Relation between Superconducting Energy Gaps and Critical Magnetic Fields

Abstract

In 1965 Toxen noted¹ that for most elemental superconductors there exists a linear relation between the initial slope of the reduced critical magnetic field curve and the zero-temperature energy gap:

$$-({{T}_{0}}/{{H}_{0}}){{(d{{H}_{c}}/dT)}_{T}}{{_{=}}_{T}}_{_{0}}\equiv -{{(dh/dt)}_{t}}_{=1}=\vartriangle /k{{T}_{0}}$$

(1)

where T₀ is the superconducting transition temperature, H _c is the critical magnetic field [H _o ≡ H _c (T = 0)], and 2∆ is the zero-temperature energy gap. According to weak coupling BCS theory,²

$$\Delta /k{{T}_{0}}=1.016{{(dh/dt)}_{t}}{{=}_{1}}$$

(2)

where (dh/dt) _t = ₁ and ∆/kT₀ are not variables but are fixed numbers, 1.737 and 1.764, respectively. The importance of Toxen’s observation is that the weak coupling BCS expression is found to hold for moderate and strong coupling superconductors provided (dh/dt) _t=1 and ∆/k T₀ are treated as variables. One can therefore determine ∆ if T ₀ , H ₀ , and ${\left( {d{H_c}/dT} \right)_{T = {T_0}}}$ are known.