Dirac Bogoliubov De Gennes Equation Research Articles

Effects of strong spin-orbit coupling on Shiba states from magnetic adatoms using first-principles theory

Magnetic impurities at surfaces of superconductors can induce bound states referred to as Yu–Shiba–Rusinov states (i.e. Shiba states) within superconducting (SC) gaps. For superconductors with strong spin–orbit coupling (SOC), Shiba states arising from even single magnetic adatoms are too complex to be fully understood using effective models alone because SOC cannot be treated perturbatively and multiple orbitals are strongly mixed with spin projections. Here we investigate Shiba states of single magnetic adatoms at the surface of strongly spin-orbit coupled SC Pb, by solving the fully relativistic Dirac–Bogoliubov–de Gennes equations using multiple scattering Green’s function methods. For Fe and Co adatoms on Pb(110), we show that the Shiba states are better characterized by total angular momentum, J, and its projections on the z axis, m J . As a hallmark of the SOC effect, the Shiba states show a strong dependence of the orientation of the adatom moment. As the orientation of the Fe/Co moment changes, the deepest Shiba states merge at zero energy. This zero-energy state disappears with an additional non-magnetic adatom next to the magnetic adatom, although the other Shiba states unchange. For a Mn adatom on Pb, our Shiba states overall agree with experiments. The characteristics of our Shiba states are also observed with the similar energies and characters in the experiments. The deepest Shiba states that we compute, however, do not appear as close to the Fermi level as the experimental data. It would be interesting to compute the Shiba states with continuously varying vertical distances of the Mn adatom from the surface or with varying the charge state of the adatom, and to calculate the spatial dependence of the spectral density. Our findings will be also useful for understanding of Shiba states for dimers and longer spin chains on the Pb surface considering noncollinear magnetic structures in them.

Spin transport properties in ferromagnet/superconductor junctions on topological insulator

The spin-dependent Andreev reflection is investigated theoretically by analyzing the electronic transport in a thin-film topological insulator (TI) ferromagnet/superconductor (FM/SC) junction. The tunneling conductance and shot noise are calculated based on the Dirac–Bogoliubov–de Gennes equation and Blonder–Tinkham–Klapwijk theory. It is found that the magnetic gap in ferromagnet can enhance the Andreev retro-reflection but suppress the specular Andreev reflection. The gate potential applied to the electrode on top of superconductor can enhance the two types of reflections. There is a transition between the two types of reflections at which both the tunneling conductance and differential shot noise become zero. These results provide a method to realize and detect experimentally the intra-band specular Andreev reflection in thin film TI-based FM/SC structures.

Tunneling conductance of the s-wave and d-wave pairing superconductive graphene–normal graphene junction

Within the framework of the Blonder–Tinkham–Klapwijk formalism we calculate and analyze the conductance of the normal graphene — s-wave and independently d-wave pairing superconductive graphene junction. The eigenfunctions, the Andreev and the normal reflection rates are obtained by solving the Dirac–Bogoliubov–de Gennes equations. The Fermi velocity is believed to be different in the normal and in the superconductive regions. We consider the options of gapless and gapped graphene for both cases: s-wave and independently d-wave pairing. It is demonstrated that the characteristics of the junction considered are sensitive to the ratio vFN/vFS where vFN, vFS are the Fermi velocities in the normal and the superconductive graphene respectively. This conclusion refers to the Andreev reflection as well as to the normal one. The first of them is shown to be the dominant process for the formation of the conductivity. These results are true for an arbitrary value of the orientational angle of the d-waves. Each of four cases considered: s-, d-wave pairing and gapless and gapped graphene displays its own specific features of the conductance. The dependence of the conductance on the external electrostatic potential as well as on the Fermi energy is also analyzed in every case. The obtained results may be useful for controlling the transport properties of the normal graphene–superconductive graphene junction.

Transport properties of silicene-based ferromagnetic-insulator-superconductor junction

We study the tunneling conductance of a silicene-based ferromagnet/insulator/superconductor (FIS) junction by the use of the spin-dependent Dirac-Bogoliubov de-Gennes equation. We demonstrate that the conductance spectra are strongly affected by exchange energy h, Fermi energy EF, and external perpendicular electric field Ez. In the thin barrier limit of insulator silicene IS, the zero-bias charge conductance of the FIS silicene junction oscillates as a function of barrier strength χG. It is shown that the period of oscillations changes from π/2 to π corresponding to undoped and doped silicene. Remarkably, in contrast to that of the graphene FIS junction where the conductance only vanishes at the exchange energy h=EF, here due to the buckled structure of silicene, there is a transport gap region for the range of h values and the magnitude of such a gap region can be controlled by Ez. Moreover, it is found that by appropriate choice of h and Ez, it is possible to achieve a fully spin and valley-polarized charge conductance through the FIS silicene junction. This property suggests experimentally measuring the Fermi energy of silicene.

Dirac-fermions in graphene d-wave superconducting heterojunction with the spin orbit interaction

In this study, based on the Dirac–Bogoliubov–de Gennes equation, we theoretically investigate the interaction effect between the anisotropic d-wave pairing symmetry and the spin orbit interaction (the Rashba spin orbit interaction (RSOI) and the Dresselhaus spin orbit interaction (DSOI)) in a graphene superconducting heterojunction. We find that the spin orbit interaction (SOI) plays a critical role on the tunneling conductance in the pristine case, but minimally affecting the tunneling conductance in the heavily doped case. As for the zero bias state, in contrast to the keep intact feature in the heavily doped case, it exhibits a distinct dependence on the RSOI and the DSOI in the pristine case. In particular, the damage of the zero bias state with a slight DSOI results in the disappearance of the zero bias conductance peak. Moreover, the tunneling conductances also show a qualitative difference with respect to the RSOI when both the RSOI and the DSOI are finite. These remarkable results suggest that the SOI and the anisotropic superconducting gap can be regarded as a key tool for diagnosing the specular Andreev reflection.

Specular Andreev reflection in graphene-based superconducting junction with substate-induced spin orbit interaction

Based on the Dirac–Bogoliubov–de Gennes equation, the chirality-resolved transport properties through a ballistic graphene-based superconducting heterojunction with both the Rashba and the Dresselhaus spin orbit interaction have been investigated. Our results show that, in contrast to the retro-Andreev reflection suppressed by the spin orbit interaction (SOI), the specular Andreev reflection (SAR) can be enhanced largely by the SOI. Moreover, the Fabry–Perot interferences in the barrier region lead to the oscillating feature of the tunneling conductance. It is anticipated to apply the qualitative different results to diagnose the SAR in single layer graphene in the presence of both kinds of the SOI.

The nonlinear Dirac equation in Bose–Einstein condensates: superfluid fluctuations and emergent theories from relativistic linear stability equations

We present the theoretical and mathematical foundations of stability analysis for a Bose–Einstein condensate (BEC) at Dirac points of a honeycomb optical lattice. The combination of s-wave scattering for bosons and lattice interaction places constraints on the mean-field description, and hence on vortex configurations in the Bloch-envelope function near the Dirac point. A full derivation of the relativistic linear stability equations (RLSE) is presented by two independent methods to ensure veracity of our results. Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex solutions of the nonlinear Dirac equation, which include Anderson–Toulouse skyrmions with lifetime s. Beyond vortex stabilities the RLSE provide insight into the character of collective superfluid excitations, which we find to encode several established theories of physics. In particular, the RLSE reduce to the Andreev equations, in the nonrelativistic and semiclassical limits, the Majorana equation, inside vortex cores, and the Dirac–Bogoliubov–de Gennes equations, when nearest-neighbor interactions are included. Furthermore, by tuning a mass gap, relative strengths of various spinor couplings, for the small and large quasiparticle momentum regimes, we obtain weak-strong Bardeen–Cooper–Schrieffer superconductivity, as well as fundamental wave equations such as Schrödinger, Dirac, Klein–Gordon, and Bogoliubov–de Gennes equations. Our results apply equally to a strongly spin–orbit coupled BEC in which the Laplacian contribution can be neglected.

Charge, spin and thermal transport of graphene-based FNF multilayer

In this paper the impact of repetition of graphene-based ferromagnet-normal(FN) regions on the transport properties of system is studied, theoretically. Here, the purpose is to generalize the Fert–Grünberg FNF multijunction to a graphene-based one. Thus, parallel and antiparallel magnetization alignments of ferromagnetic regions are considered. The concept of quantum well is used to calculate the transmission probability using the Dirac–Bogoliubov–de Gennes equations. Charge, spin and thermal conductances are investigated for multilayer junctions in terms of magnetization alignments, strength of ferromagnets and thickness of ferromagnet regions. Finally charge, spin and thermal Giant magnetoresistances (GMR) are calculated for the above-mentioned structures. It is obtained that the spin conductance of antiparallel setups is zero, thus it can be used as a spin valve.

Perfect GMR effect in gapped graphene-based ferromagnetic–normal–ferromagnetic junctions

We investigate the quantum transport property in gapped graphene-based ferromagnetic/normal/ferromagnetic (FG/NG/FG) junctions by using the Dirac–Bogoliubov–de Gennes equation. The graphene is fabricated on SiC and BN substrates separately, so carriers in FG/NG/FG structures are considered as massive relativistic particles. Transmission probability, charge, and spin conductances are studied as a function of exchange energy of ferromagnets (h), size of graphene gap, and thickness of normal graphene region (L) respectively. Using the experimental values of Fermi energy in the normal graphene part (EFN ∼ 400 meV) and energy gap in graphene (260 meV for SiC and 50 meV for BN substrate), it is shown that this structure can be used for both spin-up and spin-down polarized current. The latter case has different behavior of gapped FG/NG/FG from that of gapless FG/NG/FG structures. Also perfect charge giant magnetoresistance is observed in a range of .

In-plane magnetoresistance in a topological insulator ferromagnet/barrier/ferromagnet/superconductor junction

Dirac Bogoliubov-de Gennes equation and Blonder-Tinkham-Klapwijk formalism are applied to study the conductance and magnetoresistance of a topological insulator ferromagnet/barrier/ferromagnet/superconductor (F1/B/F2/SC) junction with the in-plane magnetization. We find that the induced in-plane magnetization sets a lower critical incident angle for Andreev reflection and causes a very large magnetoresistance. We find also that both the conductance and magnetoresistance are oscillating functions of the gate voltage exerted on the middle ferromagnetic layer and the thickness of this layer.

Surface and vortex bound states in topological superconductors

In topological superconductors with strong spin–orbit coupling, we investigate quasiparticle bound states due to topological reasons. We show that the Dirac-type Hamiltonian can describe the surface and vortex bound states in these topological superconductors. We reveal that the linearized Dirac–Bogoliubov–de Gennes equations successfully reproduce the result obtained by numerical calculations of the tight-binding Hamiltonian for CuxBi2Se3 originated from the first-principles calculation, which means that the vortex bound states can be achieved by these linearized equations. We investigate the effect of the momentum-dependent mass term on the surface and vortex bound states. As a result, the momentum dependence of the mass term is not important for surface and vortex bound states in superconducting states, while it is crucial for the surface bound states in normal states.

Spin and charge zero-bias conductance peak in a graphene-based Fd junction

In the present paper, a study on graphene-based ferromagnetic/d-wave superconductor (Fd) junction using the Blonder-Tinkham-Klapwijk formalism and Dirac-Bogoliubov De-Gennes equation has been done. The effect of rotation of order parameter on the transport properties of this structure is investigated. This rotation is related to the orientation of high temperature superconductor crystal such as YBaCuO coexisting with a graphene monolayer. As the main result of this paper, we obtain the zero bias conductance peak as a fingerprint of unconventional superconductivity for both of spin conductance and charge conductance. Also, we obtain that the thermal conductance of this Fd junction is a linear function of temperature at the low temperatures. This later case is similar to the Wiedemann-Franz law for metal at small temperatures.

Coherent transmission of nodal Dirac fermions through a graphene-based superconducting double barrier junction

Transport characteristics of relativistic electrons through graphene-based d-wave superconducting double barrier junction and ferromagnet/d-wave superconductor/normal metal double junction have been investigated based on the Dirac–Bogoliubov–de Gennes equation. We have first presented the results of superconducting double barrier junction. In the subgap regime, both the crossed Andreev and nonlocal tunneling conductance all oscillate with the bias voltage due to the formation of Andreev bound states in the normal metal region. Moreover, the critical voltage beyond which the crossed Andreev conductance becomes to zero decreases with increasing value of superconducting pair potential α. In the presence of the ferromagnetism, the MR through graphene-based ferromagnet/ d-wave superconductor/normal metal double junction has been investigated. It is shown that the MR increases from exchange splitting h0=0 to h0=EF (Fermi energy), and then it goes down. At h0=EF, MR reaches its maximum 100. In contrast to the case of a single superconducting barrier, Andreev bound states also manifest itself in the zero bias MR, which result in a series of peaks except the maximum one at h0=EF. Besides, the resonance peak of the MR can appear at certain bias voltage and structure parameter. Those phenomena mean that the coherent transmission can be tuned by superconducting pair potential, structure parameter, and external bias voltage, which benefits the spin-polarized electron device based on the graphene materials.

Tunneling transport in topological insulator normal metal/insulator/d-wave superconductor junctions

We investigate the tunneling conductance in a normal metal/insulator/d-wave superconductor (NM/I/d-wave SC) junction with a barrier of thickness d and with an arbitrary gate voltage V0 applied across the barrier region, formed on the surface of a topological insulator, using the Dirac–Bogoliubov–de Gennes equation and Blonder–Tinkham–Klapwijk (BTK) formalism. We find that the tunneling conductance as a function of both d and V0 displays an oscillatory behavior whose amplitude decreases with increase of V0. We also find that when the Andreev resonant condition is met, the tunneling conductance approaches a maximum value of 2G0, independent of the gate voltage V0.

Tunneling Charge Transport in Graphene-Based Superconductor Junctions

We study the spin polarized electron and hole tunneling transport through a graphene-based ferromagnet(GF1) insulator(GI1) superconductor(GS) insulator(GI2) ferromagnet(GF2) junction. Proximity induced spin polarization and superconductivity in a graphene sheet are assumed to be created by superconducting and ferromagnetic electrodes placed on the top of the graphene. Using a four-dimensional version of the Dirac Bogoliubov de Gennes equation with appropriate boundary conditions we investigate the tunneling processes through the junctions. In particular, we present calculations of the amplitudes of normal and Andreev re ections as a function of the energy of the incident electron for a wide range of the model parameters, such as the strength and orientation of the exchange eld, the barrier strength, and the distance between the two ferromagnetic layers. The tunneling transport processes in the graphene-based double junction GF/GI/GS/ GI/GF are compared with those in non-graphene-based junctions.

Alternating current Josephson effect in superconductor–graphene–superconductor junctions

We investigate the ac Josephson effect in superconductor–graphene–superconductor (SGS) junctions by using the Floquet–Green’s function formalism to solve the Dirac–Bogoliubov–de Gennes equation. The numerical results show rich subharmonic gap structures such as the negative differential conductance (NDC) in the dc current. The tunability of the current magnitude can be controlled by the gate voltage, which determines the carriers’ densities in graphene. With increasing bias, the ac components decay in an oscillatory manner as in superconductor–normal–superconductor junctions. It is found that the higher-order components have an explicit contribution to the total current under a low bias, which leads to the deviation from a simple sine-like dependence on time for the total current. The NDC characteristics and the tunable current magnitude are excellent hints for the potential application of SGS junctions.

Spin-polarized conductance in graphene-based FSF junctions

In this paper, conductance of spin and electron in graphene-based ferromagnet—superconductor (FS) and parallel and antiparallel ferromagnet–superconductor–ferromagnet (FSF) junctions are studied. Using the Dirac–Bogoliubov–de Gennes equations, Andreev and normal reflections are obtained and then using these coefficients, conductance of spin and electrons are calculated at the FS interface(s) analytically. As a result, both the energy dependence of spin and charge differential conductances are investigated and a comparison between electron and spin transport is done in this paper. Effect of exchange energy of ferromagnet h on conductances is studied too.

Josephson effect in graphene SNS junction with a single localized defect

Imperfections change essentially the electronic transport properties of graphene. Motivated by a recent experiment reporting on the possible application of graphene as junctions, we study transport properties in graphene-based junctions with single localized defect. We solve the Dirac–Bogoliubov–de-Gennes equation with a single localized defect superconductor–normal(graphene)–superconductor (SNS) junction. We consider the properties of tunneling conductance and Josephson current through an undoped strip of graphene with heavily doped s-wave superconducting electrodes in the limit l def ⪡ L ⪡ ξ . We find that spectrum of Andreev bound states are modified in the presence of single localized defect in the bulk and the minimum tunneling conductance remains the same. The Josephson junction exhibits sign oscillations.

Dirac quasiparticle tunneling in a NG/ferromagnetic barrier/SG graphene junction

We study the tunneling conductance in a spin dependent barrier NG/F B/SG graphene junction, where NG, F B and SG are normal graphene, gate ferromagnetic graphene barrier with thickness d and the graphene s-wave superconductor, respectively. In our work, the quasiparticle scattering process at the interfaces is based on quasi particles governed by the Dirac Bogoliubov–de Gennes equation with effective speed of light v F ∼ 10 6 m/s. The conductance of the junction is calculated based on Blonder–Tinkham–Klapwijk (BTK) formalism. The oscillatory conductance under varying gate potential and exchange energy in F B and the conductance induced by specular Andreev reflection are studied. By taking into account both effects of barrier strengths due to the gate potential χ G ∼ V G d / ℏ v F and the exchange energy χ ex ∼ E ex d / ℏ v F in the F B region, we find that the zero bias conductance of junction depends only on the ferromagnetic barrier strength χ ex in F B, when the Fermi energy in SG is very much larger than that the Fermi energy in NG ( E FS ≫ E FN). The oscillatory conductance peaks can be controlled by either varying χ ex or χ G. In the limiting case, by setting E ex = 0, the conductance in a NG/N B/SG graphene junction, where SG is the s-wave superconductor, is also studied in order to compare with two earlier contradicted data. Our result agrees with what was obtained by Linder and Sudbo [J. Linder, A. Sudbo, Phys. Rev. B 77 (2008) 64507], which confirms the contradiction to what was given by Bhattacharjee and Sengupta [S. Bhattacharjee, K. Sengupta, Phys. Rev. Lett. 97 (2006) 217001].

Novel Andreev reflection and differential conductance of a ferromagnet/ferromagnet/superconductorjunction on graphene

Via numerical calculation of the spin-dependent Dirac–Bogoliubov–de Gennes equation, thedifferential conductance is obtained for a ferromagnet/ferromagnet/superconductor(F/F/S) junction on graphene where the two F layers are undoped. If the two F layers havenoncollinear magnetizations, the spin-flipped scattering at the F/F interface leads to thenovel Andreev reflection (AR), in which the spin directions of an incident electron and thereflected hole are opposite to each other. When the exchange energy is larger thanthe superconducting gap, this novel AR manifests itself as sub-gap differentialconductance peaks because of the formation of spin-flipped Andreev bound states in theintermediate F layer, whereas for the parallel and anti-parallel configurations nosuch peaks can be found. In the transitional regime with the exchange energyclose to the gap, for noncollinear configurations, the round-trip path supportingthe formation of those bound states is broken and a differential conductancedip can be found near the point where the external bias equals the exchangeenergy.