4.4 Coherence Length and Penetration Depth

Coherence length and penetration depth are key properties of superconductors. They determine how superconductors behave in magnetic fields and affect their ability to carry current. Understanding these length scales is crucial for developing practical superconducting devices.

The ratio of penetration depth to coherence length distinguishes between type-I and type-II superconductors. This affects how they respond to magnetic fields and their potential applications. Knowing these properties helps engineers design better superconducting materials for specific uses.

Coherence length and penetration depth

Defining coherence length

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• Coherence length is the characteristic distance over which the superconducting order parameter varies in a superconductor
  • The Ginzburg-Landau coherence length (ξ) describes the spatial variation of the superconducting order parameter near a boundary or interface
  • The BCS coherence length is ξ0 = $\frac{\hbar v_F}{\pi \Delta(0)}$, where $\hbar$ is the reduced Planck's constant, $v_F$ is the Fermi velocity, and $\Delta(0)$ is the superconducting energy gap at zero temperature
  • Coherence length determines the size of Cooper pairs and the spatial extent of the superconducting state
  • In type-I superconductors ($\kappa < \frac{1}{\sqrt{2}}$), the coherence length is larger than the penetration depth, leading to a complete Meissner effect and abrupt normal-superconducting transitions

Defining penetration depth

• Penetration depth is the distance over which an external magnetic field penetrates into a superconductor before being exponentially suppressed
  • The London penetration depth (λ) characterizes the distance over which the magnetic field and supercurrent density decay inside a superconductor
  • The London penetration depth is given by $\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}$, where $m$ is the electron mass, $\mu_0$ is the vacuum permeability, $n_s$ is the superconducting electron density, and $e$ is the electron charge
  • Penetration depth determines the extent to which a superconductor can screen out external magnetic fields
  • In type-II superconductors ($\kappa > \frac{1}{\sqrt{2}}$), the penetration depth is larger than the coherence length, allowing partial penetration of magnetic fields in the form of quantized vortices

Significance of length scales

Type-I and type-II superconductors

• The ratio of the penetration depth to the coherence length ($\kappa = \frac{\lambda}{\xi}$) distinguishes between type-I ($\kappa < \frac{1}{\sqrt{2}}$) and type-II ($\kappa > \frac{1}{\sqrt{2}}$) superconductors
  • Type-I superconductors exhibit a complete Meissner effect, where the magnetic field is entirely expelled from the superconductor (lead, aluminum)
  • Type-II superconductors allow partial penetration of magnetic fields in the form of quantized vortices, enabling higher critical fields and current densities (niobium, high-temperature superconductors)
  • The Ginzburg-Landau parameter $\kappa = \frac{\lambda}{\xi}$ determines the type of superconductor and its behavior in magnetic fields

Critical parameters

• The critical magnetic field $H_c$ is related to the coherence length and penetration depth by $H_c = \frac{\Phi_0}{2\sqrt{2}\pi\lambda\xi}$, where $\Phi_0$ is the magnetic flux quantum
  • The upper critical field $H_{c2}$ in type-II superconductors is given by $H_{c2} = \frac{\Phi_0}{2\pi\xi^2}$, showing its inverse dependence on the coherence length
  • The lower critical field $H_{c1}$ in type-II superconductors is related to the penetration depth by $H_{c1} \approx \frac{\Phi_0}{4\pi\lambda^2} \ln\left(\frac{\lambda}{\xi}\right)$
  • The critical current density $J_c$ is limited by the penetration depth, as larger $\lambda$ results in a smaller $J_c$ due to increased magnetic field penetration

Calculating length scales

Ginzburg-Landau coherence length

• The Ginzburg-Landau coherence length is given by $\xi(T) = \frac{\xi(0)}{\sqrt{1 - \frac{T}{T_c}}}$, where $\xi(0)$ is the coherence length at zero temperature and $T_c$ is the critical temperature
  • The temperature dependence of the coherence length shows that it diverges as the temperature approaches the critical temperature
  • The coherence length is a measure of the spatial extent of the superconducting order parameter and determines the size of Cooper pairs

London penetration depth

• The London penetration depth is given by $\lambda(T) = \frac{\lambda(0)}{\sqrt{1 - \left(\frac{T}{T_c}\right)^4}}$, where $\lambda(0)$ is the penetration depth at zero temperature
  • The temperature dependence of the penetration depth shows that it increases as the temperature approaches the critical temperature
  • The penetration depth characterizes the distance over which the magnetic field and supercurrent density decay inside a superconductor
  • The London penetration depth is also given by $\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}$, where $m$ is the electron mass, $\mu_0$ is the vacuum permeability, $n_s$ is the superconducting electron density, and $e$ is the electron charge

Coherence length vs penetration depth

Relationship between length scales

• The Ginzburg-Landau parameter $\kappa = \frac{\lambda}{\xi}$ is the ratio of the penetration depth to the coherence length and determines the type of superconductor
  • Type-I superconductors have $\kappa < \frac{1}{\sqrt{2}}$, meaning the coherence length is larger than the penetration depth (lead, aluminum)
  • Type-II superconductors have $\kappa > \frac{1}{\sqrt{2}}$, meaning the penetration depth is larger than the coherence length (niobium, high-temperature superconductors)
  • The relative magnitudes of the coherence length and penetration depth determine the superconductor's response to magnetic fields and the formation of vortices

Implications for applications

• Materials with shorter coherence lengths and longer penetration depths are more suitable for applications requiring high critical fields and current densities
  • Type-II superconductors, such as niobium and high-temperature superconductors, are used in applications like superconducting magnets and power transmission lines
  • Shorter coherence lengths allow for higher upper critical fields ($H_{c2}$), enabling superconductivity to persist in strong magnetic fields
  • Longer penetration depths result in lower critical current densities ($J_c$) but allow for the formation of vortices, which can be pinned to enhance current-carrying capacity
  • The optimization of coherence length and penetration depth is crucial for developing superconducting materials tailored to specific applications