Quasiclassical theory of superconducting states under magnetic fields: Thermodynamic properties

We present a simple calculational scheme for superconducting properties under magnetic fields. A combination of an approximate analytic solution with a free energy functional in the quasiclassical theory provides a wide use formalism for spatial-averaged thermodynamic properties, and requires a little numerical computation. The theory covers multiband superconductors with various set of singlet and unitary triplet pairings in the presence of an impurity scattering. It is also applicable to analyze experimental results in a rotating magnetic field with help of band structure calculations. We demonstrate the application to $s$-wave, $d_{x^2-y^2}$-wave and two-band $s$-wave pairings, and discuss the validity of the theory comparing with previous numerical studies.

Figure 1

The averaged thermodynamic quantities for the two-band $s$-wave: (a) the gap magnitude scaled by ${\Delta }_0=\Delta (T=H=0)$. The thin line represents the empirical formula $\Delta \left(T,H\right)=\Delta \left(T,0\right)\sqrt{1-H∕H_{c2}}$, (b) the density of states at $T=0$, (c) the field dependence of the zero-energy DOS compared with full numerical results (Refs. 53, 54), (d) the specific heat.

Figure 2

The averaged thermodynamic quantities for the $d_{x^2-y^2}$-wave: (a) the gap magnitude. Thin line is the same empirical formula in Fig. 1, (b) the DOS at $T=0$, (c) the ZEDOS and full numerical results (Ref. 54), (d) the specific heat.

Figure 3

The averaged thermodynamic quantities for the two-band $s$-wave: (a) the gap magnitude scaled by the primary gap ${\Delta }_0={\Delta }_1(T=H=0)$, (b) the DOS at $T=0$, (c) the ZEDOS and numerical results (Ref. 11), (d) the total specific heat. The coupling matrix is given by $\widehat{\lambda }=\left(\right({\lambda }_1,\lambda ),(\lambda ,{\lambda }_2\left)\right)$ with ${\lambda }_1=0.25$, ${\lambda }_2=0.08{\lambda }_1$, $\lambda =0.26{\lambda }_1$.

Figure 4

The magnetization curves for several GL parameters. The thin lines are taken from the GL result (Ref. 56). For $\kappa ∕{\kappa }_c<1.9$ the lower transition becomes first order in the BPT approximation. The arrow indicates the hysteresis loop.