Superconducting pump manipulated by non-topologically quasi and topological interface states

We propose an adiabatic superconducting charge pump based on massive Dirac electrons. A superconductor is sandwiched in between two pumping sources, which are formed by introducing the time-dependent and out-of-phase staggered potentials in graphene as pumping parameters. The pump is shown to be characterized by not only the topological interface state (TIS) but also the non-topologically quasi interface state (QIS). Hereafter, our attention is focused on the pumping currents ILNR and ILAR from the normal and Andreev reflections, respectively, which predominate by making the electron energy reside in the effective energy gap. It is found that modulating the energy E, superconductor length L 0, and pumping source length LP results in the considerable variation of competitive behaviors between ILNR and ILAR . In particular, the reversal effect of current direction can be realized by tuning LP . More interestingly, the current-phase relationship exhibits the platform behaviors, which can be manipulated by LP and the pumping strength. All the above pumping properties are attributed to the adiabatic evolution of TIS and non-topologically QIS, particularly the conversion between each other is the crucial origin. We also obtain the quantized pumping current by adjusting LP and L 0, and present the corresponding qualitative explanation through the pumping contour circled by the two parameters. In addition, we discuss the features of pumping current based on the armchair graphene as well.


Introduction
During the past decades, quantum charge and spin pumps have been studied extensively in solid-state systems both at the theoretical [1][2][3][4][5] as well as experimental [6][7][8][9][10] levels with focus on both the adiabatic and nonadiabatic regimes.This is mainly because of its potential application for realizing novel current standards but also for characterizing many-body systems [11,12].Adiabatic quantum pump is a transport mechanism, in which low-frequency periodic modulations of two or more system parameters [13,14] with a phase difference trigger a zero bias finite dc current in meso-and nano-scale systems.This is a consequence of the time variation of the parameters of the quantum system, which clearly breaks time-reversal symmetry [1,2].Such pumping current is proved to be proportional to the geometric area circled by the timedependent parameters [15,16].Compared with nonadiabatic quantum pump, adiabatic quantum pump is of more stable and efficient quantum transportation.It has been also explored in quantum dot structure [17], Rashba nanowire [18], and Dirac systems like graphene [19][20][21][22] and topological insulator [23][24][25][26][27].
The possible quantization of pumped charge during a cycle through noninteracting open quantum systems has been investigated.It is one of the central goals in this research area, because it is expected to revolutionize the electrical metrology [3,[28][29][30][31][32].In the most celebrated quantized pump, the Thouless topological pump (TTP) [13] for the noninteracting electron system, integral charges in a cycle can be pumped out by a one-dimensional (1D) moving potential.Such a pump is topological and the pumped charge is equal to the topological invariant of the system.It has been observed in 1D optical superlattice systems [33,34].For the traditional two-parameter charge pump, a quantized charge pump based on massive Dirac electrons was proposed.Two time-dependent and out-of-phase staggered potentials introduced in zigzag graphene act as pumping parameters [35].This pump has a topological origin, the adiabatic evolution of the topologically-protected interface state (TIS) [36,37] bridging between the two pumping sources, which is formally different from the TTP mechanism.
Quantum charge pumps, using a variety of setups involving superconductors (Ss) [38][39][40][41][42][43][44], have also been of major interest in recent years.A quantized Majorana pump in semiconductor-S heterostructures [45] and half-integer quantized charge pumping induced by a Majorana fermion [46], were respectively realized, which are both based on topological insulators.Quantum charge pump through a superconducting double barrier structure in graphene [19] and a superconducting hybrid junction of silicene [47] were also respectively obtained.However, the focuses of these works are mainly on resonances of the crossed Andreev reflection (CAR) with an incident electron transmitted as a hole or/and electron transmission, particularly for the Dirac systems like graphene and silicene.Motivated by this fact, in this paper we consider a quantum pump of graphene-based normal (N)-S-N (NSN) hybrid structure, as depicted in figure 1.Two pumping sources are formed by introducing the time-dependent and out-of-phase staggered potentials in the N region as pumping parameters.Due to the proximity effect, the superconductivity is induced by directly depositing a bulk S on the graphene in the S region.Only the Andreev reflection (AR) with a reflected hole and normal reflection (NR) with a reflected electron are involved.To the best knowledge, so far, there has been no report about the NSN pumping structure based on graphene or other materials.In particular, we want to know what happens to the adiabatic evolution of the TISs bridging between the two pumping sources in the presence of the electron-hole conversions at the interfaces.It follows how the superconducting pump is manipulated.This is another main motivation of this paper.
Therefore, we study a new scheme to realize a superconducting charge pump in the zigzag and armchair graphene NSN junctions by taking two time-dependent and out-of-phase staggered potentials separated by an S as pumping parameters.Such staggered potential has not been introduced in the previous superconducting charge pumps.Combining the BPT formula [14] and lattice Green's function technique, we have obtained the scattering matrix by the Fisher-Lee [48] relation to calculate the pumping current.By making the electron energy reside in the effective energy gap, I NR L and I AR L from the NR and AR, respectively, predominate.Thus, this is different from the pumping current only from the CAR or/and electron transmission.It is shown that I NR L and I AR L versus the S length L 0 exhibit different oscillatory behaviors.Particularly, we can realize the reversal effect of current direction through modulating the competitive relationship of I NR L and I AR L by the length L P of the pumping source with the staggered potential.More interestingly, the current-phase relationship severely deviates from the sine behavior and exhibits the platform behaviors, which can be tuned by L P and the pumping strength.The current remains constant in some range of phase shift, which is called the platform behavior.The above pumping properties stem from the adiabatic evolution and conversion of the TIS and non-topologically QIS.The former is due to that two staggered potentials produce the two neighboring different quantum phases, while the latter originates from that the electron-hole conversion at N/S interfaces brings about the quantum interference.Furthermore, we obtain the quantized pumping current by adjusting the length L P and L 0 , and display the corresponding qualitative explanations through the pumping contour circled by the two parameters.Finally, we also consider a superconducting pump based on armchair graphene.It is not involved in previous works on pumping devices.Different characteristics are demonstrated between the zigzag and armchair graphene pumping models.

Formalism and theory
Consider a typical two-parameter pump model based on the graphene NSN junction as schematically shown in figure 1.The staggered lattice potentials ∆ 1 and ∆ 2 are introduced into the pristine graphene with length L p as two pumping potentials, forming two pumping sources.A bulk S with the length L 0 is deposited on the graphene between the two pumping sources.The left and right leads are connected to electrodes without any external bias.The lattice-version Hamiltonian of the above setup is given by Figure 1.A schematic sketch of the present graphene NSN setup, where N and S stand for the pumping souce region with the length LP and superconducting region with length L0, respectively.The pumping current direction is assumed along the x-axis and the transverse direction (the y-axis) is assumed homogeneous.The left and right leads are connected to two unbiased electrodes. with where H 0 describes the pristine graphene regions with t n the hopping energy of electrons between nearestneighboring sites ⟨ij ⟩ and H S depicts the superconducting region of graphene layer induced by the proximity with a nonzero pairing potential ∆e iϕ (ϕ is the superconducting phase) and an electrostatic potential U S .V(t) is the time-dependent term induced by two staggered potentials ∆ 1(2) (t), where µ i = +(−)1 for the ith site represents the carbon A(B) atom, ∆ 1 = ∆ 0 cos(ωt) and ∆ 2 = ∆ 0 cos(ωt + φ) are homogeneously introduced into graphene with the pumping strength ∆ 0 , pumping phase shift φ, and pumping frequency ω.
In order to study the pumping current in the present setup, we need to calculate the scattering matrix of the junction firstly.The Fisher-Lee relation is a formula that connects the scattering matrix and the Green's function.Thus, based on the lattice Green's function technique, the scattering matrix can be obtained by the Fisher-Lee relation [48], Here, the matrix S αβ qp denotes the scattering amplitude of the process for an incident particle β from lead p scattered into a particle α in lead q with α and β labelling the electron e or hole h as well as p and q representing the left L or right lead R. The Γ α(β) p(q) is a block of the line-width function Γ p(q) = i(Σ r p(q) − Σ a p(q) ), where the retarded/advanced self-energy function Σ r/a p(q) can be calculated numerically by the recursive method.
is the instantaneous retarded Green's function of the two-terminal setup.For adiabatic charge pump, assuming zero temperature of environment (0K) and adiabatic limit (ω → 0), the pumping current based on the BPT formula in a period T = 2π /ω is [14] where r ee , r he , t ee , t he correspond to the scattering amplitudes of the NR, AR, electron transmission, and CAR processes, respectively, which are 2 × 2 matrices obtained from equation (5).

Results and discussions
In calculation, we take the hopping energy t n = 1 as the energy unit since it is a relatively stable constant (about 2.7 eV).Moreover, assuming the Fermi energy E F = 0 leads to the specular AR, where the incident electron and the reflected hole are in different energy bands.
If the wave vector of electrons or holes in the pumping region is imaginary (E 2 < ∆ 2 1 or E 2 < ∆ 2 2 ), the probabilities of transmission and CAR will vanish in the long junction limit.In particular, the introduction of ∆ 1 and ∆ 2 in graphene can open an energy gap of Dirac electrons.Owing to nonzero φ, ∆ 1 and ∆ 2 open and close local gaps asynchronously with ωt.Thus, there exists a minimum global gap in the system, called the effective energy gap.It does not close during the whole cycle and its magnitude is also affected by the superconducting gap.In the following parts, we make the electron energy reside in the effective energy gap, which makes the NR and AR predominate.Therefore, the current flowing into the left lead mainly comes from the contributions of the NR and AR, which are respectively given by I NR L = e 2π T ´T 0 dt Im Tr[ dree dt r † ee ] and on the basis of equation ( 6).

The pumping features of NR and AR caused by superconductor
Based on the boundary's shape, quasi-one-dimensional graphene can be divided into two types, zigzag and armchair ones, which have different energy band structures.The zigzag graphene has non-degenerate K and K ′ valleys with a zero-energy band, while the armchair graphene has degenerate valleys.Thus, distinctly different pumping characteristics are induced between the pumping structures based on the two types of graphene, as shown in follows.
Consider the structure based on zigzag graphene firstly, in which the currents flowing into the left lead I L , I NR L , and I AR L versus the energy E with different electrostatic potentials U S are plotted in figure 2. For U S = 0, the currents I L , I NR L , and I AR L satisfy the antisymmetric relationship, which can be expressed as I L (E) = −I L (−E) for I L , as shown in figure 2(a).With the enhancement of E form −∆ to 0, I NR L firstly increases and then decreases, accompanied by a peak at −E 0 , which is always greater than 0. However, I AR L firstly increases from a negative to positive value, then continues to reach the maximum, and finally decreases with a trend toward zero.It follows that I L firstly increases and then decreases with E, also accompanied by a peak.In figures 2(b) and (c) for U S > 0 and U S < 0, respectively, I NR L at |E| > E 0 and I AR L keep unchanged with U S , while I NR L at |E| < E 0 is considerably changed by U S .Particularly, it no longer satisfies antisymmetric relationship, thus leading to no antisymmetric relationship of I L .Specifically, in the range of (−E 0 , E 0 ), compared with U S = 0, I NR L at any E increases for U S > 0, while decreases for U S < 0. From the variation of I L , we can see the competitive relationship of I NR L and I AR L .The current I L is predominated by the AR for I L < 0, while by the NR for I L > 0 at E ∈ (−∆, −E 0 ), and the situation is just contrary at E ∈ (E 0 , ∆).However, the contribution of the NR is much greater than that of the AR, particularly for U S ̸ = 0 and E ∈ (−E 0 , E 0 ).
According to our numerical calculations, for the armchair graphene situation with the same parameters, I AR L is far less than that for the zigzag graphene one.At relatively large pump sizes, such as L p = 700a and L 0 = 175a, I AR L is markedly increased.Particularly, the features for I L , I NR L , and I AR L as a function of E in figure 2 can be reproduced.However, the current values can reach twice that of zigzag graphene situation due to the currents carried by two same valleys in the zigzag graphene but by two different valleys in the armchair one.If |E| < E 0 , compared with U S = 0, I NR L at any E will decrease for U S > 0, whereas increase for U S < 0, which is also thoroughly different from that in the zigzag graphene situation.All these are not presented by figures for simplicity.
Next, I L , I NR L , and I AR L in the zigzag graphene situation versus the S length L 0 with different E and pumping source length L P , are plotted in figure 3, showing an oscillatory behavior.When E is smaller, as shown in figure 3(a), with the enhancement of L 0 , the amplitude of I AR L firstly increases, then reaches a maximum, finally decays to 0, while that of I NR L decreases all the time.The amplitude of I L decreases with increasing L 0 , because of I L predominated by NR.For a larger E at a fixed L P , the features for the amplitude of I NR L and I AR L are preserved, as shown in figure 3(b).However, there exists a much greater amplitude of I AR L .As a result, with L 0 increased, the amplitude of I L firstly decreases, then increases, and finally decays to 0. For a larger L P at a fixed E, the features of figure 3(b) are also basically unvaried, except for the slightly sharper oscillation, as shown in figure 3(c).For the armchair graphene situation with the same parameters, the features of I L , I NR L , and I AR L as a function of L 0 are basically unchanged in figure 4.However, their periods of oscillation are larger.In addition, for an increased E at a fixed L P , the amplitude of I L decreases all the time, as shown in figure 4(b), which is different from the zigzag graphene situation (see figure 3(b)).
Then, we turn our attentions to the effect of L P .For the zigzag graphene situation, I L , I NR L , and I AR L versus L P are plotted in figure 5.As L P increases, I NR L firstly increases, then begins to decrease at about L P = 180a, accompanied by negative values at L P > 570a.Meanwhile I AR L remains negative all the time.This leads to that I L decreases from the positive to negative value at about L P = 370a, displaying a current reversal effect, which is also exhibited in the armchair graphene situation.Now, we investigate the variation of I L , I NR L , and I AR L for the zigzag graphene situation with the pumping phase φ under the different pumping strength ∆ 0 , as illustrated in figure 6.Based on Brouwer's theory [15], usually only in a metallic phase for the system and nonzero transmission of electrons in the pumping cycle, does the current in an adiabatic pump satisfy the current-phase relationship I ∼ sin φ.Owing to the present system without fulfillment of the above conditions, it is clearly shown that the relationship severely deviates from the sine behavior.In addition, the pumping current in the adiabatic limit is proved to be proportional to the geometric area circled by time-dependent parameters, as shown in [15].It follows that at φ = nπ, the geometric area is zero, so the current is zero, as shown in figure 6.More interestingly, because of the existence of resonance level, the peak and valley are formed in I NR L only around φ = (2n + 1)π, whose values increase with the enhancement of L P .However, They are not exhibited in I AR L .In particular, with the increase of ∆ 0 , the maximum (or minimum) of I L , I NR L , and I AR L increases whereas the amplitude decreases.Moreover, there  is the platform for a larger ∆ 0 , whose width increases with the enhancement of ∆ 0 (see figures 6(b) and (c)).The platform behavior refers to that the current remains constant in some range of φ.This can be easily explained by the fact that in a period T, the two potentials can be considered two effective barriers, thus inducing the oscillation caused by φ.
Similarly, we observe the effect of L P on the current-phase relationship (see figure 7).It is found that for a larger L P , the platform looks imperfect, meaning the current corresponding to the platform is varied slightly with φ.With L p increased, both I NR L and I AR L for the platform gradually decrease, leading to the reverse of direction of I L at L P beyond certain value.It is inferred that the features for the variation of I L , I AR L , and I NR L with L p in figure 5 remain at all the φ in the range of platform in figure 7. Here, it is worth mentioning that the reversal effect of current is also exhibited in the armchair graphene situation, which is not presented by figures for simplicity.

Non-topologically 'quasi' interface state
For the pristine graphene situation as shown in [35], the pump quantization is attributed to the time-dependent adiabatic evolution of the possible TIS bridging the two pumping sources.The topologically-protected TIS appears between the two regions for ∆ 1 and ∆ 2 with different signs or vanishes for the same signs.When the left and right leads are connected to each other, the setup forms a self-closed system, that is, there are two interfaces between ∆ 1 and ∆ 2 .Thus, in the closed system, the TIS in the effective energy gap firstly resides in the left lead, then appears in the middle interface, and finally comes back to the right lead at the end of the cycle.This means that the charge complete the circling of an period along the closed setup.
In the present work, the energy-band evolution based on a self-closed setup for different electrostatic potentials (U S = 0, 25∆, and −25∆), in which the superconducting region is involved, are shown in figures 8(a)-(c), respectively.Within the effective energy gap, two isolated energy levels of the pristine graphene pump are found to be split for either E > 0 or E < 0. Two kinds of states are formed, one is the TIS, the other is the non-topologically QIS.More specifically, in some time spans, two energy levels for E > 0 and E < 0 touch at zero, forming the TIS while the other two energy levels for E > 0 and E < 0 are close to zero energy, forming the non-topologically QIS.And the split energy levels approximately overlap and evolve into the bulk state in other time spans.The origins of the TIS and non-topologically QIS can be given as follows.The staggered potential ∆ 1,2 with opposite signs introduced in graphene can make the system become the quantum valley Hall insulator with different quantum phases, which leads to the existence of TIS.However, the non-topologically QIS is attributed to the quantum interference caused by the resonance formed in the S region between two pumping sources (potential barriers), accompanied by electron-hole conversion from the AR.In the next section, we gain an insight into the physical origins of the obtained features in section 3.1 by the aid of the TIS and non-topologically QIS.

The physical origins of the above-obtained features
During the time-dependent energy-band evolution as shown in figure 8, for a fixed energy E, the electron-or hole-like quasiparticles (ELQs or HLQs) can appear at several different moments.Particularly, the probabilities not only between the different moments but also between the different sites at the same moment are not the same, which are just shown as follows.In the pumping process, the current is not only determined by the probabilities of each site at each moment, but also concerned with the specific process, such as NR and AR.This reflects the interplay between the quasiparticles or between the quasiparticle and background.More specifically, in the evolution from one moment to another one, the contribution of the NR to the pumping current embodies the interplay between ELQs, while that of the AR reflects the interplay between ELQs and HLQs.
Take I NR L and I AR L versus the energy E with different electrostatic potentials (U S = 0, 25∆, and −25∆) as an example, whose time-dependent energy-band evolutions based on the closed system have been shown in figures 8(a)-(c), respectively.Now we are focused on investigating the situation of U S = 0 for the small and slightly larger energies, respectively.
Firstly, a small energy E such as 0.15∆, is taken to stay at TIS.For the NR process involving the pump of ELQs, in a periodic time T, the energy E = 0.15∆ locates at five moments, i.e. ωt = 0.05π, 0.45π, 1.05π, 1.45π, and 2.05π.At these moments, the probabilities appearing at each site are respectively presented in figures 9(a)-(e) and weak interface states in superconducting region are exhibited.The probabilities at ωt = 0.05π, 1.05π, and 2.05π are thoroughly the same, and the ones at ωt = 0.45π and 1.45π are thoroughly identical as well.This indicates that the probabilities are unvaried as long as ωt changes by half a period.Thus, there exist four time spans which are divided into two kinds.One is the big time span, corresponding to the negative direction of I NR L , the other is the small one with the positive direction of I NR L .The total I NR L comes from contributions of the four time spans.Due to the NR process with the interplay between ELQs involved, of which the big time spans are in favor, the contribution from these time spans predominates.Therefore, we can infer that the total I NR L is negative and small due to weak interface states, which is consistent with that as shown in figure 2(a).Thus, it is characterized by the interplay between ELQs in the NR process and the great influence of big time spans.For the AR process involving the pump of HLQs, the corresponding energy is E = −0.15∆.In a periodic time T, the energy also locates at the same five moments as for E = 0.15∆, particularly with the same probabilities as in figures 9(a)-(e).They are not presented by figures for simplicity.On the contrary, the small time spans are in favor of the AR, which predominate in I AR L , thus leading to the negative direction of I AR L as well.This is characterized by the interplay between ELQs and HLQs in the AR process and the great influence of small time span.
Next, for a slightly larger energy E = 0.5∆, it stays at not only TIS but also non-topologically QIS, which corresponds to nine moments in a periodic T, namely, ωt = 0.09π, 0.41π, 0.515π, 0.991π, 1.09π, 1.41π, 1.515π, 1.991π, and 2.09π.The corresponding probabilities appearing at each site are respectively shown in figures 10(a)-(i).It is found that in both the middle and boundary regions, there exists the strong TIS.Particularly, the mutual conversions between the TIS and non-topologically QIS are exhibited.For instance, from ωt = 0.41π to 0.515π, the TIS is converted into the QIS, while from ωt = 0.991π to 1.09π, the situation is just contrary.It is found that there are three types of time spans.The first type with the smallest four time spans is divided into two cases, corresponding to the negative and positive directions of I NR L , respectively.The second (third) type owns two medium (the biggest) time spans, which both correspond to the negative (positive) direction of I NR L .As a result, we can infer that for the NR process involving the interplay between ELQs, I NR L from the contributions of the two biggest time spans predominates.This is because the big time span is conducive to the interplay between ELQs.Especially, the probabilities are greatly changed in the biggest time spans while not in the small and medium time spans.In other words, the total I NR L is positive, which is consistent with that as shown in figure 2(a).Similarly, for the corresponding AR process involving the pump of HLQs with energy E = −0.5∆,there exist the same nine moments and probabilities as for − n) of energy level E = ±0.5∆at the time moment ωt = 0.09π, 0.41π, 0.515π, 0.991π, 1.09π, 1.41π, 1.515π, 1.991π, and 2.09π for (a)-(i), respectively.And the other parameters are the same as in figure 8(a).E = 0.5∆.It follows that the total I AR L is still predominated by the two biggest time spans.This is because the probabilities are greatly changed in the biggest time spans although the medium time spans are in favor of the interplay between ELQs and HLQs.Therefore, the total I AR L is negative, which is consistent with that as shown in figure 2(a).Now, we turn our attention to the U S = 25∆ and −25∆.From figure 8(b), we can see although the time-dependent evolution of the TIS and non-topologically QIS for U S = 25∆ is a little similar with that for U S = 0, there exist some differences.For the small energy E = 0.15∆, the probabilities in the same five moments are different from those for U S = 0. Specifically, the probabilities oscillate in the S region, leading to the negative total I NR L with a slightly bigger magnitude than that for U S = 0. Similarly, for the same slightly bigger energy E = 0.5∆, there exist the same nine moments.The corresponding probabilities are basically unchanged except for the weak oscillation in the S region, thus leading to no changes in the sign and magnitude of I NR L .However, for U S = −25∆ in figure 8(c), we can see the difference between the TIS and non-topologically QIS appears more obvious.For the small energy E = 0.15∆, although the five moments are almost unvaried, their probabilities have great changes, including the weak oscillation in the superconducting region and the magnitude.On the contrary, owing to the smaller contributions of the two large time spans than the two small ones, the positive I NR L changes slightly compared with U S = 0. Physically, for the AR process, the interplay between ELQs and HLQs, both affected by U S , keeps I AR L unvaried with U S , as shown in figure 2. However, for the NR process, the interplay of quasiparticles with the same type (either ELQs (E > 0) or HLQs (E < 0)) subjected to U S leads to different features obtained.
In the following of this section, we proceed with the brief analysis and discussion of the physical originations for the features caused by ∆ 0 , φ, L 0 , and L p .For simplicity, if not necessary, we will not present the corresponding time-dependent energy-band evolutions of the TIS and non-topologically QIS, and probabilities.
Firstly, for the variation of ∆ 0 , the features of the corresponding evolutions and their probabilities are almost not changed except for the slight variation in the magnitude.This leads to no great change in I NR L and I AR L , particularly for the bigger ∆ 0 as shown in figure 6. Obviously, the effective influence of ∆ 1 = ∆ 0 cos(ωt) and ∆ 2 = ∆ 0 cos(ωt + φ) are determined by not only ∆ 0 but also φ.When ∆ 0 is considerably large and φ ̸ = nπ , the influence of φ can be almost neglected, so that there is a platform for a larger ∆ 0 , whose width increases with the enhancement of ∆ 0 (see figures 6(b) and (c)).This can be also explained by the corresponding evolutions and their probabilities which are almost not changed by φ.
Next, for the variation of L p , although the corresponding evolutions are almost unchanged in figure 11, their probabilities are considerably varied.This gives rise to the magnitude change of I NR L and I AR L without oscillation, especially the transition from positive to negative values, as shown in figure 5.
Finally, for the variation of L 0 , the corresponding evolutions have a significant change in figure 12, especially the energy bands of the TIS and non-topologically QIS no longer overlap.Naturally, the resultant corresponding probabilities are a lot varied, thus bringing about the oscillation of I NR L and I AR L , as shown in figures 3 and 4.

Matching conditions of the quantized pumping current
In this section, we observe the matching conditions of quantized pumping current in the present setup, which are focused on the superconducting region length L 0 and pumping source length L P .The matching condition means that an integer number of electrons can be pumped out in a pumping cycle under some length parameters, i.e.I L = ±e/T.The quantized pumping current is an integer-valued ordinary pumping one.In figures 13(a)-(d), I L , I T L (the current coming from the contributions of transmission and CAR processes), I NR L , and I AR L in the L 0 − L P plane are illustrated.They are shown to oscillate with L 0 but not with L P , which is qualitatively with those in figures 4 and 5.In figure 13(b), weak I T L occurs only in the range of L P with small value, which is consistent with the above-mentioned feature with the probabilities of transmission and CAR processes vanishing in the long junction limit.For I NR L in figure 13(c), we notice that obviously dark and bright stripes are mainly distributed in the range of L 0 without too large values.These stripes for smaller L 0 extend across the entire L P region, while for larger L 0 , they occur in both smaller and larger L P regions.Moreover, the stripes are always straight.For I AR L in figure 13(d), the obviously dark and bright stripes are almost shown in whole range of L 0 except for the too small values.The scope for the corresponding L P is diminished with the enhancement of L 0 .Similarly, the stripes also keep straight.Therefore, I L owns the some features of I NR L and I AR L in figure 13(a).The obvious stripes along the L P direction are mainly distributed in  the range of L 0 with not too large value, and these stripes become discontinuous in the range of smaller L P .More importantly, these bright red and dark purple stripes represent quantized current I L , which are distributed in many matching values for L 0 and L P .Apparently, the successful implementation of quantized current within a broad parameter space in our theoretical framework, opens up a convenient and efficient route for its experimental realization.The features of the quantized pumping current can be also qualitatively understood from the behaviors of NR and AR probabilities |r ee | 2 and |r he | 2 in the ∆ 1 − ∆ 2 plane.
In figure 14, we plot the probabilities of NR, AR, and total reflection (|r ee | 2 , |r he | 2 , and |r| 2 ) in the ∆ 1 − ∆ 2 plane.The circular contours a 1 , a 2 , and a 3 corresponds to ∆ 0 = 0.02t n , 0.01t n , and 0.004t n , respectively.We find that |r ee | 2 displays the same varied behaviors along the two half contour lines given by the figure 14(a).When ∆ 0 = 0.02t n corresponding to a 1 , |r ee | 2 rapidly increases to the maximum 1 from 0, then keeps unchanged, and finally decreases to 0 quickly at approaching to the end of the half contour line a 1 .This indicates I NR L is large (see figure 13(c)) and can remain unchanged as ∆ 0 reduces gradually, such as in the contour a 2 at ∆ 0 = 0.01t n .The situation of |r he | 2 for both a 1 and a 2 is just contrary to that of |r ee | 2 (see figure 14(b)), and the corresponding total reflection |r| 2 is equal to 1 in figure 14(c  (I L ( e T ) ≈ 1, see figure 13(a)).However, due to the opposite directions of I NR L and I AR L , the current I L is not quantized for a 2 , which is not presented by figures for simplicity.In addition, nonzero transmission probability is exhibited in the central region of the plane in figure 14(c).This indicates when the value of ∆ 0 is decreased to some extent, the energy E resides outside the effective energy gap, which does not make |r ee | 2 and |r he | 2 predominate along the contour any longer (see the contour a 3 at ∆ 0 = 0.004t n ).As a result, I L no longer mainly comes from the contributions of I NR L and I AR L .

Experimental feasibility
At last, we briefly discuss the experimental feasibility of our proposed charge pump based on the graphene NSN junction.It displays a larger current and more abundant current characteristics, which could be utilized to improve the pumping efficiency.The characteristics reveal the intrinsic physics of graphene-based pumping structure, leading to the potential applications in optical device [49], quantum sensing, quantum information [50] and so on.In the practical experiment, the superconductivity in graphene can be induced by proximity coupled to an s-wave S (such as AI, NbSe 2 ) [51][52][53].It is the key to this pump model that the staggered lattice potential can be generated in growing graphene sheet on SiC or h-BN substrates [54], resulting in different site energies between the A and B sublattices.Therefore, the construction of the charge pump can be basically realized in the experiment.
There also exist some considerations to be taken into account when extending this pump device to realistic experiments.First, a sufficiently low pumping frequency is necessary for the system to reach the adiabatic limit, but this leads to a greatly prolonged experiment time and a higher experiment complexity.Hence, it is essential to carefully deliberate on the selection of a suitable pumping frequency.Second, the environmental temperature needs to be kept low enough to ensure the stability of the quantum state.Note that our calculation is valid for zero temperature.For sufficiently low temperatures (T ≪ ∆), the resonant level will have a slight change due to thermal smearing.Thus, we expect that our qualitative features for the pumping current will survive in the presence of low temperatures, although they can exhibit a quantitative change in the current.Then, the size of our pump is approximately at the nanoscale.Finally, the adiabatic quantum pump requires highly precise control and measurement.However, the above problems can be solved by the current advanced technologies and equipment.

Summary
In summary, we investigate an adiabatic superconducting charge pump based on the graphene with an S sandwiched in between two pumping sources.Their time-dependent and out-of-phase staggered potentials are introduced in zigzag graphene as pumping parameters.The pump is characterized by not only TIS but also non-topologically QIS.These interface states stem from the staggered potentials ∆ 1,2 with opposite signs and electron-hole conversion at N/S interfaces, respectively.Due to the adiabatic evolution of TIS and non-topologically QIS, particularly, the conversion between each other, the superconducting pump has three features: (1) Manipulating the energy E, S region length L 0 , and pumping source length L P significantly affects the competitive behaviors between I NR L and I AR L , the currents from NR and AR to the left lead, respectively.The currents are less sensitive to L P and E, but more to L 0 .(2) Tuning L P can realize the reversal of pumping current direction, where the current flows from one lead to another, and then returns to the original lead.This can improve the performance of charge pumps and achieve a high-power friction nanogenerator.(3) The current-phase relationship exhibits the platform behaviors with current remaining constant in some range of phase shift φ, which can be controlled by the pumping strength ∆ 0 and L P .
In addition, we get the quantized pumping current by adjusting L P and L 0 , and give the corresponding qualitative explanations by the aid of the pumping contour circled by the two parameters.
The pumping current features in armchair graphene are also discussed, which agree with those in zigzag graphene.But different edge structures cause three major differences: (1) The pumping currents are carried by two different valleys, so their values can attain twice that of the zigzag graphene situation.(2) I AR L only appears at larger pumping parameters L P and L 0 .(3) The currents oscillate with L 0 more slowly and weakly.
Some extended research can be based on our work.It is worthwhile to investigate whether other external fields (such as the ferromagnetic exchange field, antiferromagnetic exchange field, strain field, etc) can also generate large pumping current and rich current modulation features.Furthermore, if the Fermi energy deviates from zero, corresponding to the occurrence of Andreev retro-reflection, it will bring different characteristics to the pumping current.

Figure 3 .
Figure 3.I NR L and I AR L under the zigzag graphene situation as a function of L0 with (a) E = −0.5∆and LP = 350a, (b) E = −0.8∆and LP = 350a, and (c) E = −0.8∆and LP = 600a.The corresponding IL is presented in the inset.Here, the other parameters are the same as in figure 2(c).

Figure 4 .
Figure 4.The same as in figure 3 expect for the structure based on the armchair graphene.

Figure 5 .
Figure 5. IL, I NR L , and I AR L under the zigzag graphene situation as a function of LP.Here, we set L0 = 125a, E = −0.8∆,and the other parameters are the same as in figure 2(c).

Figure 6 .
Figure 6.IL, I NR L , and I AR L under the zigzag graphene situation as a function of φ with (a) ∆0 = 0.005tn, (b) ∆0 = 0.01tn, and (c) ∆0 = 0.015tn.Here, Lp = 300a and the other parameters are the same as in figure 5.

Figure 7 .
Figure 7. IL, I NR L , and I AR L under the zigzag graphene situation as a function of φ at (a) Lp = 300a, (b) Lp = 350a, and (c) Lp = 400a.Here, ∆0 = 0.02tn and the other parameters are the same as in figure 6.

Figure 8 .
Figure 8. Adiabatic evolution of the energy band around E = 0 with time ωt for (a) US = 0, (b) US = 25∆, and (c) US = −25∆.The other parameters are the same as in figure 2.

Figure 11 .
Figure 11.Adiabatic evolution of the energy band around E = 0 with time ωt for (a) LP = 400a and (b) LP = 700a.The other parameters are the same as in figure 5.

Figure 12 .
Figure 12.Adiabatic evolution of the energy band around E = 0 with time ωt for (a) L0 = 100a and (b) L0 = 135a.The other parameters are the same as in figure 3(b).
).Because the currents I NR L and I AR L have the same direction for a 1 , as shown in figures 13(c) and (d), the current I L is quantized