In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states. The
Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In
Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value
Hc. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In
Type II superconductors, raising the applied field past a critical value
Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of
magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength
Hc2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called
fluxons because the flux carried by these vortices is
quantized. Most pure
elemental superconductors, except
niobium and
carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II. The most important finding from
Ginzburg–Landau theory was made by
Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux
vortices. For this and related work, he was awarded the Nobel Prize in 2003 with
Ginzburg and
Leggett.
Fluxoid quantization For
superconductors the bosons involved are the so-called
Cooper pairs which are
quasiparticles formed by two electrons. Hence
m = 2
me and
q = −2
e where
me and
e are the mass of an electron and the elementary charge. It follows from Eq. () that {{NumBlk||2m_e \vec{v}_s=\frac{h}{2\pi}\vec{\nabla}\varphi+2e\vec{A}. Integrating Eq. () over a closed loop gives {{NumBlk||2m_e\oint \vec{v}_s\cdot \mathrm{d}\vec{s} = \oint\left(\frac{h}{2\pi}\vec{\nabla}\varphi+2e\vec{A}\right) \cdot \mathrm{d}\vec{s} As in the case of helium we define the vortex strength {{NumBlk||\oint \vec{v}_s\cdot\mathrm{d}\vec{s} =\kappa| }} and use the general relation {{NumBlk||\oint \vec{A}\cdot\mathrm{d}\vec{s} = \Phi| }} where Φ is the magnetic flux enclosed by the loop. The so-called
fluxoid is defined by {{NumBlk||\Phi_v=\Phi - \frac{2m_e}{2e}\kappa.| }} In general the values of
κ and Φ depend on the choice of the loop. Due to the single-valued nature of the wave function and Eq. () the fluxoid is quantized {{NumBlk||\Phi_v = n\frac{h}{2e}.| }} The unit of quantization is called the
flux quantum {{NumBlk||\Phi_0=\frac{h}{2e} = 2.067833758(46)\times 10^{-15} \, \mathrm{Wb}.| }} The flux quantum plays a very important role in superconductivity. The earth magnetic field is very small (about 50 μT), but it generates one flux quantum in an area of 6 μm by 6 μm. So, the flux quantum is very small. Yet it was measured to an accuracy of 9 digits as shown in Eq. (). Nowadays the value given by Eq. () is exact by definition. . In both cases we consider a loop inside the material. In general a superconducting circulation current will flow in the material. The total magnetic flux in the loop is the sum of the applied flux Φa and the self-induced flux Φs induced by the circulation current
Thick ring The first case is a thick ring in an external magnetic field (Fig. 3a). The currents in a superconductor only flow in a thin layer at the surface. The thickness of this layer is determined by the so-called
London penetration depth. It is of μm size or less. We consider a loop far away from the surface so that
vs = 0 everywhere so
κ = 0. In that case the fluxoid is equal to the magnetic flux (Φv = Φ). If
vs = 0 Eq. () reduces to {{NumBlk||0=\frac{h}{2\pi}\vec{\nabla}{\varphi}+2e\vec{A}.| }} Taking the rotation gives {{NumBlk||0 = \frac{h}{2\pi}\vec{\nabla} \times \vec{\nabla}\varphi + 2e \vec{\nabla}\times\vec{A}.| }} Using the well-known relations \vec{\nabla} \times \vec{\nabla}\varphi = 0 and \vec{\nabla}\times\vec{A} = \vec{B} shows that the magnetic field in the bulk of the superconductor is zero as well. So, for thick rings, the total magnetic flux in the loop is quantized according to
Interrupted ring, weak link s Weak links play a very important role in modern superconductivity. In most cases weak links are oxide barriers between two superconducting thin films, but it can also be a crystal boundary (in the case of
high-Tc superconductors). A schematic representation is given in Fig. 4. Now consider the ring which is thick everywhere except for a small section where the ring is closed via a weak link (Fig. 3b). The velocity is zero except near the weak link. In these regions the velocity contribution to the total phase change in the loop is given by (with Eq. ()) {{NumBlk||\Delta\varphi^*=-\frac{2\pi}{h}2m_e\int_\delta \vec{v}_s\cdot\mathrm{d}\vec{s}.| }} The line integral is over the contact from one side to the other in such a way that the end points of the line are well inside the bulk of the superconductor where . So the value of the line integral is well-defined (e.g. independent of the choice of the end points). With Eqs. (), (), and () {{NumBlk||\Phi_a+\Phi_s+\Phi_0\frac{\Delta\varphi^*}{2\pi}=n\Phi_0.| }} Without proof we state that the supercurrent through the weak link is given by the so-called DC
Josephson relation The voltage over the contact is given by the AC Josephson relation {{NumBlk||V=\frac{1}{2\pi}\frac{h}{2e}\frac{\mathrm{d}\Delta\varphi^*}{\mathrm{d}t}.| }} The names of these relations (DC and AC relations) are misleading since they both hold in DC and AC situations. In the steady state (constant \Delta\varphi^*) Eq. () shows that
V=0 while a nonzero current flows through the junction. In the case of a constant applied voltage (voltage bias) Eq. () can be integrated easily and gives {{NumBlk||\Delta\varphi^* = 2\pi\frac{2eV}{h}t.| }} Substitution in Eq. () gives {{NumBlk||i_s = i_1\sin\left(2\pi\frac{2eV}{h}t\right).| }} This is an AC current. The frequency {{NumBlk||\nu = \frac{2eV}{h} = \frac{V}{\Phi_0}| }} is called the Josephson frequency. One μV gives a frequency of about 500 MHz. By using Eq. () the flux quantum is determined with the high precision as given in Eq. (). The energy difference of a Cooper pair, moving from one side of the contact to the other, is . With this expression Eq. () can be written as which is the relation for the energy of a photon with frequency
ν. :The AC Josephson relation (Eq. ()) can be easily understood in terms of Newton's law, (or from one of the
London equation's). We start with Newton's law \vec F = m \frac{\mathrm{d}\vec v_s}{\mathrm{d}t}. :Substituting the expression for the
Lorentz force \vec F = q\left(\vec E+\vec v_s\times \vec B\right) and using the general expression for the co-moving time derivative \frac{\mathrm{d}\vec v_s}{\mathrm{d}t} = \frac{\partial \vec v_s}{\partial t} + \frac{1}{2} \vec \nabla v_s^2 - \vec v_s\times \left(\vec \nabla\times \vec v_s\right) gives \frac{q}{m} \left(\vec E + \vec v_s\times \vec B\right) = \frac{\partial \vec v_s}{\partial t} + \frac{1}{2} \vec \nabla v_s^2 - \vec v_s\times \left(\vec \nabla\times \vec v_s\right). :Eq. () gives 0 = \vec\nabla\times\vec v_s + \frac{q}{m}\vec\nabla\times\vec A = \vec\nabla\times\vec v_s + \frac{q}{m}\vec B so \frac{q}{m}\vec E = \frac{\partial \vec v_s}{\partial t}+ \frac{1}{2} \vec \nabla v_s^2. :Take the line integral of this expression. In the end points the velocities are zero so the ∇
v2 term gives no contribution. Using \int \vec E\cdot\mathrm{d}\vec \ell = -V and Eq. (), with and , gives Eq. ().
DC SQUID Fig. 5 shows a so-called DC
SQUID. It consists of two superconductors connected by two weak links. The fluxoid quantization of a loop through the two bulk superconductors and the two weak links demands {{NumBlk||\Delta\varphi_a^*=\Delta\varphi^*_b+2\pi\frac{\Phi}{\Phi_0}+2\pi n.| }} If the self-inductance of the loop can be neglected the magnetic flux in the loop Φ is equal to the applied flux with
B the magnetic field, applied perpendicular to the surface, and
A the surface area of the loop. The total supercurrent is given by Substitution of Eq() in () gives {{NumBlk||i_s = i_1 \sin\left(\Delta\varphi_b^* + 2\pi\frac{\Phi}{\Phi_0}\right) + i_1\sin(\Delta\varphi_b^*).| }} Using a well known geometrical formula we get {{NumBlk||i_s = 2 i_1 \sin\left(\Delta\varphi_b^*+\pi\frac{\Phi}{\Phi_0}\right)\cos(\pi\frac{\Phi_a}{\Phi_0}).| }} Since the sin-function can vary only between −1 and +1 a steady solution is only possible if the applied current is below a critical current given by {{NumBlk||i_c = 2 i_1 \left|\cos\left(\pi\frac{\Phi_a}{\Phi_0}\right)\right|.| }} Note that the critical current is periodic in the applied flux with period . The dependence of the critical current on the applied flux is depicted in Fig. 6. It has a strong resemblance with the interference pattern generated by a laser beam behind a double slit. In practice the critical current is not zero at half integer values of the flux quantum of the applied flux. This is due to the fact that the self-inductance of the loop cannot be neglected.
Type II superconductivity Type-II superconductivity is characterized by two critical fields called
Bc1 and
Bc2. At a magnetic field
Bc1 the applied magnetic field starts to penetrate the sample, but the sample is still superconducting. Only at a field of
Bc2 the sample is completely normal. For fields in between
Bc1 and
Bc2 magnetic flux penetrates the superconductor in well-organized patterns, the so-called
Abrikosov vortex lattice similar to the pattern shown in Fig. 2. A cross section of the superconducting plate is given in Fig. 7. Far away from the plate the field is homogeneous, but in the material superconducting currents flow which squeeze the field in bundles of exactly one flux quantum. The typical field in the core is as big as 1 tesla. The currents around the vortex core flow in a layer of about 50 nm with current densities on the order of 15 A/m2. That corresponds with 15 million ampère in a wire of one mm2. == Dilute quantum gases ==