Proximity effect (superconductivity)

In superconductivity, the proximity effect describes the mutual influence of a superconductor (S) and a normal conductor (N) at their common interface . The effect is due to the penetration of Cooper pairs into the normal conductor and a simultaneous decrease in their density in the superconductor. It is particularly used in Josephson junctions, in which Cooper pairs tunnel through normally conducting layers . If an electron from the normal conductor hits the interface, its transmission into the superconductor depends on whether its energy is large enough to overcome the energy gap of the superconductor. This process is called Andreev reflection .

The critical temperature of the system made up of normal and superconductors depends largely on the thickness of the superconducting layer. If this is significantly greater than the Ginsburg-Landau coherence length (see below), the critical temperature of the overall system is close to that of the superconductor. However, if the superconducting layer is significantly smaller than the coherence length, the critical temperature of the system is close to that of the normal conductor or at . ${\displaystyle \xi }$ ${\displaystyle T=0\,\mathrm {K} }$

The effect was discovered in the 1930s with a material combination of lead and constantan and was extensively investigated by Hans Meissner in the 1950s.

Derivation in the Ginsburg-Landau theory

Course of the order parameter in the supra and normally conducting area ${\displaystyle \psi }$

The Ginsburg-Landau theory describes the behavior of a superconductor with the help of the order parameter . It can also be used to derive the characteristic coherence length of a superconductor, which indicates the length over which the density of the Cooper pairs in the superconductor varies. ${\displaystyle \psi (r)}$ ${\displaystyle \xi }$

In the following we consider a one-dimensional system made of two materials, the flat interface of which is at . In the right half space ( ) there is a superconductor and to the left of the origin ( ) there is a normal conductor (or superconductor above the critical temperature). This is illustrated in the illustration on the right. It is assumed that the order parameter assumes the value infinitely in the superconductor and does not vary further there, so that its derivative also disappears. The same applies to the normal conducting area . ${\displaystyle x=0}$ ${\displaystyle x>0}$ ${\displaystyle x<0}$ ${\displaystyle \psi (x\to +\infty )=1}$ ${\displaystyle \psi (x\to -\infty )=0}$

Course of the order parameter in the superconductor

The first Ginsburg-Landau equation without an external magnetic field is:

${\displaystyle -\xi ^{2}{\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} x^{2}}}-\psi +\psi ^{3}=0}$

It can be rewritten by simple integration and multiplication with the factor : ${\displaystyle {\frac {1}{2}}}$

${\displaystyle -\xi ^{2}\left({\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right)^{2}-\psi ^{2}+{\frac {1}{2}}\psi ^{4}=-{\frac {1}{2}}}$

The constant of integration on the right-hand side of the equation results from the boundary condition mentioned above for . If you put this intermediate result back into the Ginsburg-Landau equation, the solution of the differential equation is: ${\displaystyle x\to \infty }$

${\displaystyle \psi =\tanh \left({\frac {x-x_{0}}{{\sqrt {2}}\xi }}\right)}$

With the integration constant is taken explicitly not the simplification requires that the order parameter directly at the interface on drops, but only in the normal range. Instead, a second boundary condition is used at the interface: ${\displaystyle x_{0}\neq 0}$ ${\displaystyle 0}$

${\displaystyle \psi (x=0)=b\left({\frac {\mathrm {d} \psi }{\mathrm {d} x}}\right){\bigg |}_{x=0}}$

${\displaystyle b}$is called the extrapolation length because it marks the point at which the course of the order parameter extended into the normally conducting area would intersect the abscissa if it kept the course that it had at the interface, coming from the superconducting area. If the hyperbolic tangent derived above is inserted into this boundary condition, this leads to an equation that describes the relationship between the parameter and : ${\displaystyle b}$ ${\displaystyle x_{0}}$

${\displaystyle \sinh \left({\frac {{\sqrt {2}}x_{0}}{\xi }}\right)=-{\frac {{\sqrt {2}}b}{\xi }}}$

Course of the order parameter in the normally conducting area

Superconductor with ${\displaystyle T>T_{\mathrm {c} }}$

If the normally conducting material is also a superconductor, the critical temperature of which is only slightly lower than that of the superconductor in the area , then the first Ginsburg-Landau equation - with a different sign for the parameters - also applies to the Cooper pairs im normal conducting area. For the sake of clarity, the different coherence length in the normally conducting range is referred to as . Now we start with the first Ginsburg-Landau equation mentioned above, but neglect the term that cubically depends on the order parameter with the argument that this is significantly smaller than in the normally conducting range . ${\displaystyle x>0}$ ${\displaystyle \xi }$ ${\displaystyle \xi _{\mathrm {n} }}$ ${\displaystyle \psi }$ ${\displaystyle 1}$

${\displaystyle -\xi _{\mathrm {n} }^{2}{\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} x^{2}}}+\psi +{\mathcal {O}}(\psi ^{3})=0}$

The solution results from the above boundary condition ${\displaystyle x\to -\infty }$

${\displaystyle \psi =\psi _{0}e^{-{\frac {|x|}{\xi _{\mathrm {n} }}}}}$

If the critical temperatures of the two materials in close proximity, can and its derivative are assumed to be constant, so that: . ${\displaystyle \psi }$ ${\displaystyle b\approx \xi _{\mathrm {n} }}$

Normal conductor

If the left half-space is a material that is not superconducting at any temperature, the Ginsburg-Landau equation cannot be used. Qualitatively, however, there is a similar behavior. How large the characteristic length is depends largely on the mean free path of an electron in the material. In pure metals, in which the mean free path is significantly greater than , applies ${\displaystyle \xi _{\mathrm {n} }}$ ${\displaystyle l}$ ${\displaystyle \xi _{\mathrm {n} }}$

${\displaystyle \xi _{n}={\frac {\hbar v_{\mathrm {F} }}{2\pi k_{\mathrm {B} }T}}}$

with the reduced Planck constant , the Fermi velocity and the Boltzmann constant . For the characteristic length diverges, so that the decrease in the order parameter is significantly slower than exponential. ${\displaystyle \hbar }$ ${\displaystyle v_{\mathrm {F} }}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T\to 0\,\mathrm {K} }$

For "dirty", impure metals with a mean free path, the following applies: ${\displaystyle l\ll \xi _{n}}$

${\displaystyle \xi _{n}={\sqrt {\frac {\hbar v_{\mathrm {F} }l}{6\pi k_{\mathrm {B} }T}}}={\sqrt {\frac {\hbar D}{2\pi k_{\mathrm {B} }T}}}}$

with the diffusion constant in the normally conducting range. The value of then depends on the conductivities and in the super- and normally conductive range. ${\displaystyle D}$ ${\displaystyle b}$ ${\displaystyle \sigma _{\mathrm {s} }}$ ${\displaystyle \sigma _{\mathrm {n} }}$

${\displaystyle b\approx {\frac {\sigma _{\mathrm {s} }}{\sigma _{n}}}\xi _{\mathrm {n} }}$

insulator

If the left half-space is an insulator, then for the order of magnitude : ${\displaystyle b}$

${\displaystyle b\approx {\frac {\xi _{0}^{2}}{a_{0}}}}$

Here is the mean expansion of the Cooper pairs and the atomic distance in the insulator. ${\displaystyle \xi _{0}}$ ${\displaystyle a_{0}}$

Individual evidence

1. ^ Meissner, Hans: Superconductivity in contacts with interposed barriers . In: Phys. Rev. . 117, 1960, pp. 672-680.
2. ↑ Werner Buckel, Reinhold Kleiner: Superconductivity - Basics and Applications . 7th edition. Wiley-VCH, Weinheim 2013, ISBN 978-3-527-41139-9 , pp. 335 .
3. ↑ Werner Buckel, Reinhold Kleiner: Superconductivity - Basics and Applications . 7th edition. Wiley-VCH, Weinheim 2013, ISBN 978-3-527-41139-9 , pp. 223 f .
4. ^ Paul Müller, Alexey V. Ustinov: The Physics of Superconductors - Introduction to Fundamentals and Applications . Springer-Verlag, Berlin / Heidelberg / New York 1997, ISBN 3-540-61243-2 , pp. 54 ff .
5. ↑ Andrei Mourachkine: Room Temperature Superconductivity . Cambridge International Science Publishing, Cambridge 2004, ISBN 1-904602-27-4 , pp. 59 ff .