Critical fields: thermodynamic, orbital and Pauli scales

Scope

A critical field needs a physical definition. The thermodynamic field $H_c$ represents the zero-field condensation-energy density, $f_n-f_s=\mu_0H_c^2/2$. This energy scale exists for both type-I and type-II superconductors; it is not generally the field at which a type-II sample becomes normal. In the ideal bulk type-II description, $H_{c1}$ marks the equilibrium onset of the mixed state and $H_{c2}$ its upper boundary. The vortex construction distinguishes those boundaries from one thermodynamic energy scale. R1

Here $H$ is magnetic field strength in amperes per metre, while $B$ and $\mu_0H$ are expressed in tesla. Inside matter, $B=\mu_0(H+M)$; replacing $B$ by $\mu_0H$ requires a stated magnetization approximation. An applied-field value, an internal field and a flux density must not silently share the same symbol. The formulas below use SI conventions, rather than copying Gaussian-unit factors from historical papers.

Mathematical assumptions

For a bulk, isotropic, single-component Ginzburg–Landau model near $T_c$, neglecting paramagnetic pair breaking, linearization gives the orbital result

$$B_{c2}^{\mathrm{orb}}=\frac{\Phi_0}{2\pi\xi^2},\qquad \Phi_0=\frac{h}{2e},\qquad \kappa=\frac{\lambda}{\xi}.$$

The usual $\mu_0H_{c2}^{\mathrm{orb}}$ notation uses $B\simeq\mu_0H$ near this continuous boundary when normal-state magnetic response is negligible. In the same model, $H_{c2}^{\mathrm{orb}}=\sqrt{2}\kappa H_c$. The GL coherence length $\xi$ and penetration depth $\lambda$ are evaluated at the same temperature. They are not interchangeable with every microscopic pair-size or fitted length. The isotropic GL type-II boundary is $\kappa=1/\sqrt{2}$. R1

A useful independent check follows from the linearized kinetic operator: the lowest Landau-level energy of a pair with charge magnitude $2e$ is $\hbar(2e)B/(2m^\ast)$. Equating it to $|\alpha|$, with $\xi^2=\hbar^2/(2m^\ast|\alpha|)$, gives the flux-over-area expression above. This is a model derivation, not a measured field or an all-temperature interpolation.

For a separate, purely paramagnetic comparison at zero temperature, take an isotropic weak-coupling singlet with $g=2$, negligible spin susceptibility in the paired state, no orbital cost and no competing modulated state. Equating its condensation energy to the normal-state polarization-energy gain gives

$$B_P=\frac{\Delta_0}{\sqrt{2}\mu_B},\qquad \frac{B_P}{T_c}=\frac{\pi e^{-\gamma}k_B}{\sqrt{2}\mu_B} \simeq 1.86\;\mathrm{T\,K^{-1}}.$$

This is the Clogston–Chandrasekhar reference scale. The commonly quoted $1.84\;\mathrm{T\,K^{-1}}$ uses rounded weak-coupling gap constants; neither coefficient is an exact universal experimental bound. The numerical value here uses the same modern weak-coupling gap convention as the BCS entry. The energy comparison is supported by the cited perspective; the original Clogston paper's identity is checked, but its full text was not available in this review. R4 R5 R6

Experimental criteria

Report the probe and the rule used to locate a field line. Resistance onset, a stated percentage of the normal-state resistance and a voltage-resolution-limited zero need not coincide. Neither a magnetic irreversibility line nor a first flux-entry signal should automatically be renamed $H_{c2}$ or equilibrium $H_{c1}$. Demagnetizing geometry, surface barriers, vortex motion, pinning and inhomogeneous current paths can affect the observed signal.

Record temperature, field orientation and angular uncertainty, sample geometry, pressure where relevant, current density, sweep direction and rate, and the normal-state baseline. Compare transport with an appropriate bulk thermodynamic or magnetic measurement on comparable specimens. Preserve hysteresis and transition widths instead of choosing the most favorable boundary. These are measurement-reporting requirements from the pinned chapter, not a claim that each possible artifact is present in a particular sample.

Limits and disputes

Werthamer–Helfand–Hohenberg theory treats orbital and spin effects within a specified microscopic approximation, including spin–orbit impurity scattering. Its publisher abstract explicitly notes that such scattering counteracts paramagnetic limiting. A fitted WHH curve therefore needs its dirty-limit, spin and scattering assumptions; merely naming the fit does not establish them. This entry's WHH check is abstract-scoped. R2

Multiband structure also changes extrapolation. Gurevich's dirty two-gap model allows $H_{c2}(T)$ shapes controlled by different intraband diffusivities, including deviations from the one-gap slope estimate near $T_c$. That calculation is a counterexample to treating a one-band extrapolation coefficient as universal. It does not validate an arbitrary multiband fit or supply a field value for an unmeasured sample. R3

Exceeding the simple Pauli estimate does not uniquely identify triplet pairing. The spin response and spin–orbit energy scales may differ from the assumed reference model; the cited Ising-superconductivity perspective discusses a concrete route to a strongly anisotropic paramagnetic response. Orbital and spin limitations must be treated together when both matter. No material-specific critical-field record or pairing assignment is introduced here. R5

Discovery implications

Use a field measurement to discriminate between declared models. First decide whether the observation concerns an equilibrium boundary, vortex dynamics or a transport threshold. Then test orientation and temperature dependence over a range where the model applies. Retain the fitting window, parameter uncertainty and alternative descriptions, particularly when extrapolating far below the measured temperatures.

A high critical field may motivate further study, but a useful discovery argument identifies which assumption failed and which new observable could distinguish the remaining explanations. Material comparisons should keep pressure, field direction and criterion attached to each value. A large fitted field alone establishes neither a microscopic pairing mediator nor topology. This is an editorial evaluation protocol; it is not a newly verified experimental result.

Related topics

Read about the Meissner effect, zero-resistance criteria, the BCS baseline, and multiband pairing.

Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.