We study the transport properties of junctions of normal and superconducting Weyl semi-metal with tilted dispersion, in the presence of magnetization induced by magnetic strips. The sub gap tunnelling conductance shows robust signatures in the presence of different orientation and strength of magnetization of the magnetic strips. We obtain the analytical results for the normal-magnetic-superconducting junction in the thin barrier limit and demonstrate that these results have no analogues to their conventional counterparts and junctions with Dirac electrons in two-dimensions. We discuss possible experimental setups to test our theoretical predictions.
We present a first-principles investigation to show that the contact conductance of a half conductance quantum ($G_0/2$) found previously does not generally hold for single-atom magnetic junctions composed of a tip and an adatom dsorbed on a surface. The contact conductance of the Ni-Co/Co(111) junction is approximately $G_0/2$, while for the Co-Co/Co(111), Ni-Ni/Ni(111), and Ni-Ni/Ni(001) junctions the contact conductances are 0.80$G_0$, 1.55$G_0$, and 1.77$G_0$, respectively. The deviation from $G_0/2$ is mainly caused by the variation of the spin-down conductance largely determined by the m…
We report detailed atomistic simulations combined with high-fidelity conductance calculations to probe the structural origins of conductance fluctuations in thermally evolving Au-benzene-1,4-dithiolate-Au junctions. We compare the behavior of structurally ideal junctions (electrodes with flat surfaces) to structurally realistic, experimentally representative junctions resulting from break junction simulations. The enhanced mobility of metal atoms in structurally realistic junctions results in significant changes to the magnitude and origin of the conductance fluctuations. Fluctuations are larg…
Effects of impurity scatterings on the conductance in normal-metal / $d$ wave superconductor junctions are discussed by using the single-site approximation. So far, the split of the zero-bias conductance peak has been believed to be an evidence of the broken time reversal symmetry states at the surface of high-$T_c$ superconductors. In this paper, however, it is shown that the impurity scattering near the interface also causes the split of the zero-bias conductance peak. Typical conductance spectra observed in experiments at finite temperatures and under external magnetic fields are explained …
We calculate the zero-temperature differential conductance $dI/dV$ of a voltage-biased one-dimensional junction between a nontopological and a topological superconductor for arbitrary junction transparency using the scattering matrix formalism. We consider two representative models for the topological superconductors: (i) spinful $p$-wave and (ii) $s$-wave with spin-orbit coupling and spin splitting. We verify that in the tunneling limit (small junction transparencies) where only single Andreev reflections contribute to the current, the conductance for voltages below the nontopological superco…
We study the relationship between the differential conductance and the local density of states in tight-binding tunnel junctions where the junction' geometry can be varied between the point-contact and the planar-contact limits. The conductances are found to differ significantly in these two limiting cases. We also examine how the matrix element influences the tunneling characteristics and produces contrast in a simple model of scanning tunneling microscope (STM). Some implications regarding the interpretation of STM spectroscopic data in the cuprates are discussed. The calculations are carrie…
The electronic transport properties through junctions in nanographite ribbons are investigated using the Landauer approach. In the low-energy regime ribbons with zigzag boundary have a single conducting channel of edge states. The conductance as a function of the chemical potential shows a rich structure with sharp dips of zero conductance. Each zero-conductance resonance is connected with a resonant state which can be interpreted as the superposition of two degenerate flux states with Kekulé-like current patterns. These zero-conductance dips are connected with a pronounced negative magneto re…
Based on the spin-pumping theory and first-principles calculations, the spin-mixing conductance (SMC) is theoretically studied for Pt/Permalloy (Ni$_{81}$Fe$_{19}$, Py) junctions. We evaluate the SMC for ideally clean Pt/Py junctions and examine the effects of interface randomness. We find that the SMC is generally enhanced in the presence of interface roughness as compared to the ideally clean junctions. Our estimated SMC is in good quantitative agreement with the recent experiment for Pt/Py junctions. We propose possible routes to increase the SMC in Pt/Py junctions by depositing a foreign m…
A method for the calculation of the conductance of nanoscale electrical junctions is extended to ab-initio electronic structure methods and applied to realistic models of metallic wires and break-junctions of sodium and gold. The method is systematically controllable and convergeable, and can be straightforwardly extended to include more complex processes and interactions. Important issues about the order in which are taken both the thermodynamic and the static (small field) limits are clarified, and characterized further through comparisons to model systems.
Amine terminated molecules show well behaved conductance in the scanning tunneling microscope break-junction experimental measurements. We performed density functional theory based electron transport calculations to explain the nature of this phenomenon. We find that amines can be adsorbed only on apex Au atom, while thiolate group can be attached equally well to undercoordinated and clean Au surfaces. Our calculations show that only one adsorption geo metry is sterically and energetically possible for amine anchored junction whereas three different adsorption geometries with very distinct tra…
Devices made of few molecules constitute the miniaturization limit that both inorganic and organic-based electronics aspire to reach. However, integration of millions of molecular junctions with less than 100 molecules each has been a long technological challenge requiring well controlled nanometric electrodes. Here we report molecular junctions fabricated on a large array of sub-10 nm single crystal Au nanodots electrodes, a new approach that allows us to measure the conductance of up to a million of junctions in a single conducting Atomic Force Microscope (C-AFM) image. We observe two peaks …
We calculate the tunneling conductance of a graphene normal metal-insulator-superconductor (NIS) junction with a barrier of thickness $d$ and with an arbitrary voltage $V_0$ applied across the barrier region. We demonstrate that the tunneling conductance of such a NIS junction is an oscillatory function of both $d$ and $V_0$. We also show that the periodicity and amplitude of such oscillations deviate from their universal values in the thin barrier limit as obtained in earlier work [Phys. Rev. Lett. {\bf 97}, 217001 (2006)] and become a function of the applied voltage $V_0$. Our results reprod…
The Andreev bound state spectra of multi-terminal Josephson junctions form an artificial band structure, which is predicted to host tunable topological phases under certain conditions. However, the number of conductance modes between the terminals of multi-terminal Josephson junction must be few in order for this spectrum to be experimentally accessible. In this work we employ a quantum point contact geometry in three-terminal Josephson devices. We demonstrate independent control of conductance modes between each pair of terminals and access to the single-mode regime coexistent with the presen…
The conductance of a junction between a normal metal and a superconductor having the symmetry proposed by Berezinskii is studied theoretically. The main feature of this symmetry is the odd frequency dependence of the anomalous Green function, which makes possible the s-wave triplet superconducting state (the Berezinskii superconductor). The Andreev reflection (which links positive and negative energies) is sensitive to the energetic symmetry; as a result, the conductance of the junction involving the Berezinskii superconductor is qualitatively different from the case of a conventional supercon…
We revisit the expression for the conductance of a general nanostructure -- such as a quantum point contact -- as obtained from the linear response theory. We show that the conductance represents the strength of the Drude singularity in the conductivity $σ(k,k';iω\to 0)$. Using the equation of continuity for electric charge we obtain a formula for conductance in terms of polarization of the system. This identification can be used for direct calculation of the conductance for systems of interest even at the {\it ab-initio} level. In particular, we show that one can evaluate the conductance from…
We have developed an efficient scalable kernel method for thermal transport in open systems, with which we have computed the thermal conductance of a junction between bulk silicon and silicon nanowires with diameter up to 10 nm. We have devised scaling laws for transmission and reflection spectra, which allow us to predict the thermal resistance of bulk-nanowire interfaces with larger cross sections than those achievable with atomistic simulations. Our results indicate the characteristic size beyond which atomistic systems can be treated accurately by mesoscopic theories.
We study the zero temperature conductance and magnetoconductance of ballistic \textit{p-n} junctions in Weyl semimetals. Electron transport is mediated by Klein tunneling between \textit{n}- and \textit{p}- regions. The chiral anomaly that is realized in Weyl semimetals plays a crucial role in the magnetoconductance of the junction. With the exception of field orientations where the angle between $\mathbf{B}$ and the junction plane is small, magnetoconductance is positive and linear in $B$ at both weak and strong magnetic fields. In contrast, magnetoconductance in conventional \textit{p-n} jun…
We study the zero-temperature transport properties of one-dimensional normal-superconductor (NS) junctions with topological superconductors across their topological transitions. Working within the Blonder-Tinkham-Klapwijk (BTK) formalism generalized for topological NS junctions, we analytically calculate the differential conductance for tunneling into two models of a topological superconductor: a spinless intrinsic $p$-wave superconductor and a spin-orbit-coupled $s$-wave superconductor in a Zeeman field. In both cases we verify that the zero-bias conductance is robustly quantized at $2e^2/h$ …
We present a theoretical study of the transport characteristics of molecular junctions, where first-row diatomic molecules are attached to (001) gold and platinum electrodes. We find that the conductance of all of these junctions is of the order of the conductance quantum unit $G_0$, spelling out that they belong to the transparent regime. We further find that the transmission coefficients show wide plateaus as a function of the energy, instead of the usual sharp resonances that signal the molecular levels in the tunneling regime. We use Caroli's model to show that this is a rather generic pro…
The low conductance of nickel atomic junctions in the hydrogen environment is studied using the nonequilibrium Green's function theory combined with first-principles calculations. The Ni junction bridged by a $H_2$ molecule has a conductance of approximately 0.7 $G_0$. This conductance is contributed by the anti-bonding state of the $H_2$ molecule, which forms a bonding state with the $3d$ orbitals of the nearby Ni atoms. In contrast, the Ni junction bridged by the two single H atoms has a conductance of approximately 1 $G_0$, which is weakly spin-polarized. The spin-up channels were found to …
Energy dispersion and spin orientation of the protected states at interfaces between topological insulators (TIs) and non-topological materials depend on the charge redistribution, strain, and atomic displacement at the interface. Knowledge of these properties is essential for applications of topological compounds, but direct access to them in the interface geometry is difficult. We show that conductance of a gated double junction at the surface of a topological insulator exhibits oscillations and a quasi-linear decay as a function of gate voltage in different regimes. These give the values fo…
We calculate the conductance through double junctions of the type M(inf.)-Sn-Mm-Sn-M(inf.) and triple junctions of the type M(inf.)-Sn-Mm-Sn-Mm-Sn-M(inf.), where M(inf.) are semi-infinite metallic electrodes, Sn are 'n' layers of semiconductor and Mm are 'm' layers of metal (the same as the electrodes), and compare the results with the conductance through simple junctions of the type M(inf.)-Sn-M(inf.). The junctions are bi-dimensional and their parts (electrodes and 'active region') are periodic in the direction perpendicular to the transport direction. To calculate the conductance we use the…
We have investigated the effect of a difference in the tunnelling resistances of the individual normal metal-insulator-superconductor (NIS) tunnel junctions in a double junction SINIS device, with particular emphasis on the impact on the conductance, cooling and thermometric performance. By solving the electrical and thermal equations of the junctions in a self-consistent way, we find that asymmetry gives rise to many new features, such as appearance of an excess sub-gap current, improved cooling performance, exhibition of negative differential resistance, and improved temperature range of the…
Understanding the properties of electronic transport across metal-molecule interfaces is of central importance for controlling a large variety of molecular-based devices such as organic light emitting diodes, nanoscale organic spin-valves and single-molecule switches. One of the primary experimental methods to reveal the mechanisms behind electronic transport through metal-molecule interfaces is the study of conductance as a function of molecule length in molecular junctions. Previous studies focused on transport governed either by tunneling or hopping, both at low conductance. However, the up…
We study the heat conductance of hybrid superconducting junctions. Our analysis involves single-channel junctions with arbitrary transmission as well as diffusive connectors and shows the influence of the superconducting gaps and phases of the contacts on the heat conductance. If the junction is diffusive, these effects are completely quenched on average, however, we find that their influence persists in weak-localization corrections and conductance fluctuations. While these statistical properties strongly deviate from the well-known analogues for the charge conductance, we demonstrate that th…
The conductance of a single metal-molecule-metal junction depends critically on the conformations of the molecule. In the simple case of a biphenyl, two phenyl rings linked together by a single C-C bond, the conductance is expected to depend on the relative twist angle between the two rings, with the planar conformation having the highest conductance. A number of different techniques have measured the conductance of metal-molecule(s)-metal junctions. However, the conductance variation from junction to junction has made it difficult to verify even the simplest predictions about how molecules sh…
Conductance is related to dynamical correlation functions which can be calculated with \textit{time-dependent} methods. Using boundary conformal field theory, we relate the conductance tensors of quantum junctions of multiple wires to static correlation functions in a finite system. We then propose a general method for determining the conductance through \textit{time-independent} calculations alone. Applying the method to a Y-junction of interacting quantum wires, we numerically verify the theoretical prediction for the conductance of the chiral fixed-point of the Y-junction and then calculate…
We report theoretical investigations on the role of interfacial bonding mechanism and its resulting structures to quantum transport in molecular wires. Two bonding mechanisms for the Au-S bond in an Au(111)/1,4-benzenedithiol(BDT)/Au(111) junction were identified by ab initio calculation, confirmed by a recent experiment, which, we showed, critically control charge conduction. It was found, for Au/ BDT/Au junctions, the hydrogen atom, bound by a dative bond to the Sulfur, is energetically non-dissociative after the interface formation. The calculated conductance and junction breakdown forces o…
In this work we have analysed the deterimental effect on conductance caused by diffusion of hydrogen atoms in interstitial voids during fabrication process of magnesium oxide barrier of an Fe MgO Fe magnetic tunnel junction, using first principle calculations. This diffusion of hydrogen atoms in interstitial voids in barrier disturbs a certain kind of symmetry possessed by magnesium oxide often termed as delta 1 state symmetry. Distortion in delta 1 state symmetry tempers the condutance through the magnetic tunnel junction specially in parallel configuration, which in turn reduces the Tunnelli…
We investigate quantum transport via surface states in a nanostep junction on the surface of a 3D topological insulator that involves two different side surfaces. We calculate the conductance across the junction within the scattering matrix formalism and find that as the bias voltage is increased, the conductance of the nanostep junction is suppressed by a universal factor of 1/3 compared to the conductance of a similar planar junction based on a single surface of a topological insulator. We also calculate and analyze the Fano factor of the nanostep junction and predict that the Fano factor sa…
Magneto-fluctuations of the normal resistance R_N have been reproducibly observed in high critical temp erature superconductor (HTS) grain boundary junctions, at low temperatures. We attribute them to mesoscopic transport in narrow channels across the grain boundary line. The Thouless energy appears to be the relevant energy scale. Our findings have significant implications on quasiparticle relaxation and coherent transport in HTS grain boundaries.
We study theoretically the length dependence of both conductance and thermopower in metal-molecule-metal junctions made up of dithiolated oligophenylenes contacted to gold electrodes. We find that while the conductance decays exponentially with increasing molecular length, the thermopower increases linearly as suggested by recent experiments. We also analyze how these transport properties can be tuned with methyl side groups. Our results can be explained by considering the level shifts due to their electron-donating character as well as the tilt-angle dependence of conductance and thermopower.…
We studied spin-dependent conductance in a normal metal (NM)/NM junction with Dresselhaus spin-orbit coupling (DSOC) and magnetization. As a reference, we also studied the spin-dependent conductance in such a junction with Rashba spin-orbit coupling (RSOC). Using a standard scattering method, we calculated the gate-voltage dependence of the spin-dependent conductances in DSOC and RSOC. In addition, we calculated the gate-voltage dependence of the conductances in a ferromagnetic metal (FM)/NM junction with spin-orbit coupling and magnetization, which we call ferromagnetic spin-orbit metal (FSOM…
The conductance of a molecular junction is commonly determined by either charge-transfer-doping, where alignment of the Fermi energy to the molecular levels is achieved, or tunnelling through the tails of molecular resonances within the highest-occupied and lowest-unoccupied molecular-orbital gap. Here, we present an alternative mechanism where electron transport is dominated by electrode surface states. They give rise to metallization of the molecular bridge and additional, pronounced conductance resonances allowing for substantial tailoring of its electronic properties via, e.g. a gate volta…
The magneto-conductance in YBCO grain boundary Josephson junctions, displays fluctuations at low temperatures of mesoscopic origin. The morphology of the junction suggests that transport occurs in narrow channels across the grain boundary line, with a large Thouless energy. Nevertheless the measured fluctuation amplitude decreases quite slowly when increasing the voltage up to values about twenty times the Thouless energy, of the order of the nominal superconducting gap. Our findings show the coexistence of supercurrent and quasiparticle current in the junction conduction even at high nonequil…
We report electronic transport measurements of graphene contacted by NbTiN electrodes, which at low temperature remain superconducting up to at least 11 Tesla. In devices with a single superconducting contact, we find a more than twofold enhancement of the conductance at zero bias, which we interpret in terms of reflectionless tunneling. In devices with two superconducting contacts, we observe the Josephson effect, bipolar supercurrents and Fraunhofer patterns.
We study the transport properties and superconducting proximity effect in NSN junctions formed by a time-reversal symmetry broken Weyl semimetal (WSM) in proximity to an $s$-wave superconductor. We find that the differential conductance and induced pairing amplitudes strongly depend on the angle between the junction direction in real space and the axis separating the Weyl nodes in momentum space. We identify the influence of a chiral chemical potential, i.e., the electron population imbalance between Weyl nodes of opposite chirality, on the transport characteristics of the junction. Remarkably…
The effect of an insulating barrier located at a distance $a$ from a NS quantum point contact is analyzed in this work. The Bogoliubov de Gennes equations are solved for NINS junctions (S: anysotropic superconductor, I: insulator and N: normal metal), where the NIN region is a quantum wire. For $% a\neq0$, bound states and resonances in the differential conductance are predicted. These resonances depend on the symmetry of the pair potential, the strength of the insulating barrier and $a $. Our results show that in a NINS quantum point contact the number of resonances vary with the symmetry of …
We propose an objective and robust method to extract the electrical conductance of single molecules connected to metal electrodes from a set of measured conductance data. Our method roots in the physics of tunneling and is tested on octanedithiol using mechanically controllable break junctions. The single molecule conductance values can be deduced without the need for data selection.
We investigated the electrical transport properties of superconductor-graphene-superconductor (SGS) Josephson junctions. In low voltage bias, we observed conventional proximity-coupled Josephson effect, such as the supercurrent flow through the graphene, sub-gap structure of differential conductance due to Andreev reflection, and periodic modulation of the critical current Ic with perpendicular magnetic field H to the graphene. In high bias above the superconducting gap voltage, however, we also observed an anomalous jump of the differential conductance, the voltage position of which is sensit…
In this paper, we study the conductance and shot noise in transport through a multi-site system in a two terminal configuration. The dependence of the transport on the number of atoms in the atomic wire is investigated using a tight-binding Hamiltonian and the nonequilibrium Green's function method. In addition to reproducing the even-odd behavior in the transmission probability at the Fermi energy or the linear response conductance in the normal-atomic wire-normal metallic(NAN) junctions, we find the following: (i) The shot noise is larger in the even-numbered atomic wire than in the odd-numb…
We present a technique to fabricate tunnel junctions between graphene and Al and Cu, with a Si back gate, as well as a simple theory of tunneling between a metal and graphene. We map the differential conductance of our junctions versus probe and back gate voltage, and observe fluctuations in the conductance that are directly related to the graphene density of states. The conventional strong-suppression of the conductance at the graphene Dirac point can not be clearly demonstrated, but a more robust signature of the Dirac point is found: the inflection in the conductance map caused by the elect…
Motivated by recent experiments, we study the variability of the thermal conductance of single dithiolated alkane molecules of varying length connecting two gold electrodes. For this purpose, we examine (i) the influence of the metal-molecule contact and of the electrode orientation on the thermal conductance of straight alkane chains, (ii) the effect of molecule-internal disorder realized through torsional gauche defects and the behavior upon stretching, and (iii) the modifications resulting from temperature-dependent dynamical variations of the geometry. While we analyze the former two aspec…
The linear conductance spectrum of a metallic graphene junction formed by interconnecting two gapless graphene nanoribbons is calculated. A strong conductance suppression appears in the vicinity of the Dirac point. We found that such a conductance suppression arises from the antiresonance effect associated with the edge state localized at the zigzag-edged shoulder of the junction. The conductance valley due to the antiresonance is rather robust in the presence of the impurity and vacancy scattering. And the center of the conductance valley can be readily tuned by an electric field exerted on t…
We present a comprehensive study of the properties of the off-resonant conductance spectrum in oligomer nanojunctions between graphitic electrodes. By employing first-principle-based methods and the Landauer approach of quantum transport, we identify how the electronic structure of the molecular junction components is reflected in electron transport across such systems. For virtually all energies within the conduction gap of the corresponding idealised polymer chain, we show that: a) the inverse decay length of the tunnelling conductance is intrinsically defined by the complex-band structure o…
The transport properties of Chern insulator junctions generated by bipolar junctions in quantum Hall graphene are theoretically studied in the coherent regime. Coherent transport across the junction exhibits two mesoscopic features: valley-isospin dependence of the quantum Hall conductance, and the Aharonov-Bohm (AB) effects with the interface channels. We demonstrate that the valley-isospin dependence can be measured in a graphene sample with perfect edge terminations, resulting in conductance oscillation for the smallest Chern number case. On the other hand, while conductance plateaus are fo…
Conductance $G$ of a 2DEG-Superconductor (S) device in a high magnetic field is studied: $G(ν)$ is calculated. When the cyclotron diameter in 2DEG is larger than the width of the 2DEG-S surface then $G(ν)$ becomes nonmonotonous function due to the Aharonov--Bohm type interference of quasiparticles at the surface. At certain parameters of the junction the conductance oscillates with $ν$.
We use the non-equilibrium Green function (NEGF) method in the ballistic limit to provide a quantitative description of the conductance of graphene pn junctions - an important building block for graphene electronics devices. In this paper, recent experiments on graphene junctions are explained by a ballistic transport model, but only if the finite junction transition width, Dw, is accounted for. In particular, the experimentally observed anamolous increase in the resistance asymmetry between nn and np junctions under low source/drain charge density conditions is also quantitatively captured by…
We analyze the Andreev spectrum in a four-terminal Josephson junction between one-dimensional topological superconductors in class D. We find that a topologically protected crossing in the space of three superconducting phase differences can occur between the two lowest Andreev bound states. This crossing can be detected through the transconductance quantization, in units of $2e^2/h$, between two voltage-biased terminals. Our prediction provides yet another example of topology in multi-terminal Josephson junctions. We discuss possible realizations of such junctions with semiconducting crossed …
We report a systematic experimental study of mesoscopic conductance fluctuations in superconductor/normal/superconductor (SNS) devices Nb/InAs-nanowire/Nb. These fluctuations far exceed their value in the normal state and strongly depend on temperature even in the low-temperature regime. This dependence is attributed to high sensitivity of perfectly conducting channels to dephasing and the SNS fluctuations thus provide a sensitive probe of dephasing in a regime where normal transport fails to detect it. Further, the conductance fluctuations are strongly non-linear in bias voltage and reveal su…
Weyl semimetals are a new paradigmatic topological phase of matter featuring a gapless spectrum. One of its most distinctive features is the presence of Fermi arc surface states. Here, we report on atomistic simulations of the dc conductance and quantum Hall response of a minimal Weyl semimetal. By using scattering theory we show that a quantized Hall conductance with a non-vanishing longitudinal conductance emerges associated to the Fermi arc surface states with a remarkable robustness to high concentrations of defects in the system. Additionally, we predict that a slab of a Weyl semimetal wi…
We report on the fabrication of Josephson junction devices with weak links utilizing the Weyl and higher-order topological semimetal $\text{WTe}_2$. We show that $\text{WTe}_2\text{/Pd}$ contact annealed at a low temperature of 80°C did not exhibit superconducting properties because neither $\text{WTe}_2$ nor Pd are superconductors in the ground state. Upon 180°C annealing, spontaneous formation of superconducting $\text{PdTe}$ due to Pd diffusion enabled us to obtain the interface between $\text{WTe}_2$ and superconductor suitable for the Josephson junction. This result is a facile technique …
A current constrained approach is proposed to calculate negative differential conductance in molecular nano-junctions. A four-site junction is considered where a steady-state current is forced by inserting only the two central sites within the circuit. The two lateral sites (representing e.g. dangling molecular groups) do not actively participate in transport, but exchange electrons with the two main sites. These auxiliary sites allow for a variable number of electrons within the junction, while, as required by the current constrained approach, the total number of electrons in the system is ke…
We study the low-energy electronic properties of a junction made of two crossed metallic carbon nanotubes of general chiralities. We derive a tight binding tunneling matrix element that couples low-energy states on the two tubes, which allows us to calculate the contact conductance of the junction. We find that the intrinsic asymmetries of the junction cause the forward and backward hopping probabilities from one tube to another to be different. This defines a zero-field Hall conductance for the junction, which we find to scale inversely with the junction contact conductance. Through a systema…
We calculate the linear conductance of a two-dimensional electron gas (2DEG)-based junction between a normal semiconductor section and a hybrid semiconductor-superconductor section, under perpendicular magnetic field. We consider two important terms often neglected in the literature, the magneto-orbital and transverse Rashba spin-orbit. The strong orbital effect due to the magnetic field yields topological phase transitions to nontrivial phases hosting Majorana modes in the hybrid section. The presence of a potential barrier at the junction interface reveals the Majorana phases as quantized pl…
