A topological theory of d-wave superconductors is derived in this thesis. Ginzburg-Landau theory describes superconductivity by defining a complex order parameter and applying Landau's theory for phase transitions. However, there is no local, gauge invariant order parameter for a superconductor and classical order is no appropriate description. Topological order has proven to be a powerful tool for describing the Quantum Hall effect and it gives also an appropriate description of s-wave superconductors. For d-wave superconductors there are gapless excitations at four points on the Fermi surface. The topological theory for superconductors exhibits a similar structure for the ground state as found in the s-wave case. However, the gapless excitations destroy the topological degeneracy of the ground state and introduce an additional degeneracy in one of the flux sectors. In search for regularities which reflect the topological order, I include a magnetic point impurity and focus on the effects it has on the Casimir energy. Indeed, it is shown that the energy shift reflects the topology to some extend. However, this is work in progress and further computations will be done.
We discuss the nodal structure of odd-parity superconductors in the presence of non-symmorphic crystal symmetries, both with and without spin-orbit coupling, and with and without time reversal symmetry. We comment on the relation of our work to previous work in the literature, and also the implications for unconventional superconductors such as UPt$_3$.
In this work, we show that a giant spin current can be injected into a nodal topological superconductor, using a normal paramagnetic lead, through a large number of zero energy Majorana fermions at the superconductor edge. The giant spin current is caused by the selective equal spin Andreev reflections (SESAR) induced by Majorana fermions. In each SESAR event, a pair of electrons with certain spin polarization are injected into the nodal topological superconductor, even though the pairing in the bulk of the nodal superconductor is spin-singlet s-wave. We further explain the origin of the spin …
Here we consider the universal heat conduction and the angular dependent thermal conductivity in the vortex state for a few nodal superconductors. We present the thermal conductivity as a function of impurity concentration and the angular dependent thermal conductivity in a few nodal superconductors. This provides further insight in the gap symmetry of superconductivity in Sr$_2$RuO$_4$ and UPd$_2$Al$_3$.
Angle-resolved photoemission data on optimally- and under-doped high temperature superconductors reveal a dichotomy between the nodal and antinodal electronic excitations. In this paper we propose an explanation of this unusual phenomenon by employing the coupling between the quasiparticle and the commensurate/incommensurate magnetic excitations.
Recently it was revealed that the whole Fermi surface is fully gapped for several families of underdoped cuprates. The existence of the finite energy gap along the $d$-wave nodal lines ("nodal gap") contrasts the common understanding of the $d$-wave pairing symmetry, which challenges the present theories for the high-$T_c$ superconductors. Here we propose that the incommensurate diagonal spin-density-wave order can account for the above experimental observation. The Fermi surface and the local density of states are also studied. Our results are in good agreement with many important experiments…
We analyze the nodal points in multi-orbital Fe-based superconductors from a topological perspective. We consider the $s^{+-}$ gap structure with accidental nodes, and the $d$-wave gap with nodes along the symmetry directions. In both cases, the nodal points can be moved by varying an external parameter, e.g., a degree of inter-pocket pairing. Eventually, the nodes merge and annihilate via a Lifshitz-type transition. We discuss the Lifshitz transition in Fe-based superconductors from a topological point of view. We show, both analytically and numerically, that the merging nodal points have win…
We demonstrate that Majorana fermions exist in edges of systems and in a vortex core even for superconductors with nodal excitations such as the d-wave pairing state under a particular but realistic condition in the case with an anti-symmetric spin-orbit interaction and a nonzero magnetic field below the upper critical field. We clarify that the Majorana fermion state is topologically protected in spite of the presence of bulk gapless nodal excitations, because of the existence of a nontrivial topological number. Our finding drastically enlarges target systems where we can explore the Majorana…
Time-reversal invariant superconductors in three dimensions may contain nodal lines in the Brillouin zone, which behave exactly as Wilson loops of 3d momentum-space Chern-Simons theory of the Berry connection. Here we study the conditions of realizing linked nodal lines (Wilson loops), which yield a topological contribution to the thermal magnetoelectric coefficient that is given by the Chern-Simons action. We find the essential conditions are the existence of torus or higher genus fermi surfaces and spiral spin textures. We construct such a model with two torus fermi surfaces, where a generic…
We demonstrate that checkerboard magnet-superconductor hybrid systems possess a rich phase diagram exhibiting both strong topological superconducting (STSC) and topological nodal point superconducting (TNPSC) phases. We show that TNPSC phases exist both for ferromagnetic and antiferromagnetic systems, yielding a plethora of qualitatively different edge mode structures. Checkerboard MSH systems also facilitate the emergence of STSC phases which can be induced even in the limit of vanishing magnetization. Our results provide a new path for the quantum engineering of topological superconducting p…
Topological nodal superconductors are generally realized based on unconventional pairings. In this work, we propose a minimal model to realize these topological nodal phases with only $s$-wave interaction. In our model the linear and quadratic spin-orbit couplings along the two directions break the isotropy in momentum space and introduce effective unconventional pairings on the Fermi surface. This model may support different nodal superconducting phases characterized by either winding number in BDI class or Pfaffian in D class at the particle-hole invariant axes. In the vicinity of the nodal …
High-precision measurements of magnetic penetration depth $λ$ in clean single crystals of LiFeAs and LiFeP superconductors reveal contrasting low-energy quasiparticle excitations. In LiFeAs the low-temperature $λ(T)$ shows a flat dependence indicative of a fully gapped state, which is consistent with previous studies. In contrast, LiFeP exhibits a $T$-linear dependence of superfluid density $\propto λ^{-2}$, indicating a nodal superconducting order parameter. A systematic comparison of quasiparticle excitations in the 1111, 122, and 111 families of iron-pnictide superconductors implies that th…
We study the spontaneous gap generation for gapless nodal fermions within an effective gauge field theory of high temperature superconductors. When superconductivity appears, the gauge boson acquires a finite mass via Anderson-Higgs mechanism. Spontaneous nodal gap generation takes place if the gauge boson mass $ξ$ is zero or less than a critical value $ξ_{c}$ but is suppressed by a larger gauge boson mass. The generated nodal gap prohibits the appearance of low-energy fermion excitations and leads to antiferromagnetic order. Using the fact that gauge boson mass $ξ$ is proportional to superflu…
Vortex entry into (110) oriented YBa$_2$Cu$_3$O$_{7-x}$ films has been studied by tunneling into Andreev - Saint-James bound states, whose energy is shifted by surface currents. At low temperatures, the characteristic field for vortex entry has been found to increase up to values several times higher than that of the Bean-Livingston entry field for conventional superconductors, in agreement with recent theoretical predictions.
The superconducting gap is a pivotal character for a superconductor. While the cuprates and conventional phonon-mediated superconductors are characterized by distinct d-wave and s-wave pairing symmetry with nodal and nodeless gap distribution respectively, the superconducting gap distributions in iron-based superconductors are rather diversified. While nodeless gap distributions have been directly observed in Ba1-xKxFe2As2, BaFe2-xCoxAs2, KxFe2-ySe2, and FeTe1-xSex, the signatures of nodal superconducting gap have been reported in LaOFeP, LiFeP, KFe2As2, BaFe2(As1-xPx)2, BaFe2-xRuxAs2 and FeSe…
The discovery of signatures of topological superconductivity in superconducting bulk materials with topological surface states has attracted intensive research interests recently. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, here, we demonstrate the existence of topological nodal-line states and drumheadlike surface states in centrosymmetric superconductor SnTaS2, which is a type-II superconductor with a critical transition temperature of about 3 K. The valence bands from Ta 5d orbitals and the conduction bands from Sn 5p orbitals cross each other, for…
We study the properties of a quasi-one dimensional superconductor which consists of an alternating array of two inequivalent chains. This model is a simple charicature of a locally striped high temperature superconductor, and is more generally a theoretically controllable system in which the superconducting state emerges from a non-Fermi liquid normal state. Even in this limit, ``d-wave like'' order parameter symmetry is natural, but the superconducting state can either have a complete gap in the quasi-particle spectrum, or gapless ``nodal'' quasiparticles. We also find circumstances in which …
We study the stability of topologically protected zero-energy flat bands at the surface of nodal noncentrosymmetric superconductors, accounting for the alteration of the gap near the surface. Within a selfconsistent mean-field theory, we show that the flat bands survive in a broad temperature range below the bulk transition temperature. There is a second transition at a lower temperature, however, below which the system spontaneously breaks time-reversal symmetry. The surface bands are shifted away from zero energy and become weakly dispersive. Simultaneously, a spin polarization and an equili…
Nodal quasiparticles of Bi2Sr2CaCu2O8 have been studied by angle-resolved photoemission spectroscopy with high momentum and energy resolution. Low-energy tunable photons have enabled us to resolve a small nodal bilayer splitting, unmasking the intrinsic single-particle scattering rate. The nodal scattering rate is abruptly suppressed upon the superconducting transition, and shows a linear energy dependence at low energies, indicating the nontrivial effect of elastic scatterings on the quasiparticles. With increasing energy, the antibonding-band scattering rate becomes higher than the bonding o…
Supercurrents in superconducting flux threaded loops are expected to oscillate with the magnetic flux with a period of hc/2e. This is indeed true for s-wave superconductors larger than the coherence length xi_0. Here we show that for superconductors with gap nodes, there is no such strict condition for the supercurrent to be hc/2e rather than hc/e periodic. For nodal superconductors, the flux induced Doppler shift of the near nodal states leads to a flux dependent occupation probability of quasi-particles circulating clockwise and counter clockwise around the loop, which leads to an hc/e perio…
We discuss the symmetry property of a nodal superconductor that hosts robust flat-band zero-energy states at its surface under potential disorder. Such robust zero-energy states are known to induce the anomalous proximity effect in a dirty normal metal attached to a superconductor. A recent study has shown that a topological index ${\cal N}_\mathrm{ZES}$ describes the number of zero-energy states at the dirty surface of a $p$-wave superconductor. We generalize the theory to clarify the conditions required for a superconductor that enables ${\cal N}_\mathrm{ZES}\neq 0$. Our results show that ${…
Motivated by the recent discovery of a new family of Chromium based superconductors, we consider a two-band model, where a band of electrons dispersing only in one direction interacts with a band of electrons dispersing in all three directions. Strong $2k_f$ density fluctuations in the one-dimensional band induces attractive interactions between the three-dimensional electrons, which, in turn makes the system superconducting. Solving the associated Eliashberg equations, we obtain a gap function which is peaked at the "poles" of the three-dimensional Fermi sphere, and decreases towards the "equ…
In BiS$_2$-based layered superconductors,the existence of gap nodes on Fermi-surface curves has been suggested from angle-resolved photoemission spectroscopy measurements, whereas the conventional $s$-wave gap has been proposed from measurements of superfluid density and thermal conductivity. To reconcile these two distinct experimental results of the gap node, we investigate nonmagnetic impurity effects in the superconductor with a disconnected pocketlike Fermi-surface structure. Then, we claim that the seemingly contradictory situation concerning the gap node is resolved by a concept of dirt…
In contrast to multigap superconductors (e.g. MgB$_{2}$), the low-temperature properties of nodal superconductors are dominated by nodal excitations. Here we extend for a variety of nodal superocnductors the earlier work by Simon and Lee and Kübert and Hirschfeld. The scaling relations seen in the thermodynamics and the thermal conductivity will provide an unequivocal test of nodal superconductivity.
Boundary surfaces of nodal gap superconductors can host Andreev bound states (ABS) which develop a paramagnetic response under external RF field in contrast to the bulk diamagnetic response of the bulk superconductor. At low temperature this surface paramagnetic response dominates and enhances the nonlinear RF response of the sample. With a recently developed photoresponse imaging technique, the anisotropy of this "paramagnetic" nonlinear Meissner response, and its current direction (angular) and RF power dependence has been systematically studied. A theoretical model describing the current fl…
How to determine the symmetry of the superconducting order parameter is one of the important issues in novel superconductors, which include charge conjugated organic superconductors. We have proposed that the angular dependence of the thermal conductivity in a planar magnetic field provides a new window to look at the symmetry of the order parameter. After a brief summary of the quasiclassical approach we describe how the symmetry of the superconducting order parameter in Sr$_2$RuO$_4$, CeCoIn$_5$ and $κ$-(ET)$_2$Cu(NCS)$_2$ is determined. Also in some of experiments the phononic thermal condu…
We present a theoretical study of the superconducting gap function in PrOs4Sb12 using a symmetry-based approach. A three-component order parameter in the triplet channel best describes superconductivity. The gap function is non-degenerate and the lower branch has four cusp nodes at unusual points of the Fermi surface, which lead to power law behaviours in the density of states, specific heat and nuclear spin relaxation rate.
Motivated by Sr2RuO4, we consider a chiral superconductor where the gap is strongly suppressed along certain momentum directions. We evaluate the thermal Hall conductivity in the gapless regime, i.e., at temperature small compared with the impurity band width γ, taking the simplest model of isotropic impurity scattering. We find that, under favorable circumstances, this thermal Hall conductivity can be quite significant, and is smaller than the diagonal component (the universal thermal conductivity) only by a factor of 1/ ln(2 Δ_M/γ), where Δ_M is the maximum gap.
A layered superconductor with a full pairing energy gap can be driven into a nodal superconducting (SC) state by inter-layer pairing when the SC state becomes more quasi-3D. We propose that this mechanism is responsible for the observed nodal behavior in a class of iron-based SCs. We show that the intra- and inter-layer pairings generally compete and the gap nodes develop on one of the {\em hole} Fermi surface pockets as they become larger in the iron-pnictides. Our results provide a natural explanation of the c-axis gap modulations and gap nodes observed by angle resolved photoemission spectr…
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 ($θ$) resolved zero energy density of states $N(θ)$. 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…
We study possible applications of high critical temperature nodal superconductors for the search for Majorana bound states in the DIII class. We propose a microscopic analysis of the proximity effect induced by d-wave superconductors on a semiconductor wire with strong spin-orbit coupling. We characterize the induced superconductivity on the wire employing a numerical self-consistent tight-binding Bogoliubov-de Gennes approach, and analytical considerations on the Green's function. The order parameter induced on the wire, the pair correlation function, and the renormalization of the Fermi poin…
We consider a quasi two-dimensional superconductor with line nodes in an in-plane magnetic field, and compute the dependence of the specific heat, $C$, and the in-plane heat conductivity, $κ$, on the angle between the field and the nodal direction in the vortex state. We use a variation of the microscopic Brandt-Pesch-Tewordt method that accounts for the scattering of quasiparticles off vortices, and analyze the signature of the nodes in $C$ and $κ$. At low to moderate fields the specific heat anisotropy changes sign with increasing temperature. Comparison with measurements of $C$ and $κ$ in C…
A topological argument is presented for nodal structures of superconducting states with time-reversal invariance. A generic Hamiltonian which describes a quasiparticle in superconducting states with time-reversal invariance is derived, and it is shown that only line nodes are topologically stable in single-band descriptions of superconductivity. Using the time-reversal symmetry, we introduce a real structure and define topological numbers of line nodes. Stability of line nodes is ensured by conservation of the topological numbers. Line nodes in high-Tc materials, the polar state in p-wave pari…
Previous works on nodal d-wave superconductors have shown that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) like modulated superconducting (SC) state can be realized with no magnetic field when quasiparticles acquire an additional linear term in the wavevector in their dispersion. In the present work, stability of such a novel modulated SC state in an artificial film against an applied magnetic field is studied. As a reflection of the presence of the two different FFLO states, one close to zero field and the other at the high field end, in a single field v.s. temperature phase diagram of thin fil…
In the last few years the gap symmetries of many new superconductors,including Sr$_2$RuO$_4$, CeCoIn$_5$, $κ$-(ET)$_{2}$Cu(NCS)$_{2}$, YNi$_{2}$B$_{2}$C and PrOs$_{4}$Sb$_{12}$, have been identified via angle-dependent magnetothermal conductivity measurements. However, a controversy still persists as to the nature of the superconductivity in Sr$_2$RuO$_4$. For PrOs$_{4}$Sb$_{12}$, spin-triplet superconductivity has recently been proposed. Here, we also propose g-wave superconductivity for UPd$_2$Al$_3$ (i.e., $Δ({\bf k})=Δ\cos(2χ), χ= ck_{z}$) based on recent thermal conductivity data.
In recent years, a number of nodal superconductors have been identified; d-wave superconductors in high T_c cuprates, CeCoIn$_5$, and κ-(ET)_2Cu(NCS)_2, 2D f-wave superconductor in Sr_2RuO_4 and hybrid s+g-wave superconductor in YNi_2B_2C. In this work we conduct a theoretical study of nodal superconductors in the presence of supercurrent. For simplicity, we limit ourselves to d-wave and 2D f-wave superconductors. We compute the quasiparticle density of states and the temperature dependence of the depairing critical current in nodal superconductors, both of which are accessible experimentally.
Coherence effects by the impurity scattering of Caroli--de Gennes--Matricon (CdGM) modes in a vortex for nodal topological superconductors have been studied. The coherence effects reflect a topological number defined on a particular momentum space avoiding the superconducting gap nodes. First, we analytically derived the eigenvalue and eigenfunction of the CdGM modes, including the zero-energy modes, in a nodal topological superconducting state without impurities, where we focused on a possible superconducting state of UPt$_3$ as an example. Then, we studied impurity effects on the CdGM modes …
We have directly imaged the anisotropic nonlinear Meissner effect in an unconventional superconductor through the nonlinear electrodynamic response of both (bulk) gap nodes and (surface) Andreev bound states. A superconducting thin film is patterned into a compact self-resonant spiral structure, excited near resonance in the radio-frequency range, and scanned with a focused laser beam perturbation. At low temperatures, direction-dependent nonlinearities in the reactive and resistive properties of the resonator create photoresponse that maps out the directions of nodes, or of bound states assoc…
We report measurements of the temperature dependence of the magnetic penetration depth in different quality polycrystalline samples of noncentrosymmetric LaNiC2 down to 0.05 K. This compound has no magnetic phases and breaks time-reversal symmetry. In our highest quality sample we observe a T^2 dependence below 0.4Tc indicative of nodes in the energy gap. We argue that previous results suggesting conventional s-wave behavior may have been affected by magnetic impurities.
Motivated by recent work on strain-induced pseudo-magnetic fields in Dirac and Weyl semimetals, we analyze the possibility of analogous fields in two-dimensional nodal superconductors. We consider the prototypical case of a d-wave superconductor, a representative of the cuprate family, and find that the presence of weak strain leads to pseudo-magnetic fields and Landau quantization of Bogoliubov quasiparticles in the low-energy sector. A similar effect is induced by the presence of generic, weak doping gradients. In contrast to genuine magnetic fields in superconductors, the strain- and doping…
Recently the kagome compounds have inspired enormous interest and made some great progress such as in the field of superconductivity and topology. Here we predict a different kagome magnesium triboride (MgB3) superconductor with a calculated Tc ~12.2 K and Tc ~15.4 K by external stress, the potentially highest among the reported diverse kagome-type superconductors. We reveal its various exotic physical properties including the van Hove singularity, flat-band, multiple Dirac points, and nontrivial topology. The system can be described by a two-band model with highly anisotropic superconducting …
We report the specific heat measurements on the newly discovered Fe-based layered superconductor LaO_0.9F_{0.1-delta}FeAs with the onset transition temperature T_c \approx 28 K. A nonlinear magnetic field dependence of the electronic specific heat coefficient gamma(H) has been found in the low temperature limit, which is consistent with the prediction for a nodal superconductor. The maximum gap value Delta_0 \approx 3.4$\pm$0.5 meV was derived by analyzing gamma(H) based on the d-wave model. We also detected the electronic specific heat difference between 9 T and 0 T in wide temperature region…
Identifying the symmetry of the wave function describing the Cooper pairs is pivotal in understanding the origin of high-temperature superconductivity in iron-based superconductors. Despite nearly a decade of intense investigation, the answer to this question remains elusive. Here we use the muon spin rotation/relaxation (muSR) technique to investigate the underlying symmetry of the pairing state of the FeSe superconductor, the basic building block of all iron-chalcogenide superconductors. Contrary to earlier muSR studies on powders and crystals, we show that while the superconducting gap is m…
Distinguishing nodal and nonunitary superconductivity in quasiparticle interference of an Ising superconductor with Rashba spin-orbit coupling: an example of NbSe$_2$
There are now many nodal superconductors in heavy fermion (HF) systems, charge conjugated organic metals, high Tc cuprates and ruthenates. On the other hand only few of them have a well established gap function. We present here a study of the angular dependent thermal conductivity in the vortex state of some of the nodal superconductors. We hope it will help to identify the nodal directions in the gap function of UPd_2Al_3, UNi_2Al_3, UBe_13 and URu_2Si_2.
This course has a dual purpose. First we review the successes of the weak-coupling BCS theory in describing new classes of superconductors discovered since 1979. They include the heavy-fermion superconductors, high-Tc cuprate superconductors, organic superconductors, Sr2RuO4, etc. Second, we present the quasiclassical approximation introduced by Volovik, which we extend to describe the thermodynamics and the thermal conductivity of the vortex state in nodal superconductors. This approach provides the most powerful tool to identify the symmetry of the energy gap function Delta(k) in these new s…
High resolution laser-based angle-resolved photoemission measurements have been carried out on Bi2Sr2CuO6+d superconductor covering a wide doping range from heavily underdoped to heavily overdoped samples. Two obvious energy scales are identified in the nodal dispersions: one is the well-known 50-80 meV high energy kink and the other is <10 meV low energy kink. The high energy kink increases monotonously in its energy scale with increasing doping and shows weak temperature dependence, while the low energy kink exhibits a non-monotonic doping dependence with its coupling strength enhanced sharp…
In-plane Nernst signal and resistivity have been measured for three $La_{2-x}Sr_xCuO_4$ single crystals (x=0.09, 0.11 and 0.145) with the magnetic field parallel to c-axis. A quadratic temperature dependence of resistivity, i.e., $ρ=ρ_0+aT^2$ is observed below a certain temperature $T_R$. It is found that the upper boundary of the Nernst signal $T_n$ coincides with $T_R$, which points to an inherent relation between the anomalous Nernst signal and the nodal quasiparticle transport in the pseudogap region. Finally a phase diagram together with the pseudogap temperature $T^*$ is presented, which…
Controllable Josephson diode effect, $0$-$π$ transition and switch effect in the superconductor/two-dimensional Weyl nodal line semimetal/superconductor junctions
The symmetry of the order parameter in iron-based superconductors, especially the presence or absence of nodes, is still a question of debate. While contradictory experiments can be explained by appropriately tuned theories of nodeless superconductivity in the iron-arsenide compounds, for LaOFeP all experiments clearly point to a nodal order parameter. We put forward a scenario that naturally explains the difference between the order-parameter character in these two sets of compounds, and use functional renormalization group (fRG) techniques to analyze it in detail. Our results show that, due …
NaAlSi is a quasi-two-dimensional semimetal with superconductivity below Tc = 6.8 K and a band structure characterized by nodal lines near the Fermi level and potential topological surface states. Electrical resistivity measurements on its superconducting transitions in magnetic fields were made using plate-like single crystals. In the magnetic field-temperature phase diagram, we observed a substantial reduction in resistivity in a pre-transitional zone above the bulk superconducting regime only when the magnetic fields were perpendicular to the plane, rather than parallel to it. Significant s…
We show that the nodal quasiparticles have significant effect on the classical phase fluctuations in a quasi-two-dimensional d-wave superconductor. They give rise to singularities in the temperature behavior of some of the coupling constants in the phase-only effective action. One of the consequences is that the classical XY-model is not adequate for the description of the superconducting fluctuations in d-wave superconductors at low temperatures.
We propose a platform to realize nodal topological superconductors in a superconducting monolayer of MoX$_2$ (X$=$S, Se, Te) using an in-plane magnetic field. The bulk nodal points appear where the spin splitting due to spin-orbit coupling vanishes near the $\pm \boldsymbol{K}$ valleys of the Brillouin zone, and are six or twelve per valley in total. In the nodal topological superconducting phase, the nodal points are connected by flat bands of zero-energy Andreev edge states. These flat bands, which are protected by chiral symmetry, are present for all lattice-termination boundaries except zi…
We study systematically the scattering processes and the conductance spectra in nodal-line semimetalsuperconductor junctions using the extended Blonder-Tinkham-Klapwijk theory. The coexistence of peculiar quadruple reflections are found, which are the specular normal reflection, the retro-normal reflection, the specular Andreev reflection and the retro-Andreev reflection. The incident angle dependence and the quasiparticle energy dependence of the double normal reflections and the double Andreev reflections are investigated under various values of parameters such as the interfacial barrier hei…
