Experiment: Superconductivity
Zero resistance and flux expulsion as measured. This chapter takes the phenomena of Superconductivity and Superfluidity and records the measurements that establish them: the Leiden helium liquefaction that made the temperature range accessible [Onnes:1908], the 1911 mercury resistance run [Onnes:1911], the persistent-current bounds that convert “very small resistance” into a quantitative upper limit [File:1963], the flux-expulsion experiment of Meissner and Ochsenfeld [Meissner:1933], the isotope effect that identified the phonon as the glue [Maxwell:1950] [Reynolds:1950], the tunnelling spectroscopy that measured the gap [Giaever:1960], the flux quantum that measured the carrier charge [Deaver:1961], the Josephson verification [Anderson:1963a], and the 1986 discovery run that broke the temperature ceiling [Bednorz:1986].
Each of these will be written up in the structured experiment
environment of the house style, with its apparatus, procedure,
observations in SI units with uncertainties, interpretation and
primary references, so that a reader can see exactly which number
carries which inference. The chapter is placed immediately after the
theory it tests and cross-references it section by section; the
low-temperature techniques it depends on belong to
Statistical Mechanics and
Classical Thermodynamics, and the metrological use of
the results feeds Measurement, SI Units, and the Theory of Errors.
Experiment: Superconductivity: all derivations and data tables of this chapter are pending.
Historical context and the claim under test
The race to liquefy the permanent gases
[Reserved: Dewar's liquefaction of hydrogen and the vacuum flask [Dewar:1898]; Kamerlingh Onnes' liquefaction of helium in 1908, reaching \(4.2\,\mathrm{K}\) at atmospheric pressure and about \(1.5\,\mathrm{K}\) by pumping [Onnes:1908]; why the new temperature decade, and not any new idea about metals, is what made the discovery possible.]
Three rival predictions for the low-temperature resistance
[Reserved: the state of the question in 1911 — Matthiessen's empirical rule, that impurity and thermal contributions to the resistivity simply add, predicted saturation at a finite residual value [Matthiessen:1864]; a rival expectation held that conduction electrons would condense onto their ions and the resistance rise without bound; a third, that it would fall smoothly to zero. The experiment discriminates, and none of the three anticipated a discontinuity.]
Mercury as the test material
[Reserved: why mercury — repeated distillation gives the purest available specimen, and freezing it in a capillary yields a thin, high-resistance thread with well-defined geometry; the sample geometry and the resistance at \(273\,\mathrm{K}\) as the normalization used throughout the 1911 papers [Onnes:1911].]
Apparatus
The Leiden cryostat and the helium cycle
[Reserved: the cascade liquefier, the Hampson–Linde regenerative cycle, the helium bath and its vapour-pressure thermometry [Onnes:1908]; the temperature scale in use and its uncertainty, which is the dominant systematic in the 1911 result; modern replacements — pumped helium-3, dilution refrigerators and adiabatic nuclear demagnetization — and the temperature ranges each reaches.]
Resistance measurement
[Reserved: the four-terminal method, separating current leads from voltage leads so that lead and contact resistance do not enter; galvanometer sensitivity and the resulting resolution floor, which is what sets the upper limit quoted rather than any property of the sample [Onnes:1911]; the modern equivalent using a superconducting quantum interference device as a null detector [Clarke:2004].]
Magnetometry for flux expulsion
[Reserved: the pick-up coils and ballistic galvanometer with which Meissner and Ochsenfeld mapped the field distribution outside tin and lead cylinders [Meissner:1933]; the modern flux transformer coupled to a SQUID [Clarke:2004]; specimen geometry and the demagnetizing factor, without which the intermediate state makes the measured susceptibility depend on shape.]
Tunnel junctions
[Reserved: the evaporated aluminium–oxide–lead sandwich, with an oxide barrier of order \(2\,\mathrm{nm}\), used by Giaever to measure the gap [Giaever:1960]; the same structure at lower barrier resistance as the Josephson junction of Anderson and Rowell [Anderson:1963a]; microwave coupling for the ac effect [Shapiro:1963]; the ring geometries used for flux quantization [Deaver:1961] [Doll:1961] and quantum interference [Jaklevic:1964].]
Procedure
The 1911 mercury run
[Reserved: cooling in steps with the resistance read at each plateau; the sequence of communications from Leiden through 1911 in which the resistance was first reported to fall faster than expected, then to vanish, then to vanish discontinuously at a sharply defined temperature [Onnes:1911]; the short-circuit hypothesis that had to be excluded, and the control measurements that excluded it.]
Field-cooled and zero-field-cooled protocols
[Reserved: the protocol that makes the Meissner experiment decisive — apply the field first, then cool through \(T_{c}\), and watch the flux leave [Meissner:1933]; contrast with cooling first and then applying the field, which a perfect conductor also passes; the reversibility check on cycling, which is the evidence that the superconducting state is a thermodynamic phase.]
Meissner and Ochsenfeld mapped the field outside tin and lead cylinders with pick-up coils and a ballistic galvanometer, under two protocols that differ only in their order of operations. Cooled through \(T_{c}\) in zero field and then exposed to a field, the specimen excludes it. Exposed to the field first and then cooled through \(T_{c}\), the specimen expels the flux it already contained, and the exterior field redistributes accordingly [Meissner:1933]. The two protocols end in the same state, and cycling the temperature or the field reproduces it. Only the second protocol discriminates, and it is the whole content of the experiment.
Derivation. Suppose the only property of the low-temperature state were infinite conductivity. A finite current density in a medium of zero resistivity requires \(\vect{E}=0\) inside it, and Faraday's law of The Maxwell Equations then gives
so the interior flux is frozen at whatever value it had at the instant the resistance vanished. Equation (131.1) is a constraint on the time derivative alone, and it therefore makes the final state depend on the path taken to reach it: cooled in zero field the specimen would keep \(B=0\), cooled in a field it would keep \(B=B_{\mathrm{applied}}\) indefinitely. The observation is that the field-cooled specimen ends with \(B=0\) in its interior, exactly as the zero-field-cooled one does. That excludes Equation (131.1) as a complete description and requires a constitutive relation constraining \(\vect{B}\) itself, which is the London relation of Superconductivity and Superfluidity. The wider consequence is thermodynamic rather than electromagnetic: a state whose properties are independent of the history by which it was prepared is a phase, so the free energies of the normal and superconducting states may be compared, and the critical field measures their difference.
∎Isotope separation and the mercury series
[Reserved: preparation of mercury samples of mean mass number from about 198 to 204 by isotope separation; measurement of \(T_{c}\) for each under otherwise identical conditions [Maxwell:1950] [Reynolds:1950]; the systematic errors that dominate — purity, strain and thermometry — and why two independent groups publishing back to back matters for a claim this consequential.]
Observations
The mercury transition
[Reserved: a table of resistance against temperature through the transition, in ohms and kelvin with uncertainties, normalized to the \(273\,\mathrm{K}\) value; the transition width of a few \(\mathrm{mK}\); the observed \(T_{c}\) of about \(4.2\,\mathrm{K}\) and the drop by more than four orders of magnitude across it [Onnes:1911]; the resistance after the transition quoted as an upper limit set by the galvanometer, not as a measurement.]
Cooling a frozen mercury thread in the Leiden helium cryostat and reading its resistance at each plateau, Onnes found the resistance to fall, near \(4.2\,\mathrm{K}\), by more than four orders of magnitude within a temperature interval of a few \(\mathrm{mK}\), to a value his galvanometer could not distinguish from zero [Onnes:1911]. The three features that make the run decisive are the sharpness, the size of the drop, and the fact that the resistance above the transition behaves normally. What the measurement establishes on its own is an upper limit on the resistance below \(T_{c}\), set by the sensitivity of the detector and not by any property of the specimen — the “zero” of the textbooks is an inference, and the honest reading of the run is that the mercury sample became a conductor whose resistance the apparatus could not detect.
The inference chain from the raw run: how a four-terminal measurement removes lead and contact resistance so that the reading belongs to the specimen; how the galvanometer sensitivity and the measuring current together convert into the quoted upper limit on the resistance; and the control observations that exclude the obvious alternative, a short circuit developing on cooling, by their dependence on measuring current and on applied magnetic field
Bounds on the residual resistivity
[Reserved: the persistent-current experiment — a superconducting solenoid short-circuited on itself, its field monitored for months; the decay-time lower bound of order \(10^{5}\,\mathrm{yr}\) reported by File and Mills [File:1963], which converts to a resistivity upper limit some seventeen orders of magnitude below that of copper at the same temperature; the honest statement that “zero” is an inference from an upper limit plus the theory of Superconductivity and Superfluidity, not a direct measurement.]
A superconducting solenoid carrying a current and then short-circuited on itself maintains that current, monitored through its magnetic field, with no detectable decay over months of observation. The bound so obtained on the decay time is of order \(10^{5}\,\mathrm{yr}\) [File:1963], which is the strongest available statement that the resistance is not merely small. Converted into a resistivity, the limit lies some seventeen orders of magnitude below that of copper at the same temperature. Note the structure of the claim: this is an experiment that reports how much of nothing happened, and it is therefore the persistence, not the sensitivity of any voltmeter, that carries the inference.
Derivation. A closed loop of self-inductance \(L\) and series resistance \(R\) carrying current \(I\) with no applied electromotive force obeys
Observing over a time \(t\) that the current has not changed by more than a fraction \(\varepsilon\) bounds the decay by \(t/\tau<-\ln(1-\varepsilon)\approx\varepsilon\), that is \(\tau>t/\varepsilon\), and therefore
The measurement is powerful because both factors on the right of Equation (131.3) work in its favour: \(\varepsilon\) is set by the sensitivity of a magnetometer to a fractional change, which is far better than any absolute voltage measurement, and \(t\) can be made arbitrarily long simply by waiting. Dividing \(R\) by the geometric factor of the winding converts Equation (131.3) into a resistivity, which is the seventeen-order-of-magnitude figure quoted. What remains true is that no experiment can measure a resistance to be exactly zero; the statement that it is exactly zero comes from the theory of Superconductivity and Superfluidity, in which a gapped condensate has no state into which to scatter, and the experiment bounds the alternative.
∎Critical temperatures and critical fields
[Reserved: a booktabs table of \(T_{c}\) and \(\mu_{0}H_{c}(0)\) for the elemental superconductors and the common alloys — aluminium, tin, lead, niobium, niobium–titanium, niobium–tin — in kelvin and tesla with uncertainties, from the critically evaluated compilation [Roberts:1976]; the upper critical field \(H_{c2}\) for the type II entries [Abrikosov:1957]; the parabolic \(H_{c}(T)\) fit and its residuals.]
The isotope exponent
[Reserved: \(T_{c}\) against mean isotopic mass for mercury on logarithmic axes, and the fitted exponent \(\alpha=0.50\pm0.03\) in \(T_{c}\propto M^{-\alpha}\) [Maxwell:1950] [Reynolds:1950]; tabulated exponents for other elements, including the anomalously small values in ruthenium, osmium and zirconium, and what those depress the naive reading to; the inference that the lattice sets the energy scale.]
Mercury samples of mean mass number from about 198 to 204, prepared by isotope separation and measured under otherwise identical conditions, have transition temperatures falling with mass. Plotted logarithmically, \(\log T_{c}\) against \(\log M\) is a straight line of slope \(-\alpha\) with
reported back to back by two groups working independently [Maxwell:1950] [Reynolds:1950]. The independence matters for a claim of this consequence, and so does the smallness of the effect: a three per cent change in mass produces a change in \(T_{c}\) of about one and a half per cent, against systematic errors from sample purity, strain and thermometry of comparable size. Several transition metals give exponents well below one half.
The inference from the measured isotope exponent: that a lattice frequency scales as the inverse square root of the ionic mass with the force constants, which are electronic, held fixed, so that an exponent of one half identifies the phonon energy as the prefactor of the pairing scale; and the corrections from the screened Coulomb repulsion, which enter the exponent and depress it below one half without altering the identification
The gap
[Reserved: three independent determinations to be tabulated together in reduced units \(2\Delta(0)/k_{B}T_{c}\) — the exponential low-temperature electronic specific heat of vanadium [Corak:1954], the far-infrared absorption edge [Glover:1957], and the tunnelling conductance [Giaever:1960], whose \(\dd I/\dd V\) maps the quasiparticle density of states directly; the BCS value \(3.53\) and the measured spread; the Hebel–Slichter peak in nuclear spin relaxation [Hebel:1959] against the fall in ultrasonic attenuation [Morse:1957] as the coherence-factor test.]
The gap of a superconductor is measured by three techniques that share no apparatus and no systematic errors: the exponential fall of the electronic specific heat at low temperature [Corak:1954], the threshold in far-infrared absorption [Glover:1957], and the threshold voltage in the tunnelling characteristic of a thin oxide junction, whose differential conductance maps the quasiparticle density of states [Giaever:1960]. Expressed in the reduced form \(2\Delta(0)/k_{B}T_{c}\) the three agree with one another, and cluster about the value \(3.5\) across the weak-coupling elemental superconductors. A related pair of measurements tests the theory more sharply than the gap does: on cooling through \(T_{c}\) the ultrasonic attenuation falls immediately [Morse:1957] while the nuclear spin relaxation rate first rises to a peak before falling [Hebel:1959]. Two processes governed by the same excitation spectrum move in opposite directions, and the sign difference is predicted.
The coherence factors: that the matrix element for a process between quasiparticle states is a sum of two amplitudes whose relative sign depends on the behaviour of the perturbation under time reversal, so that the two combinations cancel and reinforce respectively; and the resulting opposite temperature dependences of ultrasonic attenuation and nuclear spin relaxation just below the transition, against the divergent density of states at the gap edge that both share
The flux quantum
[Reserved: trapped flux against applied field for a superconducting cylinder, showing steps of constant height; the measured quantum \(\Phi_{0}=2.07\times 10^{-15}\,\mathrm{Wb}\) to within a few percent in both 1961 experiments [Deaver:1961] [Doll:1961]; the arithmetic that turns this into a carrier charge of \(2e\), and why that is direct evidence of pairing independent of any microscopic theory; the Little–Parks periodicity as a cross-check [Little:1962].]
Trapped flux measured against applied field for a small superconducting cylinder does not follow the applied field but rises in steps of constant height. Two independent 1961 experiments, using different geometries and different detection schemes, measured the step to be \(\Phi_{0}=2.07\times 10^{-15}\,\mathrm{Wb}\) to within a few per cent [Deaver:1961] [Doll:1961]. The number is half of \(h/e\), and the factor of two is the point of the measurement.
Derivation. Single-valuedness of the condensate phase round the cylinder quantizes the enclosed flux in units of \(h/q\), where \(q\) is the charge of the entity whose phase is in question — the argument is given in Superconductivity and Superfluidity and uses no microscopic theory. Reading it backwards, the measurement determines \(q\):
using the exact 2019 values of \(h\) and \(e\) [BIPM:2019] [Mohr:2025]. At the several-per-cent accuracy of the 1961 experiments this discriminates cleanly between \(q=e\) and \(q=2e\), which differ by a factor of two, and it selects the pair. That inference is worth isolating: it establishes that the charge carriers of a superconductor have twice the electronic charge, from a measurement of a flux and a ring, four years after the pairing theory was proposed and independently of it. The periodicity of the transition temperature of a thin-walled cylinder in the enclosed flux, with the same period, is an independent check on the same quantity [Little:1962].
∎Josephson currents and Shapiro steps
[Reserved: the zero-voltage supercurrent through an oxide barrier and its Fraunhofer-pattern suppression by an applied field, which is what distinguished the effect from a metallic short [Anderson:1963a]; the constant-voltage steps induced by microwave irradiation, spaced by \(hf/2e\) [Shapiro:1963]; frequency-to-voltage conversion at \(483.6\,\mathrm{MHz}/\mu\mathrm{V}\) and the agreement of \(2e/h\) so obtained with other determinations.]
A tunnel junction irradiated with microwaves of frequency \(f\) develops, in its current–voltage characteristic, a series of vertical steps at which the voltage does not change while the current does. The steps sit at
and their spacing depends on the microwave frequency and on nothing else — not the material, the junction area, the temperature or the barrier thickness [Shapiro:1963]. That the zero-voltage current seen without irradiation is a genuine supercurrent and not a metallic short is established separately, by its suppression in a magnetic field in a diffraction-like pattern [Anderson:1963a]. A frequency, which is the most accurately measurable quantity in physics, is thereby converted into a voltage by fundamental constants alone.
Derivation. The second Josephson relation, \(\dd\varphi/\dd t=2eV/\hbar\), makes the phase difference across the junction advance at a rate set by the voltage, so a constant voltage \(V\) produces a supercurrent \(I_{c}\sin\varphi\) alternating at the Josephson frequency \(f_{J}=2eV/h\). Under irradiation the junction is driven at \(f\) as well, and the average current is non-zero only when the two are commensurate, \(f_{J}=nf\); the junction then phase-locks and holds that average voltage over a finite range of bias current, which is the step. Setting \(2eV_{n}/h=nf\) gives Equation (131.6). The conversion factor is the Josephson constant
exact in the present SI because \(h\) and \(e\) are exact [BIPM:2019] [Mohr:2025]. Two properties make Equation (131.6) a metrological standard rather than a curiosity. The step voltage is a ratio of a frequency to a constant, so it inherits the accuracy of the frequency reference; and nothing about the junction appears in it, so thousands of junctions may be placed in series to reach a usable voltage without introducing a single sample-dependent parameter.
∎Interpretation
What each measurement establishes
[Reserved: an explicit inference table — resistance run [Onnes:1911] plus persistent currents [File:1963] give dissipationless flow; flux expulsion [Meissner:1933] gives a thermodynamic phase, not a memory of history; the isotope effect [Maxwell:1950] [Reynolds:1950] gives a phonon-mediated interaction; gap spectroscopy [Giaever:1960] [Glover:1957] gives a gapped excitation spectrum; flux quantization [Deaver:1961] gives carriers of charge \(2e\); the Josephson effects [Anderson:1963a] [Shapiro:1963] give a single coherent phase across a macroscopic sample. Together these are the observational content of the BCS theory [Bardeen:1957].]
What the 1986 discovery did not settle
[Reserved: the Bednorz–Müller run in Ba–La–Cu–O, its resistive onset near \(35\,\mathrm{K}\) and the cautious “possible” in its title [Bednorz:1986]; the confirmation by Meissner measurement, which is the step that turns a resistive onset into a superconductivity claim; the yttrium compound above the boiling point of nitrogen at \(93\,\mathrm{K}\) [Wu:1987]; the honest statement that these measurements establish the phenomenon and its \(d\)-wave order parameter [Wollman:1993] [Tsuei:1994] while leaving the mechanism open — see What We Observe but Do Not Understand.]
A ceramic oxide of barium, lanthanum and copper — not a metal by any of the criteria that had guided the search — shows a resistive onset near \(35\,\mathrm{K}\), above any transition temperature then known [Bednorz:1986]; within months the related yttrium compound reached \(93\,\mathrm{K}\), above the boiling point of liquid nitrogen [Wu:1987]. The methodological point is contained in the cautious “possible” of the 1986 title: a resistive drop alone is not a superconductivity claim, and what converted the observation into one was the subsequent Meissner measurement, the flux expulsion of Phenomenon 131.1 being the property a percolating filament or a measurement artefact cannot imitate. Phase-sensitive experiments later established that the order parameter changes sign between crystallographic directions [Wollman:1993] [Tsuei:1994], so the pairing is not of the simple form that Equation (131.4) describes in mercury. What remains unexplained is the mechanism: no theory predicts the transition temperature of any of these compounds.
What can and cannot be derived here: the demonstration that flux expulsion on field cooling requires a bulk, thermodynamic superconducting phase and excludes a percolating filamentary path; and the reading of the half-integer flux quantum trapped at a tricrystal ring as a sign change of the order parameter between directions. The mechanism itself is not derived anywhere in this treatise and is recorded as an open problem
Modern repetitions and precision
Metrology: the Josephson volt
[Reserved: the programmable Josephson voltage standard, in which thousands of junctions in series give a voltage fixed by a microwave frequency and the exact constant \(K_{J}=2e/h=483597.8484\,\mathrm{GHz}/\mathrm{V}\) [BIPM:2019]; reproducibility at the part in \(10^{10}\) level; consistency with the quantum Hall resistance of Experiment: The Quantum Hall Effect through the CODATA adjustment [Mohr:2025] [Tiesinga:2021]; how this closed the loop from a 1911 curiosity to a base of the SI [BIPM:2019].]
SQUID magnetometry and the London moment
[Reserved: modern SQUID sensitivity and its applications [Clarke:2004]; the London moment — the magnetic moment of a rotating superconductor, proportional to the angular velocity — used as the gyroscope readout in Gravity Probe B, whose measurement of geodetic and frame-dragging precession is reported in The Einstein Field Equations [Everitt:2011]; a superconducting instrument as the enabling technology for a test of general relativity.]
High-pressure hydrides and the standard of proof
[Reserved: the diamond-anvil cell measurement of sulfur hydride with \(T_{c}=203\,\mathrm{K}\) near \(155\,\mathrm{GPa}\), including its isotope shift on deuteration, which is the evidence that the pairing is phonon-mediated [Drozdov:2015]; lanthanum superhydride above \(250\,\mathrm{K}\) [Somayazulu:2019]; the methodological lesson — in a sample of picolitre volume a resistive drop alone is not sufficient, and the community now requires a magnetic-susceptibility signature and raw, unprocessed data, after several near-ambient claims failed replication and were retracted.]
Datasets
[Reserved: the digitized resistance-versus-temperature points of the 1911 mercury run [Onnes:1911], the isotope series [Maxwell:1950] [Reynolds:1950] and the critical-parameter compilation [Roberts:1976] will be stored in the evidence library as comma-separated files keyed to their bibliography entries, so that every figure and fitted exponent in this chapter can be recomputed from the numbers as published.]