Andreev reflections modulated by the breaking of space-inversion symmetry in graphene-typed materials

We theoretically perform a comprehensive analysis about the influences of the space-inversion symmetry breaking in graphene-based materials on the Andreev reflections (AR) in the normal-metal/superconductor (NS) and NSN heterojunctions. It is found that in the NS junction, the AR can be suppressed or be enhanced by the enhancement of space-inversion symmetry breaking, depending on the relationship among the coherence parameters. Following this result, the AR properties in the NSN structure are evaluated. It is readily observed that the local AR can be weakened for low space-inversion symmetry breaking, and can be enhanced for high space-inversion symmetry breaking. Alternatively, the efficiency of the crossed AR can be improved to a great degree, with the increase of space-inversion symmetry breaking. One can therefore understand the special role of space-inversion symmetry in modulating the AR, especially for the enhancement of crossed AR.


Introduction
Heterojunctions between normal-metal (N) and superconducting (S) materials have attracted enormous attentions in the field of condensed matter physics [1][2][3][4][5]. The main reason is that Andreev reflection (AR) occurs within the superconducting gap Δ, in which an electron in the normal metal is reflected at the NS interface as a hole accompanied by a Cooper pair entering into the superconductor [6]. This phenomenon exactly embodies the interplay between quantum coherence in the normal metal and the intrinsic coherence of the superconductivity [1]. Moreover, when two normal metals couple to one superconductor to build the NSN junction, one special AR mechanism, i.e. crossed Andreev reflection (CAR) has an opportunity to come into play, if the junction width is of the order of the superconducting coherence length [7], or even exceeds the coherence length, which has been reported recently [8,9]. Interpretively, an electron in one normal metal that hits the NS interface forms Cooper pair in the superconductor by capturing the electron from the other normal metal [10][11][12]. In other words, via the CAR process, the two electrons of one Cooper pair are allowed to be spatially separated, realizing the splitting of Cooper pair. This nonlocal quantum effect can be considered to be one typical solid-state entanglement which is certainly important for quantum physics and quantum computation [13,14].
During the past years, many groups have dedicated themselves to the investigation about the CAR process, for the purpose to achieve the high-efficiency Cooper-pair splitting [15][16][17][18][19][20][21][22][23][24]. And various Cooper-pair splitters have been designed, by considering different materials or structures. On the one hand, the quantum-dot (QD) Cooper-pair splitter has attracted much attention. As demonstrated in the previous reports [19][20][21][22][23][24], the strong Coulomb repulsion in the QDs benefits the CAR process. Another advantage of the QD Cooper-pair splitters consists in the independent level shift of respective QDs [20]. As a result, lots of theoretical researchers are encouraged to discuss the Andreev transport properties of the QD Cooper-pair splitters from different aspects [25][26][27][28][29][30][31][32][33]. For instance, the problems of coherence and entanglement of split Cooper pairs and their probing have been concerned [28][29][30][31], accompanied by the discussion about the noise correlations [25,26] and Cooper pair microwave spectroscopy [27,28]. In addition, the property of the metallic leads has been considered. When the approach, the space-inversion symmetry of graphene can be broken, and a sizable band gap can be induced both in graphene and S, as shown in figure 1(c).
To study quantum transport properties of the NS junction, we should employ the Bogoliubov-de Gennes (BdG) equation, which can be written as [53] H )are two component spinors of electron and hole, respectively. In the presence of space-inversion symmetry breaking, H can be written as where σ x , σ y , σ z are the Pauli matrixes, v F is the Fermi velocity, and I is the unit matrix. m is Dirac mass of charge carriers, which reflects the magnitude of space-inversion symmetry breaking. μ corresponds to the chemical potential. Both the S order parameter Δ and electrostatic potential U is spatially dependent, which can be expressed as The dispersion for quasiparticles at a given incident energy E and transverse wave vector q in the N region can be directly written as ) . Before the calculation about the AR process, it is necessary to present the band structure of this heterojunction. According to the dispersion relation, it is easy to find that the inversion-symmetry breaking gives rise to the appearance of band gap with width 2m at the Dirac point of graphene, and an extra band gap labeled X and Y in the S region, as shown in figure 1(c). Now we consider an electron of the conduction band with σ (σ denotes atom A) incidents from the left part, and it will be reflected as a hole in the conduction band with σ, or in the valence band with s (s denotes atom B) [51]. According to the energy dispersion relation, for m<μ, it can be divided into two situations: (i) when E m 0  m < -, the hole will be retro-reflected, as shown in the top panel of figure 1(c); (ii) when E>μ+m, the hole can be specularly reflected, as shown in the bottom panel of figure 1(c). Next, we continue to evaluate the AR properties with the help of scattering matrix method. The first step is to solve the wavefunctions of two parts, as shown in the appendix. Consider an electron incidents with energy E and wave vector q, the quasipartles can not run through the semi-infinite S region, thus the wavefunctions in the two regions can be respectively expressed as where I, II denotes the N and S regions. With respect to the NSN heterojunction, we consider the width of region S to be L, as shown in figure 1(b). And then, two boundary conditions form at the interfaces x=0 and x=L, respectively: . 9 x x x L x L Region III denotes the right N region, r ee and r eh are electron reflection and LAR coefficients, respectively. And t ee and t eh are ET and CAR coefficients. c and d are coefficients of the left moving quasiparticle modes in the S region.
Region III has an opportunity to be set as n-type or p-type, by applying gate voltages in different manners. If it is p-type, the corresponding dispersion relation for electron and hole can be written as Meanwhile, the wavefunctions also have their alternative forms, as shown in the appendix. As a result, the electron reflection and LAR coefficients R ee and R eh , and the ET and CAR coefficients T ee and T eh can be given as e cos e cos , 1 1

Numerical results and discussion
Following the theory in the above section, we proceed to study in detail the AR conductance of NS junction, and LAR and CAR conductances of NSN junction in both nSn and nSp configurations via the scattering matrix method. Prior to calculation, relevant parameters are taken to be ÿ=1 and v F =1 during the whole calculation process. For the system's temperature, we consider it to be zero in the context.

NS heterojunction
In the NS heterojunction, the AR characteristics can be well differentiated according to the dispersion relation.
To be specific, when m<μ, the retro-AR process will occur if 0<E<μ−m, whereas the specular-AR will take place if E>μ+m. On the other hand, when mμ, there only exists specular-AR if E>μ+m, but the retro-AR will be forbidden.
In figure 2, we present the curves of the AR conductance of NS heterojunction. The superconducting order parameter is considered to be the energy unit by setting Δ=1. In order to study the properties of the retro-AR and specular-AR, we would like to pay attention to two cases, respectively, i.e. μ=10Δ and μ=0.1Δ. It is evident that retro-AR will be dominant in the former case, and specular-AR will make leading contribution in the latter case. The results of μ=10Δ are shown in figures 2(a), (b) and those of μ=0.1Δ are in figures 2(c), (d), when the space-inversion symmetry breaking is taken into account with m  m. Figures 2(a) and (c) show the dependence of the AR conductance (i.e. G AR ) on the space-inversion symmetry breaking m, for different bias voltage eV. And figures 2(b) and (d) show the relation between the bias voltage eV and G AR for different values of m. It can be seen that in both the images of retro-AR and specular-AR, the magnitude of G AR decreases with the increase of m, independent of the value of eV, as shown in figures 2(a) and (c). The difference mainly lies in that the decline amplitude of retro-AR is much larger than that of specular-AR. In figure 2(b), it shows that the curve of G AR decreases monotonically with the increment of the bias voltage eV for our model. However, for the specular-AR, the situation becomes different. In figure 2(d), it shows that the AR conductance magnitude first increases and then slightly decreases with the increase of eV.
These results reflect two aspects. One is that G AR always decreases with the increase of m, independent of the type of AR. The other consists in that G AR is much more sensitive to the change of m for the retro-AR, in comparison with the specular-AR. These two facts can be explained by paying attention to the band structure of our system shown in figure 1(c). For the retro-AR, the incident electron and the reflected hole are both from the conduction band. When increasing m, one can see the decrease of the density of states of the hole in the conduction band for the invariance of μ and eV. As a result, G AR of retro-AR decreases. Similarly, the hole is reflected to the valence band for specular-AR. When the values of eV and μ are fixed and m increases, the density of states of hole in the valence band decreases, thus G AR of specular-AR decreases. In addition, the chemical potential μ is set to be only 0.1Δ for specular-AR dominates, thus the density of states of the hole in the valence band changes little with the development of m. Accordingly, G AR of specular-AR is less sensitive to the variation of m compared with retro-AR.
In figure 3, we plot the AR conductance as a function of m for different bias voltages. In this case, μ is taken to be the energy unit, i.e. μ=1, the other parameters are the function of μ. As shown in the figure, the value of m/ μ is varied from 0.01 to 100, in the cases of different eV. According to the energy dispersion in the N region, the retro-AR occurs when m/μ<1/(1+eV/m), and the specular-AR comes into being as m/μ>1/(eV/m−1). Besides, for the case of 1/(1+eV/m)<m/μ<1/(eV/m−1), there will emerge one band gap which certainly suppresses the AR process. For the retro-AR, it can be seen that the conductance magnitude decreases with the increase of m, and the smaller G AR corresponds to the higher bias voltage. Furthermore, G AR decreases more rapidly than the case in figure 2 with the increase of m, except for the zero bias voltage case. This phenomenon is easy to understand. With the increment of m, eV increases accordingly but μ remains unchanged, which accelerates the decrease of the density of states of the hole in the conduction band. If m is further increased, G AR will decrease and finally reach zero at the position of m/μ=1/(1+eV/m), because the density of states of the hole in the conduction band decreases to zero.
However for the specular-AR, the situation is quite different. For the case of zero-bias voltage, the specular-AR is eliminated. In the presence of the bias voltage, G AR first increases and then almost stabilizes at its maximum, which originates from the increase of the density of states of the hole in the valance band along with  the increment of m. And when the value of m is significantly larger than μ, the density of states almost remains unchanged with the increase of m. Moreover, one can find that in the cases of eV/Δ=0.5 and 0.8, the maximal G AR is able to arrive at 1.0 at the large-m limit. However if eV/Δ=1.0, the value of G AR only reaches 0.65 in this process. It means that G AR first increases and then decreases by increasing eV for large space-inversion symmetry breaking, which is consistent with figure 2(d). It can also be found that the variation of G AR of the specular-AR is contrast to the result in figure 2(c). This means that the trend of the specular-AR is determined by the relationship among the bias voltage, chemical potential and the inversion-symmetry breaking.

NSN heterojunction
In this subsection, we continue to investigate the AR scattering of the NSN heterojunction, by considering two configurations, i.e. nSn and nSp. In the conventional NSN junction, there exists two kinds of ARs, i.e. LAR and CAR, in which the electron enters into the S region accompanied by the appearance of one hole in the same and the other metal side, respectively. In this part, we only consider the case of in the energy unit of μ=1, because in this case the conductance versus the space-inversion symmetry breaking is more sensitive.
We first pay attention to the nSn configuration and investigate the LAR and CAR conductances versus the space-inversion symmetry breaking, for different bias voltages. The results are shown in figure 4(a), (b) respectively. In this junction, according to the energy dispersion relationship, one can know that the retro-LAR and specular-LAR occur at the same situation with those in the NS junction, respectively. In figure 4, it can be obviously seen that both the curves of G LAR and G CAR show oscillation behaviors in the retro-AR process, but these oscillations almost disappear in the specular-AR process. To be specific, in figure 4(a) the curve of G LAR shows the same variation behavior in figure 3, expect for the oscillation and a lower value, because it can be viewed as one NS junction with finite length of the S region. Next in figure 4(b), we can see that for the retro-CAR, G CAR decreases with the increase of m in the presence of the bias voltage. It is because the density of states of the hole in the conduction band decreases with the increase of m. However, the value of m has an apparent effect on G CAR at the case of zero-bias voltage. When m/μ=0.85, the maximum of G CAR occurs at the position of eV/Δ=0, almost equal to 0.12, as marked in blue circle. We choose the maximum of G CAR , and calculate the dependence of reflection and transmission coefficients on the incident angle, as shown in the inset of figure 4(b). It shows that R ee increases, and the rest coefficients decrease by increasing α, and the maximal value of T eh reaches 0.12 approximately. On the other hand, the value of G CAR is relatively small and can be ignored for the specular-CAR process. In this process, it is proven that the ET and LAR are dominant, while the electron reflection and CAR are suppressed.
Next, we would like to evaluate the LAR and CAR conductances in the nSp configuration by supposing the parameters to be identical with figure 4. The corresponding results are shown in figure 5. In such a case, the energy dispersion relationship suggests that the retro-(specular-)LAR occur at the same situation with the retro-AR (specular-AR) in the NS junction. While for the CAR, there only exists the specular-CAR, without the restriction of critical angle and bias voltage, but the retro-CAR is forbidden completely. Further more, it is important to note that in the band gap region eV m m eV m ) , the ET and the LAR are forbidden, there only exists the CAR and the electron reflection. In this case, the incident electrons have a chance to be completely converted into CAR. Firstly, in figure 5(a) we see that G LAR behaves in a similar way with the nSn configuration (See figure 4(a)). However, G CAR exhibits an alternative result in comparison with the case of nSn junction, as shown in figure 5(b). One can see that the curve of G CAR exhibits the oscillatory behavior throughout all the values of m in the presence of the bias voltage, indicating that there only exists one type of hole transmission, i.e. specular-CAR. While for the case of zero-bias voltage, the specular-CAR is eliminated after m>μ. In addition, it shows that with the increment of m, G CAR first increases with oscillations and then decreases to zero for a large space-inversion symmetry breaking in the presence of the bias voltage. The maximum value of G CAR is able to reach 0.83 when eV/Δ=0.3 and m/μ=0.42 in the band gap region, as marked in blue circle. In view of this result, we choose the maximum point of G CAR and present the dependence of reflection and transmission coefficients on the incident angle, as shown in the inset of figure 5(b). It can be readily found that there only exist the electron reflection and hole refraction, i.e. R ee and T eh in the scattering process, because the incident angle exceeds the critical angle. It is worth mentioning that in this case, the maximum of T eh has an opportunity to arrive at 1.0 for an appropriate angle. This indicates that the incident electrons can be switched to the CAR without any electron reflection. We then perform the investigation about the influence of the length of S region, i.e. L, on the property of G CAR in the nSn and nSp junctions, as shown in figure 6. For describing the role of space inversion symmetry, the results of m=0 are presented as well. Figures 6(a), (b) are the results of the nSn configuration in the cases of m=0 and m=1.1, whereas figures 6(c), (d) correspond to the nSp case. With respect to the bias voltage, its value is taken to be eV=0 in figures 6(a), (b) and eV=μ−m in figures 6(c), (d), so the critical angle α c is 2 p and 0, respectively. Under this condition, G CAR is allowed to reach its maximum in these two junctions. By comparing the results in this figure, the effects of m can be well differentiated, in addition to L. Firstly, for the junction of nSn, we can see that with the increase of L, the curve of G CAR oscillates seriously, and the CAR conductance reaches its maximum in the vicinity of L≈0.5ξ. Thereafter, the conductance amplitude decays rapidly. With respect to the effect of m, it is only to magnify the amplitude of the conductance. In the case of m/ μ=1.1, the conductance maximum is about three times the case of m=0, which is consistent with the result of figure 4(b). On the other hand, for the nSp junction shown in the bottom panel, the effect of m is relatively distinct. It shows that in the case of m=0, the amplitude of G CAR decreases monotonically by increasing L, accompanied by the weak oscillation of the conductance curve. Instead in the case of m=μ/1.1, the conductance amplitude first grows up with the increment of L until L≈1.0ξ, then it falls down solwly with the further lengthening of L. What is important is that the value of G CAR in the m=μ/1.1 case is far larger than the m=0 case, and it can almost be close to 1.0 near the position of L=ξ, and when L=5ξ, it can be 0.5 approximately. Thus, we know that the breaking of the space-inversion symmetry is an important mechanism for enhancing the CAR process. Furthermore, in the two junctions the effects of finite m exhibit notable difference. As shown in figure 6(b), the maximum value of G CAR can arrive at 0.18, in the presence of the breaking of space-inversion symmetry, for the nSn junction, but in this process the CAR conductance amplitude can almost be equal to 1.0 in the nSp junction. Therefore, the role of space-inversion symmetry breaking is dependent on the junction configuration.
Before concluding, it is necessary for us to compare our results with those of partial existence of spaceinversion symmetry breaking in the NS and NS junctions, despite the complications of their experimental realization. It is known that except the case of m 0 ¹ in the N part, the space-inversion symmetry breaking is also allowed to only exist in the S part. In figure 7, we present the numerical results of three cases, i.e. m S =m and m N =0 for Case I, m N =m and m S =0 for Case II, and m N =m S =m for Case III, i.e. our case (The subindexes of m correspond to the cases of partial existence of space-inversion symmetry breaking in the junctions). The relevant parameters are set to Δ=1, μ=10Δ and L=0.5ξ. For the NS junction, it can be seen in figure 7(a) that in these three cases, the amplitudes of G AR decrease with the increase of space-inversion symmetry breaking (i.e. m/μ). Meanwhile, their difference can be clearly observed. In Case I, the AR amplitude is smaller than the other two cases, in the region of m 0.97  m . For G AR in Case III, it is lager than Case I, except for the points that m/μ=0 and m/μ=0.97. Also when m/μ>0.7, it is more sensitive to the increase of m, due to the rapid decrease of the AR amplitude. Next with respect to the NSN junction of nSn configuration, it shows in figure 7(b) that the LAR properties exhibit much difference for the three cases. In Case I, the amplitude of G LAR decreases monotonously, whereas it shows oscillation in Case II following the increment of m/μ. However in Case III, the LAR conductance seems to be independent of the increment of m in the region of m<0.8. As for the CAR results, one can find that with the enhancement of space-inversion symmetry breaking, they all first go up and then fall down. It seems that in Case II, the CAR enhancement is coincident with Case III before m/μ=0.9, though it is advantageous to the other cases. What is notable is that the CAR ability in Case I is weaker than others. Therefore, it can be found that space inversion symmetry breaking in the N part tends to only suppress the LAR behaviors in the NS and NSN junctions, whereas in the S part it plays an alternative role. And in our considered systems, all these results can be observed. We can then conclude that our systems are more helpful for understanding the role of space inversion symmetry breaking in modulating the AR behaviors.

Summary
In Summary, we have theoretically investigated the influences of the space inversion-symmetry breaking in graphene-based materials on the ARs in the NS junction and the NSN heterojunction with its nSn and nSp configurations. It has been found that in the NS junction, the AR can be suppressed or be enhanced by the enhancement of space-inversion symmetry breaking, depending on the relationship among the coherence parameters. Following this result, the LAR and CAR properties in the NSN structure for both n-type and p-type are also evaluated. We readily observe that the LAR can be weakened for low inversion-symmetry breaking, and can be enhanced for high inversion-symmetry breaking. As for the CAR, it can be greatly enhanced to a great degree, with the increase of inversion symmetry breaking. One can therefore understand the special role of spatial inversion-symmetry in modulating the AR properties.
Our obtained results have also been analyzed by comparing the cases of space-inversion symmetry breaking in respective parts of the NS and NSN junctions, respectively. It has shown that our considered NSN junction is advantageous to the case of space-inversion symmetry breaking only in the N parts, in driving the CAR. Besides, due to the higher experimental feasibility, it can be believed that our considered NSN junction is a promising candidate of the Cooper-pair splitter.
where α is the incident and reflection angle of elections in region I, and also the refraction angle of electrons (holes) in region III for nSn (nSp) configuration. a¢ is the reflection angle of holes in region I and also the refraction angle of elections in region III for nSp configuration. k (k¢) is the wave vector for n-(p-)type election and p-(n-)type hole, the transverse vector q remains unchanged in scattering process satisfies the Snell Descartes, and σ indicates whether the holes in region I belong to the conduction band or the valence band. When α>α c , the wave vector of holes in region I and elections in region III in nSp configuration k x ¢ is an imaginary number, and the corresponding scattering angle a¢ is a complex angel. Thus the wave functions in the three regions are still expressed as equation (15)  where S e q and S h q are the scattering angles for electronlike and holelike quasiparticles.