Unconventional superconductors under rotating magnetic field I: density of states and specific heat

On the basis of the microscopic quasi-classical Eilenberger theory, we analyze the recent angle-resolved specific heat experiment carried out at low temperature for Sr$_2$RuO$_{4}$ to identify the superconducting gap symmetry, comprising either horizontal or vertical line nodes relative to the tetragonal crystal symmetry. Several characteristics, in particular, the landscape of the in-plane oscillation amplitude $A_4(B, T)$ with a definite sign for almost the entire $B$-$T$ p...

Effects of Pauli paramagnetism on thermodynamic quantities in a vortex state, such as the specific heat $C$ and magnetization $M$, are studied using the quasiclassical Eilenberger formalism. We demonstrate that with an increase of paramagnetic depairing effect, the sigh of the curvature of the field dependence of $C$ changes from negative to positive, and that the Maki parameter $\kappa_2$ becomes an increasing function of temperature. Our results provide a natural explanatio...

We propose a spectroscopic method to identify the nodal gap structure in unconventional superconductors. This method best suits for locating the horizontal line node and for pinpointing the isolated point nodes by measuring polar angle ($\theta$) resolved zero energy density of states $N(\theta)$. This is measured by specific heat or thermal conductivity at low temperatures under a magnetic field. We examine a variety of uniaxially symmetric nodal structure, including point a...

The gap function in unconventional superconductors may vanish at points or lines in momentum space, permitting electronic excitations, termed nodal quasiparticles, to exist at temperatures well below the superconducting transition. In the vortex phase, the presence of nodal quasiparticles should be directly observable through the variation of the heat capacity with the angle between a magnetic field and the location of the zeroes of the gap. The heat capacity of candidate non...

We develop a quasi-classical theory for the superconducting proximity effect in a ballistic ferromagnetic layer in contact with a d-wave superconductor. In agreement with recent experiments we find that the density of states oscillate around the normal state value with varying the thickness of the ferromagnetic layer. We show that the phase, the amplitude, and the period of these oscillations depend on the orientation of the superconductor. This effect reveals spatial oscilla...

The influence of a rotating magnetic field (in the $ab-$plane) on the density of states has been investigated in the superconducting state of the nematic FeSe superconductor using high sensitivity specific heat measurements. As expected for (quasi-)nodal superconductors, oscillations in the specific heat ($C$) associated to the Doppler energy shift of Cooper pairs with momenta close to the gap minima are observed. In the $T_c = 9$~K crystal, $C(\phi$) displays a twofold symme...

We consider the problem of the vortex contribution to thermal properties of dirty d-wave superconductors. In the clean limit, Volovik has argued that the main contribution to the density of states in a d-wave superconductor arises from extended quasiparticle states which may be treated semiclassically, giving rise to a specific heat contribution \delta C(H)\ltsim H^{1/2} . We show that the extended states continue to dominate the dirty limit, but lead to a H \log H behavior a...

We study the vortex state of a layered superconductor with vertical line nodes on its Fermi surface when a magnetic field is applied in the ab-plane direction. We rotate the magnetic field within the plane, and analyze the change of low-energy excitation spectrum. Our analysis is based on the microscopic Bogoliubov-de Gennes equation and a convenient approximate analytical method developped by Pesch and Dahm. Both methods give a consistent result. Near the upper critical fi...

We develop an alternative method to solve the Eilenberger equations numerically for the vortex-lattice states of type-II superconductors. Using it, we clarify the magnetic-field and impurity-concentration dependences of the magnetization, the entropy, the Pauli paramagnetism, and the mixing of higher Landau levels in the pair potential for two-dimensional $s$- and $d_{x^2-y^2}$-wave superconductors with the cylindrical Fermi surface.

Using the path-integral formulation we derive microscopically a Ginzburg-Landau free energy with a Zeeman coupling between the magnetic field and the orbital angular momentum of the Cooper pairs in a superconductor with singlet pairing in the $d_{x^{2}-y^{2}}$- and the sub-dominant $d_{xy}$- channels. The Zeeman coupling induces a time-reversal-symmetry-breaking pairing state. Based on careful examinations of the energy gain due to the Zeeman coupling, the energy lost due to ...