Specular Andreev Reflection Research Articles

Characteristic energies, transition temperatures, and switching effects in clean S|N|S graphene nanostructures

We study proximity effects in clean nanoscale superconductor-normal metal-superconductor (S$\mid$N$\mid$S) graphene heterostructures using a self-consistent numerical solution to the continuum Dirac Bogoliubov-de Gennes (DBdG) equations. We obtain results for the pair amplitude and the local density of states (DOS), as a function of doping and of the geometrical parameters determining the width of the structures. The superconducting correlations are found to penetrate the normal graphene layers even when there is extreme mismatch in the normal and superconducting doping levels, where specular Andreev reflection dominates. The local DOS exhibits peculiar features, which we discuss, arising from the Dirac cone dispersion relation and from the interplay between the superconducting and Thouless energy scales. The corresponding characteristic energies emerge in the form of resonant peaks in the local DOS, that depend strongly on the doping level, as does the energy gap, which declines sharply as the relative difference in doping between the S and N regions is reduced. We also linearize the DBdG equations and develop an essentially analytical method that determines the critical temperature $T_c$ of an \sns nanostructure self-consistently. We find that for S regions that occupy a fraction of the coherence length, $T_c$ can undergo substantial variations as a function of the relative doping. At finite temperatures and by manipulating the doping levels, the self consistent pair amplitudes reveal dramatic transitions between a superconducting and resistive normal state of the structure. Such behavior suggests the possibility of using the proposed system as a carbon-based superconducting switch, turning superconductivity on or off by tuning the relative doping levels.

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.

Parity of specular Andreev reflection under a mirror operation in a zigzag graphene ribbon

It is known that the parity of reflection amplitude can either be even or odd under the mirror operation. Up to now, all the parities of reflection amplitude in the one-mode energy region are even under the mirror operation. In this paper, we give an example of odd parity for Andreev reflection (AR) in a three-terminal graphene-supercondutor hybrid systems. We found that the parity is even for the Andreev retroreflection (ARR) and odd for specular Andreev reflection (SAR). We attribute this remarkable phenomenon to the distinct topology of the band structure of graphene and the specular Andreev reflection involving two energy bands with different parity symmetry. As a result of odd parity of SAR, the SAR probability of a four-terminal system with two superconducting leads (two reflection interfaces) can be zero even when the system is asymmetric due to the quantum interference of two ARs.

Andreev spectroscopy of doped HgTe quantum wells

We investigate the Andreev reflection process in high-mobility HgTe/CdTe quantum wells. We find that Andreev conductance probes the dynamics of massive $2+1$ Dirac fermions, and that both specular Andreev reflection and retroreflection can be realized even in presence of a large mismatch between the Fermi wavelengths at the two sides of the normal/superconducting junction.

A Green function approach to graphene–superconductor junctions with well-definededges

This work presents a novel approach to describing spectral properties of graphene layerswith well-defined edges. We microscopically analyze the boundary problem for thecontinuous Bogoliubov–de Gennes–Dirac equations and derive the Green functions fornormal and superconducting graphene layers. Importing the idea used in tight-bindingmodels of a microscopic hopping that couples different regions, we are able to set up andsolve an algebraic Dyson equation describing a graphene–superconductor junction. For thiscoupled system we analytically derive the Green functions and use them to calculate thelocal density of states and the spatial variation of the induced pairing correlationsin the normal region. Signatures of specular Andreev reflections are identified.

Theory of large tunneling magnetoresistance in a gapped graphene-based ferromagnetic superconductor F/(FS) junction

Abstract Coexistence of superconductivity and ferromagnetism in a gapped graphene-based system (FS) is theoretically investigated. The center-of-mass momentum, P, of a Cooper pair in FS is found to be P ∼ 2 E ex / ℏ v F 1 - ( m / E FS ) 2 , where m, Eex, EFS are the rest mass energy of the Dirac electron, exchange energy and the Fermi energy in the superconductor FS, respectively. It is unlike the nature in a conventional FFLO state where P ∼ 2 E ex / ℏ v F . This work studies the magneto effect on the transport property of a F/(FS) junction where F is a ferromagnetic gapless graphene. In this work, FS is achieved by depositing a conventional ferromagnetic s-wave superconductor on the top of gapped graphene sheet. The Zeeman splitting in FS induces spin-dependent Andreev resonance. The conductances effected by both spin-dependent specular Andreev reflections and spin-dependent Andreev resonances are investigated. The interplay between the spin-dependent specular Andreev reflection in the F region and the spin-dependent Andreev resonance in the FS region causes a very large tunneling magnetoresistance |TMR| ∼ 3000% for m → EFS, possibly valuable in the graphene-based spintronic devices. This is because of the coexistence of the superconductivity and ferromagnetism in FS and the relativistic nature of electrons in graphene.

Tunneling conductance of graphene ferromagnet-insulator-superconductor junctions

We study the transport properties of a graphene ferromagnet-insulator superconductor (FIS) junction within the Blonder-Tinkham-Klapwijk formalism by solving spin-polarized Dirac-Bogoliubov-de-Gennes equation. We find that the retro and specular Andreev reflections in the graphene FIS junction are drastically modified in the presence of exchange interaction and that the spin-polarization ($P_T$) of tunneling current can be tuned from the positive to negative value by bias voltage ($V$). In the thin-barrier limit, the conductance $G$ of a graphene FIS junction oscillates as a function of barrier strength $\chi$. Both the amplitude and phase of the conductance oscillation varies with the exchange energy $E_{ex}$. For $E_{ex}<E_F$ (Fermi energy), the amplitude of oscillation decreases with $E_{ex}$. For $E_{ex}^{c}>E_{ex}>E_F$, the amplitude of oscillation increases with $E_{ex}$, where $E_{ex}^{c}=2E_{F}+U_{0}$ ($U_{0}$ is the applied electrostatic potential on the superconducting segment of the junction). For $E_{ex} > E_{ex}^{c}$, the amplitude of oscillation decreases with $E_{ex}$ again. Interestingly, a universal phase difference of $\pi/2$ in $\chi$ exists between the $G-\chi$ curves for $E_{ex}>E_F$ and $E_{ex}<E_F$. Finally, we find that the transitions between retro and specular Andreev reflections occur at $eV=|E_{F}-E_{ex}|$ and $eV=E_{ex}+E_{F}$, and hence the singular behavior of the conductance near these bias voltages results from the difference in transport properties between specular and retro Andreev reflections.

Controllable Andreev Retroreflection and Specular Andreev Reflection in a Four-Terminal Graphene-Superconductor Hybrid System

We report the investigation of electron transport through a four-terminal graphene-superconductor hybrid system. Because of the quantum interference of the reflected holes from two graphene-superconductor interfaces with a phase difference theta, it is found that the specular Andreev reflection vanishes at theta=0 while the Andreev retroreflection disappears at theta=pi. This means that retroreflection and specular reflection can be easily controlled and separated in this device. In addition, because of the diffraction effect in the narrow graphene nanoribbon, the reflected hole can exit from both graphene terminals. As the width of nanoribbon increases, the diffraction effect gradually disappears and the reflected hole eventually exits from a particular graphene terminal depending on the type of Andreev reflection.

Perfect switching of the spin polarization in a ferromagnetic gapless graphene/superconducting gapped graphene junction

With the fabrication of gapped graphene, interest in the tunneling spectroscopy in graphene-based FG/SG junctions in which one side consists of a gapless ferro-magnetic graphene (FG) and the other side, of a gapped superconducting graphene (SG) has arisen. The carriers in the gapless (gapped) graphene are 2D relativistic particles having an energy spectrum given by E = ℏ 2 v F 2 k 2 + ( mv F 2 ) 2 (where mv F 2 is the gap and v F is the Fermi velocity). The spin currents in this FG/SG junction are obtained within the framework of the extended Blonder–Tinkham–Klapwijk (BTK) formalism. The effects of the superconducting energy gap in SG, of the gap mv F 2 which opened in the superconducting graphene, of the exchange field in FG, of the spin-dependent specular Andreev reflection, of the effective Fermi energy ( E FF ) of FG and of the bias voltage across the junction ( V) are simulated. It is seen that by adjusting E FF or V, the spin polarization (defined as SP(%) = 100% × ( G ↑ − G ↓)/( G ↑ + G ↓)) can be switched from a pure spin up (SP = +100%) state to pure spin down (SP = −100%) state.

Graphene Andreev billiards

We studied the energy levels of graphene-based Andreev billiards consisting of a superconductor region on top of a monolayer graphene sheet. For the case of Andreev retroreflection we show that the graphene-based Andreev billiard can be mapped to the normal-metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene-based Andreev billiards. The exact and the semiclassically obtained spectrum agree very well both for the case of Andreev retroreflection and specular Andreev reflection.

Andreev levels in a graphene–superconductor surface

In order to consider the Dirac-like spectrum of graphene we employ the Bogoliubov de Gennes–Dirac formalism to determine the quasiparticle Andreev levels in an NS surface (normal–superconductor). The normal region is characterized by a width L while the superconducting region is semi-infinite and both regions are made of doped graphene. The quasiparticle energy spectrum is originated by the Andreev reflections that occur in the NS interface. It is shown that this spectrum depends on the width of the normal region and the Fermi energy in each region. When the Fermi energy in the normal metal is lower than the gap of the superconductor region, the spectrum is affected by specular Andreev reflections. The equation that is obtained to find the spectrum is very general and we solve it for some particular cases. We find that the energy spectrum oscillates when the Fermi energy in graphene is changed. Finally we obtain under some approximations an equation for the energy spectrum which is similar in structure as those obtained for an INS conventional junction.

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].

Signals for Specular Andreev Reflection

We report a theoretical investigation of the spin-dependent Andreev reflection at the interface of a graphene-based ferromagnet/superconductor junction. It is found that the ferromagnetic exchange interaction in the ferromagnet can suppress Andreev retroreflection but enhance the specular Andreev reflection. There is a transition between the specular Andreev reflection and Andreev retroreflection at which the shot noise vanishes and the Fano factor has a universal value. The present work provides a new method of detecting the specular Andreev reflection, which can be experimentally tested within the present-day technique.

Specular Andreev reflection and magnetoresistance in graphene-basedferromagnet–superconductor hybrid systems

Spin-polarized transport of relativistic electrons through graphene-basedferromagnet/insulator/superconductor single and double junctions has been investigated onthe basis of the Dirac–Bogoliubov–de Gennes equation. We have presented acomparative study on two kinds of cases: in the presence and in the absence of specularAndreev reflection. Although both the magnetoresistance (MR) and conductance areoscillating functions of the effective barrier potential and the thickness of thesuperconductive graphene (SG) layer for the two kinds of cases, some differences infeatures have also been found. In the presence of specular Andreev reflection,the MR decreases quickly with an increase of the thickness of the SG layer, andnegative MR can be observed, which is in contrast to the case for the absence ofspecular Andreev reflection. It is interesting that the resonance peak of the MRcan appear at a certain bias voltage due to the retroreflection crossing over tospecular Andreev reflection. This means that the MR can be tuned by an externalbias voltage, which benefits spin-polarized electron devices based on graphenematerials.

Specular Andreev reflection and magnetoresistance in graphene-based ferromagnet-superconductor double junctions

Spin-polarized transports of relativistic electrons through graphene-based ferromagnet-superconductor double junctions have been investigated theoretically. In the presence of specular Andreev reflection, some unusual properties are found. The oscillating negative magnetoresistance (MR) is not only observed, but the resonance peak of the MR also appears at a certain bias voltage due to the retroreflection crossing over to the specular Andreev reflection. This means that the MR can be tuned by external bias voltage, which benefits the spin-polarized electron device based on the graphene materials.

Pure spin current in graphene normal-superconductor structures

We demonstrate theoretically the possibility of producing a pure spin current in graphene by filtering the charge from a spin-polarized electric current. To achieve this effect, which is based on the recently predicted property of specular Andreev reflection in graphene, we propose two possible device structures containing normal-superconductor junctions.

Excitation gap of a graphene channel with superconducting boundaries

We calculate the density of states of electron-hole excitations in a superconductor--normal-metal--superconductor (SNS) junction in graphene, in the long-junction regime that the superconducting gap ${\ensuremath{\Delta}}_{0}$ is much larger than the Thouless energy ${E}_{T}=\ensuremath{\hbar}v∕d$ (with $v$ the carrier velocity in graphene and $d$ the separation of the NS boundaries). If the normal region is undoped, the excitation spectrum consists of neutral modes that propagate along the boundaries---transporting energy but no charge. These ``Andreev modes'' are a coherent superposition of electron states from the conduction band and hole states from the valence band, coupled by specular Andreev reflection at the superconductor. The lowest Andreev mode has an excitation gap of ${E}_{0}=\frac{1}{2}(\ensuremath{\pi}\ensuremath{-}\ensuremath{\mid}\ensuremath{\phi}\ensuremath{\mid}){E}_{T}$, with $\ensuremath{\phi}∊(\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$ the superconducting phase difference. At high doping (Fermi energy $\ensuremath{\mu}⪢{E}_{T}$) the excitation gap vanishes $\ensuremath{\propto}{E}_{0}{({E}_{T}∕\ensuremath{\mu})}^{2}$, and the usual gapless density of states of Andreev levels is recovered. We use our results to calculate the $\ensuremath{\phi}$ dependence of the thermal conductance of the graphene channel.

Specular Andreev Reflection in Graphene

By combining the Dirac equation of relativistic quantum mechanics with the Bogoliubov-de Gennes equation of superconductivity we investigate the electron-hole conversion at a normal-metal-superconductor interface in graphene. We find that the Andreev reflection of Dirac fermions has several unusual features: (1) the electron and hole occupy different valleys of the band structure; (2) at normal incidence the electron-hole conversion happens with unit efficiency in spite of the large mismatch in Fermi wavelengths at the two sides of the interface; and, most fundamentally: (3) away from normal incidence the reflection angle may be the same as the angle of incidence (retroreflection) or it may be inverted (specular reflection). Specular Andreev reflection dominates in weakly doped graphene, when the Fermi wavelength in the normal region is large compared to the superconducting coherence length.

Correlation between specular Andreev reflection and zero-energy states in normal-metal/d-wave-superconductor junctions

We report the effects of interfacial roughness on a zero-bias conductance peak (ZBCP) in normal-metal/d-wave-superconductor junctions. The roughness spoils to specularity of the junction interface, and causes a diffuse Andreev reflection of a quasiparticle. The ZBCP decreases linearly with a decreasing rate of the specular Andreev reflection at the interface. A sensitivity of the ZBCP to the roughness strongly depends on a transparency of the junctions. In low transparent junctions, the ZBCP disappears even in the regime of weak roughness. We also find a splitting of the ZBCP owing to the interfacial roughness in low transparent junctions.