On this part, the bodily outcomes of our numerical evaluation on the transmission and conductance, given by Eqs. (29) and (30) are offered. The primary curiosity is to analyse the results of the incident angle, power of the magnetic subject, widths of the magnetic limitations and wells, vitality, static and oscillating scalar potentials on the conduct of the transmission and conductance by way of the system proven in Fig. 1. The numerical outcomes are generalized to the previous and latter profiles in “Uniform static scalar potential” part and “Alternative static scalar potential” part, respectively. The resonance and Klein tunneling results are studied in “Resonance and Klein” tunneling part and the conductance is investigated in “Conductance” part, for each configurations. The phrases of the collection in Eq. (29) are saved as much as (n=5). It has been thought-about (B_0=0.1) T for the scaling worth of the magnetic subject. On this case, the size, angular frequency and vitality scales are obtained to be (l_0=81.13) nm, (v_{mathrm{F}}/l_0=12.325times 10^{12}) rad/s and (hbar v_{mathrm{F}}/l_0=8.113) meV, respectively. The entire coming figures are colour on-line.

### Uniform static scalar potential

On this subsection, we consider the graphene-based system with the magnetic profile given by Eq. (1) and the scalar potential profile described by Eqs. (2) and (3), proven in Fig. 1a–c. In Fig. 2, the incident angle dependence of the transmission likelihood is depicted for various values of (d_B), non-zero magnetic flux ((d_Bne d_{-B})) and (V_1=0, 1.98). The used parameters are (B=1), (E=1), (d_{-B}=2) and (V_0=12) within the case of 1 magnetic block, i.e. (N=1). There may be an angular confinement for the quasiparticles in absence of the oscillating scalar potential, i.e. (V_1=0). Within the case of (d_B<d_{-B}), it’s specified by the important worth (varphi _c=-sin ^{-1}(1+(ND/E))) for the incident angle^{3,37}. The confinement is eliminated for non-zero values of (V_1). This limitation can be considered for the vitality of the quasiparticles as (E>mid NBD/2mid). So, eradicating the oscillating scalar potential, turns off the transmission for (varphi <varphi _c) or (Ele mid NBD/2mid). Then, it’s attainable to restrict the DW quasiparticles by turning off (V_1) within the described magnetic construction. An ideal transmission is noticed in (theta =30^circ), for (d_B=1), nevertheless the second lobe could be very small. For the remainder of this part, we consider the case (d_B=d_{-B}).

Determine 3a reveals the angular dependence of transmission within the, so known as, equal-barrier case, i.e. (E=V_0), for one magnetic block with (B=1) and (d_B=d_{-B}=1). The curves are deflected in direction of detrimental angles. The transmission lobes change into sharper by rising the vitality *E*. In Fig. 3b, it has been zoomed on the angular dependence of the transmission within the case of (E=V_0=12) and (V_1=0) for (N=1, 6). The increment of the variety of magnetic blocks makes the beams shorter and thinner. This induces a robust wave vector filtering and suppression within the transmission for the construction.

Determine 4a depicts that for (E=1) and within the absence of the oscillating scalar potential, (V_1=0), by making use of the static potential (V_0) the transmission will increase and the Klein tunneling impact activates. For the particular worth of (V_0=12), an ideal transmission is noticed in a variety of the incident angle. Within the absence of (V_0), turning on (V_1), the transmission will increase. Quite the opposite, the transmission decreases when (V_1) is turned on, within the presence of (V_0). The Klein tunneling is noticed for (V_0ne 0). For (E=5), some adjustments seem within the talked about behaviors (see Fig. 4b). Within the absence of the scalar potential ((V_0=V_1=0)) there’s a vast angular vary with the right transmission. Making use of (V_0), (V_1) or each of them, the transmission decreases over some components of this vary. The Klein tunneling is noticed right here, furthermore, there are some angles indicating the resonance results^{6}.

The width of the magnetic blocks is likely one of the necessary components within the system which might have an effect on the transmission and the Klein tunneling results. Determine 5 reveals the angular dependence of the transmission for (N=1), (E=5), (V_0=12), (V_1=0) and completely different values of (d_B). Rising (d_B), the transmission sample turns into narrower and shorter within the detrimental angle aspect. The beam disappears round a important worth given by (d_B=V_0/(ev_{mathrm{F}}B)) which, within the dimensionless notation, reads to (d_B=V_0/B). For (V_0=0), this important width is given by (d_B=2E/(ev_{mathrm{F}}B)) which is the diameter of the cyclotron orbit or (d_B=2E/B) within the dimensionless type. The important worth of (d_B) decreases by turning (V_1) on. So as to observe the response of the system to the Klein tunneling, we consider the conventional transmission. In Fig. 6 the conventional transmission is plotted versus (d_B) for (E=5), (V_0=12), (V_1=1.98) and completely different values of *N*. For small values of (d_B) the Klein tunneling is ruled within the system, whereas for (d_B) better than a important worth the conventional transmission falls to zero. This important worth of (d_B) decreases by rising the variety of blocks.

### Various static scalar potential

On this subsection, we contemplate the graphene-based system with the magnetic profile given by Eq. (1), just like the case thought-about in “Results and discussion” part, whereas the scalar potential profile is described by Eqs. (2) and (4) which is proven in Fig. 1a, b and d. In Fig. 7, the impact of non-zero magnetic flux is proven on the incident angle dependence of the transmission likelihood for (V_1=0, 1.98). The entire parameters are the identical as in Fig. 2 within the earlier configuration. The transmission is decreased compared to the uniform static scalar potential case. There is similar limitation for the incident angle as within the first profile, i.e. (varphi >-sin ^{-1}(1+(NBD/E))) for (d_B<d_{-B}), (varphi <sin ^{-1}(1-(NBD/E))) for (d_B>d_{-B}) and (E>mid NBD/2mid) for the vitality in absence of (V_1), as a result of conservation of (k_y). Once more, turning (V_1) on, this confinement is eliminated. In opposite with the primary configuration, the transmission will increase by turning (V_1) on. Any further, we contemplate (d_B=d_{-B}).

Comparable to Fig. 3 within the earlier configuration, Fig. 8 is obtained for the angular dependence of transmission at (E=V_0), on this profile. Right here, additionally, increment of the vitality makes the lobes sharper in detrimental angles and making use of (V_1) makes the lobes wider, which is clear in Fig. 8a. The lobes change into sharper for (N>1), as it’s seen in Fig. 8b. So, a robust wave vector filtering could be achieved by selecting the big values for the vitality within the case of multiblock magnetic system.

Now we analyze the angular dependence of the transmission with respect to the variation of (V_0) and (V_1). The identical outcomes proven earlier than in Fig. 4 is, precisely, obtained for (V_0=0). Once more, there’s a small vary of the incident angles with low transmission for (E=1) and a large excellent transmission vary for (E=5) and (V_1=0). In Fig. 9, the angular dependence of the transmission is depicted various (V_0) (non-zero) and (V_1) for (E=5). The Klein tunneling is noticed for all values of (V_0) and (V_1). There’s a vast angular vary with excellent transmission for (V_1=0). Turning (V_1) on, the angular vary of the transmission is just not modified however some resonance results are created within the edges of this vary and the transmission is decreased. The identical angular profile of the transmission is obtained for (V_0=0) and (V_0=10) within the absence of (V_1), which could be justified by invariance of the cyclotron radius (r_c=vert E-V_0vert /ev_{mathrm{F}}B). Determine 10 reveals the angular dependence of the transmission for various values of (d_B=d_{-B}). Once more, the transmission lobes are deflected in direction of detrimental angles. The transmission of the magnetic system decreases by rising the thickness of the only magnetic block system within the absence of the oscillating scalar potential, (V_1). The transmission lobes disappear at a important worth given by the cyclotron orbit diameter, (d_B=2E/B).

### Resonance and Klein tunneling

On this subsection, we examine the dependence of the full sideband transmission likelihood on the parameters of the system because the vitality, the scalar potential and power of the magnetic subject. Our purpose is investigation of resonances and Klein tunneling impact within the regular incidence for 2 profiles investigated in “Results and discussion” part and “Conclusion” part. In Fig. 11, the conventional transmission is plotted versus the vitality for the previous (Fig. 11a and c) and latter (Fig. 11b and d) configurations, respectively. In each profiles, within the absence of (V_1), there’s a transmission drop area within the vitality interval, given by (vert E-V_0(x)vert <vert k_y+Bvert), inside which the transmission experiences some resonance peaks^{34}. This limitation could be expressed by way of the vitality, magnetic subject or incident angle by sharing between this situation for various values of vector potential. This is because of the truth that the vector potential, *A*, takes values between zero and (Bd_B), for (d_B=d_{-B}). The eigenstates within the limitations are evanescent they usually propagate within the nicely to type quasibound states, for (Nge 2). If the incident vitality coincide with this certain state energies within the nicely, the transmission resonances happens^{43}. So, on this area, the transmission reduces besides in resonance peaks. The degenerate eigenlevels within the wells break up due to the coupling between the wells by way of tunneling within the limitations and it results in the ((N-1))-fold resonance splittings for *N* magnetic blocks (see Fig. 11a and b). Removed from this area, the eigenstates are propagating states and the DW quasiparticles can transmit completely, so the transmission approaches to unity and the Klein tunneling is ruled within the magnetic system. There are, additionally, ((N-1)) distinct peaks, related to the Fabry-P(acute{e})rot interference, due to interplay between the static and magnetic limitations and wells. The resonance splitting impact exists within the magnetic superlattice (the magnetic system with (Nge 2)) versus vitality, by way of an electrostatic barrier with the suppression of Klein tunneling. Making use of (V_1) reduces peak of the resonance peaks from unity and by rising (V_1) different units of ((N-1))-fold resonances are appeared within the neighborhood of (E=V_0). The ((N-1))-fold resonance peaks, additionally, are noticed for (k_yne 0).

Within the first configuration (uniform static scalar potential) the curve of transmission versus vitality displays a cusp at (E=V_0) for (B=1) and non-zero (V_1). The cusp is eliminated for (V_1=0) the place the transmission is suppressed (see Fig. 11c). The cusps, induced by the oscillating potential, for a couple of magnetic block develop up by the amplitude of the oscillating potential and the variety of blocks. They’re attributed to the Fabry-P(acute{e})rot fringes. The transmission round (E=V_0) could be turned on (off), for a number of magnetic blocks with (Nge 4), simply by turning on (off) (V_1). The transmission in (E=V_0) decreases by reducing (V_1) and it approaches zero for (V_1=1.98) and enormous variety of blocks, i.e. (N>40). By rising the distinction between (V_0) and *E*, the Klein tunneling seems and this happens in increased variations by rising (V_1). Rising *B* also can flip off the conventional transmission (see the inset of Fig. 11a within the absence and the inset of Fig. 11c within the presence of (V_1)). Determine 11b and d present comparable results for the second profile (various static scalar potential). Right here, not like the primary configuration, the vitality *E* and the static scalar potential (V_0) haven’t the identical roles and the transmission reveals utterly non-symmetric conduct in two sides of (E=V_0). Within the absence of (V_1), the transmission round (E=V_0) reduces to zero by rising the variety of blocks ((Nge 6)) or *B*. There aren’t any resonance peak for (E<V_0). For (V_0ge 5), a Klein tunneling area is appeared, centered round (E=V_0/2), which its width will increase by rising (V_0). Right here, the barrier and nicely modes are comparable to one another as a result of reality (vert E-V_0vert =E) which was, additionally, seen within the first configuration. The transmission decreases from unity in two areas in either side of this area, given by (vert E-V_0(x)vert <vert k_y+Avert). Out of this vary, for (E>V_0), the Klein tunneling seems once more and likewise various units of ((N-1))-fold resonances are noticed. Turning (V_1) on, the transmission round (E=V_0) will increase and the resonance peaks are modified to the same old peaks. The concerns studied right here, can be utilized in designing switching on or off devices for the transmission of the cost carriers within the graphene based mostly methods.

The counter plots for the conventional transmission for the primary configuration are sketched in Fig. 12 versus the vitality and the static scalar potential for (N=1, 4) in each circumstances of absence and presence of the oscillating scalar potential. The vitality *E* and the static scalar potential (V_0) have comparable roles on this configuration, which isn’t the case in second profile. Within the single block case (Fig. 12a and b), there are ribbon bands in the primary diameters, i.e. (E=V_0), which signifies the non-zero minimal for the transmission which its width will increase by rising (V_1). Transferring away from the primary diameter, the distinction between *E* and (V_0) will increase and the conventional transmission grows in direction of the unity which ends up in the Klein tunneling impact. From the semiclassical perspective^{44}, the Dirac fermions subjected to a perpendicular magnetic subject are rotating round a round orbit with the radius of cyclotron radius. Rising the distinction between vitality and static scalar potential will increase the cyclotron radius and the Dirac fermions can simply full their cyclotron orbits, so the transmission and conductance are elevated. Within the case of 4 magnetic blocks (Fig. 12c and d) the 3-fold resonance strains are appeared shut the primary diameters. Right here, the minimal transmission is zero for multiblock magnetic system, within the absence of (V_1) (see Fig. 12c). So, the likelihood for switching the full regular transmission to zero is supplied by selecting (V_0) and *E* shut to one another for (N>3) within the absence of (V_1). The variety of Fabry-P(acute{e})rot fringe patterns (resonances and peaks) will increase within the presence of (V_1) for the a number of magnetic blocks, symmetrically on either side of the ribbon band. The road within the middle of the primary diameter in Fig. 12d corresponds to a cusp for (E=V_0) in Fig. 11 which is created and grows by rising *N* for (B=1). The looks of the second ((N-1))-fold resonance strains are, additionally, noticed in either side of the primary diameter, in Fig. 12d, by making use of (V_1=1.98).

In Fig. 13a dependence of the conventional transmission to (V_0) is investigated within the various static scalar potential case. The transmission drop area for (V_0) is (E-vert k_y+Bvert<V_0<E+vert k_y+Bvert) which consists of the fluctuations as ((N-1))-fold resonances centered at (V_0=E). Within the presence of (V_1) the resonance, asymmetrically, alters to the peaks and the fluctuations exterior this area are exacerbated. For even variety of blocks, the central peak doesn’t embody excellent transmission just like the others. Rising *N*, will increase the amplitude of the transmission drop. These behaviors for the transmission are strongly affected by the vitality. Relying on the worth of the vitality, it reveals two varieties of behaviors. In Fig. 13a the vitality was (E=5) and, in its inset, the conduct of the transmission is proven for the energies (E=4, 7) with (N=3) and (V_1=0). For these energies, two ((N-1))-fold resonances are noticed in either side of (V_0=E) and the transmission drops in (V_0=E). In Fig. 13b the conduct of the transmission is studied versus (V_0=E). The transmission, for one magnetic block, will increase as much as the energies round (V_0=E=3) and, then, goes to be glad in (tau =0.8). For a number of blocks, the transmission takes oscillatory conduct as a result of multireflections within the partitions of the magnetic limitations. The interval of this oscillations will depend on (d_B=d_{-B}). The ((N-1))-fold resonances are, additionally, noticed as earlier than. So, not like the primary configuration, the transmission is just not decreased for (V_0=E) however the excellent transmissions are, additionally, noticed within the resonance peaks.

Determine 14 reveals the full regular transmission of the Dirac fermions versus the magnetic subject power for various values of vitality, oscillating scalar potential and variety of magnetic blocks within the former and latter profiles. Within the former configuration, as is proven in Fig. 14a, the Klein tunneling is noticed for weak magnetic fields and it’s suppressed within the first important magnetic subject with the magnitude (B_{c1}). The transmission decreases and vanishes round a second important subject (B_{c2}). The radius of the cyclotron orbit decreases with the increment of *B* and the transmission vanishes^{16}. The important fields (B_{c1}) and (B_{c2}) depend upon *E*, (V_0), (V_1) and *N*. They each lower with the increment of the vitality, for (E<V_0), as a result of spreading of the scalar potential barrier in the entire magnetic area. Making use of (V_1), the primary (second) important subject decreases (will increase) and the distinction (B_{c2}-B_{c1}) will increase. (B_{c2}) decreases by rising *N*, as is proven in Fig. 14c. The identical evaluation is carried out for the latter configuration in Fig. 14b and d with the identical parameters of the primary profile. The primary important magnetic subject is sort of zero however the second is decreased, compared with the previous configuration. (B_{c2}) will increase by the increment of the vitality and is impartial of (V_1). In each profiles, by including variety of the magnetic blocks, two units of ((N-1))-fold resonances are appeared; the primary between (B_{c1}) and (B_{c2}) and the second after (B_{c2}). It has been noticed that they’re remodeled, within the presence of (V_1), to the fluctuations with non-perfect transmission. The proper and 0 transmission areas are separated from one another by a drop area which its width decreases by increment of *N*. For giant variety of magnetic blocks this area turns into narrower and the zero transmission area is obtained by the situation (vert E-V(x)vert <vert k_y+Bvert) which yields (B>7) and (B>5) within the first (Fig. 14c) and second (Fig. 14d) configurations, respectively.

The resonance splitting angle areas are proven in Fig. 15 for various values of vitality and variety of the magnetic blocks with (V_0=10) within the former and latter profiles. Within the angular interval given by (vert sin varphi +A/Evert <vert 1-V_0(x)/Evert), for each configurations, the right transmission is noticed as a result of propagating states contained in the blocks. It experiences some fluctuations due to the Fabry-P(acute{e})rot resonances and the Klein scattering induced by the electrostatic barrier. It yields (-90^circ<varphi <53.1^circ) for (E=5) and (-41.8^circ<varphi <30^circ) for (E=6). For the particular worth of (E=V_0/2) for the vitality, the angular profiles of the previous and latter profiles coincide (see Fig. 15a), whereas they’re separated for different values of the vitality, as proven in Fig. 15b. The ((N-1))-fold resonances are noticed out of the angular interval of the propagating state’s lobe. It has been noticed that, by making use of (V_1), some fluctuation peaks are created with non-perfect transmission. The presence and absence of the oscillating scalar potential can be utilized as a attribute of the transport properties of the studied magnetic methods. In the entire figures, thus far, the frequency has been set on (omega =2). Altering (omega) doesn’t have an effect on the entire mentioned bodily outcomes. In Fig. 16, the conventional transmission has been plotted for (omega =2, 10) in each configurations. The ((N-1))-fold peaks are noticed for each frequencies. To keep away from the divergency as a result of oscillatory nature of the Bessel features mentioned earlier than, it needs to be famous that (omega >V_1/n).

### Conductance

The results mentioned thus far associated to the transmission is, additionally, mirrored within the complete sideband conductance of the studied magnetic system. In Fig. 17 the full sideband conductance of the system is plotted versus the vitality for (N=2), (V_0=10), (V_1=0) and completely different worth of *B* for the previous (Fig. 17a) and latter (Fig. 17b) configurations, respectively. Within the first profile, the conductance experiences a non-zero minimal in (E=V_0) like the conventional transmission. This minimal conductance approaches to zero by rising the variety of blocks as much as (Nsim 6) within the absence of (V_1) whereas this occurs, within the presence of (V_1), for (Nsim 40). Within the second configuration, as is proven in Fig. 17b, the conductance has two minimums round (E=0) and (E=V_0) that are attributed to the nicely and barrier modes and the second has the identical behaviors talked about for the primary configuration. The conductance experiences a resonance peak after (E=V_0). The conductance reaches to a most round (E=V_0/2) which decreases within the presence of (V_1) and by rising *B*. The conductance decreases by rising *B* however there’s a area round (E=V_0) and after (E=V_0) within the former and latter configurations, respectively, the place the conductance is sort of insensitive to the increment of the magnetic subject. Making use of (V_1), removes the insensitivity of the conductance, as is proven within the insets in Fig. 17. The oscillations of the conductance, significantly proven in Fig. 17a, have already been noticed in the same experimental work^{45}.

Determine 18 reveals the dependence of the conductance to the static scalar potential for the choice static scalar potential profile. As it’s clear in Fig. 18a, the conductance involves a minimal in (V_0=E) apart from some particular values for the vitality with a interval relying on (d_B). For (E=2, 5, 8, 11, cdots), the conductance reaches to a most at (V_0=E). Switching (V_1) on, the conductance will increase round (V_0=E) and reduces out of this area. In Fig. 18b the conduct of the conductance is studied for the equal-barrier case, i. e. (V_0=E). For a single block construction, the conductance reaches uniformly to a most in low energies after which reduces to zero. For a number of blocks, the conductance has an oscillatory conduct with damping amplitude and a interval relying on (d_B). Rising *N*, creates peaks within the maxima which was seen in Fig. 13 for regular transmission. The conductance will increase by introducing (V_1) for all values of (V_0=E). Within the inset of Fig. 18b, the impact of *B* and (d_B) is studied for equal-barrier case. Rising *B* doesn’t have an effect on the place of the peaks until the amplitude of the primary most is lowered however rising (d_B) reduces the amplitude and interval of the conductance oscillations. Regardless of the damping conduct of the transmission and conductance within the equal-barrier case within the first configuration and different perviously studied researches^{16}, they’ve oscillatory behaviors within the second profile of the current work. This rising conduct could be useful in controlling and interrupting the transport.

Determine 19 illustrates the conductance as a perform of the power of the magnetic subject for the primary (Fig. 19a) and second (Fig. 19b) configurations, respectively. In accordance with the conduct of regular transmission, studied by way of Fig. 14, the conductance is suppressed after the second important magnetic subject (B_{c2}) for each profiles. Its magnitude is about 1.55 T and 0.69 T by consideration of (B_0=0.1) T for (N=1), (E=1) and (V_1=0) for the previous and latter configurations, respectively. It decreases by increment of *N*, as is proven within the insets of Fig. 19. The important subject (B_{c2}) for the latter profile is lower than the case for the previous configuration with the identical parameters. Regardless of the truth that the conventional transmission is unity for weak magnetic fields ((B<B_{c1})) within the first configuration, the conductance decreases by making use of (V_1) which could be helpful in designing of graphene-based nanostructures. The oscillations within the insets are related to the Fabry-P(acute{e})rot fringes.