Double Andreev reflections and double normal reflections in nodal-line semimetal-superconductor junctions

First, we present the reflection probabilities as functions of the incident angle $\theta$, which are always symmetric about $\theta=0$. Fig.2 shows the results for different incident energy $E$ at $E_{z}=0.3\Delta$ and $\mu_{N}=\mu_{S}=0.5\Delta$. In this instance, the band gap for ELQs spans from $-0.8\Delta$ to $-0.2\Delta$ and that for HLQs spans from $0.2\Delta$ to $0.8\Delta$. When $0<E<0.2\Delta$, ELQs on the conduction band $E_{e}^{+}$ and HLQs on the conduction band $E_{h}^{-}$ participate in the scattering processes which are schematically presented in Figs.1(d) and (f). Double Andreev reflections and double normal reflections exist simultaneously. The four reflection probabilities are all non-zero as shown in Fig.2(a) when $E=0.1\Delta$. As $\theta$ increases from $0$ to $\pi/2$, SNR $R_{n1}$ is enhanced while RNR $R_{n2}$ and SAR $R_{a1}$ are weakened. As for RAR $R_{a2}$, it exhibits a small oscillation. The curve “sum” gives the summation of the reflection probabilities. Its value is always $1$ for $E<\Delta$ according to the conservation relations in Eq.(14).

When $0.2\Delta<E<0.8\Delta$, only ELQs on the conduction band $E_{e}^{+}$ take part in the scattering processes. For the incident energy $E=0.5\Delta$ in Fig.2 (b), the energy lying in the gap for HLQs leads to the vanishing SAR and RAR and $R_{a1}=R_{a2}=0$ exactly. Double normal reflections still exist as two inverse phase oscillating curves. SNR $R_{n1}$ and RNR $R_{n2}$ reach their valley value and peak value at $\theta=0$, respectively. When $E>0.8\Delta$, HLQs on the valence band $E_{h}^{+}$ will be activated and participate in the scattering processes which are similar to Figs.1(c) and (e). Double Andreev reflections and double normal reflections reappear as shown in Fig.2(c). In the four reflections, RAR $R_{a2}$ becomes the dominant process in a large angle range around $\theta=0$. If one continuously increases the value of $E$ until $E>\Delta$ as given in Fig.2 (d) with $E=1.2\Delta$, the reflections will be weakened since quasiparticle transmissions begin to happen. The conservation relations in Eq.(14) should be revised. In this case, the summation of the reflection and transmission probabilities equals to $1$. From the above results, it is found that the reflection styles and magnitudes can be controlled by adjusting the incident quasiparticle energy.

Fig.3 shows the reflection probabilities as functions of the incident angle $\theta$ for different chemical potentials $\mu_{N/S}$. The incident quasiparticle energy is taken as $E=0.6\Delta$ and $E_{z}$ is fixed at $0.3\Delta$. For $\mu_{N}=\mu_{S}=0$, ELQs and HLQs in NLSM have the same energy dispersions. The isoenergetic circles for ELQs and HLQs in the $k_{x}$-$k_{y}$ plane coincide with each other as presented in Fig.1 (c). The energy gap is symmetric about $E=0$. The ELQs on the conduction band $E_{e}^{+}$ and HLQs on the valence band $E_{h}^{+}$ participate in the scattering processes which are shown in Figs.1(c) and (e). When $E<E_{z}$, all reflections disappear due to the absence of the incident states in the gap. But, as long as $E>E_{z}$, there will be double Andreev reflections and double normal reflections (see Fig.3 (a)). The variations of SNR $R_{n1}$ and RAR $R_{a2}$ are complementary. That is to say, SNR will acquire its maximum (minimum) value if RAR gets its minimum (maximum) value. Actually, this is an universal character when four types of reflections coexist. RNR $R_{n2}$ exhibits oscillating behaviour and obtains its peak value near $\theta=\pi/2$. In contrast, the magnitude of SAR $R_{a1}$ is higher near $\theta=0$.

For $\mu_{N}=\mu_{S}>0$, the energy bands for ELQs and HLQs split. The energy for ELQs is lowered and that for HLQs is raised. The energy gap is asymmetric about $E=0$. When $\mu_{N}=\mu_{S}=0.2\Delta$, the gap for ELQs spans from $-0.5\Delta$ to $0.1\Delta$ and that for HLQs spans from $-0.1\Delta$ to $0.5\Delta$. The bands involved in the scattering processes are still the conduction band $E_{e}^{+}$ and valence band $E_{h}^{+}$ since $\mu_{N}<E_{z}$. The four reflections are presented in Fig.3(b). Comparing with Fig.3(a), the changes of double Andreev reflections are not obvious while SNR and RNR are slightly elevated and reduced, respectively. The probability for SAR $R_{a1}$ is obviously larger than RNR $R_{n2}$ in a wide angle range around $\theta=0$.

For $\mu_{N}=\mu_{S}=0.4\Delta$, the gap for ELQs is from $-0.7\Delta$ to $-0.1\Delta$ and that for HLQs is from $0.1\Delta$ to $0.7\Delta$. The incident energy $E=0.6\Delta$ is just in the gap of HLQs and crosses the conduction band $E_{e}^{+}$ of ELQs. Therefore, only ELQs will be reflected and the Andreev reflections vanish with $R_{a1}=R_{a2}=0$ exactly. The probabilities of the double normal reflections are given in Fig.3(c), which exhibit inverse phase oscillations. When we increase the chemical potentials, the lower edge of the HLQs gap moves up while the upper edge of the ELQs gap moves down. If $\mu_{N}=\mu_{S}=0.9\Delta$, the energy $E=0.6\Delta$ still crosses the conduction band $E_{e}^{+}$ and at the same time it is tangent to the lower edge of the HLQs gap. For the lager values of the chemical potentials such as $\mu_{N}=\mu_{S}=1.4\Delta$ in Fig.3(d), the energy $E=0.6\Delta$ will simultaneously intersect the conduction bands $E_{e}^{+}$ and $E_{h}^{-}$ and double Andreev reflections reoccur. In this case, both the incident ELQs and outgoing HLQs are in the conduction bands. Here RNR $R_{n2}$ reaches its peak value at $\theta=0$ and RAR $R_{a2}$ is suppressed slightly around $\theta=0$ comparing to Figs.3(a) and (b). From the above analyses, it is found that the positions of band gaps and hence the reflection processes in NLSM-SC junctions can be tuned by the chemical potential which can be controlled by an gate voltage.

Fig.4 gives the incident angle dependence of reflection probabilities for different interfacial barrier heights. We have taken $E_{z}=0.4\Delta$, $\mu_{N}=\mu_{S}=0.2\Delta$ and $E=0.7\Delta$. Under these parameters, the conduction band $E_{e}^{+}$ and valence band $E_{h}^{+}$ both participate in the scattering. However, for the transparent interface with $V_{0}=0$, there are only retro-reflections (RNR and RAR) as shown in Fig.4(a). In particular, the SNR disappears also, which is similar to the ordinary normal-metal-SC junction.Blonder The probabilities are approximatively independent of the incident angle. In fact, the parameters $\mu_{N}$, $\mu_{S}$, the incident energy $E$, and $E_{z}$ in Fig.4(a) are much smaller than $E_{0}=200\Delta$, leading to the wave vectors $k_{x}^{e\pm}$ and $k_{x}^{h\pm}$ in NLSM and $p_{x}^{\pm}$ and $q_{x}^{\pm}$ for quasiparticles in SC can well be simplified into $\sqrt{\frac{2m}{\hbar^{2}}E_{0}-k_{y}^{2}}=\frac{2m}{\hbar^{2}}E_{0}\cos{\theta}$. Under this simplification and $V_{0}=0$, the RNR and RAR coefficients can be analytically obtained as

which are independent of the incident angle $\theta$, and SNR and SAR coefficients $r_{n1}=r_{a1}=0$ exactly. As a fact, a finite barrier height is the necessary condition for the formation of the specular reflections. Even for a small barrier height with $V_{0}=2$ (see Fig.4(b)), the specular reflections will immediately appear although they are not dominant in a large angle range centered at $\theta=0$. When the barrier height is raised, SNR $R_{n1}$ and SAR $R_{a1}$ are markedly enhanced and the curve of SAR $R_{a1}$ exceeds that of RNR $R_{n2}$ as shown in Fig. 4(c) with $V_{0}=5$ in a large range. When $V_{0}=10$, the probabilities of SNR and SAR at $\theta=0$ can achieve about $34\%$ and $12\%$, respectively. The presence of finite barrier height is beneficial to the specular reflections and detrimental to the retro-reflections. The reflection styles and magnitudes can be regulated by the interfacial transparency.

Fig.5 gives the reflection probabilities as functions of the incident angle $\theta$ for different $E_{z}$ (i.e., the incident wave vector $k_{z}$). For $E_{z}=0$, the orbital coupling in the Hamiltonian $\check{H}_{N}$ (see Eq.(1)) and the gaps for ELQs and HLQs vanish. The energy dispersions degenerate to $E_{e}^{\pm}=\pm\epsilon_{k}-\mu_{N}$ and $E_{h}^{\pm}=\pm\epsilon_{k}+\mu_{N}$. When ELQs on the conduction band $E_{e}^{+}$ are injected from NLSM to the NLSM-SC interface, they will be normal reflected as ELQs on $E_{e}^{+}$ and Andreev reflected as HLQs on the valence $E_{h}^{-}$. In other words, only two reflections, SNR and RAR, can happen as shown in Fig.5(a). In this situation, the scattering processes in NLSM-SC junctions are the same with those in the ordinary normal metal-SC junctions.Blonder For $E_{z}\neq 0$, the coupling of two orbits leads to the appearances of RNR and SAR. Double normal reflections and double Andreev reflections are realized as shown in Figs.5(b) and (c). The conduction band $E_{e}^{+}$ and the valence band $E_{h}^{+}$ are involved in the scattering. With the increase of $E_{z}$, RNR $R_{n2}$ and SAR $R_{a1}$ are enhanced. However for $E_{z}=0.7\Delta$ and $\mu_{N}=\mu_{S}=0.2\Delta$, the situation is different. Under these parameters, the energy $E=0.8\Delta$ lies in the gap of HLQs and crosses the band $E_{e}^{+}$ of ELQs. The Andreev reflections (RAR and SAR) disappear and only the normal reflections of ELQs are possible. The curves of SNR and RNR oscillate with the incident angle $\theta$, which have two valley values and two peak values respectively as shown in Fig.5(d).

Now, we turn to the incident energy dependence of reflection probabilities. Fig.6 presents the results for different $\theta$ at $\mu_{N}=\mu_{S}=0.7\Delta$ and $E_{z}=0.4\Delta$. The gap for HLQs spans from $0.3\Delta$ to $1.1\Delta$. As a result, the Andreev reflections, both RAR and SAR, are absent in the energy range $0.3\Delta<E<1.1\Delta$. The double Andreev reflections can happen in the other energy range, $E<0.3\Delta$ or $E>1.1\Delta$, as shown in Figs.6 (a)-(d). For $E<0.3\Delta$, the double Andreev reflections originate from the involved conduction band $E_{h}^{-}$. On the other hand, while for $E>1.1\Delta$, the double Andreev reflections originate from the involved valence band $E_{h}^{+}$. For $E<0.3\Delta$, the changes of RAR and SAR are not dramatic. When the incident energy $E$ increases from $1.1\Delta$, the curves of RAR and SAR rise first and then gradually decrease. Note, the summation of the reflection probabilities for $E>\Delta$ will not equal to $1$ due to the emergence of the quasiparticle transmissions. As for the normal reflections, SNR $R_{n1}$ and RNR $R_{n2}$ oscillate dramatically in the range $0.3\Delta<E<\Delta$ while the changes in the range $E<0.3\Delta$ and $E>1.1\Delta$ are not obvious. In addition, the Andreev reflections, both RAR and SAR, will tend to zero as $\theta$ approaches to $0.5\pi$ as shown in Fig.6 (d). This is consistent with the incident angle dependence of reflections discussed above.

Fig.7 presents the incident energy dependence of reflections for different $E_{z}$ at $\mu_{N}=\mu_{S}=0.4\Delta$ with the fixed incident angle $\theta=0.2\pi$. For $E_{z}=0$, there are only SNR and RAR in the scattering processes due to the orbital coupling is absent (see Fig.7(a)). For $E_{z}=0.2\Delta$, the gap for HLQs is from $0.2\Delta$ to $0.6\Delta$. Correspondingly, the Andreev reflections are absent in the range $0.2\Delta<E<0.6\Delta$ (see Fig.7(b)). In this energy range, the behaviours of the normal reflections are similar to the results in Fig.6. In the energy range $0.6\Delta<E<\Delta$, the probability of SNR drops sharply and RAR $R_{a2}$ dominates the scattering processes. For $E_{z}=0.4\Delta$, the gap for HLQs is from $0$ to $0.8\Delta$ and there are only normal reflections in the energy range $E<0.8\Delta$ (see Fig.7 (c)). For $E_{z}=0.6\Delta$, the gap for HLQs spans from $0$ to $\Delta$, the Andreev reflections will be absent inside the whole gap (see Fig.7 (d)).

Fig.8 gives the reflection probabilities for different $\mu_{N}$ and $\mu_{S}$ at $E_{z}=0.4\Delta$ and $\theta=0.2\pi$. For $\mu_{N}=\mu_{S}=0$, the gaps for ELQs and HLQs are both from $-0.4\Delta$ to $0.4\Delta$ which are symmetric about $E=0$. For $E<0.4\Delta$, the energy of the incident quasiparticles lies in the gaps and there is no traveling wave for ELQs. The reflections, both the normal ones and Andreev ones, are not present as shown in Fig.8 (a). When $E>0.4\Delta$, ELQs on the conduction band $E_{e}^{+}$ and HLQs on the valence band $E_{h}^{+}$ participate in the scattering processes. Double normal reflections and double Andreev reflections exist simultaneously. It is worth noting that the summation of reflection probabilities equals to $1$ from $E=0.4\Delta$ to $E\approx 1.077\Delta$ not to $\Delta$ (see Fig.8(a)). This indicates no quasiparticle transmissions in the energy range $\Delta<E<1.077\Delta$ and the superconducting gap in SC is larger than $\Delta$ in this case. Actually, the quasiparticle dispersion in SC is $E=\sqrt{\epsilon_{k}^{2}+E_{z}^{2}+\Delta^{2}}$ when $\mu_{S}=0$. The effective gap in SC is $\sqrt{E_{z}^{2}+\Delta^{2}}$ not $\Delta$ which is consistent with the results in Fig.8(a).

When $\mu_{S}\neq 0$, the quasiparticle dispersion in SC becomes $E=\sqrt{[\sqrt{\epsilon_{k}^{2}+E_{z}^{2}}-\mu_{S}]^{2}+\Delta^{2}}$. When $\mu_{S}<E_{z}$, the effective gap in SC is $\sqrt{[|E_{z}|-\mu_{S}]^{2}+\Delta^{2}}$. For example, if $E_{z}=0.4\Delta$ and $\mu_{S}=0.2\Delta$ as considered in Fig.8(b), the effective gap will approximate to $1.02\Delta$ (see also Fig.7(d)). At the same time, there will be a gap from $-0.6\Delta$ to $0.2\Delta$ for ELQs in NLSM. Hence, there are no reflections in the range $0<E<0.2\Delta$ and the sum of probabilities for four reflections equals to $1$ in the range $0.2\Delta<E<\sqrt{[|E_{z}|-\mu_{S}]^{2}+\Delta^{2}}\simeq 1.02\Delta$. When $\mu_{S}$ is raised to $\mu_{S}=E_{z}$, the gap in SC reverts to its intrinsic value $\Delta$ and the gap for ELQs in NLSM completely locates below $E=0$, which leads to the reflections probabilities shown in Fig.8(c). Here the Andreev reflections disappear at $E<0.8\Delta$ due to the gap for HLQs in NLSM. When $\mu_{S}$ continues to rise, the gap in SC will not be changed and keep the value $\Delta$. For $\mu_{N}=\mu_{S}=0.6\Delta$ in Fig.8(d), ELQs and HLQs involved in the scattering are from the conduction bands $E_{e}^{+}$ and $E_{h}^{-}$ in the energy range $0<E<0.2\Delta$ while for $0.2\Delta<E<\Delta$, only ELQs from $E_{e}^{+}$ participate in the scattering.

Fig.9 shows the incident energy dependence of reflections for different interfacial barriers at $\mu_{N}=\mu_{S}=0.7\Delta$ and $E_{z}=0.4\Delta$. Since the relation $\mu_{N}>E_{z}$ is fixed, the conduction band $E_{e}^{+}$ is always involved in the scattering processes while the dispersions for HLQs have a gap from $0.3\Delta$ to $1.1\Delta$. For $0.3\Delta<E<1.1\Delta$, there will be no Andreev reflections as given in Fig.9 (a)-(d). When the interface is transparent, the specular reflections, SNR and SAR, are also absent. Therefore, there will be only RNR in the energy range $0.3\Delta<E<1.1\Delta$ for $V_{0}=0$ with $R_{n2}=1$, as shown in Fig.9(a). In this case, the incident ELQs are completely reflected back along the retro-reflected direction, which phenomenon is very rare. In the energy range $E<0.3\Delta$, the conduction band $E_{h}^{-}$ also participates in the scattering and two retro-reflections coexist. This is consistent with the results in Fig.4(a). When $V_{0}\neq 0$, the specular reflections are activated. For $0.3\Delta<E<1.1\Delta$, there will be double normal reflections and for $E<0.3\Delta$ or $E>1.1\Delta$, there will be four types of reflections. From Figs.9 (b)-(d), we can find the specular reflections are enhanced as the opacity of the interface is increased. In other words, the presence of finite barrier height is beneficial to the specular reflections and detrimental to the retro-reflections. This conclusion is consistent with the results in Figs.4 (b)-(d).

Here, we discuss the conductance of the NLSM-SC junctions. In the expression of conductance in Eq.(15), there is an integral of $E_{z}$ from $-eV_{b}$ to $eV_{b}$ with $V_{b}$ the bias. Since the probabilities are even functions of $E_{z}$, it is enough to analyze the conductance supposing $E_{z}>0$. Fig.10(a) gives the normalized conductance for the transparent interface with the barrier potential $V_{0}=0$. According to the discussions in the last subsection, the specular reflections in this situation are absent and only RNR and RAR exist. For $\mu_{N}=0$, the gaps for ELQs and HLQs in NLSM are symmetric about the energy $E=0$ and span from $-E_{z}$ to $E_{z}$. The expression of conductance requires $eV_{b}>E_{z}$, which indicates the conduction band $E_{e}^{+}$ and the valence band $E_{h}^{+}$ always participate in the scattering processes. Inside the gap, the total RAR will lead to the normalized conductance being $2$. This is the same value with the conductance of the ordinary normal-metal-SC junctions with the transparent interface Blonder .

For $\mu_{N}=\mu_{S}\neq 0$, HLQs are not always involved in the scattering. The gap for HLQs is no longer symmetric about $E=0$ and spans from $\mu_{N}-E_{z}$ to $\mu_{N}+E_{z}$. The condition for RAR is $eV_{b}+E_{z}<\mu_{N}$ or $eV_{b}-E_{z}>\mu_{N}$ (Note, SAR is absent due to $V_{0}=0$.). In the former situation, HLQs from the conduction band $E_{h}^{-}$ are responsible for RAR. For $eV_{b}<\mu_{N}$, when $eV_{b}$ is increased, the value range of $E_{z}$ satisfying the former condition becomes narrow. Correspondingly, the $E_{z}$ range for RAR also narrows and the conductance decreases. In the latter situation, HLQs from the valence band $E_{h}^{+}$ are responsible for RAR. For $eV_{b}>\mu_{N}$, when $eV_{b}$ is increased, the value range of $E_{z}$ satisfying the latter condition becomes wide. The $E_{z}$ range for RAR also widens and the conductance rises. For the special case of $eV_{b}=\mu_{N}$, no value of $E_{z}$ under the supposed relation $E_{z}>0$ meets the two conditions for RAR. There is only the total RNR in the scattering processes which will cause the zero conductance. Our analysis can be immediately demonstrated by the numerical results with $\mu_{N}=\mu_{S}=0.1\Delta$, $0.5\Delta$ and $1.0\Delta$ in Fig.10(a). For $\mu_{N}=\mu_{S}>\Delta$, the zero conductance phenomenon will not disappear, since quasiparticle transmissions happen at the incident energy larger than $\Delta$.

Fig.10(b) gives the normalized conductance for $V_{0}=10$. According to the discussions in the last subsection, the specular reflections will be activated. For $\mu_{N}=0$, double normal reflections and double Andreev reflections coexist. The presence of the normal reflections will weaken the conductance and its value can not reach $2\sigma_{0}$ inside the gap. For $\mu_{N}\neq 0$, the condition for the Andreev reflections, both RAR and SAR, is also $eV_{b}+E_{z}<\mu_{N}$ or $eV_{b}-E_{z}>\mu_{N}$. For $eV_{b}=\mu_{N}$, the conductance becomes zero. On the both sides of $eV_{b}=\mu_{N}$, the variations of conductance are similar to the curves in Fig.10(a). The conductance for $\mu_{N}=\mu_{S}=5\Delta$ at $V_{0}=10$ is also presented in Fig.10(b) and its properties are similar to the curves with $\mu_{N}=\mu_{S}=0$. Although the impressive vanishing conductance at $eV_{b}=\mu_{N}$ are reminiscent of graphene-SC junctions Beenakker , the chemical potential dependence of conductance spectra is very different. For the case of graphene, the single Andreev reflection, RAR or SAR, happens depending on the value of the chemical potential.