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An Abrikosov vortex has a core where the superconducting order parameter is suppressed, surrounding circulating electric currents, and a winding of its quantum phase. A singly wound vortex carries one superconducting flux quantum in the appropriate bulk flux-quantization limit. The exact contour statement is fluxoid and flux quantization in a superconductor; the enclosed magnetic flux alone need not equal a flux quantum on a contour with appreciable electric current.

An ideal superconducting loop supports persistent current that preserves linked magnetic flux when resistance vanishes.

For a nonvanishing superconducting order parameter $\psi =\sqrt{n_s}{e}^{i\theta }$, $j=(q{n}_s/m)(\hslash \nabla \theta -qA)$. Single-valuedness gives phase winding $2\pi N$ around a closed curve, hence the fluxoid and flux quantization in a superconductor identity above. Magnetic flux alone is $Nh/q$ when the current integral vanishes. If the entire spanning surface is in a region where $D\psi =0$, the field vanishes there and the winding is zero; nonzero quantized flux requires a hole or a vortex core where that condition does not hold.

For a chosen homogeneous composition ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_{4-\delta }$, assign formal oxidation states $+3$ to lanthanum, $+2$ to barium and $-2$ to oxygen. If $v$ is the mean formal copper oxidation state, charge neutrality requiresThe excess formal positive charge per copper relative to $+2$ is therefore $p_{\mathrm{formal}}=x-2\delta$. For stipulated $x=0.12$ and $\delta =0.01$, it is $0.10$; at $\delta =0.06$, it is zero. These are bookkeeping examples, not measurements of mobile carriers. The symbol $x$ here belongs to this declared formula, not the original mixed-phase recipe. A whole-pellet elemental ratio cannot simply be assigned to each constituent phase.

Charge neutrality alone does not locate holes. The lecture reviews oxygen-related carrier evidence and charge-$2e$ pairing from the Josephson effect and fluxoid and flux quantization in a superconductor. Pair charge does not establish phonon-mediated BCS theory for cuprate superconductors.^[ref]

For a bulk type-II superconductor with negligible demagnetizing field, raising the applied magnetic field intensity past the lower critical value $H_{c1}$ favors a superconducting mixed state containing quantum vortex lines; superconductivity survives until the upper critical value $H_{c2}$. The magnetic field extends around each vortex over a penetration scale, not solely inside its core. Thus a nonzero average magnetic flux density can coexist with superconductivity. Below $H_{c1}$, a type-II superconductor also supports a Meissner state; a vortex-free state above the equilibrium entry field may instead be metastable. ^[ref]

Real preparation histories may leave quantum vortex lines held in place by material inhomogeneities, retaining magnetic flux and producing hysteresis. This flux pinning restricts access to equilibrium: it is different from the arbitrary frozen interior magnetic field of the ideal perfect conductor model. Harvard's vortex-imaging work illustrates strong pinning to bulk disorder in a type-II superconductor. ^[ref] Separately, flux conservation in a superconducting loop and fluxoid and flux quantization in a superconductor explain why a hole can retain magnetic flux. Neither bore retention, vortex penetration nor weak screening in a thin sample should be silently identified with the equilibrium Planar London screening calculation.

Rearrange the source's condensate-current expression and integrate around a closed path lying in the superconducting material:Single-valuedness of the superconducting wavefunction requires $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$, and Stokes theorem gives $\oint \mathbf{A}\cdot d\mathbf{l}=\Phi$. Thus the exact result is fluxoid and flux quantization in a superconductor:Reversing the winding orientation absorbs the sign of $q$. Since a Cooper pair has $|q|=2e$, the positive superconducting flux quantum isFor a thick ring, choose the path deep enough that the supercurrent is negligible; then the enclosed magnetic flux is $n{\Phi }_0$. This is the flux quantization usually intended in the question.

Qualification of the unqualified flux claim: a thin ring can have a nonzero kinetic contribution around every available path, so it is the fluxoid that is exactly quantized. For example, with zero applied flux and geometric and kinetic inductances $L_g=L_k$, a metastable winding-$1$ state has $(L_g+L_k)I={\Phi }_0$ but $\Phi =L_{g}I={\Phi }_0/2$. This concrete counterexample satisfies the given current law while its magnetic flux is not an integer multiple of ${\Phi }_0$.

There is a sign inconsistency in the printed starting energy: its kinetic term uses $\nabla +i{q}_{s}A/\hslash$, while the requested field equation, current and later zero-covariant-derivative condition use $\nabla -i{q}_{s}A/\hslash$. Both conventions can be derived explicitly. Set $D_{\eta }=\nabla -i\eta q_{s}A/\hslash$ for $\eta =\pm 1$; the intended later equations take $\eta =1$, whereas the literal printed energy takes $\eta =-1$.

The magnetic term is $B^2/(2{\mu }_0)$, because $A_{k,j}(A_{k,j}-A_{j,k})=|\nabla \times A{|}^2$. Vary ${\psi }^{\ast }$ independently with compactly supported variations. Integration by parts in the kinetic term givesIn particular $\eta =1$ gives the requested equation. Its conjugate follows from variation in $\psi$.

For variation in $A$, $\delta D_{\eta }\psi =-i\eta q_s\delta A\psi /\hslash$. The kinetic variation is $-j_{\eta }\cdot \delta A$, whereThe magnetic variation is $(\nabla \times B)\cdot \delta A/{\mu }_0$ after integration by parts. ThereforeFor $\eta =1$ this current equals the two displayed forms in the question. For the literal plus-sign energy, $\eta =-1$, the phase-gradient current term has the opposite sign and the field equation contains $D_{-1}^2$. Thus the requested set follows after correcting that one energy sign, not from silently varying inconsistent expressions. This is the gauge-coupling sign in Ginzburg-Landau superconductivity.

Now use the intended convention and write $\psi =\sqrt{n_s}{e}^{i\theta }$ with $n_s>0$. Dividing $D_1\psi =0$ by $\psi$ separates real and imaginary parts:Thus $n_s$ is constant on each connected region. The remaining field equation requires $\alpha +\beta n_s=0$, so for $T<T_c$,The current is $j=(q_s{n}_s/m_s)(\hslash \nabla \theta -q_{s}A)=0$. Taking the curl locally gives $B=0$, hence also solves the Maxwell equation.

For a closed curve on which the order parameter is nonzero and $D_1\psi =0$, Stokes' theorem givesSingle-valuedness of $\psi$ permits phase change $2\pi N$, so the flux quantum magnitude is $h/|q_s|$ (for Cooper pairs, $h/(2|e|)$). There is an important geometric qualification: if the entire spanning surface lies in the smooth zero-field superconducting region assumed above, its flux and the total boundary winding are zero. Nonzero quantized flux occurs through a hole or vortex core where those assumptions do not hold over the spanning surface. More generally, without $j=0$ along the curve, the exact quantized quantity is the fluxoidThus fluxoid and flux quantization in a superconductor are related but are not interchangeable without the current condition.

Write the condensate order parameter as $\Psi =|\Psi |e^{i\theta }$. Away from vortex cores, single-valuedness around a closed loop gives $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. For neutral condensed objects of mass $M$, ${\mathbf{v}}_s=\hslash \nabla \theta /M$, hence the quantized circulationHere $M$ is the atomic mass for a simple bosonic atomic condensate, or the pair mass for a paired-fermion condensate. A nonzero winding can surround a vortex core where the order-parameter amplitude falls to zero. The result concerns collective condensate phase winding, not separate orbit quantization of every constituent particle.

For charge $q$ per condensed object, the gauge-invariant relation is $M{\mathbf{v}}_s=\hslash \nabla \theta -q\mathbf{A}$. Integrating givesThis is fluxoid and flux quantization in a superconductor. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude isThe orientation fixes its sign. A loop taken deep in a thick superconductor has negligible superflow and therefore encloses an integer flux quantum. In a thin loop or one with appreciable screening current, it is the fluxoid, including the circulation term, that is quantized; the magnetic flux alone need not be exactly an integer multiple.

Use a complex pair order parameter $\Psi =|\Psi |e^{i\theta }$ with signed charge $q$, $|q|=2e$. A local isotropic Ginzburg-Landau theory of superconductivity has free energyFor $\alpha <0$, $\beta >0$, the uniform amplitude is $|{\Psi }_0{|}^2=-\alpha /\beta$. Variation gives the static field equation and supercurrentThe vector potential obeys Maxwell's equations, including $\nabla \times \mathbf{B}={\mu }_0({\mathbf{j}}_s+{\mathbf{j}}_n)+{\mu }_0{ϵ}_0{\partial }_t\mathbf{E}$ and $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$, with $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$. The static approximation drops the displacement current. A common dissipative near-transition dynamical extension is the time-dependent Ginzburg-Landau equationIt is a phenomenological relaxation law in its applicable regime, not a universal microscopic Schrödinger equation for every superconductor. Its covariant time derivative and the covariant spatial derivative ensure consistent phase and electromagnetic gauge transformations.

The amplitude and screening lengths areFor $\kappa >1/\sqrt{2}$ the ideal model is type II. An Abrikosov vortex has a $2\pi n$ phase winding, a core with suppressed $|\Psi |$ of size $\xi$, and circulating screening currents over $\lambda$. Single-valuedness implies fluxoid and flux quantization in a superconductor:The sign of $n$ depends on the chosen circulation and charge convention. Far from the core, where the contour current term is negligible, each singly wound vortex carries one flux quantum in magnitude.

The equilibrium lower critical magnetic field marks favorable entry of the first vortex; for $\kappa \gg 1$, ${\mu }_0{H}_{c1}\simeq [{\Phi }_0/(4\pi {\lambda }^2)](\ln \kappa +C)$ with an order-one core correction $C$. The upper critical magnetic field follows by linearizing the order-parameter equation in the field:Between these fields lies the superconducting mixed state, often with an Abrikosov vortex lattice. The ideal equilibrium entry and exit fields need not yield large irreversible loops. Flux pinning and surface barriers impede vortex motion; field sweeps then retain history-dependent flux and produce magnetic hysteresis and dissipative losses.

A superconducting weak link, such as a tunnel junction or narrow constriction, couples the phases of two superconductors. With consistent choices of phase and voltage orientation, the Josephson effect givesThe Gauge-invariant Josephson phase includes the line integral of $\mathbf{A}$ across the link. A zero voltage can sustain a phase-dependent steady current; a constant nonzero voltage makes that contribution oscillate. Microwave irradiation can lock this oscillation to an applied frequency, giving voltage steps at $V_n=nhf/(2e)$, an experimental test of coherent pair transport rather than an ordinary resistive current law.

A DC SQUID places two weak links in a loop. For identical junction critical currents $I_0$ and negligible self-inductance, flux quantization fixes their relative phase difference to $2\pi {\Phi }_{\mathrm{ext}}/{\Phi }_0$. Adding the two sinusoidal currents and maximizing over their common phase givesThe flux-periodic response provides sensitive magnetic measurements. Junction asymmetry or appreciable loop inductance changes the ideal contrast and can introduce additional hysteresis; the phase-coherent interference principle remains. The Jaklevic Josephson interference experiment is a motivating experimental realization of this link between a superconducting loop, weak links and magnetic flux.

For a one-component superfluid with nonzero order-parameter magnitude on a closed loop, the phase defines a map from the loop into the circle. Its winding number isThe fundamental group ${\pi }_1(S^1)=\mathbb{Z}$ means that smooth deformations preserving nonzero amplitude cannot change this integer. With superfluid velocity ${\mathbf{v}}_s=\hslash \nabla \theta /m_B$, it gives quantized circulation $\oint {\mathbf{v}}_s\cdot d\mathbf{l}=nh/m_B$. A vortex core permits the order parameter to vanish and carries the winding; a superfluid phase slip changes a loop's winding when a core crosses it. For paired neutral fermions, the relevant condensed-object mass is the pair mass.

For charged pairs, gauge-covariant phase rigidity gives fluxoid and flux quantization in a superconductor:The magnitude of the pair flux quantum is $h/(2e)$. Magnetic flux alone approaches an integer multiple of this value when the circulating-velocity term on a bulk contour is negligible. The topological condition is not a claim that an arbitrary finite loop has exactly quantized magnetic flux independent of its screening current.

There are also momentum-space invariants, distinct from real-space vortex winding. A fully gapped two-dimensional Bogoliubov--de Gennes Hamiltonian can have a Chern number obtained from its occupied quasiparticle bands' Berry curvature. In a chiral paired model it can protect edge modes while the bulk gap and relevant symmetry class are maintained. Such an invariant can change only when those assumptions fail, for example through a bulk gap closing. A nodal d-wave state requires a different analysis; it is not automatically a fully gapped Chern superconductor.

For a neutral condensate with order parameter $\psi =\sqrt{n_s}{e}^{i\varphi }$, the superfluid velocity away from singularities is ${\mathbf{v}}_s=\hslash \nabla \varphi /m$. Single-valuedness around a closed contour gives $\oint \nabla \varphi \cdot d\mathbf{l}=2\pi n$, hence quantization of circulation:The mass is that of the condensed object: an atom for a bosonic atomic superfluid, or a pair for a paired-fermion condensate. A nonzero winding cannot be continuously removed while the order parameter remains nonzero throughout a spanning region. A vortex allows its phase to wind while its amplitude vanishes in a core.

For a charged condensate, minimal coupling gives $m{\mathbf{v}}_s+q\mathbf{A}=\hslash \nabla \varphi$. ThusUsing ${\mathbf{j}}_s=n_{s}q{\mathbf{v}}_s$ and ${\lambda }_L^2=m/({\mu }_0{n}_s{q}^2)$ gives the fluxoid and flux quantization in a superconductor relation $\Phi +{\mu }_0{\lambda }_L^2\oint {\mathbf{j}}_s\cdot d\mathbf{l}=nh/q$. The signs of oriented flux and winding depend on the signed pair charge; the magnitude of the elementary superconducting flux quantum is ${\Phi }_0=h/(2e)$. Flux alone is quantized when the current term is negligible on a contour deep in the bulk. A thin ring or a contour near a core can instead require the full fluxoid.

In a neutral superfluid, vortex flow carries kinetic energy and angular momentum. Rotation can favor vortices because equilibrium minimizes the rotating-frame quantity $E-\mathbf{\Omega }\cdot \mathbf{L}$. The core size is set by a coherence or healing scale. In a type-II superconductor, an Abrikosov vortex carries quantized magnetic flux and circulating supercurrent; its core scale is the coherence length and its magnetic screening scale is the penetration depth. Between the lower and upper critical fields, a vortex lattice can be favored. Flux pinning affects how that structure responds to current. Type-I bulk superconductors instead favor macroscopic normal/superconducting separation rather than the usual stable dilute Abrikosov-vortex regime.

A superconducting persistent current is an electric current circulating in a closed path with superconductivity after the external driving electromotive force has ceased. Its associated magnetic field makes the circulation observable without inserting a measuring instrument into the path. This article concerns macroscopic circulation in superconductivity. A finite experiment establishes persistence over its observation window with a stated measurement error, rather than directly observing an infinite lifetime.

The ideal circuit connection is flux conservation in a superconducting loop. In a fixed loop with self-inductance $L$, a single circulating electric current $I$, and externally imposed linked magnetic flux ${\Phi }_{\mathrm{ext}}$, the lumped circuit approximation givesWhen the electrical resistance $R$ vanishes, integrating gives $LI+{\Phi }_{\mathrm{ext}}=\text{constant}$. Removing the external linked magnetic flux therefore changes the circulating electric current; holding it fixed afterwards preserves that electric current. This is a circuit approximation with a specified self-inductance, not a derivation of the microscopic state of superconductivity. For thin paths where the carrier contribution matters, fluxoid and flux quantization in a superconductor supplies the more careful description.

For constant $L>0$, constant effective electrical resistance $R\ge 0$, and no changing external linked magnetic flux, the RL circuit instead predictsThis exponential decay follows by separating $dI/I=-(R/L)dt$. A long relaxation time tests extremely small electrical resistance without trying to resolve an equally tiny steady electric potential difference. The stored magnetic energy is $U=L{I}^2/2$, so the same model gives $dU/dt=-R{I}^2$, exactly the Joule heating loss. The finite energy stored in the loop supplies the magnetic readout; the circulating electric current is not a continuously powered source of useful work.

The magnitude of one magnetic flux unit for an electron-pair order parameter, whose electric charge has magnitude $2e$. The enclosing contour must satisfy the flux condition in fluxoid and flux quantization in a superconductor; it is not a statement that every finite area inside the material contains an integer number of these units.