Andreev reflection in a patterned graphene nanoribbon superconducting heterojunction

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Andreev reflection in a patterned graphene nanoribbon superconducting heterojunction Chunxu Bai a,b,∗ , Yanling Yang c , Yongjin Jiang a , H.-X. Yang a,∗∗ a b c

Key Lab of Magnetic Materials and Devices, Ningbo Institute of Industrial Technology, Chinese Academy of Sciences, Ningbo, 315201, People’s Republic of China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang, 464000, People’s Republic of China School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 4 October 2018 Received in revised form 21 December 2018 Accepted 23 December 2018 Available online xxxx Communicated by R. Wu Keywords: Graphene nanoribbon Andreev reflection Tight-binding model

a b s t r a c t In the study, an improved superconducting heterojunction is made up of a zigzag graphene nanoribbon, which is patterned by a triangle and supports localized edge mode. Since all the localized edge modes stem from a pattern operation, the structure features of the pattern exert an enormous function on the coherent quantum transport. Especially, the patterned modes can enhance the Andreev reflection largely both in the ferromagnetic nanoribbon edge and the antiferromagnetic nanoribbon edge. The spin resolved zero bias conductances, in sharp contrast to its counterpart in the infinite width superconducting heterojunction, exhibit the different dependence on the patterned ferromagnetic interaction. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In recent years, two dimensional (2D) materials, exemplified by the well known graphene, exhibiting many exotic properties that are distinctly different from those of their bulk counterparts, have attracted intensive attention [1–5]. Making use of the promising electronic properties, such as, high carrier mobility and long spin diffusion length, graphene is considered to be a post silicon material for the future nanoelectronics and spintronics devices. From the theoretical point of view, the strictly 2D material-graphene can be regarded as a cornucopia of new physics. Owing to its Dirac like electronic spectrum, graphene has resulted in a plenty of unusual quantum relativistic phenomena, some of which are strenuous in high energy physics, can now be realized in a much cheap and simple table-top experiment. Besides those relativistic phenomena, there is currently increasing interest in graphene based superconducting heterojunction [1,6], especially in diagnosing the specular Andreev reflection (SAR) in graphene material [7–11]. Retro–Andreev reflection (RAR), a phase coherent two quasiparticles transport process in the subgap regime at the conventional normal metal superconducting heterojunction, was first revealed by Andreev in 1964 [12]. Forty two

*

Corresponding author at: Key Lab of Magnetic Materials and Devices, Ningbo Institute of Industrial Technology, Chinese Academy of Sciences, Ningbo, 315201, People’s Republic of China. Corresponding author. E-mail addresses: [email protected] (H.-X. Yang), [email protected] (C. Bai).

**

https://doi.org/10.1016/j.physleta.2018.12.038 0375-9601/© 2019 Elsevier B.V. All rights reserved.

years latter, SAR was first discovered in the graphene based superconducting heterojunction by Beenakker in 2006 [7]. In contrast to the conventional intraband RAR, SAR is an interband tunneling process where the incident electron and the reflected hole stem from the conduction band and the valence band, respectively, or reversely. The idea for detecting the SAR has been extended to the ferromagnet (nonferromagnet)/superconductor two terminal or three terminal heterojunction. Using the ferromagnetic exchange interaction in the ferromagnet lead to distinguish the information of SAR was first proposed in Ref. [8] in the ferromagnet/superconductor two terminal heterojunction structure. Moreover, by surveying the magnetoresistance and the conductance, the SAR can be tuned by external bias voltage in a two terminal ferromagnet/superconductor double heterojunction [9]. Additionally, the first proposal to use the nonlocal conductance and the shot noise cross correlations to distinguish the SAR was given in Ref. [10] where Benjamin et al. showed that normal/insulator/superconductor three terminal heterojunction can be used to test SAR process. Very recently, Yang et al. theoretically proposed that SAR can be experimentally diagnosed by the spin orbit interaction in a single layer graphene van der Waals heterojunction [11]. Exhilaratingly, in experiment, a breakthrough progress has been steadily made in an unprecedentedly clean bilayer graphene based superconducting heterojunction where a smoking gun for the SAR has been observed [13]. On the other hand, the surge of interest in testing the SAR in single layer graphene has not realized up to now [14,15]. On the other hand, in ambient condition, graphene nanoribbon with defined edges is proposed as a promising platform for the realization of the future valleytronics, nanoelectronics, and spin elec-

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tronics [16–18]. To the practical application of valleytronics, several proposals in a variety of systems, such as the external strain and magnetic field, have been reported [19–23]. However, with the time reverse symmetry, the valley freedom would make no sense on the superconducting heterojunctions. Meanwhile, to investigate the SAR in the graphene nanoribbon, many authors have proposed a variety of superconducting heterojunctions [24–26]. It is already well known that pseudoparity, a quantum number characterizing the graphene nanoribbon eigenstate (depends on whether the number of sites along the transverse direction is even or odd), can lead to a fatal impact on the RAR and SAR of the graphene nanoribbon/superconductor heterojunction [27–29]. It is shown that, for the even zigzag graphene nanoribbon, the injected electron like quasiparticle has different pseudoparity from that of the reflected hole like quasiparticle, thereby the Andreev reflection (AR) is prohibited; i.e., the two electrons with different parity cannot compose a Cooper pair tunneling into the superconducting lead. While, for the odd case, the injected electron like quasiparticle and the reflected hole like quasiparticle have the same pseudoparity, thus the AR is allowed [27]. The results of Ref. [27] have been extended in a magnetized graphene nanoribbon/superconductor heterojunction by considering both the edge magnetization and pseudoparity effect [30]. Apart from unique features of pseudoparity transport, graphene nanoribbon is also interesting from the point of view of the usage as a setup for detecting the SAR. Cheng et al. [24] found that the RAR and SAR can be effectively tuned by the phase difference of the two superconductor terminals in the four terminal graphene nanoribbon superconducting heterojunction. Furthermore, a graphene nanoribbon ring has also been suggested as a new structure to provide a clear signature to distinguish between the RAR and the SAR [25]. All those make graphene nanoribbon very attractive for both fundamental and applied research, and many theoretical and experimental works on graphene nanoribbon have been reported in the past few years [31–37]. In addition to the pristine graphene nanoribbon, there is currently a great interest in patterned graphene nanoribbon structure, which followed by the interplay of spin, charge, and valley transport [38–41]. Of particular interest is the localized edge mode along the patterned edge [40–42], which can significantly modify the electronic spectrum and especially the transport properties. Yang et al. also proposed that the ground state is strongly dependent on the patterned shape and size [43]. For a triangular shaped pattern, significant intrinsic ferromagnetic exchange splitting interaction can be obtained for a large patterned structure. To our best knowledge, transport properties in patterned graphene nanoribbon superconducting heterojunction are not investigated yet. Since the patterned handle is sufficiently important and also sensitive to the electronic structure, in this paper we investigate the spin-resolved transport properties of a patterned zigzag graphene nanoribbon superconducting heterojunction with and without the edge ferromagnetic interaction. The effect of the localized patterned ferromagnetic mode on the process of AR is extensively investigated. As we all know, the ground state in a narrow zigzag graphene nanoribbon is antiferromagnetic but ferromagnetic for a wide zigzag graphene nanoribbon [44], so we consider both ferromagnetic and antiferromagnetic edge state in our study. Numerical results show that both the energy band and the transport properties can be remarkably modulated in the nanoribbon with different patterned sizes (no matter which ground state is applied). As for the spin resolved zero bias conductances, they exhibit the different dependence on the intrinsic patterned edge ferromagnetic interaction. A robust distinct dependence feature singular point can be found when the Fermi energy equates to the edge ferromagnetic interaction, either the ferromagnetic edge or the antiferromagnetic edge. This different dependence appears due to the change of subgap electronic structure in the patterned nanorib-

Fig. 1. Sketch of the patterned zigzag graphene nanoribbon superconducting heterojunction. The left lead is a semi-infinite zigzag graphene nanoribbon lead. The central lead is the central scatting region circled by the red dotted rectangle. The blue dashed triangular denotes the triangular-shaped patterned structure. The right lead represents a semi-infinite metal superconducting lead without the honeycomb lattice structure. P top denotes the top patterned zigzag site line respect to the top zigzag edge. P down denotes the bottom patterned zigzag site line respect to the bottom zigzag edge. N and L represent the width and length of the central scatting region, respectively. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

bon. In contrast to the proximity ferromagnetic effect, the localized edge ferromagnetic interaction is intrinsic and can be tuned precisely by the special patterned structure. Therefore, the present two-terminal structure with a high quality patterned zigzag edge graphene nanoribbon gives us an effective experimental setup to study the localized edge ferromagnetic mode and the spin-resolved transport properties. The rest of the paper is organized as follows: In Sec. 2 we briefly introduce the tight binding model theory and the specific structure which is used to calculate the transport properties of our patterned samples. The results of numerical calculation and discussion are given in Sec. 3. Finally, in Sec. 4 we give a brief summary. 2. Model and methods In this study, we mainly focus on the transport properties through a zigzag graphene nanoribbon superconducting heterojunction with the triangular-shaped patterned structure. The geometry sketch of the heterojunction is shown in Fig. 1. The growth direction is along the x axis. The semi-infinite zigzag graphene nanoribbon lead extends from x = −∞ to x = 0, the central patterned region, circled by the red dotted rectangle, extends from x = 0 to x = L, and the semi-infinite superconducting lead occupies x > L. Note that, the central transitional region in the red dotted rectangle is termed as the scattering region where the patterned operation is applied on the lattices. Here the superconducting lead can be realized by a normal metal superconducting lead, and it is not necessary to have the same honeycomb lattice structure as graphene. It means that we assume the graphene nanoribbon is directly coupled to a normal metal superconductor lead. In experiment, the Al and Pb leads were applied at initial stage [45–47], and recently the layered dichalcogenide NbSe2 was selected as the superconductor material [48–51]. In the central lead region, the integer number P top denotes the top patterned zigzag site counting from the top zigzag edge. While the integer number P down denotes the bottom patterned zigzag site counting from the bottom zigzag edge. The patterned site number of the bottom zigzag edge can be counted in the following form: ( N − P top − P down ) × 2 + 1. By setting the integer number P top and P down , we can directly modulate the triangular-shaped patterned size. The integer number N and L represent the width and the length of the central scattering region, respectively. Throughout this study, we restrict ourselves to a single particle picture and overlook the electron–electron interaction effect.

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The low energy excitation quasiparticles propagation in the present zigzag graphene nanoribbon superconducting heterojunction can be described by the following tight binding Hamiltonian [52]

H = H GN + H S + H T

(1)

where H GN , H S and H T are the tight binding Hamiltonians of the graphene nanoribbon region, normal metal superconducting lead and tunneling of the graphene nanoribbon and the semi-infinite normal metal superconducting lead, respectively. For the graphene nanoribbon (both the left lead and the central lead), H GN can be given as [52–55]

H GN =

  (εi + hi ,σ )a+ a −t a+ a i ,σ i ,σ i ,σ j ,σ

(2)

i , j ,σ

i ,σ

where a+ (ai ,σ ) is the creation (annihilation) operator at the site i ,σ i with spin σ (σ =↑ or ↓), εi is the on-site energy in the graphene nanoribbon and equals to zero for the undoped graphene, which can be controlled experimentally by the gate voltage and the doping. h i ,σ denotes the edge magnetism effect which is pinned along the zigzag graphene nanoribbon edges. For a ferromagnetism case, the two edge magnetism equates to each other. While the two edge magnetism strengths equate to each other but antiparallel in the antiferromagnetism configuration. t is the nearest-neighbor hopping integral and assumed to be all the same in the zigzag graphene nanoribbon lattice. i , j  denotes the summation over the nearest neighbor graphene nanoribbon lattice sites. The second term H S describes the semi-infinite normal metal superconducting lead. Here we assume the superconducting lead has a very good contact with the graphene nanoribbon because it can be realized easily in experiment [13–15,48–51]. In general, a conventional s-wave metal superconducting lead is employed, such as Al, and NbSe2 etc. [45–51]. The semi-infinite superconducting lead thus does not have a honeycomb structure and can be described by a continuum model in the following form [52]

HS =

 k,σ

   εk b+ b +  b+ b+ + b−k↓ bk↑ kσ kσ k↑ −k↓

(3)

k

where b+  and bkσ are the creation and the annihilation operators kσ

with spin σ and momentum k = (kx , k y ) in the normal metal superconducting lead. εk and  denote the on-site energy in superconducting lead and the real superconducting pair potential (the macroscopic superconducting phase is neglected because a single superconducting lead has been considered), respectively. The Hamiltonian H T is the tunneling between the graphene nanoribbon and the semi-infinite normal metal superconducting lead and can be given as [52]

HT =



t T a+ b ( y i ) + H .c . i ,σ i ,σ

(4)

i ,σ

where t T is the hopping term connecting the graphene nanoribbon and the metal superconducting lead. We set t T = t and assume that the spin is conserved when quasiparticles tunnel between the two different leads for the simple. Though the Hamiltonian H S is given by a continuum model, the lattice version b i ,σ ( y i ) can be obtained by the operator bk,σ via a Fourier transform b i ,σ ( y i ) = 1 2π



e ik y y bkσ [52]. k The spin-resolved current flowing through the present superconducting heterojunction can be obtained by using the Heisenberg equation of motion I σ = e ∂ ∂Ntσ = ieh¯ [ N σ , H ] where N σ is the total number operator at the surface sites of the graphene nanoribbon. By a dull and a directly algebraic operation, the spin-resolved conductance thus can be given as [30,52]:

Gσ =

2e 2 h



3



T r ΓLσ G r Γ R G a

  11

+

e2 h



T r Γ L σ G r12σ Γ L σ¯ G a21σ



(5)

where the first term T A = T r [Γ L σ [G r Γ R G a ]11 ] is the AR coefficient and the second term T = T r [Γ L σ G r12σ Γ L σ¯ G a21σ ] is the normal tunneling coefficient. Note that, the subscript 1 and 2 denote the electron and the hole component of the Numbu space, while σ denotes the spin component of the quasiparticles. In particular, the former dominates in the superconducting gap |eV | <  while the latter mainly contributes to the conductance when |eV | > . Γ L ( R )σ is the linewidth function and given by Γ L ( R )σ = i (Σ Lr ( R )σ − Σ La( R )σ ). Σ r and Σ a is the retarded and advanced self-energy of the two semiinfinite lead. The retarded self-energy of the left lead + is related by Σ Lr σ = T GN g 0 T GN which can be obtained by the recur0 sive Green method. g is the surface Green function of the semiinfinite graphene lead, that can be numerically calculated [52]. T GN denotes the hopping matrix between the two adjacent longitudinal chains in the graphene nanoribbon. The advanced self-energy Σ a can be obtained by a hermitian conjugate operation Σ a = (Σ r )+ . As the self-energy of the right semi-infinite metal superconducting lead, Σ Rr can be obtained straightforwardly as in the follow 1 / E ing form [52]: Σ Rr = −i πρ |t T |2 J 0 (k F ( y i − y j ))β( E ) / E 1 where ρ is the constant density of state in the energy space, J 0 is the first-kind

√ Bessel function, k F is the Fermi wavevector, | E |/ √E 2 − 2 , | E | > . and β( E ) = G r and G a are the retarded − i E / 2 − E 2 , | E | < . and the advanced Green function of the central scattering region, which can be obtained directly by using the Dyson equation, G r = [ E I − H central − Σ Lr − Σ Rr ], where I is a unit matrix and H central is the Hamiltonian of the scattering region described by H GN . Similar to the self-energy, G a = (G r )+ . Through the above formulas, the tunneling coefficients for the present superconducting heterojunction can be obtained easily by the numerical calculations. 3. Results and discussions In the numerical calculations, the hopping energy is taken as t = t T = 2.75 eV, the nearest-neighbor carbon–carbon distance is set as a0 = 0.142 nm in a graphene nanoribbon lead, the Fermi wave vector in the metal superconducting lead is set as k F = 1 Å−1 , and the s-wave pair superconducting potential is set as  = 1 meV [52]. Note that, we always set the on site energy εi = 0 in the intrinsic graphene nanoribbon case (unless otherwise specified). 3.1. Nonmagnetic case Here we first investigate the case of the nonmagnetic zigzag graphene nanoribbon superconducting heterojunction. In this case, the new features mainly stem from the local patterned edge states. The reason is that the patterned sizes can induce a remarkable modulation on the energy band in the central nanoribbon lead, thereby improving electrical transport properties in the present junction. In Fig. 2, the Andreev reflection (AR) coefficients and the normal tunneling (NT) coefficients are plotted. We can see that the AR dominates in the superconducting gap |eV | <  while the NT mainly contributes to the conductance at |eV | > . In Fig. 2, the top panel and the bottom panel correspond to the cases of the even and the odd width of the nonmagnetic graphene nanoribbon lead, respectively. And these figures correspond to the AR (a and c) and the NT (b and d), respectively. In the absence of the patterned effect, the results in Fig. 2 (the solid lines) are in good agreement with theoretical predictions in Refs. [27,30]. At superconducting subgap regime, conservation of pseudoparity gives rise to a zero AR when the graphene nanoribbon has an even number

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Fig. 2. The AR (a and c) and NT (b and d) for the superconducting heterojunction as a function of the incident energy E for the different patterned sizes. The parameters used in the calculation are shown in the figure.

of sites in the transverse direction and a non-zero AR when the graphene nanoribbon has an odd number of sites. For the case of the even width (Fig. 2(a)), as the patterned size increases from zero, it is clear to see that the AR demonstrates a huge enhancement at first and then yields an oscillating characteristic. For the even case, due to different pseudoparity of the injected electron and the reflected hole, the AR is prohibited [27]. The intriguing features suggest that the conservation of pseudoparity is destroyed by the patterned operation and the subgap energy band also can be effectively tuned by the patterned operation. Especially, the AR can survive even at a very large patterned size. For the case of the odd width (Fig. 2(c)), the AR is more sensitive to the patterned size as compared to that for the even width case, i.e., the AR decays quickly to zero as the patterned size increases. Moreover, in contrast to the even width case where the dominating peaks show a non-monotonous feature with the patterned size, it is very clear to see that the dominating peaks shrink and move towards to the superconducting gap edge E = ±. Those features all can be ascribed to the evolution of the band structure by the patterned operation. To interpret those phenomena, in Fig. 3, the energy band structure of the quasiparticles in nonmagnetic patterned graphene nanoribbon is plotted. Though the central region is a single unit in the present system, we suppose a period structure with the central region as a unit and give an energy band structure of the quasiparticle in nonmagnetic patterned graphene nanoribbon. Based on the hypothesis, we can find a direct and clear physical picture of the features. For the even case in Fig. 3(a) and (b), a small patterned size, i.e., P top = 2 and P down = 7, has destroyed the zero energy zigzag edge states and gives rise to a slight energy gap around the Fermi energy. Physically, the patterned structure in the graphene nanoribbon may induce the local edge states around the patterned structure. Through the local edge states, the top and the bottom zigzag edge states interact with each other and an energy gap can be given. On the other hand, the pseudoparity of the quasiparticle has been also destroyed by the patterned operation. The AR thus is permitted almost in the intact superconducting gap regime except the induced energy gap regime. However, the effect of the patterned operation in the even graphene nanoribbon case is non-monotonous. In the case of P top = 2 and P down = 4, it

Fig. 3. Two energy levels of the nonmagnetic patterned graphene nanoribbon near Fermi energy as a function of the wave vector ka0 . The black line (E < 0) denotes the valance band, while the red line (E > 0) stands for the conduct band. ((a) and (b)) N = 10 for the even graphene nanoribbon and ((c) and (d)) N = 11 for the odd graphene nanoribbon. The other parameters used in the calculation are shown in the figure.

is clearly shown that the perfect zero energy edge states are reforming again. Thereby the AR can emerge in the superconducting gap and even give rise to a slight peak at the Fermi energy. In contrast to that for the even graphene nanoribbon case, a monotonous effect of the patterned operation is found in the odd graphene nanoribbon, as shown in Fig. 3(c) and (d). The allowed states in the superconducting gap shrink sharply by the patterned size. Note that the suppression effect is monotonous and rapid, no allowed states in the superconducting gap exist just at P top = 2 and P down = 6. Thereby, the AR is prohibited rapidly with respect to the patterned size. 3.2. Antiferromagnetic case In last section where we focus on a nonmagnetic structure case, the obtained results suggest a considerable role of the local patterned edge states on the tunneling coefficient. In the following, we will turn to the magnetic effect. In physics, edge magnetism has been predicted theoretically and observed experimentally in zigzag graphene nanoribbon [56–61]. Here we first study the magnetic effect of a graphene nanoribbon with antiferromagnetic zigzag edges, as shown in Fig. 4. Note that hedge and hpattern denote the edge magnetism strength and the patterned magnetism strength, respectively. It should be pointed that we just only give the results of a spin up incident quasiparticle in the figure. The results of a spin down incident quasiparticle just symmetrical to that for the spin up with respective to E = 0. The AR coefficients T A are shown in Fig. 4 for both the even and the odd graphene nanoribbon. As expected, T A exhibits a tunneling gap around the Fermi energy for both cases due to the antiferromagnetic insulating band gap. In comparison with the nonmagnetic case, an asymmetry structure of T A arise by the antiferromagnetic zigzag edge states. Those phenomena are found a good agreement with the early study in a perfect zigzag graphene nanoribbon [27,30]. For the even case (N = 12), similar to the above, the local patterned edge states give rise to a non-monotonous feature with the patterned size. While for the ferromagnetic local patterned edge states, a ferromagnetic resolved tunneling feature can be obtained. Clearly, the position of the peak shows a significant dependence on the hpattern in the superconducting subgap regime, i.e., the position

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Fig. 5. The density of state in the central scattering region with antiferromagnetic zigzag edge for the even graphene nanoribbon case (a and c) and the odd graphene nanoribbon case (b and d). The parameters used in the calculation are the same as the Fig. 4. Fig. 4. The AR for the antiferromagnetic zigzag edge superconducting heterojunction as a function of the incident energy E for the even graphene nanoribbon case (a and c) and the odd graphene nanoribbon case (b and d). The parameters used in the calculation are shown in the figure.

can be tuned largely by the local patterned ferromagnetic edge states. Since the modulation increases with the field hpattern , a large net ferromagnetic effect around the patterned edge may be expected to arise a considerable shift in the experiment. For the odd case (N = 13), with increasing the patterned size as above to the fixed hedge , we also find that the SAR coefficients T A monotonously decrease. When the patterned size reaches P top = 4 and P down = 6, the AR coefficients T A decay to a negligible value in the superconducting subgap regime except the superconducting gap edge. The interesting thing is that the tunneling peak at the superconducting gap edge can also be tuned by the local patterned ferromagnetic edge state. However, this local patterned ferromagnetic edge state just modulates the height of the peak and has no effect on the position of the peak. Physically, the net ferromagnetic effect around the patterned edge is in proportion to the pattern size. To get a large value of hpattern , we may employ a large enough patterned structure in the experiment. Nevertheless, a large enough patterned structure yields a negligible T A coefficient in the superconducting subgap regime. Thus it is suggested that the effect of the local patterned ferromagnetic edge state may not suit for observation in the odd graphene nanoribbon case. The presence of those tunneling phenomena is confirmed by the calculation of the density of state in the central region, as shown in Fig. 5. This density of state is only weakly sensitive to the patterned size for the case of an even graphene nanoribbon, as local patterned edge state leads to the slight oscillation of the allowed states. While, in an odd graphene nanoribbon, the density of state exhibits an exponential decay characteristics with the patterned size, thereby the AR coefficients. Those behaviors are shown in Fig. 5(a) and (b). When the magnetic effect in the local patterned edge states becomes open, the density of state in superconducting subgap regime is considerably shifted, though it still keeps at the specified incident energy, as shown in Fig. 5(c) (in an even graphene nanoribbon). This shift effect increases with increasing hpattern (the pattern size increases), and for a sufficiently large value of hpattern the shift effect becomes totally suppressed due to the antiferromagnetic insulating band gap and the superconducting gap edge. In Fig. 5(d) where the density of state in an odd graphene nanoribbon is plotted, a slight value change can be seen at the superconducting gap edge (it remains the same in the superconducting subgap

regime with the variety of hpattern ), so that a weak change of the strength of the AR at the superconducting gap edge is allowed. The results explain the shift effect and the modulation effect of the peak of the tunneling coefficient when the magnetic effect in the local patterned edge states is modulated, and offer an explanation for the dependence of the pattern operation on the transport properties in a zigzag graphene nanoribbon superconducting heterojunction. 3.3. Ferromagnetic case In general, the antiferromagnetic ground-state orders mean that the magnetic coupling between opposite nanoribbon edges is antiferromagnetic, while the magnetic coupling along each of the two zigzag edges of graphene nanoribbon are ferromagnetic. However, the inter-edge superexchange interaction of such antiferromagnetic states in zigzag graphene nanoribbon rapidly weakens ∼ W −2 as the ribbon-width W increases [62]. Moreover, upon increasing the ribbon width, a semiconductor (the antiferromagnetic edge states)-to-metal (the ferromagnetic edge states) transition is revealed, even at room temperature [44]. We thus now proceed to study the case of the ferromagnetic zigzag edge of the present junction. In this section, we will also consider the influence of the patterned operation (both the induced exchange field hpattern and the local edge state) on the density of state and the transport properties. In Fig. 6, the AR coefficients T A are plotted, and the AR can be enhanced in the whole superconducting subgap regime for the even graphene nanoribbon case in comparison with Fig. 4. While T A for the odd graphene nanoribbon case has more difference from that of the antiferromagnetic junction in Fig. 4(b), the AR is allowed in the superconducting subgap regime even at a very large patterned size ( P top = 4 and P down = 4). As for the effect of hpattern , apart from the peak at the superconducting gap edge, the tunneling peak shifts are also found in the odd graphene nanoribbon case, though they all show negligible shifts in Fig. 6(d). Such peaks are absent in the antiferromagnetic junction (Fig. 4(d)). However, T A exhibits a very similar feature as that of the even graphene nanoribbon superconducting heterojunction with an antiferromagnetic zigzag edge. The results presented above are for the graphene nanoribbon with the ferromagnetic zigzag edge. Since the transport properties for quasiparticles in the superconducting subgap regime are strongly determined by the density of state, one may expect that these novel transport characteristics also can be elucidated by the density of state in the central scattering region. Indeed, this is

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zigzag edge case, the density of state exhibits asymmetry and oscillation characteristics with respect to the patterned size in an odd graphene nanoribbon, thereby the AR tunneling coefficient T A . Turn on the magnetic effect of the local pattern edge state, and it is clearly seen that the density of state has a peak shifts with respect to hpattern in the odd graphene nanoribbon. On the other hand, something similar also holds for the case of ferromagnetic zigzag edge to that of the antiferromagnetic zigzag edge case. As a result, the features of the density of state give the novel tunneling properties a sound elucidation. 3.4. Zero bias case

Fig. 6. The AR for the ferromagnetic zigzag edge superconducting heterojunction as a function of the incident energy E for the even graphene nanoribbon case (a and c) and the odd graphene nanoribbon case (b and d). The parameters used in the calculation are shown in the figure.

Fig. 7. The density of state in the central scattering region with ferromagnetic zigzag edge for the even graphene nanoribbon case (a and c) and the odd graphene nanoribbon case (b and d). The parameters used in the calculation are the same as the Fig. 6.

the case as shown in Fig. 7, where the density of state is plotted (N = 12 is for even and N = 13 for odd). This figure reveals a nonzero density of state through the whole superconducting subgap regime. Such a feature is absent in the antiferromagnetic zigzag edge case where the density of state becomes totally suppressed at the energy interval [−hpattern , hpattern ] around the Fermi level (Fig. 4). Importantly, in contrast to the antiferromagnetic

In the last, we consider the influence of Fermi energy in the graphene nanoribbon region. To grasp the essential physics, we assume a zero bias voltage is employed through the present superconducting heterojunction. Fig. 8 shows the zero bias tunneling coefficient T A versus the induced magnetic effect hpattern and the Fermi energy εi . The parameters used in the calculation are N = 10, L = 11, P top = 4, P down = 3, and hedge = 0.20 . From Fig. 8(a), it is found that T A is completely suppressed in [0, hedge ), but yields a considerable value beyond hedge and then also suppressed for larger εi . Besides, at εi = hedge , the zero bias tunneling coefficient T A displays a decrease characteristic with respect to the induced magnetic effect hpattern . This behavior has been well elucidated in the conventional superconducting heterojunction with AR [63]. Beyond hedge , T A increases with increasing hpattern for AR, which is well consistent with the former infinite width superconducting heterojunction [8]. While, as compared with the case of the former infinite width superconducting heterojunction [8], a complete different feature of the spin-resolved zero-bias conductances can be found in the regime [0, hedge ]. The AR processes can find a physical background in Fig. 9. The field hedge opens a gap around the Fermi level in the spectrum, and the width of this gap increases linearly with the field. As seen in Fig. 9, there are allowed local pattern edge states in the spin up subband. However, no allowed states are alive in the energy interval [0, hedge ). As a result, the AR process shuts down since a s-wave superconducting potential is applied. Note that the local pattern edge state of spin down electron is lifted up by the field hpattern . In this manner, an electron with spin up in the conduction band combining a spin down electron in the valance band jumps into the superconducting lead, and gives rise to a spin up resolved AR. At εi = hedge , the AR is allowed for both spin up and spin down electrons, as the two species band edge just align. For the ferromagnetic zigzag edge case, our data indicates a very similar result as the above. But in the energy interval [0, hedge ), since the allowed states are alive in the spin down band, as seen in Fig. 10(b), the zero bias tunneling coefficient T A of spin down resolved AR survives and decreases with increasing hpattern . Therefore, we can find a clear criterion for distinguishing the spin-resolved AR conductance based on the zero bias transport properties in the present patterned zigzag graphene nanoribbon superconducting heterojunction.

Fig. 8. The zero bias tunneling coefficient T A in the present heterojunction with the antiferromagnetic zigzag edge (a) and the ferromagnetic zigzag edge (b). The parameters used in the calculation are shown in the figure.

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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11504005 and 11474255). This work was also partially supported by the 1000 Talents Program for Young Scientists of China and Ningbo 3315 Program. C.B. also acknowledges partial support by University Talents’ Science and Technology Innovation Planning Project of Henan Province, China (No. 16HASTIT045) and Nanhu Scholars Program for Young Scholars of XYNU. References

Fig. 9. The energy levels of the antiferromagnetic zigzag edge patterned graphene nanoribbon as a function of the wave vector ka0 . ((a) and (c)) for the spin up and ((b) and (d)) for spin down. The other parameters used in the calculation are the same as the Fig. 8(a).

Fig. 10. The energy levels of the ferromagnetic zigzag edge patterned graphene nanoribbon as a function of the wave vector ka0 . ((a) and (c)) for the spin up and ((b) and (d)) for spin down. The other parameters used in the calculation are the same as the Fig. 8(b).

4. Conclusions Based on the tight-binding model and the Green function method, the parity-resolved transport properties through a patterned graphene superconducting heterojunction with a straight zigzag edge have been investigated. Our results show that, the energy band and the intrinsic magnetic properties can be easily tuned by the patterned size and the Fermi energy. Consequently, the AR and transport properties not only can be modulated by the Fermi energy, but also by the patterned structure of the nanoribbon. Thus, we suggest that the proposed pattern operation can be used to design spin-resolved quantum device based on a zigzag graphene nanoribbon.

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