A Basic Review on Superconducting Materials

You are currently viewing A Basic Review on Superconducting Materials

Three years after achieving the liquefaction of helium and thus the ability to achieve the temperatures of few degrees of Kelvin, in 1911, H. Kamerlingh Onnes found the first experimental evidence of superconductivity, when he observed that liquid helium has a sharp drop in its resistivity when cooled beyond 4K. Later, he also found that specific materials like tin, lead and mercury offered no resistance to the current flowing in the coil after being cooled beyond a certain critical temperature, Tc, which is specific to each material. Zero resistance could be identified by no apparent change in the strength of the magnetic field induced by the coil for a very long amount of time. The small decay of supercurrents (the current flowing through superconducting samples) was analysed by File and Mills 7 using precision Nuclear Magnetic Resonance (NMR) techniques to detect faint changes in the magnetic fields due to supercurrents, it was found that the lower bound of decay of currents in superconducting coils was at least 105 years. Hence, an electrical property of negligible resistance was the first observed phenomenon of superconductors. For his hallmark experiment (Fig. 1), Onnes was conferred the Nobel Prize in 1913. Later, the superconducting critical temperature Tc for several other materials were found.

Coming to magnetic properties, Meissner and Orchenfeld in 1933, discovered the phenomenon of perfect diamagnetism in superconducting materials where they tend not to let the magnetic flux into the system, and also, expel the existing magnetic hold when cooled through the Tc, thus a transition from normal to superconducting state. Such a reversible process also later implied the existence of a critical magnetic field (Hc) after which the superconductivity gets destroyed.

Similarly, when the current density exceeds a certain value (Jc), it also lets the magnetic field exceed its critical value, thus resulting in loss of superconductivity, otherwise known as ‘quenching’. The parameter Jc is usually dependent on the dimensions of the sample considered. We can express the existence of superconductivity by using these 3 parameters, Tc, Hc, Jcas per Fig 2.

Figure 1: Existence of superconductivity discovered by H. Kammerlingh Onnes in 1911
Figure 2: Superconductivity as expressed by its critical parameters17

Vanishing magnetic field in the Meissner-Ochsenfeld effect can’t be explained by perfect conductivity alone. Since, according to Ohm’s Law, E =  ρ . J, and for superconductors ρ = 0 for a non-zero value of current density J, it means that E = 0 which implies ×E = B/t = 0. It doesn’t necessarily imply a vanishing magnetic field. Hence, perfect diamagnetism is an independent property of superconductors along with perfect conductivity.

Inspired from the experimental results of the former two, in their paper “The Electromagnetic Equations of the Supraconductor,” Heinz and Fritz London proposed a simple model of superconductors, where charges were influenced only by the Lorentz force from an external field, and not by any dissipative interactions. These resulted in 2 equations, first of which described perfect conductivity and the 2nd, when used along with Maxwell’s equations, described that external magnetic fields decrease exponentially within a length λ inside the superconductor. This screens the ‘core’ of the sample from external magnetic fields and thus the core acts as a perfect diamagnet!

The London Equations motivated Fritz London to believe that there is some sort of rigidity in the state of the superconductor in a local regime, as it might not show much variance in its state upon variation of electromagnetic fields. When Pippard tried to perform a generalisation in the non-local regime of the same, he emphasized on a parameter, known as intrinsic coherence length ξ0, which represented the measure of distance within which the electron concentration of a pure superconducting electron doesn’t vary drastically. It is somewhat analogous to the mean free path of an electron in a normal metal. This also represents the thickness of the transition layer when a superconducting and a ‘normal’ phase are kept in contact (Figure 3).

Figure 3: Meissner-Oschenfeld effect where we could see the core being shielded from external magnetic field by the penetration depth λ 1

Parallel to the theoretical explanations, superconductivity was also tested for many materials in their elemental, compound or alloy forms (Fig 4). In the early stages of research in superconductivity, the materials tended to exclude magnetic flux till the applied magnetic field exceeded its corresponding Hc, after which the material lost its property of superconductivity at the expense of allowing flux. These are known as Type I superconductors, where λ << ξ. Since we know that Bint = Bext0M for a sample of magnetisation M, for a superconducting sample, Bint = 0 implying M/Bext= -1/μ0, thus giving Fig. 5

Figure 4: Bernd T. Matthias and John K. Hulm started an extensive systematic search that delivered a number of new compounds in the early 1950s. He even went on to establish a set of ‘rules’ to obtain ‘good’ superconductors 3.

But in 1935, Rjabinin and Shubnikov discovered a certain type of superconductivity where the material excluded the magnetic field upto a certain threshold (Hc1), and beyond that, flux ‘leaked’ into it through some points on the sample, while the material still behaved like a superconductor electrically. At a much higher field (Hc2), the flux penetrates the sample completely and thus the superconductivity is lost. These samples were called Type II superconductors. They are observed with ξ >> λ.

Upon investigating the theory behind this new-found phenomenon, Abrikosov gave the concept of ‘Vortex state’, building up on the works of Lars Onsager and Richard Feynman on quantum vortices in superfluids. The ‘Vortex state’ is the state of a superconductor where it behaves as a superconductor while allowing flux through it. Abrikosov claimed the formation of stable ‘normal’ regions surrounded by superconducting currents allowed the flux through it, while the purely superconducting region was being shielded by a distance of the London penetration depth λ from these normal flux points (Fig 6).

Apart from electrical and magnetic properties of a superconductor, it also exhibited peculiar thermodynamic properties. As first experimentally verified by Corak et.al., electronic specific heat well below Tc was dominated by an exponential factor of e-Δ/kBT , indicating the existence of a gap, or energy interval with no allowed energy states in between in the energy spectrum. The idea was that when there is a gap, only an exponentially small number of particles have enough thermal energy to be promoted to the available unoccupied states above the gap. But any strong theory of solid state physics was yet to arrive to confirm this indication 2.

Figure 5: Magnetisation vs applied magnetic field curves for Type I and Type II superconductors, it could be easily seen that Type I superconductors quench at a lower magnetic field than the Type II superconductors. Hence the former is also called a low-field superconductor and the latter is known as a high-field superconductor 2.

Due to the atomic bomb development, single isotope superconducting samples like mercury had become available. It was observed that the critical temperature Tc varied with the isotopic mass (MαTc = constant) which pointed to a phonon-driven mechanism. John Bardeen realized that phonons introduce an attractive interaction between electrons close to the Fermi surface 2,3.

Figure 6: Abrikosov vortices as found in YBCO samples 6

Then came the epoch-making microscopic electron-pairing BCS 9 (Bardeen-Cooper-Schrieffer) theory where even a tiny attractive interaction between these electrons lead to the formation of bound electron pair states (“Cooper pairs”). These Cooper pairs were the result of attractive electron-electron interaction mediated by phonon, where one electron distorts the crystal lattice for another electron of opposite momentum and spin to feel an effective attractive interaction, thus forming a weakly-interacting electron pair, which is strong enough to overwhelm the coulombic repulsion. These paired electrons no longer have to abide by Pauli’s exclusion principle, and thus they all occupy the same ground state, and form a macroscopic ground state, acting like a boson. The condensation of these single electrons into pairs requires energy, which is nothing but the much speculated energy gap as discussed before. The involvement of the phonon interaction mechanism also explained why isotope effects in superconducting samples could be observed. BCS theory had an impact not only on solid state physics but also on elementary particle physics where it was further developed to the idea of the Higgs mechanism of elementary particle mass generation.

Figure 7: Qualitative representation of variation of superconductor’s specific heat. The dashed line represents the specific heat of a normal material. It can be observed that the specific heat of a superconductor shows exponential nature well below Tc and at Tc it is much higher than its normal state counterpart. When the sample quenches at Tc, a steep discontinuity is observed, for the sample to fall back to the ‘normal’ line 15.

Prior to BCS theory, V. L. Ginzburg and Lev Landau gave their Ginzburg-Landau equation of superconductivity which primarily focussed on the superconducting electrons rather than their excitations, unlike BCS theory. They introduced a macroscopic quantum wave-function of these electrons, and upon usage of variational principle, they derived a nonlinear Schrodinger’s like equation which explained the non-linear effects due to the strong fields. But back then it didn’t get much importance, until Gorkov in 1957 proved that the GL equations are a limiting condition of the BCS theory with temperature range near the Tc. In fact, the GL wave function was later found to be the behaviour centre-of-mass of the Cooper pairs. Its near-Tc approximation was successful in explaining in-phase transitions of a sample from superconducting to normal state and vice-versa. Due to its simplicity and a macroscopic picture of superconductivity, it went on to become a very popular equation in superconductivity.

In 1962, B. D. Josephson at Cambridge University predicted the flow of current between two superconducting materials, even though they have a normal or insulating material placed between them. His prediction of this tunneling effect won him the Nobel prize in Physics in 1973 and the phenomenon was known as the Josephson effect. It has been applied to electronic devices such as the SQUID (Superconducting QUantum Interference Device), capable of detecting even the faintest of magnetic fields 10.

In 1964, Bill Little of Stanford University had suggested the possibility of organic (carbon-based) superconductors. The intuition was that conductive polymer chains with polarizable molecular groups may provide for electrons running along the polymer chains a highly effective Cooper pair coupling by means of an energy exchange via localized excitons  But upon exploration, it turned out to be different, since superconductivity in these systems originated from pi-electrons in stacked rings.

Figure 8: The quest to find superconductivity at higher temperature over the course of time. It can be easily noticed how the discovery of superconductivity in YBCO bloomed the search for further high-Tc superconductors! 5

In 1986, a remarkable discovery was made in regards to the limit of Tc that could be achieved for a superconducting sample. A. Muller and G. Bednorz12 of IBM Research Lab, Switzerland, shared the Nobel Prize in Physics for creating a brittle ceramic (a cuprate of lanthanum and barium) compound which gave a Tc of 30K, the highest known value till then. Ceramics, normally being insulators, superconducting till a relatively higher limit was a marvel. A small portion of lead added to it for calibration purposes gave an even peculiar result, a Tc as high as 58K!

This phenomenal discovery led to a hot spree of trial-and-error experiments to find higher and higher value of Tc possible. Later in 1987, a research team re- placed lanthanum by ytterbium in the previous cuprate prepared by Muller and Bednorz (the compound came to be commonly called as YBCO), which was found to superconduct at a temperature greater than liquid nitrogen, somewhere near 93K! It boosted the search of high-Tc superconductors manifolds.

But interestingly, a theoretical backdrop to high-Tc superconductors are yet to be found, given that BCS theory struggles to explain superconductivity beyond 30 K, since a temperature greater than that is efficient enough in destroying Cooper pairs. Also, high-Tc superconductors exhibit other phenomena which couldn’t be explained by BCS theory. Discussion has shifted to whether there are pairing mechanisms other than phonon mediated interaction. But any conclusive and satisfactory model is yet to be found! 

Recently, the fabled “room temperature superconductivity” has been claimed with Tc near to 300 K but only when a pressure in the range of GPa has been applied to the system 13. In 2018, Thapa and Pandey14 from IISc Bangalore claimed an Ag-Au implanted sample showing superconductivity. Some theoretical physicists have hinted at this feat being a manifestation of Hubbard’s model of solids.

Research on superconductors currently spans over Cuprate High-Temperature Superconductors, other Oxide Superconductors, Iron-Based Superconductors, other Chalcogenide Superconductors, Heavy Fermion Superconductors, Nitride Superconductors, Organic and Other Carbon-and Silicon-Based Superconductors, Borides and Borocarbides.

Superconductors have found great practical importance in many fields. Superconducting coils have been used to produce very strong magnetic fields for biomedical purposes like that of MRI and also in particle accelerators. Superconducting coils and wires have started being used at several precision instruments. Niobium thin films with Josephson junctions are used in SQUID magnetometers. Superconducting circuits are well suited for quantum computers as well 3. Many research groups also are working to fabricate superconducting devices which could be used to understand several physical phenomena as well. With further advancements, there may come a time when achieving all these may not need cryogenic cooling, and maybe, someday, we can actually sit on a magnetically levitated train while travelling from Bhubaneswar to Delhi in a few minutes!

References

  • [1] – M. Tinkham, Introduction to superconductivity, (Dover Publications, INC, New York, 1975).
  • [2] – Kittel, C. (2004). Introduction to solid state physics (8th ed.). John Wiley Sons.
  • [3] – Hott, R., Kleiner, R., Wolf, T. and Zwicknagl, G. (2016). Review on Superconducting Materials. In digital Encyclopedia of Applied Physics, Wiley-VCH Verlag GmbH  Co. KGaA (Ed.). https://doi.org/10.1002/3527600434.eap790
  • [4] – B. Douine, K. Berger and J. Leveque, ”Critical current density determination of  superconducting  material,”  2014 International  Conference  on  Electrical Sciences and Technologies in Maghreb (CISTEM), Tunis, 2014, pp. 1-5, doi:10.1109/CISTEM.2014.70769679
  • [5] – https://medium.com/the-physics-arxiv-blog/metamaterial-superconductor-raises-critical-temperature-of-zero-resistance- 3da6c4657baa
  • [6] – Wells,   F.,   Pan,   A.,   Wang,   X.   et   al.   Analysis of low-field isotropic vortex glass containing vortex groups in YBa2Cu3O7x thin films visualized by scanning SQUID microscopy.   Sci   Rep   5,    8677   (2015). https://doi.org/10.1038/srep08677
  • [7] – J. File and R. G. Mills,  Observation of Persistent Current in a Superconducting Solenoid, Phys. Rev. Lett. 10, 93 Published 1 February 1963
  • [8] – Shubnikov L.V.,  Khotkevich V.I.,  Shepelev Yu.D.,  Riabinin Yu.N. (1937). Magnetic properties of superconducting metals and alloys. Zh. Exper. Teor. Fiz. (USSR), V.7, No 2, p.221-237;
  • [9] – J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev. 108, 1175 – Published 1 December 1957
  • [10] – P. W. Anderson and J. M. Rowell, Probable Observation of the Josephson Superconducting Tunneling Effect, Phys. Rev. Lett. 10, 230 – Published 15 March 1963
  • [11] – Bednorz,  J.G.,  M ̈uller,  K.A.  Possible  highT  c  superconductivity  in  the BaLaCuO  system. Z. Physik B – Condensed  Matter  64,  189–193  (1986). https://doi.org/10.1007/BF01303701
  • [12] – Snider, E., Dasenbrock-Gammon, N., McBride, R. et al. Room-temperature superconductivity  in  a  carbonaceous sulfur  hydride.  Nature  586,  373–377 (2020).
  • [13] – Thapa  et.al.,  Coexistence  of  Diamagnetism  and  Vanishingly  Small  Electrical Resistance at Ambient Temperature and Pressure in Nanostructures, eprint 1807.08572, arXiv, 2019
  • [14] – Dunlap R.A, Other properties of superconductors, electrons in solids, Morgan and Claypool publishers, 2053-2571, ch 8, 2019
  • [15] – Dunlap R.A, Other properties of superconductors, electrons in solids, Morgan and Claypool publishers, 2053-2571, ch 8, 2019
  • [16] – https://physicstoday.scitation.org/doi/10.1063/1.3490499
  • [17] – https://www.slideshare.net/DebiPrasadDash3/superconductivity-68227517