Fluxoid quantization

For superconductors the bosons involved are the so-called Cooper pairs which are quasiparticles formed by two electrons.^[12] Hence m = 2m_e and q = −2e where m_e and e are the mass of an electron and the elementary charge. It follows from Eq. (8) that

$${\displaystyle 2m_{e}{\vec {v}}_{s}={\frac {h}{2\pi }}{\vec {\nabla }}\varphi +2e{\vec {A}}.}$$

(15)

Integrating Eq. (15) over a closed loop gives

$${\displaystyle 2m_{e}\oint {\vec {v}}_{s}\cdot \mathrm {d} {\vec {s}}=\oint \left({\frac {h}{2\pi }}{\vec {\nabla }}\varphi +2e{\vec {A}}\right)\cdot \mathrm {d} {\vec {s}}}$$

(16)

As in the case of helium we define the vortex strength

$${\displaystyle \oint {\vec {v}}_{s}\cdot \mathrm {d} {\vec {s}}=\kappa }$$

(17)

and use the general relation

$${\displaystyle \oint {\vec {A}}\cdot \mathrm {d} {\vec {s}}=\Phi }$$

(18)

where Φ is the magnetic flux enclosed by the loop. The so-called fluxoid is defined by

$${\displaystyle \Phi _{v}=\Phi -{\frac {2m_{e}}{2e}}\kappa .}$$

(19)

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. (16) the fluxoid is quantized

$${\displaystyle \Phi _{v}=n{\frac {h}{2e}}.}$$

(20)

The unit of quantization is called the flux quantum

$${\displaystyle \Phi _{0}={\frac {h}{2e}}=2.067833758(46)\times 10^{-15}\,\mathrm {Wb} .}$$

(21)

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. (21). Nowadays the value given by Eq. (21) is exact by definition.

Fig. 3. Two superconducting rings in an applied magnetic field

1. thick superconducting ring. The integration loop is completely in the region with v_s = 0;
   
2. thick superconducting ring with a weak link. The integration loop is completely in the region with v_s = 0 except for a small region near the weak link.

In Fig. 3 two situations are depicted of superconducting rings in an external magnetic field. One case is a thick-walled ring and in the other case the ring is also thick-walled, but is interrupted by a weak link. In the latter case we will meet the famous Josephson relations. 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 Φ_a and the self-induced flux Φ_s induced by the circulation current

$${\displaystyle \Phi =\Phi _{a}+\Phi _{s}.}$$

(22)

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 v_s = 0 everywhere so κ = 0. In that case the fluxoid is equal to the magnetic flux (Φ_v = Φ). If v_s = 0 Eq. (15) reduces to

$${\displaystyle 0={\frac {h}{2\pi }}{\vec {\nabla }}{\varphi }+2e{\vec {A}}.}$$

(23)

Taking the rotation gives

$${\displaystyle 0={\frac {h}{2\pi }}{\vec {\nabla }}\times {\vec {\nabla }}\varphi +2e{\vec {\nabla }}\times {\vec {A}}.}$$

(24)

Using the well-known relations ${\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}\varphi =0}$ and ${\displaystyle {\vec {\nabla }}\times {\vec {A}}={\vec {B}}}$ 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

$${\displaystyle \Phi =n\Phi _{0}.}$$

(25)

Interrupted ring, weak links

Fig. 4. Schematic of a weak link carrying a superconducting current i_s. The voltage difference over the link is V. The phases of the superconducting wave functions at the left and right side are assumed to be constant (in space, not in time) with values of φ₁ and φ₂ respectively.

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. (15))

$${\displaystyle \Delta \varphi ^{*}=-{\frac {2\pi }{h}}2m_{e}\int _{\delta }{\vec {v}}_{s}\cdot \mathrm {d} {\vec {s}}.}$$

(26)

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 v_s = 0. So the value of the line integral is well-defined (e.g. independent of the choice of the end points). With Eqs. (19), (22), and (26)

$${\displaystyle \Phi _{a}+\Phi _{s}+\Phi _{0}{\frac {\Delta \varphi ^{*}}{2\pi }}=n\Phi _{0}.}$$

(27)

Without proof we state that the supercurrent through the weak link is given by the so-called DC Josephson relation^[13]

$${\displaystyle i_{s}=i_{1}\sin(\Delta \varphi ^{*}).}$$

(28)

The voltage over the contact is given by the AC Josephson relation

$${\displaystyle V={\frac {1}{2\pi }}{\frac {h}{2e}}{\frac {\mathrm {d} \Delta \varphi ^{*}}{\mathrm {d} t}}.}$$

(29)

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 ${\displaystyle \Delta \varphi ^{*}}$) Eq. (29) shows that V=0 while a nonzero current flows through the junction. In the case of a constant applied voltage (voltage bias) Eq. (29) can be integrated easily and gives

$${\displaystyle \Delta \varphi ^{*}=2\pi {\frac {2eV}{h}}t.}$$

(30)

Substitution in Eq. (28) gives

$${\displaystyle i_{s}=i_{1}\sin \left(2\pi {\frac {2eV}{h}}t\right).}$$

(31)

This is an AC current. The frequency

$${\displaystyle \nu ={\frac {2eV}{h}}={\frac {V}{\Phi _{0}}}}$$

(32)

is called the Josephson frequency. One μV gives a frequency of about 500 MHz. By using Eq. (32) the flux quantum is determined with the high precision as given in Eq. (21).

The energy difference of a Cooper pair, moving from one side of the contact to the other, is ΔE = 2eV. With this expression Eq. (32) can be written as ΔE = hν which is the relation for the energy of a photon with frequency ν.

The AC Josephson relation (Eq. (29)) can be easily understood in terms of Newton's law, (or from one of the London equation's^[14]). We start with Newton's law

$${\displaystyle {\vec {F}}=m{\frac {\mathrm {d} {\vec {v}}_{s}}{\mathrm {d} t}}.}$$

Substituting the expression for the Lorentz force

$${\displaystyle {\vec {F}}=q\left({\vec {E}}+{\vec {v}}_{s}\times {\vec {B}}\right)}$$

and using the general expression for the co-moving time derivative

$${\displaystyle {\frac {\mathrm {d} {\vec {v}}_{s}}{\mathrm {d} t}}={\frac {\partial {\vec {v}}_{s}}{\partial t}}+{\frac {1}{2}}{\vec {\nabla }}v_{s}^{2}-{\vec {v}}_{s}\times \left({\vec {\nabla }}\times {\vec {v}}_{s}\right)}$$

gives

$${\displaystyle {\frac {q}{m}}\left({\vec {E}}+{\vec {v}}_{s}\times {\vec {B}}\right)={\frac {\partial {\vec {v}}_{s}}{\partial t}}+{\frac {1}{2}}{\vec {\nabla }}v_{s}^{2}-{\vec {v}}_{s}\times \left({\vec {\nabla }}\times {\vec {v}}_{s}\right).}$$

Eq. (8) gives

$${\displaystyle 0={\vec {\nabla }}\times {\vec {v}}_{s}+{\frac {q}{m}}{\vec {\nabla }}\times {\vec {A}}={\vec {\nabla }}\times {\vec {v}}_{s}+{\frac {q}{m}}{\vec {B}}}$$

so

$${\displaystyle {\frac {q}{m}}{\vec {E}}={\frac {\partial {\vec {v}}_{s}}{\partial t}}+{\frac {1}{2}}{\vec {\nabla }}v_{s}^{2}.}$$

Take the line integral of this expression. In the end points the velocities are zero so the ∇v² term gives no contribution. Using

$${\displaystyle \int {\vec {E}}\cdot \mathrm {d} {\vec {\ell }}=-V}$$

and Eq. (26), with q = −2e and m = 2m_e, gives Eq. (29).

DC SQUID

Fig. 5. Two superconductors connected by two weak links. A current and a magnetic field are applied.

Fig. 6. Dependence of the critical current of a DC-SQUID on the applied magnetic field

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

$${\displaystyle \Delta \varphi _{a}^{*}=\Delta \varphi _{b}^{*}+2\pi {\frac {\Phi }{\Phi _{0}}}+2\pi n.}$$

(33)

If the self-inductance of the loop can be neglected the magnetic flux in the loop Φ is equal to the applied flux

$${\displaystyle \Phi =\Phi _{a}=BA}$$

(34)

with B the magnetic field, applied perpendicular to the surface, and A the surface area of the loop. The total supercurrent is given by

$${\displaystyle i_{s}=i_{1}\sin(\Delta \varphi _{a}^{*})+i_{1}\sin(\Delta \varphi _{b}^{*}).}$$

(35)

Substitution of Eq(33) in (35) gives

$${\displaystyle i_{s}=i_{1}\sin \left(\Delta \varphi _{b}^{*}+2\pi {\frac {\Phi }{\Phi _{0}}}\right)+i_{1}\sin(\Delta \varphi _{b}^{*}).}$$

(36)

Using a well known geometrical formula we get

$${\displaystyle i_{s}=2i_{1}\sin \left(\Delta \varphi _{b}^{*}+\pi {\frac {\Phi }{\Phi _{0}}}\right)\cos(\pi {\frac {\Phi _{a}}{\Phi _{0}}}).}$$

(37)

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

$${\displaystyle i_{c}=2i_{1}\left|\cos \left(\pi {\frac {\Phi _{a}}{\Phi _{0}}}\right)\right|.}$$

(38)

Note that the critical current is periodic in the applied flux with period Φ₀. 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.^[15]

Type II superconductivity

Fig. 7. Magnetic flux lines penetrating a type-II superconductor. The currents in the superconducting material generate a magnetic field which, together with the applied field, result in bundles of quantized flux.

Type-II superconductivity is characterized by two critical fields called B_c1 and B_c2. At a magnetic field B_c1 the applied magnetic field starts to penetrate the sample, but the sample is still superconducting. Only at a field of B_c2 the sample is completely normal. For fields in between B_c1 and B_c2 magnetic flux penetrates the superconductor in well-organized patterns, the so-called Abrikosov vortex lattice similar to the pattern shown in Fig. 2.^[16] 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 15×10¹² A/m². That corresponds with 15 million ampère in a wire of one mm².