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Register a new userKramer-Pesch approximation for analyzing field-angle-resolved measurements made in unconventional superconductors: A calculation of the zero-energy density of states
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By measuring angular-oscillation behavior of the heat capacity with respect to the applied field direction, one can detect the details of the gap structure. We introduce the Kramer-Pesch approximation (KPA) as a new method to analyze the field-angle-dependent experiments quantitatively. We calculate the zero energy density of states for various combinations of typical Fermi surfaces and superconducting gaps. The KPA yields a merit that one can quantitatively compare theoretical calculations with experimental results without involving heavy numerical computations, even for complicated Fermi surfaces. We show an inadequacy of the frequently-used Doppler-shift technique, which is remedied by application of the KPA.
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We theoretically discuss the magnetic-field-angle dependence of the zero-energy density of states (ZEDOS) in superconductors. Point-node and line-node superconducting gaps on spherical and cylindrical Fermi surfaces are considered. The Doppler-shift (DS) method and the Kramer-Pesch approximation (KPA) are used to calculate the ZEDOS. Numerical results show that consequences of the DS method are corrected by the KPA.
We investigate the field-angle-dependent zero-energy density of states for YNi2B2C with using realistic Fermi surfaces obtained by band calculations. Both the 17th and 18th bands are taken into account. For calculating the oscillating density of states, we adopt the Kramer-Pesch approximation, which is found to improve accuracy in the oscillation amplitude. We show that superconducting gap structure determined by analyzing STM experiments is consistent with thermal transport and heat capacity measurements.
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 and/or line nodes with linear and quadratic dispersions, by solving Eilenberger equation in vortex states. It is found that (A) the maxima of $N(theta)$ continuously shift from the anti-nodal to the nodal direction ($theta_{rm n}$) as a field increases accompanying the oscillation pattern reversal at low and high fields. Furthermore, (B) local minima emerge next to $theta_{rm n}$ on both sides except for the case of linear point node. These features are robust and detectable experimentally. Experimental results of $N(theta)$ performed on several superconductors, UPd$_2$Al$_3$, URu$_2$Si$_2$, Cu$_x$Bi$_2$Se$_3$, and UPt$_3$, are examined and commented in light of the present theory.
Over the past two decades, unconventional superconductivity with gap symmetry other than s-wave has been found in several classes of materials, including heavy fermion (HF), high-T_c, and organic superconductors. Unconventional superconductivity is characterized by anisotropic superconducting gap functions, which may have zeros (nodes) along certain directions in the Brillouin zone. The nodal structure is closely related to the pairing interaction, and it is widely believed that the presence of nodes is a signature of magnetic or some other exotic, rather than conventional phonon-mediated, pairing mechanism. Therefore experimental determination of the gap function is of fundamental importance. However, the detailed gap structure, especially the direction of the nodes, is an unresolved issue in most unconventional superconductors. Recently it has been demonstrated that the thermal conductivity and specific heat measurements under magnetic field rotated relative to the crystal axes are a powerful method for determining the shape of the gap and the nodal directions in the bulk. Here we review the theoretical underpinnings of the method and the results for the nodal structure of several unconventional superconductors, including borocarbide YNi$_2$B$_2$C, heavy fermions UPd$_2$Al$_3$, CeCoIn$_5$, and PrOs$_4$Sb$_{12}$, organic superconductor, $kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$, and ruthenate Sr$_2$RuO$_4$, determined by angular variation of the thermal conductivity and heat capacity.
Field-angle dependent specific heat measurement has been done on the heavy-fermion superconductor CeCoIn5 down to ~ 0.29 K, in a magnetic field rotating in the tetragonal c-plane. A clear fourfold angular oscillation is observed in the specific heat with the minima (maxima) occurring along the [100] ([110]) directions. Oscillation persists down to low fields H << Hc2, thus directly proving the existence of gap nodes. The results indicate that the superconducting gap symmetry is most probably of dxy type.
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