Abrikosov vortex

<h2 id="overview">Overview</h2>
The solution is a combination of fluxon solution by <a href="/facts/Fritz_London/9NvJ4MR7">Fritz London</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> combined with a concept of core of quantum vortex by <a href="/facts/Lars_Onsager/V1rDtqkQ">Lars Onsager</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
In the quantum vortex, <a href="/facts/Supercurrent/RgxuAVEd">supercurrent</a> circulates around the normal (i.e. non-superconducting) core of the vortex. The core has a size 
 
 
 
 ∼
 ξ
 
 
 {\displaystyle \sim \xi }
 
 — the <a href="/facts/Superconducting_coherence_length/DKy1eoqN">superconducting coherence length</a> (parameter of a <a href="/facts/Ginzburg%25E2%2580%2593Landau_theory/VAHPgSzY">Ginzburg–Landau theory</a>). The supercurrents decay on the distance about 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 (<a href="/facts/London_penetration_depth/lTmmbTS0">London penetration depth</a>) from the core. Note that in <a href="/facts/Type-II_superconductors/8V4UTGjY">type-II superconductors</a> 
 
 
 
 λ
 >
 ξ
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle \lambda >\xi /{\sqrt {2}}}
 
. The circulating <a href="/facts/Supercurrent/RgxuAVEd">supercurrents</a> induce magnetic fields with the total flux equal to a single <a href="/facts/Magnetic_flux_quantum/ywtHmBQW">flux quantum</a> 
 
 
 
 
 Φ
 
 0
 
 
 
 
 {\displaystyle \Phi _{0}}
 
. Therefore, an Abrikosov vortex is often called a <a href="/facts/Fluxon/RVAJetQE">fluxon</a>.
The magnetic field distribution of a single vortex far from its core can be described by the same equation as in the London's fluxoid <a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

B
 (
 r
 )
 =
 
 
 
 Φ
 
 0
 
 
 
 2
 π
 
 λ
 
 2
 
 
 
 
 
 
 K
 
 0
 
 
 
 (
 
 
 r
 λ
 
 
 )
 
 ≈
 
 
 
 λ
 r
 
 
 
 exp
 ⁡
 
 (
 
 −
 
 
 r
 λ
 
 
 
 )
 
 ,
 
 
 {\displaystyle B(r)={\frac {\Phi _{0}}{2\pi \lambda ^{2}}}K_{0}\left({\frac {r}{\lambda }}\right)\approx {\sqrt {\frac {\lambda }{r}}}\exp \left(-{\frac {r}{\lambda }}\right),}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
where 
 
 
 
 
 K
 
 0
 
 
 (
 z
 )
 
 
 {\displaystyle K_{0}(z)}
 
 is a zeroth-order <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a>. Note that, according to the above formula, at 
 
 
 
 r
 →
 0
 
 
 {\displaystyle r\to 0}
 
 the magnetic field 
 
 
 
 B
 (
 r
 )
 ∝
 ln
 ⁡
 (
 λ
 
 /
 
 r
 )
 
 
 {\displaystyle B(r)\propto \ln(\lambda /r)}
 
, i.e. logarithmically diverges. In reality, for 
 
 
 
 r
 ≲
 ξ
 
 
 {\displaystyle r\lesssim \xi }
 
 the field is simply given by

B
        (
        0
        )
        ≈
        
          
            
              Φ
              
                0
              
            
            
              2
              π
              
                λ
                
                  2
                
              
            
          
        
        ln
        ⁡
        κ
        ,
      
    
    {\displaystyle B(0)\approx {\frac {\Phi _{0}}{2\pi \lambda ^{2}}}\ln \kappa ,}

where κ = λ/ξ is known as the Ginzburg–Landau parameter, which must be 
 
 
 
 κ
 >
 1
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle \kappa >1/{\sqrt {2}}}
 
 in <a href="/facts/Type-II_superconductor/8V4UTGjY">type-II superconductors</a>.
Abrikosov vortices can be trapped in a <a href="/facts/Type-II_superconductor/8V4UTGjY">type-II superconductor</a> by chance, on defects, etc. Even if initially <a href="/facts/Type-II_superconductor/8V4UTGjY">type-II superconductor</a> contains no vortices, and one applies a magnetic field 
 
 
 
 H
 
 
 {\displaystyle H}
 
 larger than the <a href="/facts/Upper_critical_field/z4rEQR1f">lower critical field</a> 
 
 
 
 
 H
 
 c
 1
 
 
 
 
 {\displaystyle H_{c1}}
 
 (but smaller than the <a href="/facts/Upper_critical_field/z4rEQR1f">upper critical field</a> 
 
 
 
 
 H
 
 c
 2
 
 
 
 
 {\displaystyle H_{c2}}
 
), the field penetrates into superconductor in terms of Abrikosov vortices. Each vortex obeys London's magnetic flux quantization and carries one quantum of magnetic flux 
 
 
 
 
 Φ
 
 0
 
 
 
 
 {\displaystyle \Phi _{0}}
 
.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> Abrikosov vortices form a lattice, usually triangular, with the average vortex density (flux density) approximately equal to the externally applied magnetic field. As with other lattices, defects may form as dislocations.

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Macroscopic_quantum_phenomena/upqTn4Rp">Macroscopic quantum phenomena</a></li>
<li><a href="/facts/Nielsen%25E2%2580%2593Olesen_vortex/vbQBQwba">Nielsen–Olesen vortex</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Abrikosov, A. A. (1957). "The magnetic properties of superconducting alloys". Journal of Physics and Chemistry of Solids. 2 (3): 199–208. Bibcode:1957JPCS....2..199A. doi:10.1016/0022-3697(57)90083-5. <a href="/wiki/Journal_of_Physics_and_Chemistry_of_Solids" target="_blank">/wiki/Journal_of_Physics_and_Chemistry_of_Solids</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">London, F. (1948-09-01). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">London, Fritz (1961). Superfluids (2nd ed.). New York, NY: Dover. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Onsager, L. (March 1949). "Statistical hydrodynamics". Il Nuovo Cimento. 6 (S2): 279–287. Bibcode:1949NCim....6S.279O. doi:10.1007/BF02780991. ISSN 0029-6341. S2CID 186224016. <a href="http://link.springer.com/10.1007/BF02780991" target="_blank">http://link.springer.com/10.1007/BF02780991</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Feynman, R.P. (1955), Chapter II Application of Quantum Mechanics to Liquid Helium, Progress in Low Temperature Physics, vol. 1, Elsevier, pp. 17–53, doi:10.1016/s0079-6417(08)60077-3, ISBN 978-0-444-53307-4, retrieved 2021-04-11 {{citation}}: ISBN / Date incompatibility (help) <a href="978-0-444-53307-4" target="_blank">978-0-444-53307-4</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">London, F. (1948-09-01). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">London, Fritz (1961). Superfluids (2nd ed.). New York, NY: Dover. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">de Gennes, Pierre-Gilles (2018) [1965]. Superconductivity of Metals and Alloys. Addison Wesley Publishing Company, Inc. p. 59. ISBN 978-0-7382-0101-6. <a href="978-0-7382-0101-6" target="_blank">978-0-7382-0101-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">London, F. (1948-09-01). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">London, Fritz (1961). Superfluids (2nd ed.). New York, NY: Dover. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Abrikosov vortex open-in-new

Abrikosov vortex