London Equations of superconductivity

Meissner effect implies a magnetic susceptibility χ=−1. It does not account for the flux penetration observed in thin films. On the basis of Maxwell’s equations alone, it has not been possible to deduce the Meissner effect and field penetration in superconductors. On the basis of two fluid model London brothers ( F.London and H.London) first examined the magnetic aspects quantitatively and showed that it was necessary to introduce two additional equations to explain them completely.

Let us suppose that a small transient electric field E arises within a superconductor which freely accelerates the superelectrons. If vs is the average velocity, m is the mass and e is the charge of super electron, then the equation of motion can be written as

md v sdt

=−e E …….(1)

and the current density of super electrons is

Js=−ens vs …….(2)

From equations (1) and (2), we obtain

d J sdt

=ns e

2

mE …….(3)

This is called first London equation. This shows that it is possible to have steady current in a superconductor in the absence of an electric field, because in equation (3) for E=0, J is finite and constant or vice versa. The corresponding expression for normal current density is

J=σ E

Which shows that no current is possible in the absence of an electric field, a usual behaviour of materials in the normal state.

The conclusion E=0 leads to another important result when combined with the Maxwell’s equation

∇× E=−( d Bdt ) ……..(4)

that is ( d Bdt )=0∨B=aconstant . ……..(5)

Equation (5) tells us that in steady state B is constant inside a superconductor irrespective of its temperature. This result is not in agreement with the Meissner effect, according to which a superconductor expels magnetic flux completely for all temperatures below T c. In order to remove this discrepancy, London suggested some modifications in the above formalism.

Let us take curl of equation (5) and then substitute the value of curl E from equation (4). we obtain

∇×( d J sdt )=−ns e2

m.d Bdt

……..(6)

Integration equation (6) w.r.t.time and putting the constant of integration equal to zero ( so that it remains consistent with the Meissner effect ), we have

∇×J s=−ns e

2

mB …….(7)

This is called the second London equation and leads to results that are in agreement with the experiment.

Maxwell’s equations is

∇×B=μ0 J s ……..(8)

Taking curl of the above equation, we obtain

∇×∇×B=μ0∇×J s

i.e., ∇ (∇ .B )−∇2B=μ0∇×J s ………(9)

But we have∇ .B=0∧¿equation (7 ) ,∇×J s=−ns e

2

mB, equation (9) becomes

−∇2B=μ0nse

2

mB= B

λL2 ………(10)

where λL=( mμ0nse

2 )12 and is called the London penetration depth.

Let us suppose that the specimen is semi-infinite with its surface lying in the yz-plane and the field applied in the z-direction. Then one dimensional form of equation (10) can be written as

∂2B z∂ x2 = 1

λ2 Bz ………(11)

The solution of equation (11) can be assumed in the form

Bz ( x )=B z (0 ) exp(−xλ ) ……….(12)

The graphical representation of equation (12) is shown below. This indicates that

the flux density decreases exponentially inside the superconductor, falling to 1e of

its initial value at a distanceλ, the London penetration depth. It further indicates that the flux inside the bulk of superconductor is zero and hence is in agreement with the Meissner effect.

Hence London brothers successfully explained the Meissner effect and zero resistivity by adding two new equations to four Maxwell’s equations used in electrodynamics. According to London equations, the flux does not suddenly drops to zero at the surface of Type I superconductors, but decreases exponentially.

Reference

Solid State Physics-M A Wahab