CND-10

London penetration depth

λ_L = √(m / (μ₀ n_s e²)). How far a magnetic field sneaks into a superconductor.

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SuperconductivityLondon

Governing equation

\lambda_L=\sqrt{m/(\mu_0 n_s e^2)}

$n_s \in \left[0.01, 10\right]$

where

n_s: Supercarrier density (10²⁸/m³)
m^*/m: Effective mass (—)
\lambda_L: Penetration (nm)

The figure, part by part

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①

Governing law — red

\lambda_L=\sqrt{m/(\mu_0 n_s e^2)}

$n_s \in \left[0.01, 10\right]$

$n_s = 0.01$ $\lambda_L = 531.4$

Along the curve

②

Domain of the variable — blue

n_s_{\min} = 0.01,\quad n_s_{\max} = 10

$n_s \in \left[0.01, 10\right]$

$n_s = 0.01$ $n_s = 0.01$

Along the curve

③

Response curve — green

\lambda_L=\sqrt{m/(\mu_0 n_s e^2)}

$n_s \in \left[0.01, 10\right]$

$n_s = 0.01$ $\lambda_L = 531.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\lambda_L}{\mathrm{d}n_s}

$n_s \in \left[0.01, 10\right]$

$n_s = 0.073$ $\lambda_L^{\prime} = -3080.8$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-10 — London penetration depth) is the form associated with London. Working symbols: $n_s$ , $m^*/m$ $\rightarrow$ $\lambda_L$ . The London equation ∇×j = −n_s e² A / m. λ_L is tens to hundreds of nm.

Purpose

Purpose: compute

\lambda_L

from

n_s

m^*/m

in Condensed matter via

\lambda_L=\sqrt{m/(\mu_0 n_s e^2)}

λ_L = √(m / (μ₀ n_s e²)). How far a magnetic field sneaks into a superconductor. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n_s = 1.000\,\mathrm{10^{2}⁸/m^{3}}

m^*/m = 1.000\,\mathrm{—}

, the governing relation

\lambda_L=\sqrt{m/(\mu_0 n_s e^2)}

yields

\lambda_L = 53.1\,\mathrm{nm}

. A slab, an exponentially dying B. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Supercarrier density n_s1 10²⁸/m³Effective mass m^*/m1 —

Outputs

Penetration \lambda_L53.1 nm

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Narration of this film

A slab, an exponentially dying B.

The London equation ∇×j = −n_s e² A / m. λ_L is tens to hundreds of nm.

Reading speed

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