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In superconductors, the London penetration depth (usually denoted as $\lambda$ or $\lambda _{L}$ ) characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to $e^{-1}$ times that of the magnetic field at the surface of the superconductor.^[1] Typical values of λらむだ_L range from 50 to 500 nm. It was first derived by Geertruida de Haas-Lorentz in 1925, and later by Fritz and Heinz London in their London equations (1935).^[2]

The London penetration depth results from considering the London equation and Ampère's circuital law.^[1] If one considers a superconducting half-space, i.e. superconducting for x>0, and weak external magnetic field B₀ applied along z direction in the empty space x<0, then inside the superconductor the magnetic field is given by^[1]

B(x)=B_{0}\exp \left(-{\frac {x}{\lambda _{L}}}\right),

\lambda _{L}

can be seen as the distance across in which the magnetic field becomes

e

times weaker. The form of

\lambda _{L}

is found by this method to be^[1]

\lambda _{L}={\sqrt {\frac {m}{\mu _{0}nq^{2}}}},

for charge carriers of mass

m

, number density

n

and charge

q

.

The penetration depth is determined by the superfluid density, which is an important quantity that determines T_c in high-temperature superconductors. If some superconductors have some node in their energy gap, the penetration depth at 0 K depends on magnetic field because superfluid density is changed by magnetic field and vice versa. So, accurate and precise measurements of the absolute value of penetration depth at 0 K are very important to understand the mechanism of high-temperature superconductivity.

There are various experimental techniques to determine the London penetration depth, and in particular its temperature dependence. London penetration depth can be measured by muon spin spectroscopy when the superconductor does not have an intrinsic magnetic constitution. The penetration depth is directly converted from the depolarization rate of muon spin in relation which σしぐま(T) is proportional to λらむだ²(T). The shape of σしぐま(T) is different with the kind of superconducting energy gap in temperature, so that this immediately indicates the shape of energy gap and gives some clues about the origin of superconductivity.

References[edit]

^ ^a ^b ^c ^d Kittel, Charles (2004). Introduction to Solid State Physics. John Wiley & Sons. pp. 273–278. ISBN 978-0-471-41526-8.
^ Fossheim, Kristian, and Asle Sudbø. Superconductivity: physics and applications. John Wiley & Sons, 2005.

[Kittel8thEd-1] Kittel, Charles (2004). Introduction to Solid State Physics. John Wiley & Sons. pp. 273–278. ISBN 978-0-471-41526-8.

[2] Fossheim, Kristian, and Asle Sudbø. Superconductivity: physics and applications. John Wiley & Sons, 2005.

[1]

[2]

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+{{short description|Distance to which a magnetic field penetrates into a superconductor}}
-In [[superconductors]], the '''London penetration depth''' (usually denoted as <math>\lambda</math> or <math>\lambda_L</math>) characterizes the distance to which a [[magnetic field]] penetrates into a superconductor and becomes equal to''' 1/e''' times that of the magnetic field at the surface of the superconductor.<ref name=Kittel8thEd/> Typical values of &lambda;<sub>L</sub> range from 50 to 500&nbsp;nm.
+In [[superconductors]], the '''London penetration depth''' (usually denoted as <math>\lambda</math> or <math>\lambda_L</math>) characterizes the distance to which a [[magnetic field]] penetrates into a superconductor and becomes equal to <math>e^{-1}</math> times that of the magnetic field at the surface of the superconductor.<ref name=Kittel8thEd/> Typical values of λらむだ<sub>L</sub> range from 50 to 500&nbsp;nm. It was first derived by [[Geertruida de Haas-Lorentz]] in 1925, and later by [[Fritz London|Fritz]] and [[Heinz London]] in their [[London equations]] (1935).<ref>Fossheim, Kristian, and Asle Sudbø. ''Superconductivity: physics and applications''. John Wiley & Sons, 2005.</ref>
-The London penetration depth results from considering the [[London equations|London equation]] and [[Ampère's circuital law]].<ref name=Kittel8thEd/> If one considers a superconducting medium occupying ''x''<sub>0</sub>, and weak external magnetic field '''B'''<sub>0</sub>> applied along ''z'' <direction in the empty space ''x''<sub>0</sub>, then inside the superconductor the magnetic field is given by
+The London penetration depth results from considering the London equation and [[Ampère's circuital law]].<ref name=Kittel8thEd/> If one considers a superconducting [[Half-space (geometry)|half-space]], i.e. superconducting for x>0, and weak external magnetic field '''B'''<sub>0</sub> applied along ''z'' direction in the empty space ''x''<0, then inside the superconductor the magnetic field is given by<ref name=Kittel8thEd/>
-:<math>B(x)=B_0\exp\left(-\frac{x}{\lambda_L}\right),</math><ref name=Kittel8thEd/>
+<math display="block">B(x) = B_0\exp\left(-\frac{x}{\lambda_L}\right),</math>
-<math>\lambda_L</math> can be seen as the distance across in which the magnetic field becomes <math>e</math> times weaker. The form of <math>\lambda_L</math> is found by this method to be
+<math>\lambda_L</math> can be seen as the distance across in which the magnetic field becomes [[e (math)|<math>e</math>]] times weaker. The form of <math>\lambda_L</math> is found by this method to be<ref name=Kittel8thEd/>
-:<math>\lambda_L=\left(\frac{m}{\mu_0 n q^2}\right)^{\frac{1}{2}}</math>,<ref name=Kittel8thEd/>
+<math display="block">\lambda_L=\sqrt{\frac{m}{\mu_0 n q^2}},</math>
-for charge carriers of mass <math>m</math>, number density <math>n</math> and charge <math>q</math>.
+for [[charge carrier]]s of [[mass]] <math>m</math>, [[number density]] <math>n</math> and [[Electric charge|charge]] <math>q</math>.
-The penetration depth is determined by the superfluid density, which is an important quantity that determines ''T''<sub>c</sub> in high-temperature superconductors. If some superconductors have some node in their energy gap, the penetration depth at 0&nbsp;K depends on magnetic field because superfluid density is changed by magnetic field and vice versa. So, accurate and precise measurements of the absolute value of penetration depth at 0&nbsp;K are very important to understand the mechanism of high-temperature superconductivity.
+The penetration depth is determined by the [[superfluid]] density, which is an important quantity that determines ''T''<sub>c</sub> in high-temperature superconductors. If some superconductors have some node in their [[energy gap]], the penetration depth at 0&nbsp;K depends on magnetic field because superfluid density is changed by magnetic field and vice versa. So, accurate and precise measurements of the absolute value of penetration depth at 0&nbsp;K are very important to understand the mechanism of high-temperature superconductivity.
-London penetration depth can be measured by [[muon spin spectroscopy]] when the superconductor doesn't have an intrinsic magnetic constitution. The penetration depth is directly converted from the depolarization rate of muon spin in relation which
-''σしぐま''(''T'') is proportional to ''λらむだ''<sup>2</sup>(''T''). The shape of ''σしぐま''(''T'') is different with the kind of superconducting energy gap in temperature, so that this immediately indicates the shape of energy gap and gives some clues about the origin of superconductivity to us.
+There are various experimental techniques to determine the London penetration depth, and in particular its temperature dependence. London penetration depth can be measured by [[muon spin spectroscopy]] when the superconductor does not have an intrinsic magnetic constitution. The penetration depth is directly converted from the depolarization rate of muon spin in relation which
-==Further reading==
+''σしぐま''(''T'') is proportional to ''λらむだ''<sup>2</sup>(''T''). The shape of ''σしぐま''(''T'') is different with the kind of superconducting energy gap in temperature, so that this immediately indicates the shape of energy gap and gives some clues about the origin of superconductivity.
-*http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000094000012127001000001&idtype=cvips&gifs=yes
 ==References==
 {{reflist|refs=
-<ref name=Kittel8thEd>{{Cite book | last = Kittel | first = Charles | title = Introduction to Solid State Physics | publisher = John Wiley & Sons | year = 2004 | pages = 273–278 | isbn = 978-0471415268}}
+<ref name=Kittel8thEd>{{Cite book | last = Kittel | first = Charles | title = [[Introduction to Solid State Physics]] | publisher = John Wiley & Sons | year = 2004 | pages = [https://archive.org/details/isbn_9780471415268/page/273 273–278] | isbn = 978-0-471-41526-8 }}
 </ref>
+}}{{Superconductivity}}
-}}
-==See also==
-*[[London equations]] 1935
-*[[Ginzburg-Landau theory]] 1950
 [[Category:Superconductivity]]
-[[Category:Physics]]
-[[ja:磁場じば侵入しんにゅう長ちょう]]
-[[pl:Głębokość wnikania Londonów]]
-[[uk:Лондонівська глибина проникнення]]