Zero-energy Density Of States Research Articles

Microscopic Eilenberger theory of Fulde-Ferrell-Larkin-Ovchinnikov states in the presence of vortices

We theoretically investigate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state by using the microscopic quasi-classical Eilenberger equation. The Pauli paramagnetic effects and the orbital depairing effects due to vortices are treated in an equal footing for three dimensional spherical Fermi surface model and $s$-wave pairing. The field evolution of the LO state is studied in detail, such as the $H$-$T$ phase diagram, spatial structures of the order parameter, the paramagnetic moment, and the internal filed. Field-dependences of various thermodynamic quantities: the paramagnetic moment, entropy, and the zero-energy density of states are calculated. Those quantities are shown to start quickly growing upon entering the LO state. We also evaluate the wave length of the LO modulation, the flux line lattice form factors for small angle neutron scattering, and the NMR spectra to facilitate the identification of the LO state. Two cases of strong and intermediate Pauli paramagnetic effect are studied comparatively. The possibility of the LO phase in Sr$_2$RuO$_4$, CeCoIn$_5$, CeCu$_2$Si$_2$, and the organic superconductors is critically examined and crucial experiments to identify it are proposed.

Fate of superconductivity in disordered Dirac and semi-Dirac semimetals

The influence of weak disorder on the superconductivity in ordinary metals can be formally described by the Abrikosov-Gorkov diagrammatic approach. The vertex correction is ignored in this approach because an inequality kF l ≫ 1, where kF is the Fermi momentum and l mean free path, is satisfied in ordinary metals with a large Fermi surface. In a Dirac semimetal that has discrete Fermi points, this inequality may break down even for arbitrarily weak disorder since , and thus the vertex correction could be important. We incorporate the vertex correction into the self-consistent equations of the superconducting gap and the disorder scattering rate, and then apply the generalized approach to study how s-wave superconductivity is affected by random chemical potential in two- and three-dimensional Dirac semimetals, as well as two-dimensional semi-Dirac semimetal. In the clean limit, superconductivity is formed only when the pairing interaction strength is greater than some critical value in these materials. Adding random chemical potential to the system promotes superconductivity by generating a finite fermionic density of states at the Fermi level. In three-dimensional Dirac semimetal, the critical attraction strength is reduced by weak disorder, but remains finite. In the other two cases, superconductivity is induced by arbitrarily weak attraction. Including the vertex correction does not change these qualitative results, and actually could further promote superconductivity in the weak-attraction regime. Bilayer graphene is quite special in that its zero-energy density of states is nonzero despite the existence of Fermi points. Due to this peculiar property, superconductivity is always slightly suppressed by random chemical potential, and the impact of vertex correction is nearly negligible.

Absence of zero-energy surface bound states in CuxBi2Se3studied via Andreev reflection spectroscopy

Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ has been proposed as a potential topological superconductor characterized by an odd-parity full bulk superconducting gap and zero-energy surface Andreev bound states (Majorana fermions). A predicted consequence of such Majorana fermions is a peak in the zero-energy density of states, which should lead to a persistent zero bias conductance peak (ZBCP) in Andreev reflection (AR) or tunneling experiments. Here, we employ a newly developed nanoscale AR spectroscopy method to study normal metal/superconductor (N-S) devices featuring Cu${}_{x}$Bi${}_{2}$Se${}_{3}$. The results show that a ZBCP can be tuned in or out from Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ samples depending on the N-S barrier strength. While the appearance of ZBCP may be traced to different origins, its absence under finite barrier strength represents the absence of zero-energy Majorana fermions. The present observations thus call for a reexamination of the intriguing superconductivity in Cu${}_{x}$Bi${}_{2}$Se${}_{3}$.

Using controlled disorder to distinguishs±ands++gap structure in Fe-based superconductors

We reconsider the effect of disorder on the properties of a superconductor characterized by a sign-changing order parameter appropriate for Fe-based materials. Within a simple two band model, we calculate simultaneously $T_c$, the change in residual resistivity $\Delta \rho_0$, and the zero-energy density of states, and show how these results change for various types of gap structures and assumptions regarding the impurity scattering. The rate of $T_c$ suppression is shown to vary dramatically according to details of the impurity model considered. We search therefore for a practical, experimentally oriented signature of a gap of the $s_\pm$ type, and propose that observation of particular evolution of the penetration depth, nuclear magnetic resonance relaxation rate, or thermal conductivity temperature dependence with disorder would suffice.

Analysis of field-angle dependent specific heat in unconventional superconductors: A comparison between Doppler-shift method and Kramer–Pesch approximation

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.

Quasiclassical Theory for Thermodynamic Properties of the Two-Band Superconductor MgB2 under Magnetic Fields

Consistent account is provided of strongly anisotropic properties in thermodynamics of MgB$_2$ under magnetic field $H$ using approximate analytic solution in the quasiclassical formalism. With the strength of the pairing interactions and the Fermi velocities of two bands as fitting parameters, it is possible to reproduce in wide regions of the phase diagram the $H$- and temperature dependences of relevant quantities such as the zero-energy density of states (ZEDOS), two energy gaps, the specific heat and the $H_{c2}$ anisotropy. It is demonstrated that the velocity ratio of two bands affects considerably the low-field behavior of the ZEDOS and energy gaps. The ratio also affects the high-temperature behavior of the $H_{c2}$ anisotropy. The anisotropy of the coherence length in the $\sigma$ band depends weakly on $H$ and $T$, whereas the coherence length in the $\pi$ band remains isotropic even in the presence of the inter-band coupling.

Theory of extracting superconducting gap structure from field-angle rotational experiments

We study the relation between the superconducting gap structure and field-orientational dependence of zero-energy density of states (ZEDOS), on the basis of an approximate analytical solution of the Eilenberger equation. Firstly, we quantitatively estimate the crossover magnetic field below which the result of the Doppler-shift method becomes correct. Secondly, we show that field-orientational dependence of ZEDOS has rather broad minima when the shape of node is point-like, in contrast to the result of the Doppler-shift method. Finally, we propose a novel mechanism leading to the cusp-like minima found in YNi 2 B 2 C in terms of a two-band model. This two-band model also provides a typical example in which the Doppler-shift predominant region is suppressed to much lower fields.

Effects of Fermi surface and superconducting gap structure in field-rotational experiments: A possible explanation for the cusplike singularity inYNi2B2C

We have studied the field-orientational dependence of zero-energy density of states (FODOS) for a series of systems with different Fermi surface and superconducting gap structures. Instead of phenomenological Doppler-shift method, we use an approximate analytical solution of Eilenberger equation together with self-consistent determination of order parameter and a variational treatment of vortex lattice. First, we compare zero-energy density of states (ZEDOS) when a magnetic field is applied in the nodal direction ($\nu_{node}(0)$) and in the antinodal direction ($\nu_{anti}(0)$), by taking account of the field-angle dependence of order parameter. As a result, we found that there exists a crossover magnetic field $H^*$ so that $\nu_{anti}(0) > \nu_{node}(0)$ for $H < H^*$, while $\nu_{node}(0) > \nu_{anti}(0)$ for $H > H^*$, consistent with our previous analyses. Next, we showed that $H^*$ and the shape of FODOS are determined by contribution from the small part of Fermi surface where Fermi velocity is parallel to field-rotational plane. In particular, we found that $H^*$ is lowered and FODOS has broader minima, when a superconducting gap has point nodes, in contrast to the result of the Doppler-shift method. We also studied the effects of in-plane anisotropy of Fermi surface. We found that in-plane anisotropy of quasi-two dimensional Fermi surface sometimes becomes larger than the effects of Doppler-shift and can destroy the Doppler-shift predominant region. In particular, this tendency is strong in a multi-band system where superconducting coherence lengths are isotropic. Finally, we addressed the problem of cusp-like singularity in YNi$_2$B$_2$C and present a possible explanation of this phenomenon.

Quasiparticle density of states in layered superconductors under a magnetic field parallel to theabplane: Determination of the gap structure ofSr2RuO4

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 invented by Pesch and developed by Dahm et al. Both methods give consistent results. Near the upper critical field ${H}_{c2}$, we observe a larger zero-energy density of states (ZEDOS) when the magnetic field is applied in the nodal direction, while much below ${H}_{c2}$, larger ZEDOS is observed under a field in the anti-nodal direction. We give a natural interpretation to this crossover behavior in terms of contributions of quasiparticles propagating parallel and perpendicular to the applied field in the plane. We examine the recent field angle variation experiments of thermal conductivity and specific heat. Comparisons with our results suggest that special care should be taken to derive the position of line nodes from the experimental data. Combining the experimental data of the specific heat and our analyses, we conclude that ${\mathrm{Sr}}_{2}\mathrm{Ru}{\mathrm{O}}_{4}$ has a vertical-line-node-like structure in the direction of the $a$ axis and the $b$ axis.

Entropy and spin susceptibility ofs-wave type-II superconductors nearHc2

A theoretical study is performed on the entropy ${S}_{\mathrm{s}}$ and the spin susceptibility ${\ensuremath{\chi}}_{\mathrm{s}}$ near the upper critical field ${H}_{c2}$ for s-wave type-II superconductors with arbitrary impurity concentrations. The changes of these quantities through ${H}_{c2}$ may be expressed as $[{S}_{\mathrm{s}}(T,B)\ensuremath{-}{S}_{\mathrm{s}}(T,0)]/[{S}_{\mathrm{n}}(T)\ensuremath{-}{S}_{\mathrm{s}}(T,0)]=1\ensuremath{-}{\ensuremath{\alpha}}_{S}(1\ensuremath{-}{B/H}_{c2})\ensuremath{\approx}{(B/H}_{c2}{)}^{{\ensuremath{\alpha}}_{S}},$ for example, where B is the average flux density and ${S}_{\mathrm{n}}$ denotes entropy in the normal state. It is found that the slopes ${\ensuremath{\alpha}}_{S}$ and ${\ensuremath{\alpha}}_{\ensuremath{\chi}}$ at $T=0$ are identical, connected directly with the zero-energy density of states, and vary from $1.72$ in the dirty limit to $0.5--0.6$ in the clean limit. This mean-free-path dependence of ${\ensuremath{\alpha}}_{S}$ and ${\ensuremath{\alpha}}_{\ensuremath{\chi}}$ at $T=0$ is quantitatively the same as that of the slope ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(T=0)$ for the flux-flow resistivity studied previously. The result suggests that ${S}_{\mathrm{s}}(B)$ and ${\ensuremath{\chi}}_{\mathrm{s}}(B)$ near $T=0$ are convex downward (upward) in the dirty (clean) limit, deviating substantially from the linear behavior $\ensuremath{\propto}{B/H}_{c2}.$ The specific-heat jump at ${H}_{c2}$ also shows fairly large mean-free-path dependence.

Quasiparticle Structure in Antiferromagnetism around the Vortex and Nuclear Magnetic Relaxation Time

On the basis of the Bogoliubov–de Gennes theory for the two-dimensional extended Hubbard model, the vortex structure in d -wave superconductors is investigated including the contribution of the induced incommensurate antiferromagnetism around the vortex core. As the on-site repulsive interaction U increases, the spatial structure of charge and spin changes from the antiferromagnetic state with checkerboard modulation to that with the stripe modulation. By the effect of the induced antiferromagnetic moment, the zero-energy density of states is suppressed, and the vortex core radius increases. We also study the effect of the local density of states (LDOS) change on the site-dependent nuclear relaxation rate T 1 -1 ( r ). These results are compared with a variety of experiments performed on high T c cuprates.

Orientational field dependence of low-lying excitations in the mixed state of unconventional superconductors

Orientational field-dependence of the zero energy density of states (ZEDOS) is calculated for superconductors with the polar state (line node), axial state (point node) and 3D d-wave state. Depending on the gap topology and relative field direction the field dependencies of ZEDOS sensitively differ, providing us a useful and practical method to identify the gap topology. It is also demonstrated that for d-wave state the field rotation in the basal plane shows a sizable oscillation ~3% of ZEDOS. This is directly measurable in low-T specific heat experiment in the mixed state.