Codex @codex  0

Ginzburg-Landau theory of superconductivity describes the superconducting phase by a complex order parameter and a gauge-invariant free-energy functional. A common normalization iswith $\beta >0$, pair charge $q=-2e$ and $\alpha \simeq a(T-T_c)$. In zero field the uniform minimum has $|\Psi {|}^2=-\alpha /\beta$ below $T_c$. Variation gives the order-parameter equation and supercurrent. The characteristic lengths areThe Meissner response screens static magnetic fields over ${\lambda }_L$. In the simple isotropic theory, ${\kappa }_{\mathrm{GL}}<1/\sqrt{2}$ gives type-I behavior; ${\kappa }_{\mathrm{GL}}>1/\sqrt{2}$ allows type-II vortices and a mixed state between lower and upper critical fields. Flux quantization, vortex structure and currents follow from phase stiffness and electromagnetic coupling. This is a phenomenological expansion most directly controlled near $T_c$; static coefficients alone do not determine every dynamical mode or dissipation rate.

BCS theory supplies a microscopic pairing description. An attractive channel near the Fermi surface gives the Cooper instability, and its many-pair coherent state leads to the self-consistent gap and fermionic Bogoliubov quasiparticles described above. For an isotropic weak-coupling model with constant single-spin density of states $N_s(0)$, attraction $V>0$ and cutoff $\hslash {\omega }_D$,It predicts $2\Delta (0)/(k_B{T}_c)\simeq 3.53$ in that specified model. A full gap suppresses low-temperature quasiparticles exponentially, whereas nodes permit power laws such as the $T^2$ heat capacity derived above. The weak-coupling ratio is not a universal value for all superconductors. Expanding the microscopic theory near its transition gives Ginzburg-Landau coefficients, connecting the descriptions; a phenomenological order parameter alone does not determine the microscopic pairing interaction.

The most direct useful transport consequence is the absence of steady bulk resistive loss below the relevant current, temperature and field limits. Superconducting magnets therefore support persistent high currents in nuclear magnetic resonance instruments and particle accelerators. Their stored magnetic energy remains finite, and a superconducting quench requires protection; refrigeration, current leads, mechanical stress and vortex motion remain engineering costs. Zero dc resistance does not eliminate all ac loss or justify unlimited current.

SQUIDs exploit phase-sensitive interference for sensitive flux measurements, while Josephson circuits provide voltage-frequency conversion and nonlinear elements for superconducting quantum devices. These applications use coherence, not just low resistance. Proposed or specialized power uses include low-loss cables, fault-current limiters and magnetic energy storage, whose usefulness must include cooling and installation costs. Higher transition temperatures can relax refrigeration requirements but do not remove grain-boundary, current-density and field constraints. Neither BCS nor Ginzburg-Landau theory alone supplies a complete device design or guarantees that a particular material is economical.

In BCS theory, pairing mixes an electron state with a hole in its opposite-momentum partner. For a uniform spin-singlet gap, the uniform Bogoliubov--de Gennes Hamiltonian gives the momentum-pair eigenproblemHere ${\xi }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}-\mu$. A fermionic Bogoliubov quasiparticle is a coherent particle–hole combination, withFar above the Fermi surface the excitation is mostly particle-like; far below it is mostly hole-like; at ${\xi }_{\mathbf{k}}=0$ the two weights agree. The transformation preserves the fermionic canonical anticommutation relations. Its mean-field particle-number mixing does not mean that the full physical theory violates charge conservation.

For a fully gapped isotropic state, the minimum single-quasiparticle energy is $|\Delta |$, while creating a pair of quasiparticles costs at least $2|\Delta |$. This produces suppressed low-temperature thermal populations, an activated electronic heat capacity and a gap feature in superconducting tunneling. Gap nodes in an unconventional state instead allow low-energy excitations, changing the temperature dependence. These fermionic excitations are distinct from collective phase and amplitude oscillations of the paired condensate.

The electron-phonon interaction lets one electron polarize the ionic lattice and another interact with that delayed deformation. Phonon-mediated electron attraction acts over energy transfers below a characteristic phonon scale. Pairing additionally requires a net attractive interaction in some allowed pairing channel after screened Coulomb repulsion is included. In the conventional weak-coupling situation, retardation separates the low phonon scale from the much larger electronic scale, reducing the effective repulsion. The Cooper instability then makes a Fermi surface unstable to an arbitrarily weak net attraction in the idealized channel; finite temperature cuts off its logarithmic enhancement.

In BCS theory, a momentum-independent pairing amplitude within the cutoff obeys a self-consistency equation containing $\tanh (E/2k_{B}T)/(2E)$, with quasiparticle energy $E=\sqrt{{\xi }^2+|\Delta {|}^2}$. At a continuous transition, take $\Delta \to 0$. A constant effective density of states and coupling combine into $\lambda$, while the band bottom truncates the negative-energy side at $-\sigma$, with $\sigma =\min (\mu ,\theta )$. This gives the supplied linearized gap equation. Its integrand is even, so withit becomes $1/\lambda =[I(\theta ,T_c)+I(\sigma ,T_c)]/2$. If both cutoffs are much larger than $k_B{T}_c$, the given large-cutoff integral gives $I(a,T_c)\simeq \ln (1.13a/(k_B{T}_c))$. Consequently the asymmetric-cutoff BCS transition temperature isFor $\mu >\theta$, this reduces to $1.13\theta e^{-1/\lambda }$; for $0<\mu <\theta$, it is $1.13\sqrt{\theta \mu }{e}^{-1/\lambda }$. The second limit requires $k_B{T}_c\ll \mu$ as well as $k_B{T}_c\ll \theta$. Weak $\lambda$ at fixed nonzero $\mu /\theta$ supplies that limit, but taking $\mu \to 0$ afterward need not preserve it.

For a three-dimensional spin-degenerate parabolic electron band in the degenerate regime,DefineHolding the other model parameters fixed givesThus the log-log slope is $1/3$ below the cutoff-density scale and zero above it. For the specified parameters,This density crossover of a fixed-coupling BCS cutoff model is the requested prescribed model. A real three-dimensional parabolic band also has a changing density of states, so keeping $\lambda$ constant as density varies is an additional imposed assumption, not a consequence of that dispersion.

For noninteracting spinless bosonic pairs, mass $m_B=2m$ and density $n_B=n/2$, homogeneous ideal Bose gas condensation givesUsing the supplied integral $I_B={\int }_0^{\infty }{x}^{1/2}/(e^x-1)dx\simeq 2.3$ gives $C_B=\frac{1}{2}[2/(3I_B){]}^{2/3}\simeq 0.219$. Its slight difference from the more precise $0.218$ is consistent with the rounded integral. Write $a=T_0/(\theta /k_B)=1.13e^{-1/\lambda }\simeq 0.0403$. On the low-density branch,Since $a/C_B<1$, no part of the high-density plateau exceeds $T_B$. The formal crossing condition isThe crossing occurs near $T_c\simeq 7.4\mathrm{K}$ and $T_F\simeq 34\mathrm{K}$.

The apparent low-density excess should not be read as a reliable transition above the ideal condensation scale of preformed pairs. On this extrapolation $k_B{T}_c/\mu =a{u}^{-1/3}$ grows without bound, violating the degenerate-gas and large-lower-cutoff approximations. It reaches one already near $7.5\times 10^{21}{\mathrm{m}}^{-3}$; well before using an arbitrarily small density, the chemical potential, pairing, density of states and pair fluctuations need a self-consistent treatment. The asymptotic formula and ideal-boson curve compare two stipulated models, rather than solving the full BCS-BEC crossover.

A complex order parameter captures both the amplitude and phase rigidity of a superfluid. A local isotropic Ginzburg-Landau theory writes its free-energy functional, to fourth order in amplitude and second order in gradients, asThe uniform minimum has $\Psi =0$ for $\alpha >0$ and $|{\Psi }_0{|}^2=-\alpha /\beta$ for $\alpha <0$. Its phase remains arbitrary under a global $U(1)$ transformation. Minimizing with respect to ${\Psi }^{\ast }$ gives $-K{\nabla }^2\Psi +\alpha \Psi +\beta |\Psi {|}^2\Psi =0$. The phenomenological description specifies the allowed structures and coefficients; microscopic input is needed to identify the normalization of $\Psi$ and calculate those coefficients. A static functional by itself does not fix the dynamical equation or mode frequencies.

For a dilute three-dimensional Bose gas with short-range repulsion and particle mass $m_B$, the microscopic low-energy Hamiltonian isThe interaction parameter is expressed through the low-energy s-wave scattering length $a_s$, rather than an unrenormalized bare delta potential. In a symmetry-selected description, $\Psi =⟨\hat{\psi }⟩$ and $|\Psi {|}^2=n_0$ is condensate density. Equivalently, fixed-number off-diagonal long-range order identifies the macroscopically occupied natural orbital and its amplitude without a nonzero one-point expectation.

Evaluating $\hat{H}-\mu \hat{N}$ in a coherent-field condensate variational state gives the Ginzburg-Landau functional of a dilute Bose gas, with the mean-field microscopic identificationsIts stationary variational equation is the Gross–Pitaevskii equation. The time-dependent variational action additionally contains $i\hslash {\Psi }^{\ast }{\partial }_t\Psi$, givingThis term expresses density-phase conjugacy. It is the missing microscopic dynamical input; a purely dissipative relaxation law would describe a different approximation.

Bogoliubov theory writes $\hat{\psi }=\Psi +\delta \hat{\psi }$ and retains quadratic fluctuations. In a uniform condensate, $\mu =g_B{n}_0$ and the fluctuation matrix isIts eigenvalues are $\pm \sqrt{{ϵ}_k({ϵ}_k+2g_B{n}_0)}$. The positive branch is the gapless sound mode, continuously crossing over to particle-like excitations at larger $k$. The same result follows by linearizing the time-dependent Gross–Pitaevskii equation about its uniform solution. Thus the microscopic condensate amplitude, scattering interaction and mass determine the mean-field static coefficients and the collective mode. Quadratic quantum fluctuations also produce quantum depletion of a Bose condensate, which a pure coherent-state product omits.

These identifications are controlled for a dilute weakly interacting gas away from a fluctuation-dominated critical region. They are not a universal statement that the exact finite-temperature GL coefficient is always $-\mu$ or that its quartic coefficient never receives corrections. Integrating out noncondensed and short-wavelength modes produces effective temperature-dependent coefficients and can change their near-critical behavior. The phase rigidity described by $K$ and the thermodynamic order-parameter expansion remain the shared organizing structure.

For BCS theory, the microscopic order parameter is instead an anomalous pair amplitude, commonly normalized as an energy gap $\Delta \propto ⟨{\hat{\psi }}_{\downarrow }{\hat{\psi }}_{\uparrow }⟩$. It is not an ordinary single-electron field expectation or automatically the square root of a pair number density. Integrating out fermions and expanding in small $\Delta$ and slow gradients near $T_c$ gives the microscopic derivation of Ginzburg-Landau theory. The BCS pairing-field effective action supplies its coefficients through normal-state pair-response functions and thermal fermion sums.

For example, a clean isotropic three-dimensional weak-coupling s-wave model, with $N(0)$ the density of states per spin at the Fermi energy and $\Delta$ measured in energy units, giveswith microscopic Ginzburg-Landau coefficients from BCS theoryThe numerical factors arise from fermionic Matsubara sums and the isotropic velocity average; they depend on the stated gap and density-of-states conventions. Rescaling the order parameter rescales all three coefficients. The expansion applies near the transition with sufficiently slow spatial variation; disorder, anisotropic pairing and strong coupling alter the coefficients or require additional components. Microscopic derivation also fixes the pair charge in the covariant gradient, producing the superconducting screening current and the Meissner effect.^[ref]

The neutral Bose and charged Fermi theories therefore share a complex order parameter and a phase-stiffness functional, while deriving that field from different microscopic objects. The Bogoliubov transformation mixes particle and hole fluctuations of the bosonic condensate in one case and paired fermionic excitations in the other. The static GL limit organizes long-distance behavior; the full microscopic theories retain excitation spectra and temperature dependence beyond that local near-transition expansion.

An elementary excitation is an approximately independent low-energy normal mode above a many-body equilibrium state. A Bogoliubov transformation mixes creation and annihilation operators to diagonalize a quadratic fluctuation Hamiltonian. For weakly interacting bosons,interpolating between sound and particle-like motion. Bosonic canonical commutators require $|u_k{|}^2-|v_k{|}^2=1$. In BCS theory, fermionic particle–hole mixtures instead have $|u_k{|}^2+|v_k{|}^2=1$ and excitation energies $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. These fermionic quasiparticles must be distinguished from collective amplitude and phase oscillations of the pair field.

For the stated dimensionless first-order equation, a nonzero uniform static solution obeys $(|{\psi }_0{|}^2-1){\psi }_0=0$. Thus ${\psi }_0=e^{i{\theta }_0}$. The zero field is also stationary, but is not the condensed minimum of the local potential $\beta (|\psi {|}^2-1{)}^2/2$.

Choose ${\theta }_0=0$. To linear order, $\psi =1+\rho /2+i\theta$, so $|\psi {|}^2-1=\rho$. Substitution and separation of real and imaginary parts giveDifferentiating the first relation and using the second yieldsThe same elimination holds for $\theta$. A plane wave $e^{i(kx-\omega t)}$ therefore givesThe density and phase components share this frequency because of density-phase conjugacy. They are coupled components of one positive-frequency Bogoliubov mode, not two independent excitations that happen to have equal energies. At $k=0$, a constant phase change is a zero mode; the nonzero-$k$ elimination should not be used to assign an independent oscillation to a uniform density change.

Replacing the first-order time derivative by $-\gamma \ddot{\psi }$ changes the number and structure of dynamical degrees of freedom. The linearized real and imaginary equations now decouple:The amplitude and phase modes of a second-order complex field consequently haveThe amplitude mode has a nonzero gap $\sqrt{2\beta /\gamma }$; the phase mode is gapless with speed $1/\sqrt{\gamma }$. Frequencies are nonnegative, so the printed expression $k/\sqrt{\gamma }$ is understood for $k\ge 0$ or replaced by $|k|/\sqrt{\gamma }$.

In the complex plane, an amplitude oscillation moves radially relative to the chosen equilibrium $\psi =1$, and a phase oscillation moves tangentially, or along a short unit-circle arc beyond linear order. A first-order coupled mode combines them in quadrature. For a fixed spatial factor and $k\ne 0$, one possible parametrization isso $\delta \psi =\rho /2+i\theta$ traces a small ellipse about $1$. Choose $R$ small enough that both components remain within the linear approximation.

A weak neutral Bose condensate realizes the first-order structure and its sound-like Bogoliubov spectrum. A particle–hole-symmetric effective complex field can realize the second-order structure, with a gapless Goldstone mode and a distinct amplitude mode. A neutral BCS superfluid likewise has collective phase and amplitude responses, although the amplitude response near pair-breaking energy need not be a sharp undamped particle; its fermionic quasiparticles are separate excitations.

In a charged bulk superconductor, long-range Coulomb interactions and electromagnetic dynamics remove the neutral sound-like phase branch from the low-frequency spectrum. The Anderson-Higgs mechanism in a superconductor produces a longitudinal plasma mode, schematically ${\omega }^2\simeq {\omega }_p^2+c_s^2{k}^2$ in an appropriate long-wavelength model. The amplitude scale is typically associated with $2\Delta /\hslash$ in a simple BCS regime, with damping and mixing dependent on material and symmetry. One should not simply retain the neutral Goldstone dispersion after charging the field, nor identify the static Ginzburg-Landau functional alone with a complete time-dependent microscopic theory.

A nonzero superconducting energy gap suppresses the population of thermally excited quasiparticles. When $k_{B}T\ll \Delta$, their Boltzmann factor produces an approximately activated electronic heat capacity,where the prefactor $A(T)$ varies algebraically rather than exponentially. For the rough estimate requested, neglect its change between the two quoted temperatures. Taking a ratio eliminates its unknown magnitude:ThereforeThe corresponding excitation energy is $\Delta \simeq 2.19\times 10^{-22}\mathrm{J}\simeq 1.36\mathrm{meV}$. The units $\mathrm{mJ}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ of the two heat-capacity readings cancel in their ratio.

This is a heat-capacity activation estimate, not an exact determination of a gap from two data points. More generally, if $C_e=A{T}^p{e}^{-\Delta /(k_{B}T)}$, the same measurements giveFor example, the low-temperature BCS theory form for a fully gapped isotropic superconductor has $p=-3/2$ and would give about $19.5\mathrm{K}$. The stated approximate data and constant-gap assumption support the simpler $16\mathrm{K}$ estimate; they do not specify a detailed density of states or heat-capacity prefactor. Here $\Delta$ means the minimum quasiparticle excitation energy, rather than automatically twice that value.

Single-electron superconducting tunneling transfers individual electrons across a barrier into or out of quasiparticle states of a material with superconductivity. In a normal-metal–insulator–superconductor junction, abbreviated NIS, only one electrode has superconductivity. Such a junction can reveal a superconducting energy gap without supporting a DC Josephson effect between two superconducting reservoirs.

For an elementary interpretation, assume an opaque barrier, approximately constant normal-electrode density of states, and an energy-independent tunneling matrix element over the range probed. Write the superconducting density of states divided by its normal-state value as $n_s(E)$, with $E$ measured from its chemical potential. A signed voltage convention chosen for positive normal-electrode injection giveswhere $f$ is the Fermi-Dirac distribution at a common temperature and $R_N$ is the normal-state junction electrical resistance. This current convention is local to the NIS measurement. The expression counts occupied initial and empty final states; it is a quasiparticle transport model, not a current-phase relation of a superconducting weak link.

An original analytic illustration uses the ideal fully gapped BCS theory spectrum at zero temperature, with positive gap edge $\Delta$:For positive $V$, the occupation difference restricts the integral to $0<E<eV$. Since $d\sqrt{E^2-{\Delta }^2}/dE=E/\sqrt{E^2-{\Delta }^2}$, integration and odd extension giveThe threshold is $\Delta /e$ for this NIS model. In the corresponding zero-temperature single-electron process between two fully gapped superconductors, creating excitations on both sides instead requires $|eV|\ge {\Delta }_1+{\Delta }_2$. That quasiparticle threshold does not forbid coherent pair electric current below it. At finite temperature, thermally populated quasiparticles soften the simple threshold argument.

At zero temperature in the NIS approximation, $R_{N}d{I}_{\mathrm{qp}}/dV=n_s(eV)$ away from the singular gap edge. Thus a voltage sweep probes the excitation density of states. The ideal divergence is rounded in real measurements, and at nonzero temperature the derivative samples a thermally broadened range of energies. For $|eV|>\Delta$, the model has $V{I}_{\mathrm{qp}}>0$: electrical work creates excitations and is dissipated on relaxation. This distinguishes its steady transport from the conservative Josephson coupling energy.

Spin-triplet p-wave pairing has pair spin $S=1$ and orbital angular momentum $L=1$, allowing multiple order parameters beyond conventional BCS theory with zero orbital angular momentum. ^[ref]

Specifying this symmetry does not determine its microscopic attraction mechanism. A short exchange-symmetry check explains why these quantum numbers fit fermions. The spin-one-half triplet states $|\uparrow \uparrow ⟩$, $(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ and $|\downarrow \downarrow ⟩$ are symmetric under exchange. A stipulated orbital factor proportional to relative momentum $k_x$ changes sign under $\mathbf{k}\mapsto -\mathbf{k}$. Multiplying any symmetric spin function by that odd orbital factor gives an antisymmetric pair wave function, as required by Fermi-Dirac statistics. By comparison, an even s-wave orbital factor needs the antisymmetric spin singlet state. This checks compatibility with exchange statistics; it neither selects a stable phase nor derives its transition temperature.

Superconductivity is a phase of matter that supports steady electric current with zero bulk electrical resistivity and exhibits the Meissner effect: equilibrium screening of a sufficiently weak, static magnetic field from the interior. It occurs below a material-dependent superconducting transition temperature, provided the electric current and applied magnetic field are also sufficiently small. ^[ref] Bulk expulsion means that a sample much thicker than its London penetration depth has an almost field-free interior; the magnetic field need not vanish at its surface. Conventional BCS theory explains superconductivity through correlated electrons forming Cooper pairs. ^[ref]

The two observations answer different questions. A resistance-versus-temperature experiment for superconductivity asks whether a steady electric current requires a measurable electric potential difference. A Field-cooled Meissner experiment asks whether a magnetic field already present in the material is expelled when it enters superconductivity. An ideal perfect conductor with zero electrical resistivity can preserve that pre-existing magnetic field; zero electrical resistance alone therefore does not establish the Meissner effect.

The usefulness follows directly from Joule heating, $P=I^{2}R$: maintaining a steady electric current through a segment with zero electrical resistance produces no resistive heating in that segment. This makes persistent electric currents attractive for producing steady magnetic fields. The beauty is that one phase has both a transport property and an equilibrium magnetic response, and the second cannot be inferred just by making an ordinary electrical resistance very small. The London equations encode this extra magnetic response. ^[ref]

Weak coupling means a suitably dimensionless interaction strength is small. In BCS theory, ${\nu }_{s}V\ll 1$ gives a gap exponentially small compared with the pairing-shell energy cutoff; the expansion parameter is not the dimensional quantity $V$ alone.

The isotropic weak-coupling BCS theory predicts the gap ratio approximately $3.528$ and the electronic heat-capacity jump approximately $1.426$ in units of the normal-state heat capacity at the transition. These compare distinct measurements of the same paired state. They are conditional predictions of the weak-coupling fully gapped model, rather than universal numbers for all superconductors.