Signature of Anomalous Andreev bound states in magnetic Josephson junction of noncentrosymmetric superconductor on a topological insulator

First, we consider the ABS corresponding to the different angles of incidence of the injected particles. The results are displayed in Fig. 3 for different misalignment angles considering two different length scales viz., $L/L_{0}=0.01$ and $0.05$. We consider $Z_{0}=0.1$, $Z_{R}=0$ and $\delta=1$ for this analysis. We find a gapless dispersion with $4\pi$ periodic ABS for perpendicular incidence as represented by the solid curves of Fig. 3(a). It indicates the presence of Majorana at $\phi=\pi$ for perpendicular incidence. More interestingly result in Fig. 3(a) signifies that Majorana is present in a spin-active barrier. However, for oblique incidence, a gap is always observed for all cases, as shown by the dotted and dashed lines of Figs. 3(a)-3(j). It is to be noted that the gap opens up further for larger the angle of incidence as seen for $\theta=\frac{\pi}{3}$ indicated by the dotted lines. The presence of a gap at oblique incidence is due to non-zero backscattering at a finite angle. To realize the impact of barrier magnetic moment, we have plotted ABS in Figs. 3(a)-3(j) for different values of $\chi_{m}$ with $\rho=0.01$ and $\xi_{m}=0.5\pi$. It is seen that the Andreev levels split corresponding to spin helicity $\sigma$ and for non-vanishing values of $\chi_{m}$. The wave vector mismatch occurs for the misalignment of the polar angle of magnetic moments. Thus the two degenerate energy bands split into two more branches, as seen in Figs. 3(b)-3(d) and Figs. 3(g)-3(i). We also find that the two sets of ABS depart away from each other with the increase in misalignment angle $\chi_{m}$. This results in the formation of a $\phi$ junction for non-vanishing values of $\chi_{m}$ corresponding to normal incidence. Similar behavior is also observed for oblique incidence. However, the gap remains constant in this situation also. It is seen that the $4\pi$ periodicity of the ABS will not present for non-vanishing values of $\chi_{m}$ and finally for anti-parallel orientation $\chi_{m}=\pi$, the system resides in $2\pi$-ABS branches as seen from Figs. 3(e) and 3(j). Thus, interaction with the barrier magnetic moments results in the system to reside in the lower Andreev energy branches. The Andreev levels are found to display similar characteristics in Figs. 3(f)-3(j) for $L/L_{0}=0.05$ in normal incidence condition. However, for oblique incidence, the gap is significantly reduced for this length scale. It may be due to the reason that with increasing length, induced magnetic proximity on the surface of 3D TI gets weaker and hence the effect of barrier and bulk moment is reduced. Thus, the system is found to display $4\pi$ periodicity for the parallel alignment while it shows anomalous characteristics and $\phi$ periodicity for the misalignment of the barrier and bulk moments.

In Figs. 4(a)-4(j), we show the dependence of ABS on the ratio of barrier magnetic to non-magnetic moment ($\rho$) in perpendicular incidence condition. We study the ABS spectra for different polar angles ($\chi_{m}$). The plots in the top panel are for azimuthal angle $\xi_{m}=0.5\pi$ while the plots in bottom panel are for $\xi_{m}=\pi$. We find that the Majorana modes appear at $\phi=\pi$ in parallel configuration for all choices of $\rho$ and $\xi_{m}$ as seen in Figs. 4(a) and 4(f). Andreev levels shrinks with the increasing values of $\rho$ and maximum contraction is observed for $\rho=1$ as seen from the dashed curves of Figs. 4(a) and 4(b). However, for $\chi_{m}=0.5\pi$, $0.75\pi$ and $\pi$ the ABS grows more for $\rho=1$ and a opposite characteristic is seen. It signifies that for anti-parallel alignment ABS energy becomes maximum when magnitude of barrier magnetic moment approaches to non-magnetic moment. This behaviour is seen in Figs. 4(c)-4(e) for $\xi_{m}=0.5\pi$. The ABS depart again for non-vanishing values of $\chi_{m}$. However, the Andreev levels for $\rho=0.01$ and $0.5$ are found to display nearly same characteristics for $\chi_{m}=0.5\pi$. The ABS shrinks further for $\chi_{m}=0.75\pi$ with $\rho=0.01$ as seen from Fig. 4(d). Though a similar characteristic is seen for $\chi_{m}=0$ and $0.25\pi$ with $\xi_{m}=\pi$, but for $\chi_{m}=0.5\pi$ the Andreev energy levels are found to be independent of $\rho$ as seen from Fig. 4(h). For $\chi_{m}=0.75\pi$ and $\pi$ with $\xi_{m}=\pi$, the Andreev levels grows for all choices of $\rho$. Thus, although the Andreev energy levels becomes anomalous for nonzero values of $\chi_{m}$ but the magnitude of the Andreev levels can significantly change for different azimuthal angle as seen from Fig. 4. This indicate that the ABS not only depends on the polar angle $\chi_{m}$ but also on azimuthal angle $\xi_{m}$.

We study the effect of barrier transparency on ABS as shown in Figs. 5(a)-5(e) considering a $\delta$-like barrier at $x=0$ and $x=L$. We make this analysis by considering a quasi-magnetic barrier with $\rho=0.01$ and the barrier moment is misaligned by an angle $\chi_{m}=\xi_{m}=0.3\pi$ for all plots of Fig. 5. We also set $L/L_{0}=0.01$, $Z_{R}=0$ and $\delta=1$. For a nearly transparent barrier the Andreev levels are found to be nearly similar to that obtained in Figs. 4(b), 4(g), 5(a) and 5(b). In this scenario, the ABS depart again due to non-vanishing values of $\chi_{m}$. For $Z_{0}=0.5$, the Andreev levels shrinks but the splitting is still present as seen in Fig. 5(b). With the further increase in barrier transparency $Z_{0}$, the energy of the ABS significantly decreases as seen from the Figs. 5(c)-5(e). It is seen that the pattern of ABS is found to be significantly different in Figs. 5(a) and Fig. 5(e) characterize by an anticrossing at $\phi=0$ and is due to finite value of $Z_{0}$. However, the splitting of the Andreev levels still present in the system.

The plots in the middle panel of Fig. 5 display the effect of RSOC on Andreev energy levels. In the presence of RSOC, the four degenerate energy bands of the TI further split into four branches, as seen in Figs. 5(f)-5(j). In this situation, the wave vectors of the NCSC are different from that of TI. Due to the mismatch of the wave vector, an additional gap appears at $\phi=\pi$. However, it is seen from Figs. 5(g)-5(i) that the four energy branches grow while the other four shrinks simultaneously with an increase in the strength of RSOC. For $Z_{R}=1$, four Andreev energy branches grow to their maximum value while the other four branches shrink completely, as seen in Fig. 5(j). In this scenario, thus the supercurrent is carried by only one branch of Andreev energy levels. Furthermore, it is noted that though the RSOC splits the energy bands, it does not create a phase shift. It indicates that there exist Majorana modes at $\phi=\pi$ for normal incidence even in the presence of RSOC. Thus, the main effect of RSOC on the Andreev levels are, (i) shift of energy bands and (ii) presence of an additional gap at $\phi=\pi$. So, it can be concluded that RSOC plays a very significant role on the ABS spectrum.

We show ABS spectra for different values of singlet-triplet mixing parameter $\delta$ in the bottom panel of Fig. 5. We set $L/L_{0}=0.01$, $Z_{R}=0$ and $Z_{0}=0$ for this analysis. The mixing parameter $\delta$ plays a very significant role in NCSC with $\delta=0$ correspond to majority of singlet and $\delta=1$ correspond to majority of triplet correlations. We find that the system exhibit exactly same behaviour for $\delta=0$ and $1$ as seen from Figs. 5(k) and 5(o). We encounter band splitting again for unequal mixing of the singlet-triplet components. For unequal mixing, the Andreev levels corresponding to $\Delta_{-}$ shrinks while the levels corresponding to $\Delta_{+}$ remains unchanged as seen from Fig. 5(l) and 5(n). Moreover, the helical ABS spectra for $\delta=0.25$ is found to be exactly same as that of $\delta=0.75$. For equal mixing, the Andreev energy levels corresponding to $\Delta_{-}$ die out completely while the levels corresponding to $\Delta_{+}$ remains unchanged as seen from Fig. 5(m). Thus the supercurrent is only due to the $\Delta_{+}$ component, while the contribution of the supercurrent from $\Delta_{-}$ is zero for equal mixing conditions. We find that the magnitude of ABS energy is dependent on gap $\Delta_{-}$ while it is totally independent of $\Delta_{+}$. Furthermore, the crossing of the energy levels and the phase are totally independent of the gap parameter $\Delta_{\pm}$. It signifies that Majorana modes can exist at the surface of 3D TI in a $\pi$-NCSC Josephson junction.

The magnetic tunability of the Andreev energy levels and its effects on the superconducting phase indicates the presence of a non-trivial Josephson current in the proposed setup. Experimentally, Josephson junctions made up of Superconductor-Topological Insulator-Superconductor (S$|$TI$|$S) has been realized recently veldhorst . The work indicates the anomalous characteristics of the supercurrent. Moreover, presently there is a great interest in the experimental realization of ferromagnetic topological insulator hybrids (FTI). It can be achieved by random doping of transition metal elements like Cr or V on the surface of 3D TI chang11 ; chang12 . This result in magnetic proximity on the surface of TI. Recently it is found that due to the presence of magnetic material in the NCSC Josephson junction, the system displays anomalous Josephson current zhang11 . It arises due to the different orientation of the exchange field. However, the Josephson current in an NCSC corresponding to an HM ferromagnet is not studied yet. Also, as NCSC$|$HM$|$NCSC Josephson junctions hold Majorana modes at $\phi=\pi$ in parallel orientation for half-metallic limit, so it is necessary to understand the behaviour of supercurrent flowing at the surface of 3D TI. We initially calculated the Josephson supercurrent for an HM with bulk magnetic moments $\mathbf{m}=(0,0,|m|)$ and the barrier magnetic moments $\mathbf{V}_{m}=(V_{x},V_{y},V_{z})=(\rho V_{0}\cos\xi_{m}\sin\chi_{m},\rho V_{0}\sin\xi_{m}\sin\chi_{m},\rho V_{0}\cos\chi_{m})$. The results in the half-metallic limit are displayed in Figs. 6, 7, 8 and 9. However, we also plotted the supercurrent for a general ferromagnet in Fig. 10. For simplicity of our calculations, we ignore the continuum states of asymmetric contact junction and we only focus on the discrete states. This approximation does not drastically change the effective current phase relation and hold good to simplify the calculations as seen from the previous works klam .

In Fig. 6, we have plotted the Josephson supercurrent for different values of the effective ratio of magnetic and non-magnetic barrier potentials. We find that for parallel orientation of the barrier and bulk moments with $\xi_{m}=0.01\pi$, the magnitude of supercurrent is maximum for $\rho=0.01$. It decreases with the increasing values of $\rho$ as seen from Fig. 6(a). This signifies that in this situation, a non-magnetic barrier offers more supercurrent than the magnetic barrier. It is also noted that the supercurrent is found to be 2$\pi$ periodic. If the barrier and the bulk moments are misaligned to an angle $\chi_{m}=0.25\pi$, then the supercurrent significantly reduces for all choices of $\rho$. However, in this condition also non-magnetic barrier offer more supercurrent than the magnetic barrier and the supercurrents still found to display $2\pi$ periodic phase as seen from Fig. 6(b). This is due to the presence of symmetric branches in ABS spectra in HM limit, which is significantly different for a bulk ferromagnet zhang11 . The magnitude of the supercurrent further decreases for the perpendicular misalignment of the barrier and bulk moments characterize by $\chi_{m}=0.5\pi$. In this case, all the magnetic barrier corresponding to non-zero values of $\rho$ display exactly the same characteristics. As the misalignment angle $\chi_{m}$ increases further to $0.75\pi$ and $\pi$, the supercurrents display $\pi$ phase shift. But in this case, the component $\rho=1$ display critical current while it is found to be minimum for $\rho=0.01$. Thus the magnetic barriers enhance the supercurrent while the non-magnetic barriers suppress the supercurrent, as seen from Figs. 6(d) and 6(e). It is to be noted that the magnitude of the supercurrent is found to be maximum for the anti-parallel misalignment of the barrier and the bulk moments. To understand the effect of azimuthal angle on the supercurrent, we plotted the CPR for $\xi_{m}=0.5\pi$ in Figs. 6(f)-6(j). It is seen that the nature of the supercurrent remains quite similar for the $\chi_{m}=0$ case as seen from Fig. 6(f). However, for $\chi_{m}=0.25\pi$ we find that the supercurrent corresponding to $\rho=0.01$ and $0.2$ enhances while it suppressed further for $\rho=0.8$ and $0.1$ as seen from Fig. 6(g). Supercurrent is found to be independent of the strength of the magnetic barrier for $\chi_{m}=0.5\pi$. However, it is still found to be maximum for non-magnetic barrier that is seen from Fig. 6(h). The CPR for $\chi_{m}=0.75\pi$ from Fig. 6(i) is found to drastically different from Fig. 6(d). In this condition, the magnitude of the supercurrent corresponding to the magnetic barrier drastically increases while it suppressed for a non-magnetic barrier. It indicates that CPR is also dependent on the azimuthal angle of the barrier moment. For the anti-parallel misalignment, we find exactly similar characteristics from Figs. 6(e) and 6(j). Thus these results indicate that supercurrent is dependent on the misalignment of barrier and bulk magnetic moments. Depending upon the orientations, both magnetic and non-magnetic barriers offer critical current in certain orientations.

We study the CPR of the total supercurrent for different singlet-triplet correlations in Fig. 7. We set $\rho=0.3$, $\xi_{m}=0.5\pi$, $Z_{0}=0$, $Z_{R}=0$ and $L/L_{0}=0.0$1 for this analysis. We find that the CPR corresponding to $\delta=0$ coinciding with that for $\delta=1$ which is in accordance with Fig. 5. It is to be noted that the critical current is observed for the majority of either singlet or triplet components represented by $\delta=0$ and $1$ respectively. The critical current is found to be minimum for equal mixing of the singlet-triplet parameter for all possible misalignment angles. It is due to the reason that for equal mixing, the supercurrent is only due to the $\Delta_{+}$ component while it is completely suppressed for the $\Delta_{-}$ component represented by solid black lines of Fig. 7. However, the critical current display a moderate value for unequal mixing of the singlet-triplet parameter represented by solid orange lines of Fig. 7. In this condition, though the ABS is suppressed due to the $\Delta_{-}$ component, some supercurrent still presents due to the non-vanishing value of $\Delta_{-}$. Moreover, the CPR is found to execute nearly sinusoidal characteristics as the contribution from the higher harmonics are very much suppressed.

From Figs. 6 and 7, we have seen that the characteristics of Josephson supercurrent are dependent on the effective ratio between magnetic and non-magnetic moments, misalignment angle between the barrier and bulk magnetic moment and singlet-triplet mixing parameter. So to understand the interplay between them, we have plotted supercurrent as a function of $\rho$ in Fig. 8 for different singlet-triplet mixing parameter. We find that for $\chi_{m}=0$, the critical current is observed at $\rho=0$ and the supercurrent decreases monotonically with the increasing values of $\rho$ for all choices of singlet-triplet mixing parameter as seen from Fig. 8(a). The maximum critical current is observed for $\delta=0$ and $1$ while the minimum critical current is observed for $\delta=0.5$ as seen earlier from Figs. 6 and 7. The characteristics of the critical current remain the same for $\chi_{m}=0.25\pi$ as seen from Fig. 8(b). However, the critical current is found to be decreased in all mixing scenarios for $\chi_{m}=0.25\pi$. For $\chi_{m}=0.5\pi$, the Josephson current is found to be totally independent of $\delta$ for all $\rho$ values as seen from Fig. 8(c). We find an exactly opposite behaviour characterized by the negative critical current for anti-parallel misalignment. In this case also $\delta=0$ and $\delta=1$ display maximum critical current as seen from Fig. 8(d).

We study the dependence of the critical Josephson current on the length of the Josephson junction in Fig. 9. It is a quantity that represents the maximum supercurrent and often measured in magnetic Josephson junctions linder12 ; snelder1 ; snelder2 . We consider $\xi_{m}=0.5\pi$, $Z_{0}=0$ and $Z_{R}=0$ for this analysis. We find that the critical current monotonically decays for all choices of mixing parameter $\delta$. The misalignment angle can only split the energy bands but not transfer any finite centre of mass momentum to the Cooper pairs. This results in a monotonous decay of the critical current. From Fig. 9 we see that $\rho=0.1$ provides the maximum critical currents in all three scenarios, while minimum critical current is observed for $\rho=1$. It is also seen that the magnitude of critical current is dependent on the mixing parameter $\delta$. The critical current is found on decaying rapidly in an equal mixing scenario as seen from Fig. 9(a). For unequal mixing $\delta=0.75$ and $1$, the critical current decays comparatively slower than that for $\delta=0.5$ as seen from Figs. 9(b) and 9(c). The decay rate of critical current is found to be minimum for $\delta=1$. Thus it can be concluded that the decay rate of the critical current is dependent on the singlet-triplet mixing parameter.

Figs. 4, 5 and 6 indicate that the presence of a spin-active barrier drastically changes the characteristics of Andreev energy levels and Josephson supercurrent in an HM ferromagnet. For all our analysis, we consider the magnetization of the bulk HM ferromagnet is along z-direction and contribution due to xy-component is zero, i.e., $h_{xy}=0$. However, it is natural to address the scenario of a general ferromagnet which can be represented by random orientation of the bulk moments. Though many similar works tanaka1 ; zhang11 ; linder12 is done before but the interplay of the bulk moment with spin-active barrier moments on the surface of a TI is yet to be studied. So, we have plotted ABS energy levels as a function of $\theta$ and $\phi$ in Fig. 10. We consider three different bulk moments. In Fig. 10(a), we consider the scenario where no barrier and bulk moment is present, i.e., $h_{xy}=0$, $h_{z}=V_{z}=0$. In this scenario, we observe a 4$\pi$ periodic ABS and a band crossover at $\phi=-\pi$ and $\pi$ specif by the darker region of the density plot of Fig. 10(a). In this condition, the Andreev energy levels are found to be symmetric with respect to $\phi=0$ and Majorana modes are observed at $\phi=\pi$. However, for $h_{xy}=0.1V_{z}$, the band crossing will not appear at $-\pi$ and $\pi$. Thus the ABS spectra are found to be asymmetric in this condition seen from Fig. 10(b). In Fig. 10(c), we display the ABS for $h_{xy}=V_{z}$. In this condition, the crossover of the Andreev energy bands will appear at $\phi=0$, which signify the $2\pi$-periodic ABS. The ABS spectra and CPR for all the choices of $h_{xy}$ are shown in Figs. 10(d) and 10(e). We find that there exists a phase shift in Josephson supercurrent for random orientation of the bulk magnetic moments.

It is very worthwhile to mention that experimentally transport properties of heterostructures are studied in the diffusive limit. In the diffusive limit, one needs to use quasi-classical Usadel equations to obtain the Andreev levels and supercurrent. However, an analytic equation for ABS energy considering all contributing parameters is too hard to obtain for heterostructure like NCSC$|$HM$|$NCSC. So in that case also we need to use a numerical approach. Though we perform all our calculations considering BdG equations in the ballistic limit, our results are quite adequate in diffusive limits also because of two reasons: (1) The anomalous Andreev energy levels and the $\phi$-states are generated due to chiral spin symmetry breaking along with scattering from the spin-active barrier. It is to be noted that these effects pertain uniquely to both ballistic and diffusive limits. (2) We find that CPR for low and moderate interface transparency is close to sinusoidal in nature. It is due to the suppression of the higher harmonics. As in diffusive limit, first harmonics give the most important contribution. So our results will be quite adequate in diffusive limits also. There may be some difference in ballistic and diffusive limits qualitatively. But it is expected to have no dramatic alteration of the results in diffusive limit.