Andreev Reflection Coefficients Research Articles

Andreev reflection imaging and longitudinal shift of pseudospin-1 fermions in two-dimensional materials

In this work, the transverse magnetic focusing of pseudospin-1 fermions in two-dimensional materials with a superconducting contact is studied. The transmission coefficient and Andreev reflection coefficient from injector lead to collector lead are calculated based on the Landauer–Büttiker formula combined with the non-equilibrium Green's function method. The local state density diagrams of the electron and hole transmission in the center region are given, which provide a complete image of the ballistic electron–hole cyclotron orbits. Using these images, the positive longitudinal shift in Andreev reflection of the pseudospin-1 fermion can be obviously seen. Furthermore, it is found that the total offset is proportional to the number of Andreev reflection occurrence. This setup provides the possibility of detecting this shift experimentally.

Andreev reflection via the zero-energy bound states at the superconducting skyrmion crystal–normal skyrmion crystal interface

In this work, we used the double exchange model and studied the transport properties of the electronic states across the superconducting skyrmion crystal (SkX)–normal SkX interface. By calculating the energy dispersion and the real-space wave function of the electronic states we found topologically protected edge zero modes in the superconducting SkX nanoribbon. By consideration of the symmetry of the Hamiltonian of the bulk superconducting SkX and its nanoribbon, we found that they belong to the symmetry class D and support Majorana edge modes. To demonstrate the transport properties of these zero edge modes, we used the Green's function method and calculated the Andreev reflection coefficients of the electronic states across the superconducting SkX–normal SkX interface. Zero-mode-induced Andreev reflection peaks are demonstrated and also reflected in the tunneling current. Single-channel conductance is quantized at 2e2/h and 4e2/h for two-fold and four-fold degenerate Majorana bound states, respectively.

Quasi-periodic Andreev reflection in topological nodal-line semimetal–superconductor junctions

The Andreev reflection in topological nodal-line semimetal–superconductor (TNLSM–SC) junctions is studied based on the Landauer–Büttiker formula combined with the non-equilibrium Green’s function method. It is found that the Andreev reflection coefficient can reach large values of several tens for a TNLSM–SC junction. In the energy gap, two new peaks for the Andreev reflection appear. Outside the energy gap, the Andreev reflection shows a quasi-periodic variation. With the incident energy increasing, the Andreev reflection decreases periodically. This phenomenon originates from the peculiar energy spectrum of the TNLSMs. The value and quasi-period of the Andreev reflection coefficient depend on the size of the system in different directions. These anomalous Andreev reflections are helpful to study the transport properties in normal metal–superconductor junctions.

Influence of Rashba spin-orbit coupling on Josephson effect in triplet superconductor/two-dimensional semiconductor/triplet superconductor junctions

We study theoretically Josephson effect in a planar ballistic junction between two triplet superconductors with p-wave orbital symmetries and separated by a two-dimensional (2D) semiconductor channel with strong Rashba spin–orbit coupling. In triplet superconductors, three types of orbital symmetries are considered. We use Bogoliubov–de Gennes formalism to describe quasiparticle propagations through the junction and the supercurrents are calculated in terms of Andreev reflection coefficients. The features of the variation of the supercurrents with the change of the strength of Rashba spin–orbit coupling are investigated in some detail. It is found that for the three types of orbital symmetries considered, both the magnitudes of supercurrent and the current-phase relations can be manipulated effectively by tuning the strength of Rashba spin–orbit coupling. The interplay of Rashba spin–orbit coupling and Zeeman magnetic field on supercurrent is also investigated in some detail.

A Perfect Andreev Reflection Induced by Anderson Disorder in a Normal–Superconductor Junction

The electronic transport properties of a 2D normal metal superconductor are studied by a nonequilibrium Green's function method. The normal metal is the non‐inverted HgTe/CdTe quantum well doped by electrons. It is found that Andreev reflection (AR) is enhanced in the weak disorder regime while weakened in the strong disorder regime. However, the AR coefficient can be perfect () with vanished fluctuations in the moderate disorder regime, whereas other scattering processes are forbidden. The mechanism goes that Anderson disorder leads to a topological Anderson insulator with bulk states localized while edge states still extensive. Multiple ARs and spin‐momentum locking cooperate to vanish the normal reflection and leave AR only. A detailed numerical study shows that fixing the Fermi surface in conduction band, a large enough junction width, and a strong coupling between the normal and superconducting leads are necessary for perfect AR.

Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor

We study the tunneling effect in an AIII-class insulator proximity coupled with a spin-singlet $s$-wave superconductor, in which three phases are characterized by the integer topological invariant $\mathcal{N}$. By solving the Bogoliubov-de Gennes equation explicitly, we analytically obtain a normal reflection coefficient $R_{\sigma\sigma'}$ and an Andreev reflection coefficient $A_{\sigma\sigma'}$, and derive a charge conductance formula,where $\sigma(\sigma')$ is the spin index of a reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes: (i)gap structure without coherence peaks for $\mathcal{N}=0$, (ii)quantized zero-bias conductance peak (ZBCP) with height $2e^{2}/h$ for $\mathcal{N}=1$, and (iii)ZBCP spitting for $\mathcal{N}=2$. At zero bias voltage $eV=0$, $\sum_{\sigma\sigma'} R_{\sigma\sigma'} = \sum_{\sigma\sigma'} A_{\sigma\sigma'}$ is satisfied and the spin direction of an injected electron is rotated at approximately $90^\circ$ for the $\mathcal{N}=1$ state. Meanwhile, $A_{\sigma\sigma'}=0$ is satisfied for the $\mathcal{N}=2$ state, and the spin rotation angle can become $180^\circ$.

Theory of the Josephson current on a magnetically doped topological insulator

Proximity induced superconducting states in the surface of magnetically doped topological insulators can host chiral Majorana modes. We consider a Josephson junction in that system with changing the chemical potential, which drives a topological phase transition in the induced superconducting states as well as a metal-insulator transition in the surface states. The local density of states and the Josephson current are analytically calculated by McMillan's Green's function method in terms of the Andreev reflection coefficient. We show that although the magnitude of the Josephson current is greatly enhanced when the surface state changes from insulating to metallic, its temperature dependence drastically changes at the topological phase transition point, reflecting the appearance of the chiral Majorana modes

Magnetoanisotropic spin-triplet Andreev reflection in ferromagnet-Ising superconductor junctions

We theoretically study the electronic transport through a ferromagnet-Ising superconductor junction. A tight-binding Hamiltonian describing the Ising superconductor is presented. Then by combing the non-equilibrium Green's function method, the expressions of Andreev reflection coefficient and conductance are obtained. A strong magnetoanisotropic spin-triplet Andreev reflection is shown, and the magnetoanisotropic period is $\pi$ instead of $2\pi$ as in the conventional magnetoanisotropic system. We demonstrate a significant increase of the spin-triplet Andreev reflection for the single-band Ising superconductor. Furthermore, the dependence of the Andreev reflection on the incident energy and incident angle are also investigated. A complete Andreev reflection can occur when the incident energy is equal to the superconductor gap, regardless of the Fermi energy (spin polarization) of the ferromagnet. For the suitable oblique incidence, the spin-triplet Andreev reflection can be strongly enhanced. In addition, the conductance spectroscopies of both zero bias and finite bias are studied, and the influence of gate voltage, exchange energy, and spin-orbit coupling on the conductance spectroscopy are discussed in detail. The conductance reveals a strong magnetoanisotropy with period $\pi$ as the Andreev reflection coefficient. When the magnetization direction is parallel to the junction plane, a large conductance peak always emerges at the superconductor gap. This work offers a comprehensive and systematic study of the spin-triplet Andreev reflection, and has underlying application of $\pi$-periodic spin valve in spintronics.

Modeling of the Transport Properties of SNS Contacts for Strong Electron-Phonon Interactions

A modified expression for the Andreev reflection coefficient is derived taking into account strong electronphonon interactions. A dependence of the superconducting order parameter ∆σ is numerically modeled for magnesium diboride. The current-voltage characteristics and the dI/dV spectra of the dynamic conductivity of superconductor-normal metal-superconductor (SNS) contacts based on strongly coupled superconductors are calculated for the Averin–Bardas model using the modified Andreev reflection coefficient (on an example of magnesium diboride). Calculations are performed for the mode of high transparency (0.9–0.98). The special features of the SNS contacts based on strongly coupled superconductors are shown.

Quantum Transport through a Triple Quantum Dot System in the Presence of Majorana Bound States**Supported by National Natural Science Foundation of China under Grant No. 11274040, and by the Program for New Century Excellent Talents in University under Grant No. NCET-08-0044

We study the electron transport through a special quantum-dot (QD) structure composed of three QDs and two Majorana bound states (MBSs) using the nonequilibrium Green's function technique. This QD-MBS ring structure includes two channels with the two coupled MBSs being Channel 1 and one QD being Channel 2, and three types of transport processes such as the electron transmission (ET), the Andreev reflection (AR), and the crossed Andreev reflection (CAR). By comparing the ET, AR, and CAR processes through Channels 1 and 2, we make a systematic study on the transport properties of the QD-MBS ring. It is shown that there appear two kinds of characteristic transport patterns for Channels 1 and 2, as well as the interplay between the two patterns. Of particular interest is that there exists an AR-assisted ET process in Channel 2, which is different from that in Channel 1. Thus a clear “X” pattern due to the ET and AR processes appears in the ET, AR, and CAR transmission coefficients. Moreover, we study how Channel 2 affects the three transport processes when Channel 1 is tuned in the ET and CAR regimes. It is shown that the transport properties of the ET, AR and CAR processes can be adjusted by tuning the energy level of the QD embedded in Channel 2. We believe this research should be a helpful reference for understanding the transport properties in the QD-MBS coupled systems.

The Andreev reflection of zero line mode in graphene-superconductor hybrid junction

The zero line mode (ZLM) in two dimensional materials provides a quasi-one dimensional path for electronic transport. We report the theoretical investigation of the Andreev reflection of ZLM by using the staggered graphene-superconductor based models. For a two-terminal system in which the valley index is well preserved, when graphene is zigzag edged, the Andreev reflection coefficient can be either large or strongly suppressed depending on the symmetric properties of the transverse wave function in graphene ribbon. However, the Andreev reflection coefficient, independent of the staggering profile in the armchair edged model, is large due to the absence of wave function symmetry. When ZLM changes its direction in a vertical path, a perfect Andreev reflection could happen when the incident ZLM stems from a zigzag edged graphene ribbon. In a zigzag edged four-terminal hybrid model, the interference of reflected holes leads to perfect Andreev reflection with probability unity and the annihilation of the crossed Andreev reflection. For the armchair edged model, the interference effect disappears because the Andreev reflection from one of the paths is prohibited. The interference of Andreev reflections in four-terminal models is investigated by spacial local density of states in the central scattering region as well.

Minimal model of point contact Andreev reflection spectroscopy of multiband superconductors

We formulate a minimal model of point contact Andreev reflection spectroscopy of a normal- metal/multiband superconductor interface. The theory generalizes the Blonder-Tinkham-Klapwijk (BTK) formulation to a multiband superconductor and it is based on the quantum waveguides theory. The proposed approach allows an analytic evaluation of the Andreev and normal reflection coefficients and thus is suitable for a data fitting of point contact experiments. The obtained differential conductance curves present distinctive features similar to the ones measured in the experiments on multiband systems, like the iron-based pnictides and the MgB2.

Helical Andreev bound states in topological insulator f-wave Josephson junction

We study the effect of f-wave triplet superconductivity induced by proximity effect on the surface states of a three-dimensional topological insulator (3DTI). In fact, the gapless surface state for excitation spectrum of a triplet superconductor topological insulator gives rise to formation of helical Andreev bound states (ABSs) and, necessarily, suppression of Andreev reflection at the interface of a superconductor/ferromagnetic structure. By calculating the relevant form of Dirac spinors for two-dimensional time-reversal symmetric Hamiltonian, the normal and Andreev reflection coefficients and ABSs corresponding to tunneling and Josephson junctions, respectively, are obtained. It is shown, that the Andreev process vanishing occurs strongly, so that, the evanescent waves will be permissible in tunneling process. It is found that the dispersion of ABSs is an even function of electron incident angle θ for f1-wave and odd for f2-wave. In f1-order, for module of incident angle, θ>0.167π, the ABSs are independent of superconducting phase difference, ϕ, so that the Josephson current may vanish for this region. Effect of magnetization on flattening of ABSs occurs in f2 order faster than f1.

Interface currents in topological superconductor–ferromagnet heterostructures

We propose the existence of a substantial charge current parallel to the interface between a noncentrosymmetric superconductor and a metallic ferromagnet. Our analysis focuses upon two complementary orbital-angular-momentum pairing states of the superconductor, exemplifying topologically nontrivial states which are gapped and gapless in the bulk, respectively. Utilizing a quasiclassical scattering theory, we derive an expression for the interface current in terms of Andreev reflection coefficients. Performing a systematic study of the current, we find stark qualitative differences between the gapped and gapless superconductors, which reflect the very different underlying topological properties. For the fully gapped superconductor, there is a sharp drop in the zero-temperature current as the system is tuned from a topologically nontrivial to a trivial phase. We explain this in terms of the sudden disappearance of the contribution to the current from the subgap edge states at the topological transition. The current in the gapless superconductor is characterized by a dramatic enhancement at low temperatures, and exhibits a singular dependence on the exchange-field strength in the ferromagnetic metal at zero temperature. This is caused by the energy shift of the strongly spin-polarized nondegenerate zero-energy flat bands due to their coupling to the exchange field. We argue that the interface current provides a novel test of the topology of the superconductor, and discuss prospects for the experimental verification of our predictions.

Tunneling conductance in gapped graphene-based f-wave superconductor N/S and N/I/S junctions

Proximity effect by a f-wave superconductor on top of the graphene grown on a substrate-induced bandgap leads to a two-dimensional relativistic Dirac–Bogoliubov–de Gennes Hamiltonian with f-type pair coupling, that electrons are treated as massive Dirac particles. In this work, we aim to study the influence of two types of f-wave superconductivity on normal tunneling conductance in gapped graphene-based superconductor junctions. Using the explicit obtained forms of DBdG spinors for normal, insulator and superconductor regions in the N/S and N/I/S structures, the normal and Andreev reflection coefficients are exactly found. Finally, the behavior of normal conductance spectra in terms of bias voltage, electrostatic potentials and energy gap is investigated via the BTK formalism. Because of energy gap of graphene we find our results for N/I/S structure in the thin barrier limit with large voltage V0 applied across the barrier region.

P-Wave Asymmetry Pairing in Graphene-Superconductor Junction

Using the extended Blonder-Tinkham-Klapwijk formalism, the normal conductance spectra in a normal metal/superconductor gapped graphene-based junction considering the effect of p-asymmetry pair coupling in superconductor is investigated. The exact forms of Dirac spinors for Hamiltonian of graphene interacted with Cooper pair potential of superconductor are obtained via the Dirac-Bogoliubov-de Gennes theory. Actually, we obviously study the effect of the energy gap of graphene and also triple p-wave coupling on Andreev reflection coefficients and conductance spectra in the N/S structure.

The Josephson effect between triplet superconductors through a finite ferromagnetic barrier

Charge and spin transport in a junction involving two triplet superconductors and a ferromagnetic barrier are studied. We use Bogoliubov–de Gennes wavefunctions to construct the Green’s function, from which we obtain the Josephson currents in terms of the Andreev reflection coefficients. We focus on the consequences of a finite barrier width for the occurrence of 0–π transitions and for the spin currents, and examine the appropriateness of the common δ-function approximation for the tunneling region.

Effect of electron-hole inhomogeneity on specular Andreev reflection and Andreev retroreflection in a graphene-superconductor hybrid system

The electron-hole inhomogeneity in graphene has been confirmed to be a new type of charge disorder by recent experiments, and the largest energy displacement of electron and hole puddles with respect to the Dirac point can reach nearly $30$ meV. Here we focus on how the electron-hole inhomogeneity affects the specular Andreev reflection as well as the Andreev retroreflection. In a four-terminal graphene-superconductor hybrid system, we find that the Andreev coefficients can hardly be affected even under a rather large electron-hole inhomogeneity (typically $30$ meV), and the boundary distinguishing two Andreev reflections can well hold, although the strength of the charge puddles, $W=30$ meV, is much larger than the superconductor gap, $\ensuremath{\Delta}=1$ meV. Furthermore, when charge puddles are two orders of magnitude larger than the superconductor gap, a specific kind of Andreev reflection can still be obviously detected. To quantitatively describe what degree of the boundary is blurred, a quantity $D$ is introduced which measures the width of a crossover region between specular Andreev reflection and retroreflection in energy space. We confirm that the boundary blurring is much smaller than the charge puddle strength $W$. In addition, we study the effect of Anderson disorder for comparison, and we find that the boundary is held much more obviously in this case. Finally, the fluctuations of the Andreev reflection coefficient are studied. Under a typical experimental charge puddle, the fluctuations are very small when the energy of the particles is away from the boundary, again confirming that the retroreflection and specular reflection can be clearly distinguished and detected in the presence of the electron-hole inhomogeneity.

Quantum Andreev effect in two-dimensional HgTe/CdTe quantum well/superconductor systems

The Andreev reflection (AR) in 2D HgTe/CdTe quantum well-superconductor hybrid systems is studied. A quantized AR with AR coefficient equal to one is predicted, which is due to the multi-Andreev reflection near the interface of the hybrid system. Importantly, this quantized AR is not only universal, i.e., independent of any system parameters and quality of the coupling of the hybrid system, it is also robust against disorder as well. As a result of this quantum Andreev effect, the conductance exhibits a quantized plateau when the external bias is less the superconductor gap.

Signatures of d-wave symmetry on thermal Dirac fermions in graphene-based F|I|d junctions

We study theoretically behavior of thermal massless Dirac fermions inside graphene-based Ferromagnetic |Insulator| d-wave/s-wave superconductor (F|I|d and F|I|S) junctions in the ballistic regime. We employ Dirac–Bogoliubov–de Genne formalism and use Dirac–BdG wave functions within the ferromagnetic, normal, and superconducting regions and appropriate boundary conditions, to derive Andreev and normal reflection coefficients. By employing the obtained Andreev and normal reflection coefficients, thin barrier approximation and also appropriate values of parameters, we investigate characteristics of heat current through the F|I|d and F|I|S junctions. We find that for s-wave superconductors, thermal conductance (Γ) oscillates sinusoidally versus strength of barrier in the thin barrier approximation. The finding persists for all values of α, orientation of d-wave superconductor crystal in the k-space, for values of α very close to π/4, magnitude of the oscillations suppress completely. By increasing temperature, the thermal conductance increases exponentially for small values of α and for larger values of superconductor crystal orientation angle the thermal conductance modifies and interestingly for α=π/4 exhibits linear behavior, i.e., Γ∝T which is similar to Wiedemann–Franz law for metals in low temperatures. While increasing the temperature move minimum value of the thermal conductance with respect to strength of magnetization texture h/Δ0 towards smaller values of h/Δ0, increasing the superconductor crystal orientation angle, α from 0 to π/4 enhances whole values of the thermal conductance and consequently the minimum value move towards up. At last we suggest an experimental setup for verifying these findings.