the main contribution to it is made by metal ions. Mass determines many lattice properties. It is known that the frequency of lattice vibrations is related to mass:
Superconductivity, which is a property of the electronic system of a metal, turns out to be associated, due to the discovery of the isotope effect, with the state of the crystal lattice. Consequently, the occurrence of the superconductivity effect is due to the interaction of electrons with the metal lattice. This interaction is responsible for the resistance of the metal in its normal state. Under certain conditions, it should lead to the disappearance of resistance, that is, to the effect of superconductivity.
1.5 Prerequisites for the creation of the theory of superconductivity
The first theory that quite successfully described the properties of superconductors was the theory of F. London and G. London, proposed in 1935. The Londons in their theory were based on a two-fluid model of a superconductor. It was believed that when in a superconductor there are “superconducting” electrons with a concentration and “normal” electrons with a concentration, where is the total conductivity concentration). The density of superconducting electrons decreases with increasing and goes to zero at. When it tends to the density of all electrons. A current of superconducting electrons flows through the sample without resistance.
London, in addition to Maxwell's equations, obtained equations for the electromagnetic field in such a superconductor, from which its basic properties followed: the absence of resistance to direct current and ideal diamagnetism. However, due to the fact that the Londons' theory was phenomenological, it did not answer the main question of what “superconducting” electrons are. In addition, it had a number of other shortcomings, which were eliminated by V.L. Ginzburg and L.D. Landau.
In the Ginzburg-Landau theory, quantum mechanics was used to describe the properties of superconductors. In this theory, the entire set of superconducting electrons was described by a wave function of one spatial coordinate. Generally speaking, the wave function of electrons in a solid is a function of coordinates. By introducing the function, the coherent, consistent behavior of all superconducting electrons was established. Indeed, if all electrons behave in exactly the same way, in a consistent manner, then to describe their behavior, the same wave function is sufficient as to describe the behavior of one electron, i.e. functions of one variable.
Despite the fact that the Ginzburg-Landau theory, which was further developed in the works of A.A. Abrikosov, described many properties of superconductors, it could not provide an understanding of the phenomenon of superconductivity at the microscopic level.
This chapter discusses the discovery of the phenomenon of superconductivity, the first experimental facts, the first theories, as well as some properties of superconductors.
Analyzing the above, the following conclusions can be drawn:
1) This state of a conductor in which its electrical resistance is zero is called superconductivity, and substances in this state are called superconductors.
2) Foucault currents in superconductors persist for a very long time and do not fade due to the lack of Joule heat (currents up to 300A continue to flow for many hours in a row).
3) Superconductivity disappears under the influence of the following factors: an increase in temperature, the action of a sufficiently strong magnetic field, a sufficiently high current density in the sample, a change in external pressure.
4) The magnetic field is pushed out of the superconductor regardless of how this field is created - an external source or a current flowing through the superconductor itself.
5) There is a connection between the critical temperature of transition to the superconducting state and the mass of isotopes, which is called the isotope effect.
6) The isotopic effect indicated that lattice vibrations are involved in the creation of superconductivity.
Chapter 2. Theory of superconductivity
2.1 BCS theory
In 1957, Bardeen, Cooper and Schrieffer constructed a consistent theory of the superconducting state of matter (BCS theory). Long before Landau, the theory of superfluidity of helium II was created. It turned out that superfluidity is a macroscopic quantum effect. However, the transfer of Landau's theory to the phenomenon of superconductivity was hindered by the fact that helium atoms, having zero spin, obey Bose-Einstein statistics. Electrons, having half spin, obey the Pauli principle and Fermi-Dirac statistics. For such particles, Bose-Einstein condensation, which is necessary for the occurrence of superfluidity, is impossible. Scientists have suggested that electrons are grouped into pairs that have zero spin and behave like Bose particles. Regardless of them, in 1958 N.N. Bogolyubov developed a more advanced version of the theory of superconductivity.
The BCS theory refers to an idealized model in which the structural features of the metal are so far completely discarded. The metal is considered as a potential box filled with an electron gas that obeys Fermi statistics. Coulomb repulsion forces act between individual electrons, largely weakened by the field of the atomic cores. The isotope effect in superconductivity indicates the presence of interaction of electrons with thermal vibrations of the lattice (with phonons).
An electron moving in a metal deforms and polarizes the crystal lattice of the sample by electrical forces. The displacement of the lattice ions caused by this is reflected in the state of the other electron, since it now finds itself in the field of a polarized lattice, which has somewhat changed its periodic structure. Thus, the crystal lattice acts as an intermediate medium in electronic interactions, since with its help electrons realize attraction to each other. At high temperatures, sufficiently intense thermal motion pushes particles away from each other, effectively reducing the force of attraction. But at low temperatures, attractive forces play a very important role.
Two electrons repel each other if they are in empty space. In the environment, the force of their interaction is equal to:
where e is the dielectric constant of the medium. If the environment is such that<0, то одноименные заряды, в том числе и электроны, будут притягиваться. Кристаллическая решетка некоторых веществ является той средой, в которой выполняется это условие, а значит при определенных температурах возможно возникновение эффекта сверхпроводимости. Таким образом, эффект взаимного притяжения электронов не противоречит законам физики, так как происходим в некоторой среде.
Let's consider a metal at T = 0 0 K. Its crystal lattice undergoes “zero” vibrations, the existence of which is associated with the quantum-mechanical uncertainty relation. An electron moving in a crystal disrupts the vibration mode and transfers the lattice to an excited state. The return transition to the previous energy level is accompanied by the emission of energy, captured by another electron and exciting it. The excitation of the crystal lattice is described by sound quanta - phonons, therefore the process described above can be represented as the emission of a phonon by one electron and its absorption by another electron, while the crystal lattice plays an intermediate role as a transmitter. The exchange of phonons determines their mutual attraction.
At low temperatures, this attraction for a number of substances prevails over the Coulomb repulsive forces of electrons. In this case, the electronic system turns into a connected collective, and in order to excite it, the expenditure of some finite energy is required. The energy spectrum of the electronic system in this case will not be continuous - the excited state is separated from the ground state by an energy gap.
It has now been established that the normal state of a metal differs from the superconducting state in the nature of the energy spectrum of electrons near the Fermi surface. In the normal state at low temperatures, electronic excitation corresponds to the transition of an electron from an initially occupied state to (<к F) под поверхностью Ферми в свободное состояние к (>to F) above the Fermi surface. The energy required to excite such an electron-hole pair in the case of a spherical Fermi surface is equal to
Since k and k 1 can lie quite close to the Fermi surface, then.
The electronic system in a superconductor can be represented as consisting of bound pairs of electrons (Cooper pairs), and excitation as the breaking of the pair. The size of the electron pair is approximately ~10 -4 cm, the size of the lattice period is 10 -8 cm. That is, the electrons in the pair are located at a huge distance.
The most characteristic property of a metal in a superconducting state is that the excitation energy of a pair always exceeds a certain certain value 2D, which is called the pairing energy. In other words, there is a gap in the excitation energy spectrum on the low-energy side. For example, for the metals Hg, Pb, V, Nb, the value 2D corresponds to thermal energy at temperatures of 18 0 K, 29 0 K, 18 0 K and 30 0 K.
The magnitude of the pairing energy is measured directly experimentally: when studying the absorption of electromagnetic radiation, only radiation with a frequency ђш = 2Д is absorbed, when studying the exponential change in sound attenuation, etc.
If there is a gap in the energy spectrum, quantum transitions of the system will not always be possible. The electronic system will not be excited at low speeds, therefore, the movement of electrons will occur without friction, which means there is no resistance. At a certain critical current, the electronic system will be able to move to the next energy level and superconductivity will collapse.
2.2 Gap in the energy spectrum
The first indications of the existence of an energy gap were obtained from the exponential law of decay of the electronic heat capacity of a superconductor:
c es ~ g T k e - bTk / T ~ c ns e - bTk / T . (16)
The energy gap in superconductors is directly observed experimentally, and not only is the existence of the gap in the spectrum confirmed, but its magnitude is also measured. The transition of electrons through a thin non-conducting layer ~10E thick, separating the normal and superconducting films, was studied. In the presence of a barrier, there is a finite probability of an electron passing through the barrier. In a normal metal all energy levels are filled, up to the maximum e F , in a superconducting metal up to e F -D. In this case, the passage of current is impossible.
The presence of an energy gap in a superconductor leads to the absence of corresponding states between which a transition would occur. In order for the transition to occur, the system must be placed in an external electric field. In the field, the entire picture of levels shifts. The effect becomes possible if the applied external voltage becomes equal to D/e. The tunnel current appears at a finite voltage U, when eU is equal to the energy gap. The absence of a tunneling current at an arbitrarily low applied voltage is proof of the existence of an energy gap.
Currently, a number of methods have been developed to detect such a gap and measure its width. One of them is based on studying the absorption of electromagnetic waves in the far infrared region by metals. The idea of the method is as follows. If a stream of electromagnetic waves is directed onto a superconductor and their frequency u is continuously changed, then as long as the energy of the quanta V of this radiation remains less than the gap width E w (if there is one, of course), the radiation energy should not be absorbed by the superconductor. At the frequency зк, for which ђш к = Е ь, intense absorption of radiation should begin, increasing to its values in a normal metal. By measuring shk, you can determine the width of the gap E sh.
Experiments have fully confirmed the presence of a gap in the energy spectrum of conduction electrons in all known superconductors. As an example, the table shows the gap width E w at T = 0 0 K for a number of metals and the critical temperature of their transition to the superconducting state. From the data in this table it is clear that the gap E is very narrow ~ 10 -3 -10 -2 eV; There is a direct connection between the gap width and the critical transition temperature Tc: the higher Tc, the wider the gap Ec. theory
The BCS leads to the following approximate expression relating T k with E sh (0):
E sh (0) = 3.5 kT k, (17)
which is quite well confirmed by experience.
In the theory of superconductivity, most results were obtained for the isotropic model. Real metals are actually anisotropic, which is evident in many experiments. Under fairly broad assumptions, we can obtain the formula:
where is the unit vector in the direction of impulse p; and is the Fermi radius vector of the surface and the velocities on it. The magnitude depends on the direction. According to experimental data, the change. At the same time, the temperature dependence is the same for all directions, i.e. .
Table 1.
|
Substance |
|||||||
|
E sh (0),10 -3 eV |
|||||||
|
E = 3.5 kT k |
Anisotropy is already visible when comparing theoretical and experimental data for heat capacity. At low temperatures
where is the minimum gap, and according to the theoretical curve (for an isotropic model), where is some average gap. Therefore, as a rule, the theoretical curve at is lower than the experimental one.
There are various methods for more detailed determination of gap anisotropy. Thus, measuring the thermal conductivity of single-crystal single-core superconductors makes it possible to determine whether the minimum gap is located in the direction of the main axis or lies in the basal plane. The nature of the gap anisotropy can also be established from experiments with a tunnel contact if one of the superconductors is a single crystal. The most interesting results on anisotropy are obtained from experiments on sound absorption. If the frequency of sound is the binding energy of pairs, then at low temperatures absorption occurs only on excitations, i.e. proportionally. However, we must take into account that the mechanism of sound absorption is the inverse Cherenkov effect. This means that sound is absorbed only by those electrons whose velocity projection onto the direction of sound propagation coincides with the speed of sound, i.e. . But the speed of electrons in a metal is cm/sec, and the speed of sound is cm/sec; this means that, i.e. perpendicularly, in other words, sound is absorbed by electrons lying on the contour resulting from the intersection of the Fermi surface with a plane perpendicular. In view of this, low-temperature sound absorption is determined by the minimum value of the gap on this contour. By changing the direction of sound propagation, you can obtain fairly detailed information about the gap.
The anisotropy of the gap is also manifested in the fact that the change in thermodynamic quantities when defects are introduced into the superconductor is greater than for the isotropic model. For example, with a decrease compared to (for pure metal), i.e. proportional to the mean square anisotropy.
2.3 Gapless superconductivity
In the first years after the creation of the BCS theory, the presence of an energy gap in the electronic spectrum was considered a characteristic sign of superconductivity, but superconductivity without an energy gap is also known - gapless superconductivity.
As was first shown by A.A. Abrikosov and L.P. Gorkov, with the introduction of magnetic impurities, the critical temperature effectively decreases. Atoms of a magnetic impurity have spin, and therefore a spin magnetic moment. In this case, the spins of the pair appear to be in a parallel and antiparallel magnetic field of the impurity. With an increase in the concentration of atoms and magnetic impurities in a superconductor, an increasing number of pairs will be destroyed, and in accordance with this, the width of the energy gap will decrease. At a certain concentration n equal to 0.91n cr (n cr is the concentration value at which the superconducting state completely disappears), the energy gap becomes equal to zero.
It can be assumed that the appearance of gapless superconductivity is due to the fact that when interacting with impurity atoms, some pairs are temporarily broken. This temporary decay of the pair corresponds to the appearance of local energy levels within the energy gap itself. As the impurity concentration increases, the gap becomes increasingly filled with these local levels until it disappears completely. The existence of electrons formed when the pair breaks leads to the disappearance of the energy gap, and the remaining Cooper pairs ensure that the electronic resistance is zero.
We come to the conclusion that the existence of a gap in itself is not at all a necessary condition for the manifestation of a superconducting state. Moreover, gapless superconductivity, as it turns out, is not such a rare phenomenon. The main thing is the presence of a bound electronic state - a Cooper pair. It is this state that can exhibit superconducting properties even in the absence of an energy gap.
2.5 Electron pair formation
Forbidden bands in the energy spectrum of semiconductors arise due to the interaction of electrons with the lattice, which creates a field in the crystal with a periodically varying potential.
It is natural to assume that the energy gap in the conduction band of a metal in a superconducting state arises due to some additional interaction of electrons that appears during the transition of the metal to this state. The nature of this interaction is as follows.
A free conduction band electron, moving through the lattice and interacting with ions, slightly “pulls” them away from the equilibrium position (Figure 5), creating in the “wake” of its motion an excess positive charge, to which another electron can be attracted. Therefore, in a metal, in addition to the usual Coulomb repulsion between electrons, an indirect attractive force may arise due to the presence of a lattice of positive ions. If this force turns out to be greater than the repulsive force, then the combination of electrons into bound pairs, which are called Cooper pairs, becomes energetically favorable.
When Cooper pairs are formed, the energy of the system decreases by the amount of binding energy Eb of the electrons in the pair. This means that if in a normal metal the electrons of the conduction band at T = 0 K had a maximum energy E F , then upon transition to a state in which they are bound in pairs, the energy of two electrons (pairs) decreases by E St, and the energy of each of them - by E st /2, since this is exactly the energy that must be expended in order to destroy this pair and transfer the electrons to the normal state (Fig. 6a). Therefore, between the upper energy level of electrons in bonded pairs and the lower level of normal electrons there must exist a gap of width E, which is precisely what is necessary for the appearance of superconductivity. It is easy to verify that this gap is mobile, that is, capable of shifting under the influence of an external field along with the electron distribution curve among states.
In Fig. Figure 7 shows a schematic model of a Cooper pair. It consists of two electrons moving around an induced positive charge, somewhat reminiscent of a helium atom. Each electron in a pair can have a large momentum and wave vector; the pair as a whole (the center of mass of the pair) can be at rest, having zero translational speed. This explains the at first glance incomprehensible property of electrons populating the upper levels of the filled part of the conduction band in the presence of a gap (Fig. 6a). Such electrons have enormous (and) translational speeds. Since the central positive charge of the pair is induced by the moving electrons themselves, under the influence of an external field, the Cooper pair can move freely throughout the crystal, and the energy gap E will shift along with the entire distribution, as shown in Fig. 6b. Thus, from this point of view, the conditions for the appearance of superconductivity are satisfied.
Fig.5 Fig. 7
However, not all conduction band electrons are capable of bonding into Cooper pairs. Since this process is accompanied by a change in the energy of the electrons, only those electrons that are capable of changing their energy can bond in pairs. These are only electrons located in a narrow strip located near the Fermi level (“Fermi electrons”). A rough estimate shows that the number of such electrons is ~ 10 -4 of the total number, and the width of the strip is, in order of magnitude, 10 -4.
In Fig. a Fermi sphere with radius is constructed in momentum space.
There are rings of width dl on it, located relative to the p y axis at angles q1, q2, q3. electrons whose vectors end up on the area of a given ring form a group with almost the same momentum. The number of electrons in each such group is proportional to the area of the corresponding ring. Since as μ increases, the area of the rings also increases the number of electrons in their corresponding groups. Generally speaking, electrons from any of these groups can bond into pairs. The maximum number of pairs is formed by those electrons that are larger. And most of all electrons, whose momenta are equal in magnitude and opposite in direction. The ends of the vectors of such electrons are located not on a narrow strip, but along the entire Fermi surface. There are so many of these electrons compared to any other electrons that practically only one group of Cooper pairs is formed - pairs consisting of electrons having momenta of equal magnitude and opposite direction. A remarkable feature of these pairs is their momentum ordering, which consists in the fact that the centers of mass of all pairs have the same momentum, equal to zero when the pairs are at rest, and different from zero, but the same for all pairs when the pairs move along the crystal. This leads to a fairly strict correlation between the movement of each individual electron and the movement of all other electrons bound in pairs.
The electrons “move like climbers tied together with a rope: if one of them fails due to the unevenness of the terrain (caused by the thermal movement of atoms), then its neighbors bring it back.” This property makes a collective of Cooper pairs less susceptible to scattering. Therefore, if the pairs are brought into orderly motion by one or another external influence, then the electric current created by them can exist in the conductor for an indefinitely long time, even after the cessation of the action of the factor that caused it. Since such a factor can only be the electric field E, this means that in a metal in which Fermi electrons are bound into Cooper pairs, the excited electric current i continues to exist unchanged even after the cessation of the field: i=const at E=0. This is evidence that the metal is indeed in a superconducting state, possessing ideal conductivity. Roughly, this state of electrons can be compared with the state of bodies moving without friction: such bodies, having received an initial impulse, can move for as long as desired, keeping it unchanged.
Above we compared the Cooper pair with the helium atom. However, this comparison should be taken very carefully. As already noted, the positive charge of the pair is unstable and strictly fixed, like that of a helium atom, but induced by the moving electrons themselves and moving with them. In addition, the binding energy of electrons in a pair is many orders of magnitude lower than their binding energy in a helium atom. According to the data in Table 1, for Cooper pairs E light = (10 -2 -10 -3) eV, while for helium atoms E light = 24.6 eV. Therefore, the size of a Cooper pair is many orders of magnitude larger than the size of a helium atom. The calculation shows that the effective diameter of the pair is L? (10 -7 -10 -6) m; it is also called the coherence length. The volume L 3 occupied by the pair contains the centers of mass of ~ 10 6 other such pairs. Therefore, these pairs cannot be considered as some kind of spatially separated “quasi-molecules”. On the other hand, the resulting colossal overlap of the wave functions of all pairs enhances the quantum effect of electron pairing to its macroscopic manifestation.
There is another analogy, and a very deep one, between Cooper pairs and helium atoms. It consists in the fact that a pair of electrons is a system with an integer spin, just like atoms. It is known that helium superfluidity can be considered as a manifestation of the specific effect of boson condensation at the lower energy level. From this point of view, superconductivity can be considered as a kind of superfluidity of Cooper pairs of electrons. This analogy goes even further. Another helium isotope, whose nuclei have half-integer spin, does not have superfluidity. But the most remarkable fact, discovered quite recently, is that as the temperature decreases, atoms can form pairs quite similar to Cooper’s, and the liquid becomes superfluid. Now we can say that superfluidity is like the superconductivity of pairs of its atoms.
Thus, the process of electron pairing is a typical collective effect. The attractive forces that arise between electrons cannot lead to the pairing of two isolated electrons. Essentially both the entire collective of Fermi electrons and the atoms of the lattice participate in the formation of a pair. Therefore, the binding energy (gap width E w) depends on the state of the collective of electrons and atoms as a whole. At absolute zero, when all Fermi electrons are bound in pairs, the energy gap E q reaches its maximum width E q (0). With increasing temperature, phonons appear that are capable of imparting energy to electrons during scattering, sufficient to break the pair. At low temperatures, the concentration of these phonons is low, as a result of which cases of electron pair breaking will be rare. The breaking of some pairs cannot lead to the disappearance of the gap for the electrons of the remaining pairs, but makes it somewhat narrower; the boundaries of the gap approach the Fermi level. With a further increase in temperature, the concentration of phonons increases very quickly, in addition, their average energy increases. This leads to a sharp increase in the rate of electron pair breaking and, accordingly, to a rapid decrease in the energy gap width for the remaining pairs. At a certain temperature Tk the gap disappears completely, its edges merge with the Fermi level and the metal goes into the normal state.
2.5 Effective interaction between electrons due to metal phonons
Fröhlich showed that the interaction of electrons with phonons can lead to effective interaction between electrons. Below we will outline the main provisions of his theory.
In an ideal lattice, the motion of an electron in the conduction band is determined by the Bloch function
which represents a plane wave modulated by a function u k (r) satisfying the periodicity condition u k (r) = u k (r+n), where n is the grating vector, k is the wave vector; h y is a function of the spin state. We will not need its explicit form and the form of the function u k (r) further.
The electron wave function of the entire metal containing N electrons in volume V is the antisymmetric product of the N function q k,y. The ground state corresponds to the filling of states lying in k - space inside the Fermi surface. We will assume that this surface lies far from the zone boundary and is isotropic, that is, it is a sphere of radius k 0 . upon excitation, electrons from states |k|< k 0 переходят в состояния k| >k 0 .
If е k is the energy of the electron state with quasi-momentum ђk, then in the representation of secondary quantization the Hamiltonian of the electron system (up to a constant term) has the form
where a + kу, a kу are the Fermi operators of creation and annihilation of quasiparticles.
To determine the operator of interaction with phonons of the metal lattice, we take into account that when a positive ion occupying the nth place in the lattice is displaced by an amount about n, the energy of interaction of the electron with the lattice will change by the amount. Therefore, in the representation of secondary quantization, the electron-phonon interaction operator can be written in the form
where is the operator expressed through the Fermi operators a kу and Bloch functions using the equality
The ion displacement operator is defined, therefore,
Where, are Bose operators; s is the speed of longitudinal sound waves corresponding to the wave vector q, since only longitudinal waves make a contribution and for them u(q) = sq.
Taking into account that the sum, if, and is equal to zero, if, we obtain the final expression for the electron-phonon interaction operators in the representation of occupation numbers
where (1825) is an abbreviated designation for the sums of products of Fermi operators; - a small value that determines the electron-phonon interaction. Integration is carried out over one elementary cell. Letters "es." the terms Hermitian conjugate to all previous ones are indicated.
The interaction operator (24) does not depend on the spin state of the electrons, so in what follows we can omit writing the spin index y. Operator (24) was obtained under the assumption that the ions in the lattice move as a single unit, that D(q) depends only on q and does not depend on k, and that the vibrations of ions in the lattice are divided into longitudinal and transverse for all values of q, so interaction occurs only with longitudinal phonons. Without these simplifications, calculations become very complicated. Such complication is justified only if it is necessary to obtain quantitative results.
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