Crash Course in Materials Science of Superconductors

Stefan Bringuier
stefanbringuier@gmail.com

Materials Scientist

Sep 19, 2023

Some Comments

Important

These are working notes, so there are bound to be errors. Please keep this in mind while going through the notes. Feel free to email me if you want to provide corrections.

Note

Much of the notes derived from various sources, please checkout the references.

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Whats all the fuss

  • Why did anyone care to begin with?
    • They didn’t. Initially Heike Kamerlingh Onnes1 and others were just interested in cryogenics.
    • Once they achieved liquid Helium, they asked why note study conductive metals at these temperatures.
    • In 1911 Kamerlingh Onnes started with elemental Mercury, 💥 the field of superconductivity (SC) was born.
  • Physcist focused on measurement of other elemental solides and a theory.
  • Observation of SC in Nb is really what begun technological use.

Figure 1: Original plot of Hg transition temperature to SC phase1.

Technological Interest

  • Superconducting coils allow for high magnetic fields.
    • Stronger Magnets: Enhanced image quality and resolution.
    • Energy Efficiency: Lower operational costs due to zero resistance.
    • Cooling Required: Increased cost.

Figure 2: Cut-throught showing MRI machine and SC coils2.
  • Superconducting materials enable high-speed rail.
    • Efficient Levitation: Frictionless, high-speed travel.
    • High Current Capacity: Ideal for powerful electromagnets in propulsion.
    • Downside: Cryogenic systems maintain superconducting state.

Figure 3: Superconductors are onboard train which interact with propolsion rail coils3.
  • High Field Strength: Enables stronger magnetic confinement in fusion.
  • Zero Resistance: Increases efficiency in energy storage systems.
  • Cooling Trade-off: Cryogenic needs offset by gains in efficiency.

Figure 4: High-temperature SC (ex., REBCO) used in a magnetic fusion torodial device4.

Most common complaints

  • Cooling Costs: Cryogenic systems are energy-intensive and expensive.
  • Material Fragility: Mechanical stresses can degrade superconducting properties.
  • High Costs: Fabrication and maintenance of superconducting materials are costly.
  • Limited Current Capacity: Some materials can’t sustain high current densities.
  • Material Complexity: Difficulties in integration with existing technologies.

Many advances have been achieved in last 30 years or so to address these concerns.

Note

At this point I won’t discuss why these are addressible with advances in refrigeration and processing techonology, but this is how I would approach it 😄. Will touch on this at the very end.

Many others seek room-temperature (RT) superconductors. These many exist but who knows if they would have other suitable properties or current processing approaches would work.

Basic Theory: Background

  • As we saw in Figure 1, there are materials where electrical conductivity drops to “exactly”1 zero.
  • How is this achieved?
    • Well at low-temperature we have Bardeen, Cooper, and Schrieffer (BCS) to thank5.

  • What mechanism did they describe?
    • Describe microscopic superdconducting using quantum theory.
    • Solution: Electron Cooper pairs via condesnsate state.
    • Why Pairs? Blame the phonons.

Basic Theory: Cooper Pairs @ Low Temperature (1/4)

Mathematical Foundation

Hamiltonian: \(H = H_0 + H_{\text{int}}\)

  • \(H_0\): Kinetic energy term
  • \(H_{\text{int}}\): Interaction term

BCS Wave Function: \[ \left| \Psi_{\text{BCS}} \right\rangle = \prod_k (u_k + v_k c_{k \uparrow}^\dagger c_{-k \downarrow}^\dagger ) \left| 0 \right\rangle \qquad(1)\]

  • \(u_k\): Probability amplitude for unoccupied state
  • \(v_k\): Probability amplitude for occupied state
  • \(c_{k \uparrow}^\dagger\), \(c_{-k \downarrow}^\dagger\): Electron creation operators

Basic Theory: Cooper Pairs @ Low Temperature (2/4)

Role of Phonons

Electrons interact indirectly via phonons, leading to a net attractive force among pairs of \(\text{e}^{-}\). \[V(q, \omega) = \frac{{2 \omega(q)}}{q^2} \chi(q, \omega)\]

  • \(V(q, \omega)\): Electron-phonon interaction
  • \(\omega(q)\): Phonon frequency
  • \(\chi(q, \omega)\): Polarizability

Cooper Pairs

  • Formed by two electrons with opposite spins and momenta.
  • Exhibit Bose-Einstein-like condensation at low temperatures.

Figure 5: A “static” schematic of real-space cooper pair probablity distribution with coherence length \(\epsilon\). Red dots are the distorted lattice positions 2.

Basic Theory: Cooper Pairs @ Low Temperature (3/4)

Energy Gap

\(\Delta = 2 \left| V \right| \sqrt{N(0) V}\)

  • \(\Delta\): Energy gap, \(V\): Pairing potential, \(N(0)\): Density of states at Fermi level/

Critical Temperature \(T_c\)

The temperature below which a material becomes superconducting.

\(T_c=\frac{1.13\Delta}{k_B}\)

  • \(\Delta\): Energy gag, \(k_B\): Boltzmann constant
Figure 6: Critical tempature as a function of energy gap.

Basic Theory: Superconducting State Property Predictions

Meissner Effect

The expulsion of magnetic flux lines from the interior of a superconducting material.

London Equations: \[\vec{J}=-\frac{ne^2}{m}\vec{A} \qquad(2)\] \[\nabla\times\vec{J}=-\frac{ne^2}{m}\vec{B} \qquad(3)\]

  • \(\vec{J}\): Superconducting current density
  • \(\vec{A}\): Vector potential
  • \(\vec{B}\): Magnetic field
  • \(n\): Density of superconducting carriers
  • \(e\): Elementary charge
  • \(m\): Electron mass

Basic Theory: Experimental Evidence

  • Tunneling experiments
  • Specific heat measurements
  • Magnetic penetration depth

Superconducting phase

  • Exhibit perfect diamagnetism below \(T_c\) (Meissner effect)
  • Expels all magnetic fields (\(H= 0\)) due to Cooper pairing
  • Critical magnetic field \(H_c\) exists, beyond which superconductivity is destroyed
  • E.g., Aluminum (Al), Lead (Pb)

Add Figure contrasts Type I and Type II

  • Exhibit two critical magnetic fields (\(H_{c1}\) and \(H_{c2}\))
  • Allow magnetic vortices to form between \(H_{c1}\) and \(H_{c2}\) (mixed state)
  • Usually compound or alloy materials with complex structures.
    • Exception are Niobium and Vanadium elemental solids.
  • Often high-\(T_c\) materials, enabling applications near room temperature
  • E.g., Yttrium Barium Copper Oxide (YBCO), Niobium Titanium (NbTi)

Add Figure contrasts Type I and Type II

Backmatter

stefanbringuier@gmail.com

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This presentation can be viewed online at https://stefanbringuier.github.io/CrashCourseSCMaterials.

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References

1.
Delft, D. van & Kes, P. The discovery of superconductivity. Physics Today 63, 38–43 (2010).
2.
Wang, Q. L., Dai, Y. M., Zhao, B., Song, S. S., Wang, C. Q., Li, L., Cheng, J., Chen, S., Wang, H., Ni, Z., Li, Y., Cui, C., Hu, X., Lei, Y., Chan, K., Yan, L., Wen, C., Hui, G., Yang, W. C., Liu, F., Zhuo, Y., Zhou, X., Yan, Z., Chen, J. & Xu, T. A superconducting magnet system for whole-body metabolism imaging. IEEE Transactions on Applied Superconductivity 22, 4400905–4400905 (2012).
3.
Nishijima, S., Eckroad, S., Marian, A., Choi, K., Kim, W. S., Terai, M., Deng, Z., Zheng, J., Wang, J., Umemoto, K., Du, J., Febvre, P., Keenan, S., Mukhanov, O., Cooley, L. D., Foley, C. P., Hassenzahl, W. V. & Izumi, M. Superconductivity and the environment: A roadmap. Superconductor Science and Technology 26, 113001 (2013).
4.
Molodyk, A. & Larbalestier, D. C. The prospects of high-temperature superconductors. Science 380, 1220–1222 (2023).
5.
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957).
6.
Speller, S. A materials science guide to superconductors and how to make them super. (Oxford University Press, 2022).

Draft Outline

  • Slide 4: Importance of Superconducting Materials
  • Slide 5: Brief History of Superconductivity
  • Applications: Magnetics and Wires
    • MRI Machines
    • Maglev Trains
    • Energy Grids
    • Superconducting Coils
    • Limitations in Applications
  • Basics of Superconductivity (Theory)
    • Cooper Pairs
    • Meissner Effect
    • BCS Theory Overview
    • Zero Electrical Resistance
    • Critical Temperature
  • Thermodynamics and Phases
    • Type I and Type II Superconductors
    • Critical Fields
    • Diamagnetic Response
    • Phase Diagrams
    • Energy Gaps
  • Flux Pinning and Levitation
    • Vortex Lattices
    • Flux Tubes
    • Levitation Applications
    • Pinning Centers
    • YBaCuO Examples
  • Niobium-Titanium (NbTi) Alloys
    • Composition and Structure
    • Magnetic Properties
    • Mechanical Properties
    • Applications
    • Processing Challenges
  • Quantum Effects
    • Quantum Tunneling
    • Josephson Junctions
    • Macroscopic Quantum Phenomena
    • SQUIDs
    • Quantum Computing Applications
  • Microstructure and Grain Boundaries
    • Grain Boundary Impact on Properties
    • Microstructure Analysis
    • Sintering Methods
    • Weak Links
    • Influence on Flux Pinning
  • Mechanical Properties
    • Tensile Strength
    • Brittleness
    • Fatigue
    • Thermal Expansion
    • Composite Superconductors
  • High-Temperature Superconductors (HTSC)
    • YBaCuO and Other Cuprates
    • Iron-based Superconductors
    • Challenges and Advantages
    • Applications
    • Current Research Trends
  • Recent Trends and Future Directions
    • MgB2 Developments
    • Supposidly Room-Temperature Superconductors
    • Topological Superconductors
    • Commercialization Challenges
    • Research Funding and Outlook
  • Conclusion and Summary
    • Summary of Key Points

Stefan Bringuier

Footnotes

  1. Here I use exactly in that it is zero within measurement precision. If your device can only measure to \(10^{-10}\) then you show a resistivity value on that order.

  2. adapted from https://thiscondensedlife.wordpress.com/2015/09/12/draw-me-a-picture-of-a-cooper-pair.