BCS Theory of Superconductivity

 Introduction:

Bardeen-Cooper-Schrieffer theory (BCS theory) is the first microscopic theory of superconductivity. Cooper had discovered that electrons are in pairs in a superconductor and named as Cooper pairs. The theory describes superconductivity as an effect caused by the condensation (I will talk briefly about condensation later in this article) of the Cooper pair. The motion of all the Cooper pairs within a single superconductor constitutes a system that functions as a single entity. Applying an electrical voltage to the superconductor causes all Cooper pairs to move, forming a current. Even in the absence of voltage, the current flow continues indefinitely. The reason is that the Cooper pairs encounter no opposition inside the conductor. We should stop all the cooper pairs simultaneously to cease the current flow. As the superconductor is warmed, its Cooper pairs separate into individual electrons, and the material becomes normal or non-superconducting. Superconductors abruptly lose all resistance to the flow of an electric current when cooled to temperatures near absolute zero (1).

An electron moving through a conductor will attract nearby positive charges in the lattice. It causes deformation in the lattice. It causes another electron, with an opposite spin, to move to this high positive charge density region. The two electrons become correlated, and the attraction overcomes the Coulomb repulsion. Electrons stay paired together and resist all kicks, and the electron flow will not experience resistance (2).

Details

BCS theory assumes that attraction between electrons developed, which can overcome the Coulomb repulsion. The cause of this attraction is the coupling of electrons to the crystal lattice.

BCS can give the quantum-mechanical many-body state of the system of electrons inside the metal (BCS state). 

Electrons move independently in the normal state of metals.

In the BCS state, they are in Cooper pairs by the attractive interaction. 

                             Positive ions are attracted towards electrons forming a positive ion core (7)

Significant features:

• Theory supplies a means by which the energy required to separate the Cooper pairs into electrons.
• Explains isotope effect
• BCS theory has given the ability to describe what is occurring in the lattice and Fermi system.
• The theory supports the electrodynamics of the superconducting system and agrees with experimental findings. 

 

Cooper pairs:

BCS theory on the assumption that superconductivity arises when the attractive Cooper pair interaction dominates over the Coulomb repulsive force. A Cooper pair is a weak electron-electron-bound pair mediated by a phonon interaction (3).

During a flow of current in the superconductor, when an electron approaches the crystal lattice, there is a coulomb attraction between the electrons and lattice ion. It leads to a distortion in the lattice. It results in a positive ion being displaced from its mean position. This interaction is named electron-phonon interaction. The Coulomb attraction between the electron and the positively charged cores of ions in the material will leave a net positive charge in the vicinity and, when the second electron that approaches the distorted positive ion experiences the Coulomb force of attraction. 

                                                            Lattice distortion

The interaction between two electrons via lattice overcomes the repulsion. An apparent force of attraction develops, and they move in pairs. The attraction between electrons is less at ordinary temperatures, and pairing does not happen. Below the transition temperature, the apparent force of attraction reaches a maximum value for any two electrons of equal and opposite spin. These pairs of electrons formed by the interaction between two electrons with opposite spin and momentum in a phonon field are called cooper pairs.

Here is a step-by-step process of pair formation.

Overview

When the temperature is lower, electrons near the Fermi surface become unstable against the formation of the Cooper pair.

Cooper showed that even if the attractive potential is weaker, there will be binding of electrons. In superconductors, an attraction is due to the electron-lattice interaction. These pairs have some bosonic properties, and at sufficiently low temperatures, bosons can form a large Bose-Einstein condensate.

In many superconductors, the attractive interaction between electrons (necessary for pairing) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice (the phonons).

Significant features:

• An electron moving through a conductor will attract nearby positive charges in the lattice.
• It causes deformation of the lattice.
• The result is high positive charge density formation at that region.
• This condition results in, another electron, with the opposite spin moving into that region.
• The two electrons, then become bound as a single unit by overlapping the repulsive force between electrons (Cooper pair).

In a superconductor, there are a lot of such electron pairs, and these pairs form a highly collective condensate.

The electrons stay paired together and resist all kicks, and the electron flow as a whole will not experience resistance. Thus, the collective behavior of all pairs is a crucial factor necessary for superconductivity.

Do you know what the Fermi surface is?

In this article, I will give you an idea of the Fermi level, Fermi energy, and Fermi surface.

We know electrons are fermions and obey Fermi-Dirac statistics.

The highest energy level filled at absolute zero is the Fermi level. The energy of the Fermi level is Fermi energy.

Fermi surface is the surface that separates occupied from unoccupied electron states at zero temperature. [10] 

Important features:

• The pair has a total spin of zero (so they are bosons)

• Cooper pairs drift with identical velocity.

• The density of Cooper pairs is high.

• The small velocity of pairs combined with their precise ordering minimizes collision.

• Rare collision with lattice leads to vanishing resistivity (pairs sail over the lattice point without exchanging energy with infinite conductivity).

Explanation:
Consider the first electron with wave vector k distorts the lattice, emitting phonon of wave vector q. This results in the wave vector k-q for the first electron. Now, the second electron with wave vector k' seeks the lattice, it takes the energy from the lattice, and its wave vector changes to k+q. Two electrons with wave vectors k-q and k+q form a pair known as the Cooper pair (5).

The energy of the pairing interaction is weak (10^-3 eV), and thermal energy can easily break the pairs. So only at low temperatures, in metals and other substrates, are a significant number of the electrons bound in Cooper pairs.

Coherence length:

In electron-lattice-electron interactions, the electron is not in a fixed location. They move in opposite directions, and their correlation may persist over 10 ^-6m. This length is called coherence length.

                                                        Schematic of a Cooper pair (6)

                                                        

Connection to superconductivity:

The tendency for all Cooper pairs in a body to condense into the same ground quantum state is responsible for the peculiar properties of superconductivity. R. A. Ogg Jr. suggested that electrons might act as pairs coupled by lattice vibrations in the material (8,9). The isotope effect observed in superconductors shows this behavior. The isotope effect showed that materials with heavier ions had lower superconducting transition temperatures. The theory of Cooper pairing explains this phenomenon. Heavier ions are not smooth for the electrons to attract and move, which results in smaller binding energy for the pairs. 

We know that electrons in Cooper pairs are bound together, and to break a pair, we have to change the energy of all other pairs.

That is, there is an energy gap for single-particle excitation. At lower temperatures, the energy gap is higher and vanishes at the transition temperature when the material turns to normal condition.

The BCS theory gives expressions for:

• How does the energy gap grow with the strength of the attraction?
• How does the density of states change on entering the superconducting state?
• BCS theory predicted the dependence of energy gap (Δ) at temperature (T) on critical temperature (Tc).

The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature takes the universal value[11] Δ(𝑇=0)=1.764𝑘B𝑇c.

Independent of material, near the critical temperature, the relation asymptotes to[11]

Δ(𝑇→𝑇c)≈3.06𝑘B𝑇c1−(𝑇/𝑇c).

That is of the form suggested the previous year by M. J. Buckingham[12] 

• At low temperatures, the specific heat of the superconductor is suppressed strongly due to the energy gap, and no thermal excitations remain there.

The specific heat of the superconductor becomes higher than that of the conductor at a normal state before reaching the transition temperature. The ratio of these two values is universally given by 2.5.

BCS theory correctly describes:

• Explains the Meissner effect and the variation of the critical magnetic field with temperature.
• The theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi level.
• BCS gives the superconducting transition temperature Tc in terms of the electron-phonon coupling potential V and the Debye cutoff energy ED [5] 𝑘B𝑇c=1.134𝐸D𝑒−1/𝑁(0)𝑉, where N(0) is the electronic density of states at the Fermi level.
• The BCS theory reproduces the isotope effect.

Binding energy:

Binding energy is the minimum energy required to remove a particle from a system of particles or the minimum energy to disassemble a system of particles into individual parts. [11] In the former, the term is predominantly used in condensed matter physics, atomic physics, and chemistry, whereas in nuclear physics, the term is separation energy. [12]

Isotope effect

Isotope effects are a revelation of the quantum nature of nuclei. In physical organic chemistry, there is a change in the reaction rate of a chemical reaction if we replace one of the atoms in the reactants with one of its isotopes. That change in the reaction rate of a chemical reaction is the kinetic isotope effect (KIE). [13] 

Evidences:

• Evidence of a band gap at the Fermi level (a drastic change in conductivity demanded a drastic change in electron behavior. The pairs of electrons might perhaps act like bosons instead, which are in different condensate rules).

• Isotope effect on the critical temperature, suggesting lattice interactions

• The Debye frequency of phonons in a lattice is proportional to the inverse of the square root of the mass of lattice ions. The superconducting transition temperature of mercury showed the same dependence by substituting the most abundant natural mercury isotope, 202Hg, with a different isotope, 198Hg (4).

• An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting vanadium is at its critical temperature, its heat capacity increases by a few degrees, revealing an energy gap bridged by thermal energy.          

• Suggests a situation where some binding energy exists but gradually weakens as the temperature increases toward the critical temperature. 

 
Disadvantage:

• Insufficient to completely describe the observed features of high-temperature superconductors

• The theory best approximates only conventional weakly coupled superconductors 

To know more about superconductivity, please read my other article (https://retnacpn.blogspot.com/2023/10/superconductivity.html)

Condensation: 

Bose-Einstein condensate (BEC) is a state of matter in condensed matter physics. 

When we cool, a gas of bosons at very low densities to temperatures close to absolute zero (−273.15 °C or 0 K), resulting in a condensation state. Under such conditions, most bosons occupy the lowest quantum state. More generally, condensation refers to the appearance of macroscopic occupation of one or several states. For example, in BCS theory, a superconductor is a condensate of Cooper pairs. [14] 

References:

1 en.m.wikipedia.org.

2 Britanica.com, science.

3 phys.ufl.edu, sp.

4 hyperphysics.phy-astr.gsu.edu

5 scribd/Slideshare, BCS theory new. 2022.

6 what-when-how.com/electronic-properties-of-materials/electrical-conduction-in-metals-and-alloys-electrical-properties-of-materials-part-3

7 P. K. Palanisamy, SCITECH pub, 2008.

8  Ogg, Richard A., Bose-Einstein Condensation of Trapped electron pairs, Phys. Rev., APS, 1946, 69, 243-244.

9 Poole Jr, Encyclopedic Dictionary of condensed matter Physics, 2004, 576.

10  Dugdale, S B (2016). "Life on the edge: a beginner's guide to the Fermi surface". Physica Scripta. 91 (5): 053009. Bibcode:2016PhyS...91e3009D. doi:10.1088/0031-8949/91/5/053009. hdl:1983/18576e8a-c769-424d-8ac2-1c52ef80700e. ISSN 0031-8949

11 Rohlf, James William (1994). Modern Physics from α to Z°. John Wiley & Sons. p. 20. ISBN 0471572705

12 ^ Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons. p. 524. ISBN 047187373X.

13 Atkins P, de Paula J (2006). Atkins' Physical Chemistry (8th ed.). Oxford University Press. pp. 286–288, 816–818. ISBN 978-0-19-870072-2.

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