## Properties of Superconductors

Contents

In addition to zero electrical resistance, superconductors also have perfect diamagnetism. In other words, in the presence of an applied magnetic field, the net magnetic field within a superconductor is always zero (this figure). Therefore, any magnetic field lines that pass through a superconducting sample when it is in its normal state are expelled once the sample becomes superconducting. These are manifestations of the Meissner effect, which you learned about in the tutorial on current and resistance.

Interestingly, the Meissner effect is not a consequence of the resistance being zero. To see why, suppose that a sample placed in a magnetic field undergoes a transition in which its resistance drops to zero. From Ohm’s law, the current density, *j*, in the sample is related to the net internal electric field, *E*, and the resistivity \(\rho \) by \(j=E\text{/}\rho \). If \(\rho \) is zero, *E* must also be zero so that *j* can remain finite. Now *E* and the magnetic flux \({\text{Φ}}_{\text{m}}\) through the sample are related by Faraday’s law as

\(\oint EdI=-\cfrac{d{\text{Φ}}_{\text{m}}}{dt}.\)

If *E* is zero, \(d{\text{Φ}}_{\text{m}}\text{/}dt\) is also zero, that is, the magnetic flux through the sample cannot change. The magnetic field lines within the sample should therefore not be expelled when the transition occurs. Hence, it does not follow that a material whose resistance goes to zero has to exhibit the Meissner effect. Rather, the Meissner effect is a special property of superconductors.

Another important property of a superconducting material is its **critical temperature**, \({T}_{\text{c}}\), the temperature below which the material is superconducting. The known range of critical temperatures is from a fraction of 1 K to slightly above 100 K. Superconductors with critical temperatures near this higher limit are commonly known as “high-temperature” superconductors. From a practical standpoint, superconductors for which \({T}_{\text{c}}\gg 77\;\text{K}\) are very important. At present, applications involving superconductors often still require that superconducting materials be immersed in liquid helium (4.2 K) in order to keep them below their critical temperature. The liquid helium baths must be continually replenished because of evaporation, and cooling costs can easily outweigh the savings in using a superconductor. However, 77 K is the temperature of liquid nitrogen, which is far more abundant and inexpensive than liquid helium. It would be much more cost-effective if we could easily fabricate and use high-temperature superconductor components that only need to be kept in liquid nitrogen baths to maintain their superconductivity.

High-temperature superconducting materials are presently in use in various applications. An example is the production of magnetic fields in some particle accelerators. The ultimate goal is to discover materials that are superconducting at room temperature. Without any cooling requirements, the bulk of electronic components and transmission lines could be superconducting, resulting in dramatic and unprecedented increases in efficiency and performance.

Another important property of a superconducting material is its **critical magnetic field** \({B}_{\text{c}}(T),\) which is the maximum applied magnetic field at a temperature *T* that will allow a material to remain superconducting. An applied field that is greater than the critical field will destroy the superconductivity. The critical field is zero at the critical temperature and increases as the temperature decreases. Plots of the critical field versus temperature for several superconducting materials are shown in this figure. The temperature dependence of the critical field can be described approximately by

\({B}_{\text{c}}(T)={B}_{\text{c}}(0)[1-{\left(\cfrac{T}{{T}_{\text{c}}}\right)}^{2}]\)

where \({B}_{\text{c}}(0)\) is the critical field at absolute zero temperature. This table lists the critical temperatures and fields for two classes of superconductors: **type I superconductor** and **type II superconductor**. In general, type I superconductors are elements, such as aluminum and mercury. They are perfectly diamagnetic below a critical field *B _{C}*(

*T*), and enter the normal non-superconducting state once that field is exceeded. The critical fields of type I superconductors are generally quite low (well below one tesla). For this reason, they cannot be used in applications requiring the production of high magnetic fields, which would destroy their superconducting state.

### Critical Temperature and Critical Magnetic Field at \(T=0\;\text{K}\) for Various Superconductors

Material | Critical Temperature \((\text{K})\) | Critical Magnetic Field \((\text{T})\) |
---|---|---|

Type I | ||

\(\text{Al}\) | \(1.2\) | \(0.011\) |

\(\text{Ga}\) | \(1.1\) | \(0.0051\) |

\(\text{Hg}(\alpha )\) | \(4.2\) | \(0.041\) |

\(\text{In}\) | \(3.4\) | \(0.029\) |

\(\text{Nb}\) | \(9.3\) | \(0.20\) |

\(\text{Pb}\) | \(7.2\) | \(0.080\) |

\(\text{Sn}\) | \(3.7\) | \(0.031\) |

\(\text{Th}\) | \(1.4\) | \(0.00016\) |

\(\text{Zn}\) | \(0.87\) | \(0.0053\) |

Type II | ||

\({\text{Nb}}_{3}\text{Al}\) | \(18\) | \(32\) |

\({\text{Nb}}_{3}\text{Ge}\) | \(23\) | \(38\) |

\({\text{Nb}}_{3}\text{Sn}\) | \(18\) | \(25\) |

\(\text{NbTi}\) | \(9.3\) | \(15\) |

\({\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{7}\) | \(92\) | \(\text{>100}\) |

Type II superconductors are generally compounds or alloys involving transition metals or actinide series elements. Almost all superconductors with relatively high critical temperatures are type II. They have two critical fields, represented by \({B}_{\text{c1}}(T)\) and \({B}_{\text{c2}}(T)\). When the field is below \({B}_{\text{c1}}(T),\) type II superconductors are perfectly diamagnetic, and no magnetic flux penetration into the material can occur. For a field exceeding \({B}_{\text{c2}}(T),\) they are driven into their normal state. When the field is greater than \({B}_{\text{c1}}(T)\) but less than \({B}_{\text{c2}}(T),\) type II superconductors are said to be in a mixed state. Although there is some magnetic flux penetration in the mixed state, the resistance of the material is zero. Within the superconductor, filament-like regions exist that have normal electrical and magnetic properties interspersed between regions that are superconducting with perfect diamagnetism. A representation of this state is given in this figure. The magnetic field is expelled from the superconducting regions but exists in the normal regions. In general, \({B}_{\text{c2}}(T)\) is very large compared with the critical fields of type I superconductors, so wire made of type II superconducting material is suitable for the windings of high-field magnets.

## Example: Niobium Wire

In an experiment, a niobium (Nb) wire of radius 0.25 mm is immersed in liquid helium (\(T=4.2\;\text{K}\)) and required to carry a current of 300 A. Does the wire remain superconducting?

### Strategy

The applied magnetic field can be determined from the radius of the wire and current. The critical magnetic field can be determined from here, the properties of the superconductor, and the temperature. If the applied magnetic field is greater than the critical field, then superconductivity in the Nb wire is destroyed.

### Solution

At \(T=4.2\;\text{K},\) the critical field for Nb is, from here and this table,

\({B}_{\text{c}}(4.2\;\text{K})={B}_{\text{c}}(0)[1-{\left(\cfrac{4.2\;\text{K}}{9.3\;\text{K}}\right)}^{2}]=(0.20\;\text{T})(0.80)=0.16\;\text{T.}\)

In an earlier tutorial, we learned the magnetic field inside a current-carrying wire of radius *a* is given by

\(B=\cfrac{{\mu }_{0}I}{2\pi a},\)

where *r* is the distance from the central axis of the wire. Thus, the field at the surface of the wire is \(\cfrac{{\mu }_{0}{I}_{r}}{2\pi a}.\) For the niobium wire, this field is

\(B=\cfrac{(4\pi \;×\;{10}^{-7}\;\text{T m/A})(300\;\text{A})}{2\text{π}(2.5\;×\;{10}^{-4}\;\text{m})}=0.24\;\text{T}\text{.}\)

Since this exceeds the critical 0.16 T, the wire does not remain superconducting.

### Significance

Superconductivity requires low temperatures and low magnetic fields. These simultaneous conditions are met less easily for Nb than for many other metals. For example, aluminum superconducts at temperatures 7 times lower and magnetic fields 18 times lower.