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I'm an electronics student, and don't know much about some physics concepts. I was studying superconductivity and came to the London equation, Meissner effect and BCS theory.
I kind of understood these things, but am still wondering: what is superconducting coherence length?
All I understood is it's related to the Fermi length and for type I semiconductor.

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Superconductivity is about the appearance of an energy gap $\Delta$ in the excitation spectrum of the electron quasi-particles, which become paired in the form of a Cooper pair below the critical temperature $T_{c}$.

An other important energy scale in metals (superconducting or not) is the Fermi energy $E_{F}$, which represents the baseline energy of all conduction electrons. So one can generate many interesting criteria by manipulating the energy gap and the Fermi energy. For instance $\Delta/E_{F}$ represents the strength of the superconducting binding of electrons in the form of Cooper pairs.

From these energies : the energy gap $\Delta$ and the Fermi energy $E_{F}=mv_{F}^{2}/2$ with $v_{F}$ the Fermi velocity and $m$ the conduction band mass, one can generate a natural characteristic length $\ell_{b.}\sim\frac{\hbar v_{F}}{\Delta}$. This characteristic length naively represents the size of the Cooper pair, and it is called the coherence length.

This was for clean system. In diffusive systems, there is in addition the diffusion constant $D$ which represents an area explored by unit of time. So the natural length scale constructed from $\Delta$ is $\ell_{d.}\sim\sqrt{\frac{\hbar D}{\Delta}}$. Since one usually takes $D\sim v_{F}\cdot l$, with $l$ the mean-free path, this is still related to the Fermi velocity, though different exponent.

Coherence length depends on temperature (because $\Delta$ depends on temperature, as e.g. $\Delta\sim\sqrt{T-T_{c}}$ in the Ginzburg-Landau regime), and it's related to the phase rigidity of the superconducting condensate : one needs to tilt the superconducting phase over the size of the superconducting coherence length to get some current. Also, coherence length is related to the penetration length a superconductor exhibits in contact with a normal metal. Since $\Delta\rightarrow0$ as $T\rightarrow T_{c}$, the coherence length diverges at the critical temperature, a hallmark of a critical phenomena (i.e. a second order phase transition here).

Superconducting coherence length appears naturally in many circonstances, as e.g. the penetration of the superconducting properties over non-superconducting materials. For instance, the Josephson current $j$ behaves as $j\sim \ell_{b.}/x$ in ballistic systems and as $j\sim e^{-x/\ell_{d.}}$ in diffusive systems at distance $x$ from the superconducting interface.

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    $\begingroup$ So, in short, the characteristic length generated by transition of electrons from normal to superconducting state, is called the coherence length.?? $\endgroup$ – JayyM Dec 19 '17 at 11:26
  • $\begingroup$ Jay I think it is more as FraSchelle mentions above. This characteristic length naively represents the size of the Cooper pair, and it is called the coherence length. & one can generate a natural characteristic length ℓ∼ℏvF/Δ, with vF the Fermi velocity. $\endgroup$ – StarDrop9 Dec 19 '17 at 17:59
  • $\begingroup$ I forget to mention : $v_F$ is the only characteristic velocity of a normal metal, so it's natural to see it appearing in the construction of the characteristic length, and $\hbar$ mention its quantum origin. Otherwise, feel, free to construct a natural length-over-time constant in normal metal, like the atomic length scale over ... well, what ? $\endgroup$ – FraSchelle Dec 20 '17 at 17:52
  • $\begingroup$ @JayyM Yes of course ! That's the way physicists think about a new system : what is the length the new property/ies can be shared over ? Surely the coherence length come from microscopic calculations as well (as e.g. transport equation in superconductors), but it's quite cumbersome to extract it from your question. So feel free to ask for more microscopic details in an other question. Note nevertheless that the superconducting coherence length is related to mesoscopic coherence length, as I discuss in an other post here. $\endgroup$ – FraSchelle Dec 20 '17 at 18:02
  • $\begingroup$ @FraSchelle It is the characteristic component of the variations of the density of superconducting component. In the case of BCS theory, it is related to characteristic Cooper pair size.... Is this understood right by me? $\endgroup$ – JayyM Dec 20 '17 at 18:15
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I would like to add a more intuitive description to FraSchelle's nice answer. As you guessed correctly, the coherence length is the correlation length associated with the phase transition from a superconducting to the normal state. As with any other phase transition, this length describes how far fluctuations of the order parameter propagate. In a superfluid or superconductor(superfluid of cooper-pairs), the order parameter is the superfluid density or condensed cooper-pair density. Fluctuation of this density implies exciting particles out of the condensate. In a superconductor, you can have excited thermal cooper-pairs or you can have broken cooper-pairs, while for a superfluid of bosons, fluctuations of the density just excites thermal particles.

With this in mind you can image that at the boundary of a superconductor, there will be a higher density of unpaired electrons coming from the neighbouring materials. The region of significantly unpaired electrons will have a depth related to the coherence length.

Similarly, the cloud of unpaired electrons around an impurity will have a radius proportional to the coherence length and the region of unpaired electrons at the center of a vortex will again be the size of the coherence length.

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    $\begingroup$ Sorry but it the coherence length is the region of unpaired electrons, higher coherence length means more unpaired electrons. Doesn’t this spoil superconductivity as it needs Cooper pairs? $\endgroup$ – SuperCiocia Dec 19 '17 at 16:43
  • $\begingroup$ Yes, This is why at the transition from superconductivity to a normal metal, the coherence length diverges. Also why if you increase the fermi velocity, it increases( more kinetic energy to over come in "binding" cooper pairs). $\endgroup$ – Shane P Kelly Dec 19 '17 at 17:36
  • $\begingroup$ So for a good superconductor you want a small coherence length? $\endgroup$ – SuperCiocia Dec 19 '17 at 22:23
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    $\begingroup$ depends on what you mean by good. but yea, a superconductor with a higher energy gap (the energy need to break a cooper pair and excite an electron) will have a smaller coherence length. $\endgroup$ – Shane P Kelly Dec 19 '17 at 22:34
  • $\begingroup$ Does higher energy gap mean more stable superconductor? Against external perturbations I mean. So in this sense small coherence length is good I guess? $\endgroup$ – SuperCiocia Dec 19 '17 at 22:47
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"An independent characteristic length is called the coherence length. It is related to the Fermi velocity for the material and the energy gap associated with the condensation to the superconducting state. It has to do with the fact that the superconducting electron density cannot change quickly-there is a minimum length over which a given change can be made, lest it destroy the superconducting state. For example, a transition from the superconducting state to a normal state will have a transition layer of finite thickness which is related to the coherence length. Experimental studies of various superconductors have led to the following calculated values for these two types of characteristic lengths." Sourced from following Characteristic Lengths in Superconductors .

See table for various Coherence lengths of various materials in the linked article.

I have also added another great youtube video link that also explains Coherence length in greater detail. An interesting point is the way impurities added to a pure material can be inserted into a material to upset its electronic density and its superconducting waveform and cause the material to lose its superconducting properties. The more I research coherence length the more I understand it as the cooper electron pairing that takes place at superconducting temperatures and the resultant waveform that each substance gives rise to and the waveforms with wavelengths that stop the penetration of external magnetism.

Part2 Penetration Depth and Coherence Length.avi

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  • $\begingroup$ Thanks for the answer. So, as you said 'a transition from the superconducting state to a normal state' .. Is this related to Meissner Effect somewhere.?? Or just the transition is directly related to Coherence length, depending on the Fermi velocity? $\endgroup$ – JayyM Dec 19 '17 at 9:57
  • $\begingroup$ Yes it is related to the Meissner Effect. If the Cooper pairing is sufficient as a result of T->Tc superconducting will result and Δ -> 0 then the Meisner Effect or blocking of magnetism occurs. And this waveform is described by this coherence length. One could imagine the degree of order of the Cooper Pairs is maintained for a said distance of degrees of freedom. $\endgroup$ – StarDrop9 Dec 19 '17 at 18:43
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It´s he maximum difference in length of the path traveled between the two laser beams, or in other terms: it is the extension in space in which the wave has a sinusoidal shape in such a way that its phase can be safely predicted.

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  • $\begingroup$ Thanks for the answer. Is this related to Superconductivity? $\endgroup$ – JayyM Dec 19 '17 at 9:58
  • $\begingroup$ That's coherence length of a light beam. $\endgroup$ – George Herold Dec 21 '17 at 18:54

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