London equation

The London equations (named after the brothers Fritz and Heinz London ) are based on a postulate and replace Ohm's law in a superconductor . They also describe how the magnetic field behaves in such a substance. One result is, for example, that the magnetic field penetrates somewhat into the superconductor, despite different predictions ( penetration depth  λ _L ).

Experimental motivation

Due to the Meißner-Ochsenfeld effect , a superconductor is an ideal diamagnet ( magnetic susceptibility χ = −1 ) and its interior should be free of magnetic fields. However, this prediction cannot be confirmed experimentally. It has been observed that the magnetic field is not completely forced out of thin superconductor films, i.e. the interior is not entirely field-free. In addition, the occurrence of that prediction would violate the continuity condition for the magnetic field at the edge of the superconductor.

formulation

To explain this, one replaces the classical Ohm's law for the electric current density j and the electric field  E.

${\displaystyle {\vec {j}}=\sigma {\vec {E}}}$

by the London equation :

${\displaystyle {\vec {j}}={\frac {n\,q\,\hbar }{m}}{\vec {\nabla }}S-{\frac {n\,q^{2}}{m}}{\vec {A}}}$

With

• ${\displaystyle S=S({\vec {r}})}$the phase of the macroscopic wave function ,
• ${\displaystyle {\vec {A}}={\vec {A}}({\vec {r}},t)}$the vector potential of the magnetic field and
• n is the particle number density of the charge carriers .

(For derivation of the equation, see separate section.)

There are two useful transformations of this equation, sometimes referred to as the 1st and 2nd London Equations :

${\displaystyle \partial _{t}{\vec {j}}={\frac {n\,q^{2}}{m}}{\vec {E}}}$ and

${\displaystyle \operatorname {rot} {\vec {j}}={\vec {\nabla }}\times {\vec {j}}=-{\frac {n\,q^{2}}{m}}{\vec {B}}}$.

The phase S makes no contribution to these two equations - not to the first equation, because the phase is only position-dependent and thus constant over time, and not to the second equation because the following applies. ${\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}S=0}$

Attention: Although the phase component does not contribute to the last two formulas, it should not be neglected! If the phase component were not included, this would mean that the current density would have to be zero without a magnetic field. In reality, however, the phase gradient can also make a contribution to the current density, which then does not necessarily have to be zero, i.e. H. the current density is not zero, although there is no magnetic field. The approach of a macroscopic wave function is also made for superfluids . In this case, it is actually phase S that leads to the fountain effect or quantized vortices .

Theoretical explanation of the Meißner-Ochsenfeld effect

London penetration depth for selected materials (after Kittel 2002):

Superconductor	Penetration depth λ L in nm
Tin (Sn)	34
Aluminum (Al)	16
Lead (Pb)	37
Cadmium (Cd)	110
Niobium (Nb)	39

The second London equation can be paraphrased with the help of Maxwell's equation : ${\displaystyle \mu _{0}{\vec {j}}=\operatorname {rot} {\vec {B}}}$

${\displaystyle \nabla ^{2}{\vec {B}}={\frac {1}{\lambda _{\text{L}}^{2}}}\cdot {\vec {B}}}$

The solution to this equation describes an exponential decay of the magnetic field within the superconductor, as observed in experiments (see Meißner-Ochsenfeld effect ). For a homogeneous magnetic field of strength B ₀ in the z direction, which is applied to the surface of the superconductor (perpendicular to the x axis), the solution is:

${\displaystyle B_{\text{z}}(x)=B_{0}\cdot e^{-x/\lambda _{\text{L}}}}$

The magnetic field decays exponentially in the superconductor, with the penetration depth λ _L , for which applies:

${\displaystyle \lambda _{\text{L}}={\sqrt {\frac {m_{\text{e}}}{\mu _{0}nq^{2}}}}}$

It is the electron mass, q the charge, n , the number density of the superconducting charge carriers and the magnetic field constant . The shielding current density is obtained: ${\displaystyle m_{\mathrm {e} }}$ ${\displaystyle \mu _{0}}$

${\displaystyle j_{\text{y}}(x)={\frac {B_{0}}{\mu _{0}\lambda _{\text{L}}}}e^{-x/\lambda _{\text{L}}}}$

A shielding current flows in a thin outer layer of the superconductor, perpendicular to the magnetic field.

Derivation of the London equation via the macroscopic wave function

Approach: The superconducting state is a quantum mechanical state that extends over macroscopic length scales. It can therefore be described by a macroscopic wave function :

${\displaystyle \psi ({\vec {r}})=\psi _{0}\,\exp(i\,S({\vec {r}}))}$

It is assumed that it has a constant, real (!) Amplitude and that only the phase S is location-dependent. corresponds to the particle number density of the Cooper pairs . A constant amplitude therefore implies a constant particle number density. This assumption makes sense because the Cooper pairs in the superconductor are all negatively charged and repel each other. An imbalance of the particle number density would mean an electric field, which would be equalized immediately. ${\displaystyle \psi }$ ${\displaystyle \psi _{0}}$ ${\displaystyle |\psi _{0}|^{2}=\psi _{0}^{2}=n\,}$

The kinetic momentum operator in the presence of a magnetic field is:

${\displaystyle {\hat {\vec {p}}}=-i\hbar {\vec {\nabla }}-q{\vec {A}}}$

Applied to the wave function we get: ${\displaystyle \psi }$

${\displaystyle m{\vec {v}}\psi ={\hat {\vec {p}}}\psi =\left(\hbar {\vec {\nabla }}S-q{\vec {A}}\right)\psi }$

So:

${\displaystyle {\vec {v}}={\frac {\hbar }{m}}{\vec {\nabla }}S-{\frac {q}{m}}{\vec {A}}}$

With immediately follows: ${\displaystyle {\vec {j}}=qn{\vec {v}}}$

${\displaystyle {\vec {j}}={\frac {n\,q\,\hbar }{m}}{\vec {\nabla }}S-{\frac {n\,q^{2}}{m}}{\vec {A}}}$

This is the London equation given above.

Temperature dependence of the London penetration depth

The above-mentioned penetration depth is temperature-dependent because it depends on the particle number density. diverges at the critical temperature . The relationship between the penetration depth and the temperature can be described to a good approximation using an empirical formula from the Gorter-Casimir model (after Cornelis Jacobus Gorter ): ${\displaystyle \lambda _{\mathrm {L} }}$ ${\displaystyle n}$ ${\displaystyle \lambda _{\mathrm {L} }}$ ${\displaystyle T_{C}}$

${\displaystyle \lambda _{\mathrm {L} }(T)={\frac {\lambda _{\mathrm {L} }(0)}{\sqrt {1-\left({\frac {T}{T_{C}}}\right)^{4}}}}}$

literature

• Ch. Kittel: Introduction to Solid State Physics . Oldenbourg Verlag, Munich 1993

Individual evidence

1. ↑ Daijiro Yoshioka: The Quantum Hall Effect . Springer Science & Business Media, 2013, ISBN 3-662-05016-1 , p. 20 ( limited preview in Google Book search).
2. ^ Rudolf Gross, Achim Marx: Solid State Physics . 2nd Edition. De Gruyter, Berlin / Boston 2014, ISBN 978-3-11-035869-8 , pp. 795 .