Andreev reflection in graphene nanoribbons induced by d-wave superconductors

Honeycomb and square lattices are combined as a tight-binding model to examine the Andreev reflection in graphene nanoribbons induced by a superconductor. The superconducting symmetry is assumed to be the d-wave. The zero-bias tunneling conductance peak, which is generally produced by the d-wave superconductor, is absent for the nanoribbons under conditions similar to those when a quantum wire is the normal conductor. For the anisotropic superconductivity, propagating modes appear in the superconductor even for biases below the top of the superconducting energy gap. Features appear in the conductance at the subgap population thresholds of these propagating modes as a finite-size effect of the lattice system. The surface Andreev bound states responsible for the zero-bias anomaly also cause transport resonances in the vicinity of the zero bias despite the aforementioned destruction of the anomaly. The conductance spectra revealing these excitation behaviors are fairly unchanged regardless of the presence of a hopping barrier at the interface with the superconductor. The insensitivity to the interface scattering highlights the fact that barrier-less situation cannot be realized for the model due to the heterogeneous lattice. Concerning specular Andreev reflection, the wavefunction parity gives rise to its blocking for single-mode zigzag-edged nanoribbons. The blocking is suppressed when the anisotropic superconductivity is asymmetric for the nanoribbons.


Introduction
The electrons in graphene behave as massless relativistic particles, exhibiting unconventional properties such as the energy independence of the velocity and the infinite effective mass at the Fermi level [1].The conduction properties when a sheet of graphene is contacted by a superconductor Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.can be analyzed in terms of the propagation of quasiparticle excitations [2].In the transport process known as Andreev reflection, an electronlike excitation with energy µ + ε in the normal-conducting side is reflected from the superconductor as a holelike excitation with energy µ − ε and vice versa.Here, µ is the Fermi energy and ε the excitation energy.While the current is carried by Cooper pairs when the electrons enter the superconductor, the Andreev reflection in the normal conductor creates a Cooper pair at the Fermi level in the superconductor.The advantage of this approach is that the conductance of the system can be evaluated using the normal current carried by these quasiparticle excitations.For the junctions involving graphene, the gapless linear energy-dispersion relationship of the Dirac electrons and its modifications in graphene nanoribbons (GNRs) [3][4][5] play important roles in the Andreev reflection properties [6][7][8][9][10][11].
The superconducting energy gap 2∆ is independent of the wavevector k of the electrons for the s-wave superconducting state given by the Bardeen-Cooper-Schrieffer theory.The quasiparticle excitations are consequently expelled from the superconductor for ε < ∆.A number of superconductors have been suggested, on the other hand, to possess anisotropic superconducting symmetries [12].For the d x 2 −y 2wave symmetry, which is realized in, for instance, cuprate superconductors [13], the pair potential is given by where θ is the propagation angle of the electron and α the inclination of the d x 2 −y 2 wave.The sign changes in the variation of ∆ with θ generate surface bound states at zero energy with their density being determined by α [14,15].The surface Andreev bound states give rise to a zero-bias peak in the tunneling conductance [16,17].Remarkably, this zero-bias peak is suppressed when the normal conductor is a quantum wire [18,19].The quantization of the wavevector imposed by the transverse confinement is responsible for the suppression.The transport properties in the junctions of graphene and a d-wave superconductor have been investigated not only theoretically [20][21][22][23][24] but also experimentally [25,26].Here, the electronic states in the theoretical models were described using plane waves, i.e. both the sheet of graphene and the d-wave superconductor were assumed to be infinitely wide.The characteristics of the Andreev reflection are anticipated to be altered when GNRs are attached to the d-wave superconductor.The alteration of the energy dispersion by the finite width of the graphene sheet depends on the type of the GNR edges [3][4][5].In particular, nearly-dispersionless states emerge for energies around the Dirac point in the nanoribbons having zigzag edges.The transport properties associated with these quasi-flatband states are sensitive to the even-odd parity of the transverse wavefunction, giving rise to, for instance, the valley-valve effect [27][28][29][30][31].
In this paper, the Andreev reflection induced by the dwave superconductor is investigated numerically for zigzag and armchair GNRs using a tight-binding model.The absence of the zero-bias anomaly for the confined system is maintained for the GNRs.Possible remnants of the anomaly are, however, present as transport resonances.We clarify furthermore the consequences caused by the finite size of the superconductor.The interband Andreev reflection, which occurs for a gapless system like graphene, is blocked for single-mode symmetric zigzag GNRs due to the wavefunction parity.The blocking disappears when the superconductivity is not symmetrical for the GNRs.

Numerical model
Tight-binding lattices illustrated in figure 1 are employed for the simulations.As shown by the honeycomb lattices, we deal The square lattice sites for the superconductor are shown by the blue dots.The amplitude of the nearest-neighbor hopping shown by the black lines is assumed to be t throughout the composite system.The hopping amplitude across the GNR-superconductor interface shown by the green lines is t GS .The length of the C-C bonds in graphene is c.The lattice parameter of the square lattices is a.In the simulations, the number of the transverse sites was set to be 600 for the superconductor with periodic boundary condition.The transport probabilities when an electronlike excitation is incident are defined as shown in (c).
with GNRs having zigzag edges, figure 1(a), and armchair edges, figure 1(b).The dangling σ bonds at the GNR edges are assumed to be saturated by hydrogenation.We specify the width of the GNRs by the number of transverse C atoms.In the cases in figure 1, for instance, n z = 6 and n a = 5 for the zigzag and armchair GNRs, respectively.
In the numerical treatment of the GNRs, they are decomposed to slices having the monoatomic width, as marked by the dotted lines.In zigzag GNRs, two types of slices X and Y appear alternately in the direction of propagation.The equations of motion in the π-electron approximation are given within the nearest-neighbor hopping model as [32][33][34] where X j and Y j are the wavefunctions at the jth slices of X and Y, respectively, and T = tI with t and I being the hopping amplitude and the identity matrix, respectively.When the zigzag GNR is mirror-symmetrical with respect to the x axis, see figure 1(a), the internal hopping within the slices is given by Armchair GNRs, on the other hand, consist of four types of slices A, B, C and D. The equations of motion are given similarly as where We will consider both symmetric and asymmetric zigzag GNRs to investigate the influences of the wavefunction parity on the Andreev reflection properties [32].On the other hand, merely the mirror-symmetric ones shown in figure 1(b) will be considered for the armchair GNRs since the even-odd parity does not play a role in their transport properties.
The superconductor having the d-wave symmetry is simulated using square lattices in figure 1 [35].For the sake of simplicity, the nearest-neighbor hopping amplitude in the square lattices is assumed to be identical to that in graphene.The coupling between the GNR and the superconductor is t GS , as shown by the green lines.Note that the hopping amplitude for the square lattice is related to the lattice parameter a as t = h2 /(2m * a 2 ) in describing the free electron motion.Here, m * is the effective mass of the electron.For the cases with the zigzag and armchair GNRs, a = c and √ 3c, respectively, with c being the length of the C-C bonds of graphene.The fact that t = 2.5-3.0 eV for graphene may be interpreted as using a specific value for m * .The numerical results are anticipated, however, not to change qualitatively with this condition because of the following reasons.(1) Using a different hopping amplitude in the square lattice is equivalent to changing t GS (and µ and ∆ 0 of the superconductor correspondingly).The choice, therefore, affects the magnitude of the normal reflection caused by the mismatch between the honeycomb and square lattices as well as by t GS .The influence on the transport properties is merely quantitative since the normal reflection is inevitably significant for our model as a consequence of the lattice mismatch.That is, the GS interface is not barrier-less even if the interface is assumed to be ideal as t GS = t.(2) Regardless of assumptions, the transport properties will be investigated for energies in the vicinity of the Dirac point with respect to the subband structure of GNRs.The parameters will be set in such a way that only a few subbands are occupied by electrons in the simulations.
Using cos θ = k x /k F and sin θ = k y /k F , where k x and k y are the wavenumbers in the x-and y-directions, respectively, and is the Fermi wavenumber, equation ( 1) is approximated as Here, k F a ≪ 1 was assumed.Using the reverse Fourier transformation with respect to k the pair potential in real space r = (x, y) is given by [35] with The quantum propagation of the quasiparticle excitations is determined by Bogoliubov-de Gennes equation where u(x, y) and v(x, y) are the wavefunctions of the electronlike and holelike excitations, respectively.The pair potential ∆(x, y) is zero in the normal-conducting side, i.e. in the GNR.The single-particle Hamiltonian H 0 is composed as The Hamiltonian H g for the GNR is obtained based on equations ( 2) and (3) for zigzag GNRs and equations ( 5)-( 8) for armchair GNRs.To be specific, H g for zigzag GNRs is given in a matrix form as in terms of the basis indicated thereby.For the square lattice of the superconductor, where c † j,m and c j,m are the creation and annihilation operators, respectively, at the lattice site (j,m).The summation in the first term on the right-hand side of the equation is over the nearest neighbors (n.n.).While the number of the transverse lattice sites was set to n s = 600 for the superconductor in the simulations presented below, periodic boundary condition was imposed in the y direction.The coupling at the GNRsuperconductor (GS) interface is where the creation and annihilation operators are defined at the graphene and superconductor sides of the GS interface.The quasiparticle transport in the GS junction is as follows.Consider an electronlike excitation in GNRs that is incident to the interface with the superconductor.The quasiparticle is either reflected by the superconductor or transmitted into the superconductor.For the reflection, the outgoing quasiparticle excitation can be electronlike or holelike.These transport processes lead to the probability conservation with N being the number of the occupied transverse modes in the GNR, see figure 1(c).The probabilities R ee and R he are for the reflection as an electronlike and a holelike excitation, respectively.The transmission probability into the superconductor is T Se .The transport probabilities were calculated using the Green's function G(ε) = (ε − H) −1 .The recursion technique based on the divisions of the GNRs to the monoatomic slices was used to calculate G(ε).The numerical details were described previously in [32].The reflection coefficient from an incident mode n to an outgoing mode m is given, in the case of the zigzag GNRs, by [36] where v is the velocity of the channel and G GG is the Green's function at the GNR slice of the GS interface.The wavefunctions of the eigenmodes in the slices X and Y are contained in U X and U Y , respectively.For F Y defined as U Y ΓU −1 Y , Γ consists of λ n = exp(i k n c unit ) of the eigenmodes as the diagonal elements with c unit being the unit length of the GNR (= √ 3c for zigzag GNRs).See [36] for details.The transmission coefficient is given similarly as with v ′ m being the channel velocity in the superconductor and G SG the Green's function across the GS interface.It is worthy to note that G GG , for instance, may be calculated as where G G (G, G) and G S (S, S) are the Green's functions of the GNR and the superconductor, respectively, at the respect side of the GS interface.Changing the value of t GS is seen to be equivalent to scaling G S (S, S) accordingly, as we have mentioned above concerning the choice of the hopping amplitude used for the square lattice.
The differential conductance of the GS junction is given from the current I carried by the quasiparticle excitations as [2,37] where the transport probabilities are evaluated at ε = eV with V being the bias voltage.

Retro Andreev reflection
In figure 2, the excitation spectra of G are shown for the GS junction consisting of a zigzag GNR with n z = 50 for the case of t GS = t.Negative excitation energies are included in figures 2(a)-(d) to make the features at the zero bias clearly visible.It is pointed out that the curves are not symmetrical with respect to ε = 0 when T Se is not zero, i.e. the transport properties of the electronlike and holelike excitations are not identical in such a circumstance.The superconducting anisotropy that the carriers in the GNR encounter is changed between null and maximal by varying the orientation of the d x 2 −y 2 wave as α = 0, π/8 and π/4 in (a)-(c), respectively, see the inset of figure 2(e).The Fermi energy µ was set to be 0.05t for the red curves and 0.1t for the green curves.Throughout this paper, the Fermi energy is assumed to be identical in the GNRs and the superconductors.The normal reflection at the GS interface is enhanced when the Fermi energy differs between the two segments.The number of the occupied modes N in the GNR is plotted in the inset of figure 2(c) as a function of µ.Only the quasi-flatband state of the zigzag GNRs is occupied for µ = 0.05t.Three subbands are, on the other hand, occupied for µ = 0.1t.The superconducting gap is assumed to be much smaller than the Fermi energy as ∆ 0 = 0.01µ in this subsection.The number of the occupied modes thus remains unchanged when ε is varied.
For α = 0, a series of peaks emerge in the quasi-subgap regime ε < ∆ 0 .As α deviates from zero, the peaks are replaced by smeared step-like increases of G.These features are generated as a finite-size effect of the superconductor.Due to the anisotropy in the superconducting energy gap, the number of the propagating modes N s in the superconductor increases gradually with ε until its saturation for ε > ∆ 0 .In figure 2(d), the colored curves correspond to the case of α = 0.The modifications for α = π/4 are shown by the black curves.The features develop in G at the population of the propagating modes in the d-wave superconductor.The kinetic energy is almost zero for the top-most mode for energies just above its population threshold.The scattering from the GS interface hence changes enormously with the energy.The above behavior originates from the mode population in the d-wave superconductor.It is, therefore, not specific for the GNRs.The variation of G with ε is shown in figure 3 for the case where the normal conductor is a quantum wire described by the square lattice having 50 transverse sites.
Here, the single-mode occupation was assumed in the quantum wire with k F W/π = 1.7,where W (= 51a) is the width of the quantum wire.For the red curves, the GS interface is transparent as t GS = t.It will be worthy to emphasize that the normal reflection at the GS interface becomes negligible for t GS = t owing to the homogeneousness of the lattice.A potential barrier was, on the other hand, imposed at the interface for the green and blue curves by reducing t GS /t as 0.97 and 0.75, respectively.The overall suppression of G for smaller t GS has been corrected, for clarity, by a normalization using the values of G at a large ε, i.e. ε/∆ 0 = 10.Here, G becomes independent of ε in the limit of ε → ∞.
The series of features for ε < ∆ 0 are again present.In figure 3(d), corresponding curves for the s-wave superconductivity are shown for comparison.No peak is present for ε < ∆ 0 as such subgap propagating modes are not allowed for the s-wave superconductor.The simulations in figure 3 were carried out to reproduce the situations presented in [18].The values of t GS have been adjusted so that the curves for the s-wave superconductor mimic those in [18], where the interface barrier parameter Z in the Blonder-Tinkham-Klapwijk model [2] is indicated to be related to t GS approximately as Z ∝ 1 − t 2  GS for small Z.The finite-size effect of the d-wave superconductor was absent in [18] since the mode propagation was not involved in the calculation.While good agreements are achieved for the cases of the s-wave superconductor, the finite size of the d-wave superconductor is found to be able to alter the entire curve in the subgap regime considerably.
In figure 3, the zero-bias peak that the d-wave superconductor should produce is absent in the tunneling conductance, as previously demonstrated for quantum wires in [18].The guided propagation of the modes due to the transverse confinement is responsible for the destruction of the zero-bias peak [18,19,38].The confined modes are standing waves constructed by a superposition of the waves having transverse wavenumbers ±k y .For anisotropic superconductors, the Andreev reflection is not the same between the wave components associated with k y and −k y .The surface Andreev bound states, which are responsible for the zero-bias peak [15][16][17], can thus be forbidden for the quantized modes when the boundary conditions at the GS interface cannot be satisfied due to the inequivalence between the cases of k y and −k y .
Figure 4 shows the variation of the normalized conductance when the transparency at the GS interface is changed.The columns on the left-and right-hand sides correspond to the cases of N = 1 and 3, respectively.Contrary to the role that t GS plays in figure 3, the curves are roughly the same regardless of t GS .It was necessary to use t GS as small as 0.1t for the blue curves in order to make the difference of the curves substantial.Here, G in the limit of ε ≫ ∆ 0 is reduced for t GS = 0.1t by about a factor of 40 compared to that for t GS = t.The insensitivity of the spectra to t GS is likely a consequence of the hybrid lattice, i.e. the scattering at the GS interface cannot be negligible due to the mismatch between the honeycomb and square lattices.The interface is never transparent even when it is ideal as t GS = t.
Similar to the case for the quantum wire in figure 3, the zero-bias peak is absent for the GNRs due to the transverse confinement.It is pointed out that the conductance of the lattice-mismatched system corresponds to the tunneling conductance even for t GS = t, as for the red curves, due to the scattering at the GS interface.The tunneling conductance has been shown to exhibit the zero-bias peak when the d-wave superconductor is attached to a sheet of graphene [20][21][22][23][24].The peak is hence expected to be recovered when a large number of propagating modes are occupied in the GNR to resemble a two-dimensional system [19].Comparing the cases with N = 1 and 3, the conductance at ε → 0 is indeed seen to increase when multiple propagating modes are occupied in the GNR.
In the right-hand-side column in figure 2, the curves in the left-hand-side column are redrawn using a logarithmic scale for ε/∆ 0 .One finds in figures 2(e)-(g) that peaks and dips emerge in G for α ̸ = 0 in the regime of small ε (< 0.1∆ 0 ) although N s is unchanged.The appearance of these transport resonances may suggest that surface Andreev bound states are formed at some values of ε despite the general suppression of the zero-bias anomaly.As these transport resonances were absent in [18], the finite size of the superconductor may play a role for the occasional formation of the surface Andreev bound states.
Zigzag GNRs are mirror-symmetrical when n z is an even integer, as in figure 2. The Andreev reflection properties are examined in figure 5 for an asymmetric zigzag GNR having n z = 51.The results are shown only for α = 0 and π/4 since the excitation spectra are nearly the same between n z = 50 and 51 apart from the fact that the transport resonances in the regime ε/∆ 0 < 0.1 are inevitably affected by the difference in the width.
The simulations were carried out also for an armchair GNR, as shown in figure 6.For armchair GNRs, an energy gap opens at the Dirac point except for n a = 3i − 1 with i being integers.To take advantage of the disappearance of the bandgap, n a was chosen to be 44.The change of N with µ is shown in the inset of figure 6(b).The number of the occupied modes in the GNR is one and three for µ/t = 0.04 and 0.08, which  (e) and π/4 in (c) and (f).The number of the propagating modes N in the GNR is 1 and 3 for the red and green curves with µ/t = 0.04 and 0.08, respectively.The coupling strength at the GS interface is t GS = t.The change of N with µ is shown in the inset of (b).
are assumed for the red and green curves, respectively.The conductance increase associated with the population of an individual propagating mode in the superconductor for α = 0 is in a step-like manner for the armchair GNRs while a sharp peak appears in G at the threshold for the zigzag GNRs.The conductance spectra resemble the E −1/2 behavior with energy E of the quasi-one-dimensional density-of-states.
In figure 7, we compare the magnitude of R he among the three cases examined above, i.e. the zigzag GNRs with n z = 50 and 51 and the armchair GNR with n a = 44, as shown by the red, green and blue curves, respectively.The number of the occupied modes in the GNRs is N = 1 and 3 in figures 7(a) and (b), respectively.The dotted and solid curves correspond to α = 0 and π/4, respectively.One finds in all the cases that R he is reduced by orders of magnitude for the maximal superconducting anisotropy α = π/4.The conductance in this circumstance is thus approximately given as G ≈ (2e 2 /h)T Se , resulting in the smeared step-like increases of G with ε when α approaches π/4.
In general, the wavefunctions of electrons confined in channels having symmetric geometries are exactly symmetrical or antisymmetrical in the transverse direction, which changes alternately as the mode index increases.This wavefunction parity can block scattering since the overlap integral is zero for the transitions between the symmetric and antisymmetric states.The wavefunction parity is, on the other hand, not exact when the geometry of the channels is asymmetric.The indifference of the transport properties to the mirror symmetry of the GNRs manifested between the results in figures 2 and 5 suggests that the even-odd parity of the transverse wavefunction plays no role in the retro Andreev reflection.This speculation is supported by the mere quantitative changes between the cases of N = 1 and 3. Nevertheless, one finds in figure 7 that the suppression of R he for α = π/4 persists to ε → 0 only for the mirror-symmetrical GNRs with the single-mode occupation.The parity effect is apparently required to keep the suppression of R he in the limit of ε → 0 and thus to destroy the phenomenon of the zero-bias conductance peak.This can be interpreted to mean that the role of the superconducting anisotropy is insignificant if the wavefunction does not possess its symmetry properties.The zero-bias peak is barely recognizable in figure 5 since the reduction of R he with increasing ε is compensated by the increase of T Se .

Specular Andreev reflection
In the conventional situation where the bandgap of the normal conductor is much larger than ∆, both the electronlike and holelike excitations linked by the Andreev reflection are either in the conduction band or in the valence band, depending on whether the conduction in the normal conductor is of n-or ptype.In contrast, the absence of the energy gap for graphene enables a situation where an electronlike excitation in the conduction band is reflected as a holelike excitation in the valence band and vice versa [6,7].The intraband Andreev reflection in the former case is retro-reflection, i.e. the outgoing quasiparticle excitation traces back the trajectory of the incoming quasiparticle excitation.The interband reflection in the latter case is called specular Andreev reflection since the outgoing quasiparticle excitation leaves the GS interface in the direction of specular boundary scattering.In this subsection, the electrical properties of the GS junctions are investigated with respect to the specular Andreev reflection.While ∆ 0 = 0.01µ  was assumed in the previous subsection, ∆ 0 is set to be 2µ in this subsection.The Andreev reflection is consequently retro reflection for ε/∆ 0 < 0.5 and specular reflection for ε/∆ 0 > 0.5, see figure 8.
Figure 9 shows the variations of R he with ε in the subgap excitation regime.In (a)-(c), the GNRs attached to the d-wave superconductor are a symmetric zigzag GNR with n z = 200,  c), respectively, with t GS = t.For the red, green and blue curves, α was set to be 0, π/8 and π/4, respectively.Retro and specular Andreev reflections take place for ε/∆ 0 < 0.5 and > 0.5, respectively, as the condition ∆ 0 = 2µ was assumed.Here, µ/t = 0.009 and 0.006 for the zigzag and armchair GNRs, respectively.The number of the occupied modes changes between Ne = 1 and 3 for the electronlike excitation at ε/∆ 0 = 0.76-0.77 in all the cases.an asymmetric zigzag GNR with n z = 201 and a symmetric armchair GNR with n a = 179, respectively.For the red, green and blue curves, the superconducting anisotropy is α = 0, π/8 and π/4, respectively.Only the bottom mode is occupied being below the Fermi level at ε = 0 in all the cases.However, the parameters were chosen such that the number of the occupied modes increases to N e = 3 for the electronlike excitation during the course of increasing ε.To be specific, the transition in N e occurs at ε/∆ 0 = 0.76-0.77,as indicated in figure 9(c).The number of the occupied modes, on the other hand, remains to be one for the holelike excitation in the whole range of ε.
In figure 9(a), one finds a parity effect for the case of α = 0.As ε/∆ 0 exceeds 0.5 and thus the specular Andreev reflection takes place instead of the retro reflection, R he is abruptly suppressed to be almost zero.For the quasiflatband states of zigzag GNRs [3][4][5], the even-odd parity is opposite between those in the conduction and valence bands [28,30].The interband transition for the specular Andreev reflection is consequently forbidden if the GNRs are mirror-symmetrical, as for n z = 200.The blocking vanishes for ε/∆ 0 > 0.76 as the transition from the higher-lying state of the electronlike excitation is allowed.Similarly, the blocking is absent for the asymmetric GNR (n z = 201), as one finds in figure 9(b), due to the incomplete wavefunction parity.
The blocking of the specular Andreev reflection by the parity effect has been reported for the case where the superconductor has the conventional s-wave symmetry [9].This corresponds to the situation when the superconductivity of the d-wave superconductor is symmetric with respect to the GNRs as α = 0.The wavefunction parity does not play a role for α = π/4 since the scattering process is not mirrorsymmetrical.As previously shown in figure 7, R he is reduced by orders of magnitude for α = π/4.The reduction becomes of an intermediate degree for the specular Andreev reflection in comparison to that for the retro Andreev reflection.
For the armchair GNR in figure 9(c), the wavefunction parity is the same between the pairs of the states in the conduction and valence bands.It is the consequence of this fact that the valley-valve effect [27][28][29][30][31] does not take place for armchair GNRs.The suppression of R he originating from the parity effect is, therefore, absent in 0.5 < ε/∆ 0 < 0.76.The change of R he associated with the switch between the retro and specular Andreev reflections at ε/∆ 0 = 0.5 is also negligible.It will be worthy to state explicitly that, although R he changes with ε, the magnitude is so small that the variation is not recognizable in figure 9(c).Merely a gradual increase of R he resulting from the transition to N e = 3 for ε/∆ 0 > 0.76 is seen.The suppression of R he for α = π/4 is thereby partly compensated.

Conclusions
A tight-binding lattice system has been employed to investigate the Andreev reflection in the junctions of GNRs and dwave superconductors.The exact treatment of the transport processes in the composite system has revealed a finite-size effect resulting from the gradual population of the propagating modes in the d-wave superconductor that occurs while the excitation energy is raised.The anisotropy of the d-wave superconductivity reduces the Andreev reflection by orders of magnitude.Although the zero-bias conductance peak expected for the d-wave superconductors is thereby suppressed by the transverse confinement in the nanoribbons, transport resonances presumably caused by the surface Andreev bound states emerge in the vicinity of the zero bias.We have also shown that the suppression at the zero bias is influenced by the even-odd parity of the confined wavefunction.The excitation conductance spectra revealing these behaviors are fairly unaffected by the coupling strength at the GS interface.The dissimilar lattices for the normal conductor and the superconductor in the model is presumably responsible for the insensitivity.The specular Andreev reflection is blocked when the quasi-flatband states of symmetric zigzag GNRs are solely involved in the transport.The interband transition in the specular Andreev reflection brings the even-odd parity to act similar to the role in the valley-valve effect.The blocking is forbidden when an asymmetry made possible by the anisotropic superconductivity arises in the superconductor.

Figure 1 .
Figure 1.Tight-binding lattice models.Graphene nanoribbons having (a) zigzag and (b) armchair edges are attached to d-wave superconductors.The red dots indicate the C atoms of graphene.The square lattice sites for the superconductor are shown by the blue dots.The amplitude of the nearest-neighbor hopping shown by the black lines is assumed to be t throughout the composite system.The hopping amplitude across the GNR-superconductor interface shown by the green lines is t GS .The length of the C-C bonds in graphene is c.The lattice parameter of the square lattices is a.In the simulations, the number of the transverse sites was set to be 600 for the superconductor with periodic boundary condition.The transport probabilities when an electronlike excitation is incident are defined as shown in (c).

Figure 2 .
Figure 2. Variation of differential conductance with excitation energy ε.A zigzag GNR having nz = 50 is coupled to a d-wave superconductor with a strength of t GS = t.The inclination α of the d-wave anisotropy defined in (e) is 0 in (a) and (e), π/8 in (b) and (f) and π/4 in (c) and (g).The number of the propagating modes N in the GNR is 1 and 3 for the red and green curves with µ/t = 0.05 and 0.1, respectively.The change of N with µ is shown in the inset of (c).The number of the propagating modes Ns in the superconductor is shown in (d) and (h), where the colored curves correspond to α = 0 and the black curves to α = π/4.The magnitude of the superconducting energy gap is ∆ 0 = 0.01µ.

Figure 3 .
Figure 3. Excitation conductance spectra for conventional quantum wire.A quantum wire described by the square lattice having a width of W = 51a is terminated by the d-wave superconductor with α = (a) 0, (b) π/8 and (c) π/4.The coupling at the GS interface is varied as t GS /t = 1, 0.97 and 0.75 for the red, green and blue curves, respectively.The conductance is normalized by dividing it by the value at ε/∆ 0 = 10.Corresponding curves for the s-wave superconductor are shown in (d).Other parameters are k F W/π = 1.7 and ∆ 0 /µ = 0.01.

Figure 4 .
Figure 4.Effects of barrier at GS interface on excitation conductance spectra.The coupling between the zigzag GNR having nz = 50 and the d-wave superconductor is varied as t GS /t = 1, 0.75 and 0.1 for the red, green and blue curves, respectively.The conductance is normalized with respect to its value at ε/∆ 0 = 10.The inclination of the d-wave is α = 0 in (a) and (d), π/8 in (b) and (e) and π/4 in (c) and (f).The Fermi level is µ/t = 0.05 and 0.1 in the left-and right-hand-side columns, respectively, with the number of the occupied modes N in the GNR being 1 and 3.

Figure 5 .
Figure 5. Excitation conductance spectra for asymmetric zigzag GNR.The simulations are identical to those in figure 2 except that the width of the zigzag GNR is nz = 51.The inclination α of the d-wave is 0 in (a) and (c) and π/4 in (b) and (d).The number of the propagating modes in the GNR is 1 and 3 for the red and green curves with µ/t = 0.05 and 0.1, respectively.

Figure 6 .
Figure 6.Excitation conductance spectra for armchair GNR.The energy gap at the Dirac point is absent for the armchair GNR as na = 44.The inclination α of the d-wave is 0 in (a) and (d), π/8 in (b) and(e) and π/4 in (c) and (f).The number of the propagating modes N in the GNR is 1 and 3 for the red and green curves with µ/t = 0.04 and 0.08, respectively.The coupling strength at the GS interface is t GS = t.The change of N with µ is shown in the inset of (b).

Figure 7 .
Figure 7.Comparison of Andreev reflection probability R he in three GNRs.The red, green and blue curves correspond to zigzag GNRs with nz = 50 and 51 and an armchair GNR with na = 44, respectively.The parameters are as in the preceding figures of the corresponding GNRs with t GS = t.The dotted and solid curves correspond to α = 0 and π/4, respectively.The cases of the number of the occupied modes N being 1 and 3 are shown separately in (a) and (b), respectively.

Figure 8 .
Figure 8. Energy diagram for retro and specular Andreev reflections in graphene.For the assumption ∆ = 2µ, the holelike excitation generated from an electronlike excitation in the conduction band by the Andreev reflection appears in the conduction band for ε/∆ < 0.5 and in the valence band for ε/∆ > 0.5.The cases of the retro and specular reflections taking place for the excitation energies of εr and εs, respectively, are shown in red and blue.The gapless linear energy-dispersion relationship for the Dirac electrons is also illustrated, indicating the location of the Dirac point.

Figure 9 .
Figure 9. Variations of Andreev reflection probability R he with excitation energy ε for three types of GNRs.Zigzag GNRs having nz = 200 and 201 and an armchair GNR having na = 179 are attached to the d-wave superconductor in (a)-(c), respectively, with t GS = t.For the red, green and blue curves, α was set to be 0, π/8 and π/4, respectively.Retro and specular Andreev reflections take place for ε/∆ 0 < 0.5 and > 0.5, respectively, as the condition ∆ 0 = 2µ was assumed.Here, µ/t = 0.009 and 0.006 for the zigzag and armchair GNRs, respectively.The number of the occupied modes changes between Ne = 1 and 3 for the electronlike excitation at ε/∆ 0 = 0.76-0.77 in all the cases.