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Critical field
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For a given temperature, the critical field refers to the maximum magnetic field strength below which a material remains superconducting. Superconductivity is characterized both by perfect conductivity (zero resistance) and by the complete expulsion of magnetic fields (the Meissner effect). Changes in either temperature or magnetic flux density can cause the phase transition between normal and superconducting states.[1] The highest temperature under which the superconducting state is seen is known as the critical temperature. At that temperature even the weakest external magnetic field will destroy the superconducting state, so the strength of the critical field is zero. As temperature decreases, the critical field increases generally to a maximum at absolute zero.
For a type-I superconductor the discontinuity in heat capacity seen at the superconducting transition is generally related to the slope of the critical field () at the critical temperature ():[2]
There is also a direct relation between the critical field and the critical current – the maximum electric current density that a given superconducting material can carry, before switching into the normal state.[1] According to Ampère's law any electric current induces a magnetic field, but superconductors exclude that field. On a microscopic scale, the magnetic field is not quite zero at the edges of any given sample – a penetration depth applies. For a type-I superconductor, the current must remain zero within the superconducting material (to be compatible with zero magnetic field), but can then go to non-zero values at the edges of the material on this penetration-depth length-scale, as the magnetic field rises.[2] As long as the induced magnetic field at the edges is less than the critical field, the material remains superconducting, but at higher currents, the field becomes too strong and the superconducting state is lost. This limit on current density has important practical implications in applications of superconducting materials – despite zero resistance they cannot carry unlimited quantities of electric power.
The geometry of the superconducting sample complicates the practical measurement of the critical field[2] – the critical field is defined for a cylindrical sample with the field parallel to the axis of radial symmetry. With other shapes (spherical, for example), there may be a mixed state with partial penetration of the exterior surface by the magnetic field (and thus partial normal state), while the interior of the sample remains superconducting.
Type-II superconductors allow a different sort of mixed state, where the magnetic field (above the lower critical field ) is allowed to penetrate along cylindrical "holes" through the material, each of which carries a magnetic flux quantum. Along these flux cylinders, the material is essentially in a normal, non-superconducting state, surrounded by a superconductor where the magnetic field goes back to zero. The width of each cylinder is on the order of the penetration depth for the material. As the magnetic field increases, the flux cylinders move closer together, and eventually at the upper critical field , they leave no room for the superconducting state and the zero-resistivity property is lost.
Upper critical field
[edit]The upper critical field is the magnetic flux density (usually expressed with the unit tesla (T)) that completely suppresses superconductivity in a type-II superconductor at 0 K (absolute zero).
More properly, the upper critical field is a function of temperature (and pressure) and if these are not specified, absolute zero and standard pressure are implied.
Werthamer–Helfand–Hohenberg theory predicts the upper critical field (Hc2) at 0 K from Tc and the slope of Hc2 at Tc.
The upper critical field (at 0 K) can also be estimated from the coherence length (ξ) using the Ginzburg–Landau expression: Hc2 = Φ0 / (2πξ2)[3] where Φ0 is the magnetic flux quantum.
Lower critical field
[edit]The lower critical field is the magnetic flux density at which the magnetic flux starts to penetrate a type-II superconductor.
References
[edit]Critical field
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Fundamentals of Superconductivity
Historical Discovery
The discovery of superconductivity is credited to Dutch physicist Heike Kamerlingh Onnes, who in 1911 observed that the electrical resistance of mercury abruptly vanished at temperatures near 4.2 K while investigating the properties of materials at low temperatures using recently liquefied helium.[4] This breakthrough, reported in his paper "The resistance of pure mercury at helium temperatures," marked the first identification of a state where electrical conductivity becomes infinite, prompting further studies into its underlying mechanisms.[5] By 1913, Onnes had extended his investigations to the effects of magnetic fields on this new state, finding that even modest fields could completely suppress superconductivity in mercury, lead, and tin, thereby introducing the concept of a critical magnetic field beyond which the superconducting transition is destroyed.[5] These early experiments revealed that the strength of this critical field varied with temperature, decreasing as the temperature approached the critical value where superconductivity onset occurs, with Onnes plotting curves that showed a characteristic parabolic dependence peaking near absolute zero.[6] This temperature sensitivity highlighted the delicate balance of the superconducting phase and laid the groundwork for understanding field-induced transitions. A pivotal advancement came in 1933 when German physicists Walther Meissner and Robert Ochsenfeld conducted experiments on superconducting tin and lead samples, demonstrating that upon cooling below the critical temperature in the presence of an applied magnetic field, the material expels the field from its interior—a phenomenon now known as the Meissner effect.[6] This observation of perfect diamagnetism not only confirmed superconductivity as a thermodynamic equilibrium state but also directly linked it to critical field behavior, as the expulsion persists only up to the critical field strength, beyond which the material reverts to its normal state.[6] In 1935, brothers Fritz and Heinz London developed the first phenomenological theory of superconductivity, proposing equations that described the electromagnetic response of superconductors and explained both persistent currents and the Meissner effect as unified aspects of the same underlying physics.[6] Their work introduced key concepts like the penetration depth, over which magnetic fields decay inside the superconductor, providing an early framework for modeling critical field limits without relying on microscopic details.[6]Core Properties
Superconductivity is a quantum mechanical phenomenon observed in certain materials where electrical resistance drops to zero, allowing current to flow without energy loss, provided the temperature is below a critical value known as the critical temperature, $ T_c $.[7] This state also features perfect diamagnetism, meaning the material repels external magnetic fields completely from its interior.[7] The transition to superconductivity occurs as a second-order phase change at $ T_c $, where the material shifts abruptly from its normal conducting state to the superconducting state, with thermodynamic properties like specific heat showing a discontinuity at this point. A defining characteristic of the superconducting state is the Meissner effect, discovered in 1933, in which a superconductor expels all magnetic flux from its interior upon entering the superconducting phase, regardless of whether the field was applied before or after cooling below $ T_c $.[8] This perfect diamagnetism arises from the formation of persistent screening currents at the surface that precisely cancel the internal magnetic field, maintaining zero field inside the material.[8] The Meissner effect distinguishes superconductivity from perfect conductivity alone, as it implies an active expulsion rather than mere persistence of flux.[7] The presence of an external magnetic field limits the superconducting state, which persists only up to a certain field strength, broadly termed the critical field, beyond which the material reverts to its normal state. Superconductors are classified into type-I and type-II based on their response to magnetic fields: type-I superconductors exhibit a sharp, complete transition from the Meissner state to the normal state at the critical field, while type-II superconductors allow partial magnetic flux penetration in a mixed state of vortices before fully transitioning.[9] This distinction arises from differences in material parameters like the Ginzburg-Landau value, with type-I having and type-II having higher values enabling the intermediate state.[10] The critical field exhibits a temperature dependence approximated by $ H_c(T) = H_c(0) \sqrt{1 - (T/T_c)^2} $, decreasing from its maximum value at absolute zero to zero at $ T_c $. Near $ T_c $, $ H_c(T) $ is proportional to $ \sqrt{1 - T/T_c} $, reflecting the square-root dependence of the superconducting order parameter on $ 1 - T/T_c $.[1]Classification of Critical Fields
Thermodynamic Critical Field
The thermodynamic critical field $ H_c $ is defined as the applied magnetic field strength at which the Gibbs free energy densities of the superconducting and normal phases become equal, resulting in a first-order phase transition from the superconducting to the normal state.[11] This equilibrium condition implies that below $ H_c $, the superconducting phase minimizes the free energy, while above it, the normal phase does so.[12] This critical field applies primarily to type-I superconductors, in which superconductivity is destroyed abruptly when the field exceeds $ H_c $, without intermediate mixed phases.[13] The temperature dependence of $ H_c $ follows an approximately parabolic form:
where $ H_c(0) $ is the critical field at absolute zero and $ T_c $ is the critical temperature; this relation arises from thermodynamic measurements and holds well near $ T_c $.[14] The condensation energy density, representing the free energy difference between the normal and superconducting states at zero field, is given by $ \frac{1}{2} \mu_0 H_c^2 $, which quantifies the energetic favorability of the superconducting phase and links directly to the scale of $ H_c $.[15]
Examples of type-I superconductors exhibiting this behavior include pure metals such as lead (Pb), with $ H_c(0) \approx 0.080 $ T, and tin (Sn), with $ H_c(0) \approx 0.031 $ T; these values are typical for elemental type-I materials, ranging from 0.01 to 0.1 T at low temperatures.[16] Near $ H_c $ in type-I superconductors, an intermediate state forms in which normal and superconducting regions coexist spatially to minimize the total free energy, often manifesting as domains that allow partial flux penetration without fully destroying superconductivity.[17] This state was first proposed by Gorter and Casimir based on magnetization observations in cylindrical samples.[18]
Lower Critical Field
The lower critical field, denoted $ H_{c1} $, is the magnetic field strength at which magnetic flux begins to penetrate a type-II superconductor through the formation of Abrikosov vortices, transitioning from the Meissner state to the mixed state.[19] In type-II superconductors, $ H_{c1} $ satisfies $ H_{c1} < H_c < H_{c2} $, with $ H_c $ representing the thermodynamic critical field.[19] This field marks the point where the energy cost of introducing a single vortex becomes favorable compared to the Meissner state, allowing quantized flux lines to enter the material.[19] The approximate formula for $ H_{c1} $ in SI units is given by
where $ \Phi_0 = 2.07 \times 10^{-15} $ Wb is the magnetic flux quantum, $ \lambda $ is the London penetration depth, and $ \xi $ is the coherence length.[19] Below $ H_{c1} $, the superconductor expels all magnetic field (perfect diamagnetism, $ B = 0 $ inside); above $ H_{c1} $, a vortex lattice forms, with each vortex carrying one flux quantum and normal cores of radius approximately $ \xi $, surrounded by supercurrents decaying over distance $ \lambda $.[19]
The magnitude of $ H_{c1} $ depends strongly on material parameters such as $ \lambda $ and $ \xi T_c $ cuprate superconductors where defects and anisotropies raise the effective penetration threshold.[20] In such materials, pinning energies can exceed the intrinsic thermodynamic $ H_{c1} $, delaying observable flux entry and enhancing performance in applied fields.[20]
Experimentally, $ H_{c1} $ is determined from magnetization curves, where the onset of flux penetration appears as a kink or deviation from linear perfect diamagnetism ($ M/H = -1 $ in SI units) toward reversible flux entry in the mixed state.[21] These measurements, often using vibrating sample or SQUID magnetometers, reveal $ H_{c1} $ values on the order of 10–100 mT at low temperatures for conventional type-II materials like NbTi.[21]
Upper Critical Field
The upper critical field, denoted $ H_{c2} $, represents the maximum magnetic field strength beyond which superconductivity is entirely suppressed in type-II superconductors, primarily through orbital pair-breaking mechanisms that disrupt the formation and stability of Cooper pairs by inducing cyclotron motion of the paired electrons.[22] This field marks the boundary where the superconducting order parameter vanishes, transitioning the material fully to the normal state, and it is distinct from the lower critical field $ H_{c1} $, which initiates flux penetration.[23] Within the Ginzburg-Landau phenomenological framework, the upper critical field is expressed as
where $ \Phi_0 = 2.07 \times 10^{-15} $ Wb is the magnetic flux quantum, $ \mu_0 = 4\pi \times 10^{-7} $ H/m is the permeability of free space, and $ \xi $ is the superconducting coherence length, which characterizes the spatial extent over which the superconducting wavefunction varies.[24] This relation arises from the condition that the lowest Landau level energy matches the superconducting condensation energy, setting the scale for pair disruption.[25]
The temperature dependence of $ H_{c2} $ is approximately linear near the critical temperature $ T_c $, following $ H_{c2}(T) \approx \left. \frac{dH_{c2}}{dT} \right|{T_c} (T_c - T) $, where the slope $ \frac{dH{c2}}{dT} |{T_c} $ is typically negative and material-specific, reflecting the weakening of superconductivity as temperature approaches $ T_c T_c $ behavior provides a key experimental probe for coherence length estimation via $ \xi(T) \propto [H{c2}(T)]^{-1/2} $.[26]
In certain materials, particularly those with significant spin susceptibility, the Pauli paramagnetic limit imposes a constraint on $ H_{c2} $ through spin polarization of the Cooper pairs, limiting the field to below the orbital value predicted by the above formula; this limit is quantified as $ H_p \approx 1.86 , T_c $ (with $ H_p $ in tesla and $ T_c $ in kelvin).[27] For instance, in weak-coupling BCS superconductors, this paramagnetic effect can cap $ H_{c2} $ at values much lower than orbital limits, influencing the overall phase diagram.[28]
Practical examples include NbTi alloys, which exhibit $ H_{c2} $ values up to 15 T at 4.2 K, enabling their widespread use in high-field applications like MRI magnets where fields of 1.5–3 T are generated at liquid helium temperatures.[29] In high-$ T_c $ cuprate superconductors, $ H_{c2} $ displays pronounced anisotropy due to the layered crystal structure, with values often exceeding 100 T parallel to the ab-plane but dropping to 20–50 T along the c-axis, reflecting directional variations in the coherence length and pairing symmetry.[30]
Theoretical Frameworks
Ginzburg-Landau Theory
The Ginzburg-Landau theory, formulated by Vitaly L. Ginzburg and Lev D. Landau in 1950, offers a phenomenological framework for describing superconductivity, particularly effective near the critical temperature . This approach treats the superconducting state through a complex scalar order parameter , where represents the density of the superconducting component, interpreted as the density of Cooper pairs. The theory expands the Gibbs free energy in powers of and its gradients, enabling the derivation of key superconducting properties without relying on microscopic details.[31] Central to the theory is the Ginzburg-Landau free energy functional, expressed in SI units as
where with , are phenomenological coefficients, is the effective mass of Cooper pairs, is the magnetic vector potential (), and the integral is over the superconductor volume. Below , , favoring a nonzero . The theory's equations arise from minimizing this functional with respect to and , yielding the nonlinear Ginzburg-Landau equation for and a modified Maxwell equation for the current.
Minimizing the free energy in the absence of a magnetic field gives the equilibrium order parameter and a condensation energy density of . This energy balances the magnetic field energy at the thermodynamic critical field , where the superconducting and normal states have equal free energy, leading to the phenomenological relation (using below ). For type-II superconductors, characterized by the Ginzburg-Landau parameter (with the London penetration depth and the coherence length), the theory predicts distinct lower () and upper () critical fields through solutions to the full equations.
The upper critical field marks the field strength above which superconductivity vanishes entirely. Its derivation involves linearizing the Ginzburg-Landau equation near , where is small, transforming the problem into an eigenvalue equation resembling quantum mechanics in a magnetic field (Landau levels). The lowest eigenvalue solution yields , where is the flux quantum and is the coherence length. This result highlights the theory's ability to predict the onset of the normal state in high fields for type-II materials.[31]
Despite its successes, the Ginzburg-Landau theory is limited to temperatures close to , where the order parameter expansion is valid, and remains phenomenological rather than microscopic, lacking direct insight into pairing mechanisms.