Homes's law

In superconductivity, Homes's law is an empirical relation that states that a superconductor's critical temperature (T_c) is proportional to the strength of the superconducting state for temperatures well below T_c close to zero temperature (also referred to as the fully formed superfluid density, $\rho _{s0}$) multiplied by the electrical resistivity $\rho _{dc}$ measured just above the critical temperature. In cuprate high-temperature superconductors the relation follows the form

$\rho _{dc}^{\alpha }\,\rho _{s0}^{\alpha }/8\simeq 4.4\,T_{c}$ ,

or alternatively

$\rho _{s0}^{\alpha }/8\simeq 4.4\,\sigma _{dc}^{\alpha }\,T_{c}$.

Many novel superconductors are anisotropic, so the resistivity and the superfluid density are tensor quantities; the superscript $\alpha$ denotes the crystallographic direction along which these quantities are measured. Note that this expression assumes that the conductivity and temperature have both been recast in units of cm⁻¹ (or s⁻¹), and that the superfluid density has units of cm⁻² (or s⁻²); the constant is dimensionless. The expected form for a BCS dirty-limit superconductor has slightly larger numerical constant of ~8.1.

The law is named for physicist Christopher Homes and was first presented in the July 29, 2004 edition of Nature, and was the subject of a News and Views article by Jan Zaanen in the same issue in which he speculated that the high transition temperatures observed in the cuprate superconductors are because the metallic states in these materials are as viscous as permitted by the laws of quantum physics. A more detailed version of this scaling relation subsequently appeared in Physical Review B in 2005.

Francis Pratt and Stephen Blundell have argued that Homes's law is violated in the organic superconductors. This work was first presented in Physical Review Letters in March 2005.