It is well known that determining the symmetry of pairing potentials in high-T_C superconductors is crucial for elucidating the mechanism of superconductivity [1]. Theoretical and experimental studies have confirmed that cuprate superconductors have d-wave pairing symmetry [2–4], as well as p-wave pairing symmetry in certain bulk ferromagnetic SCs [5, 6]. The authors of [7] theoretically predicted that a YBa₂Cu₃O_6+x cuprate has f-wave symmetry. They even suggested the possibility of observing the f-wave symmetry of superconducting pairing potential with YBCO cuprate in future experiments. So far, many researchers have studied the superconducting tunneling structure involving f-wave pairing symmetry and revealed many important characteristics of their transport characteristics [8–10].

On the other hand, the interplay between ferromagnetism and superconductivity on the surface of topological insulators (TIs) is also highly desirable due to the unique band structure of TIs [11–15]. For example, Soodchomshom discussed the effect of magnetic gap on the tunneling conductance in a topological insulator ferromagnet/superconductor (TI FM/SC) junction [16]. Kang et al investigated the effect of magnetic field on the transport properties of FM/FM/s-wave SC junctions formed on the surface of a three-dimension TI [17]. In addition, the interaction between ferromagnetism and unconventional SC in tunnel junctions based on TI was also investigated. For example, the transport behaviors of quasi particles in TI-based FM/d-wave and f-wave superconducting tunnel structures have been discussed in [18, 19], respectively. All these studies suggest that new qualitative effects may occur when ferromagnetic and superconducting orders coexist in the TI environment. Recently, Majidi and Asgari [20] accurately studied the Andreev reflection (AR) [21] process in superconducting tunnel junctions based on TI. By analyzing the transport properties in the presence of a gate electric field, they theoretically revealed the potential to realize the intra-band specular AR in normal metal/s-wave SC tunnel structures based on TI thin-films. In our previous work, we investigated the specular AR and the conventional retro-reflection in TI-based FM/s-wave and ${d}_{{x}^{2}-{y}^{2}}+i{d}_{xy}$ mixed-wave SC tunnel structures [22, 23]. It is found that the specular AR exists in both types of SC junctions, and the AR process strongly depends on the superconducting pairing potential. However, so far, the AR process especially the specular AR process and the shot noise in FM/f-wave superconducting tunnel structures have not been discussed in detail.

In this work, we aim to theoretically study the magnetic gap effect and the lifetime of quasi particles on the transport properties in FM/f-wave SC tunnel structures based on a thin film TI. Two types of pairing symmetry for SC are chosen: f₁ and f₂ waves. Using the Dirac–Bogoliubov–de Gennes (DBdG) equation [24] and Blonder–Tinkham–Klapwijk (BTK) theory [25], expressions for the calculation of conductance and shot noise are constructed. Through the analysis of conductance spectrum and shot noise spectrum, the AR processes including specular Andreev reflection and conventional Andreev retro-reflection are discussed in detail.

The model under study is a ferromagnet/f-wave SC junction based on a three-dimensional (3D) thin film TI. The TI surface is considered as the (x, y) plane, and the tunnel current is limited to this plane. In the region of $x\leqslant 0,$ an external magnetic field is applied along the z-direction to induce the magnetization of the FM. Because the z component of magnetization can induce a magnetic energy gap of approximately 2m in the carrier state density of the FM [14], in our calculations, magnetization is perpendicular to the TI surface, i.e. $\mathop{m}\limits^{ \rightharpoonup }=(0,0,m).$ In the $x\gt 0$ region, an external gate potential with a value U is placed above the f-wave SC, resulting in the Fermi energy in the superconducting region being E_F + U, where E_F represents the Fermi energy in the ferromagnetic region.

The f₁ and f₂-wave symmetries of the superconducting pairing potential can be written as [19]where ${{\rm{\Delta }}}_{0}$ is the maximum superconducting energy gap, ${\theta }_{S}$ represents the incident angle of electrons in the superconducting region, and $\varphi $ is the phase of the order parameter.

The DBdG equation used to describe the FM/ f-wave SC junction based on TI reads [16]where ${v}_{F}$ is the Fermi velocity, $\mathop{\sigma }\limits^{ \rightharpoonup }$ represents the vector of Pauli spin matrices, and $\varepsilon $ is the excitation energy of the quasi particle. Considering the finite lifetime of quasi-particle (${\tau }_{0}$), the BCS-coherence factor is given by ${u}^{2}\,=1-{v}^{2}\,=\displaystyle \frac{1}{2}\left(1+\sqrt{1-\tfrac{{{{\rm{\Delta }}}_{f}}^{2}}{{\left(\varepsilon +{\rm{i}}{\rm{\Gamma }}\right)}^{2}}}\right)$ with ${\rm{\Gamma }}=\tfrac{\hslash }{{\tau }_{0}}$ [26].

Defining $\theta $ is the incident angle of a quasi-particle, the wave function in the FM region is written aswith ${X}_{e}={\left(1\,\displaystyle \frac{{E}_{F}-m}{\sqrt{{{E}_{F}}^{2}-{m}^{2}}}{{\rm{e}}}^{{\rm{i}}\theta }0\,0\right)}^{{\rm{T}}},$ ${X}_{h}={\left(0\,0\,1\,\displaystyle \frac{{E}_{F}-m}{\sqrt{{{E}_{F}}^{2}-{m}^{2}}}{{\rm{e}}}^{-{\rm{i}}\theta }\right)}^{{\rm{T}}}\,,$ r_h and r_e are the normal and AR scattering coefficients, respectively. The wave function in the superconducting region is given bywhere ${Y}_{e}={\left(u\,u{{\rm{e}}}^{{\rm{i}}{\theta }_{S}}-v{{\rm{e}}}^{{\rm{i}}{\theta }_{S}-{\rm{i}}\varphi }\,v{{\rm{e}}}^{-{\rm{i}}\varphi }\right)}^{{\rm{T}}},$ ${Y}_{h}={\left(v-v{{\rm{e}}}^{-{\rm{i}}{\theta }_{S}}\,u{{\rm{e}}}^{-{\rm{i}}{\theta }_{S}-{\rm{i}}\varphi }\,u{{\rm{e}}}^{-{\rm{i}}\varphi }\right)}^{{\rm{T}}},$ t_e and t_h are the coefficients of transmission as electron- and hole-like carriers, respectively. The wave functions of these two regions satisfy the continuity condition at interface x = 0. Therefore, all coefficients in equations (3) and (4) can be obtained by solving the equation ${\psi }_{FM}(0)={\psi }_{SC}(0).$

According to the modified BTK theory, the conductance of the FM/f-wave SC junction based on TI can be calculated using the following formula [16]Here ${R}_{h}={\left|{r}_{h}\right|}^{2},$ ${R}_{e}={\left|{r}_{e}\right|}^{2},$ the critical angle is given by ${\theta }_{c}={\sin }^{-1}\tfrac{1+U/{E}_{F}}{\sqrt{1-{\left(m/{E}_{F}\right)}^{2}}},$ and the non-magnetic normal conductance is written as ${g}_{0}=\tfrac{n({E}_{F})w{e}^{2}}{2\pi {\hslash }^{2}{v}_{F}}$ with $n({E}_{F})\sim \tfrac{{E}_{F}}{2\pi {(\hslash {v}_{F})}^{2}}$ the density of the normal state and w representing the width of the junction.

If I is defined as the tunneling current and $\hat{I}(t)$ is the current operator, then the zero-frequency shot noise can be expressed as $S=2\displaystyle {\int }_{-\infty }^{+\infty }{\rm{d}}t\left\langle \left[\hat{I}(t)-\left\langle I\right\rangle \right]\times \left[\hat{I}(0)-\left\langle I\right\rangle \right]\right\rangle $ [27, 28]. At the same time, considering the important role of the Andreev reflection process in the transport of superconducting tunnel junctions, the differential shot noise can be obtained using the following formula [29, 30]

Figure 1 gives the results for the influence of magnetic gap m and quasi-particle lifetime on the conductance in FM/f₁-wave SC junctions based on TI, where (a), (b) and (c) respectively show the case of ${\rm{\Gamma }}=0,$ $0.8{{\rm{\Delta }}}_{0},$ and ${{\rm{\Delta }}}_{0}.$ It can be seen from figure 1(a), when ${\rm{\Gamma }}=0,$ that is, without considering the lifetime of quasi particles, two peaks appear at $eV=\pm {{\rm{\Delta }}}_{0}$ in the case of m = 0. These conductance peaks arise due to Andreev resonance occurring at $eV=\pm {{\rm{\Delta }}}_{0}.$ As shown in figures 1(b) and (c), with the increase of ${\rm{\Gamma }},$ namely with the decrease of the finite lifetime of quasi particles, the distance between two conductance peaks gradually decreases until they merge into a zero-bias conductance peak (ZBCP). According to our calculation, further increasing ${\rm{\Gamma }}$ will smooth out the ZBCP. At the same time, one finds that when $m/{E}_{F}\gt 1,$ as ${\rm{\Gamma }}$ increases, the width of ZBCP narrows and the height decreases until it reaches zero. These features of the conductance spectrum suggest that shortening the lifetime of the quasi-Dirac particle will prevent the formation of the Andreev bound state in FM/f₁-wave SC junctions based on TI. Comparing figures 1(a)–(c), we can also see that with m increasing from zero to E_F, the value of the conductance across the entire energy range decreases to zero regardless of whether the quasi-particle lifetime is considered. When $m\gt {E}_{F},$ the conductance values in the energy range of approximately $(-\sqrt{{{{\rm{\Delta }}}_{0}}^{2}-{{\rm{\Gamma }}}^{2}},\sqrt{{{{\rm{\Delta }}}_{0}}^{2}-{{\rm{\Gamma }}}^{2}})$ increase with an increase in m. This shows that the tunneling conductance is highly dependent on the magnetic gap. If the excitation energy $\varepsilon $ of a quasiparticle is less than ${{\rm{\Delta }}}_{0},$ we can easily verify that ${R}_{e}+{R}_{h}=1,$ because particles with energy below the gap is forbidden to transmit to SC [20]. Therefore, equation (5) becomes $G=2{g}_{0}\sqrt{1-{(m/{E}_{F})}^{2}}\displaystyle {\int }_{-{\theta }_{c}}^{{\theta }_{c}}{R}_{h}\,\cos \,\theta d\theta $ in the case of $\varepsilon \lt {{\rm{\Delta }}}_{0}.$ Correspondingly, the tunneling conductance in the energy range of $(-{{\rm{\Delta }}}_{0},{{\rm{\Delta }}}_{0})$ represents the amplitude of the Andreev reflection. It can be reasonable to infer that in the $m\lt {E}_{F}$ case, the Andreev retro-reflection process mainly determines the characteristics of the tunneling conductance spectrum. However, in the case of $m\gt {E}_{F},$ the specular reflection becomes dominant. This feature indicates that increasing the magnetic gap effect of the TI-based FM/f₁-wave SC junction will enhance the specular Andreev reflection but suppress the conventional Andreev retro-reflection.

Figure 2 plot the interplay between the magnetic gap effect and the quasi-particle lifetime on the conductance of the FM/f₂-wave SC junction based on TI, in which (a), (b) and (c) are the cases of ${\rm{\Gamma }}=0,$ $0.2{{\rm{\Delta }}}_{0}$ and $0.4{{\rm{\Delta }}}_{0},$ respectively. Unlike figure 1(a), when ignoring the lifetime of quasi particles, a ZBCP is obtained for $m\lt {E}_{F}$ as shown in figure 2(a). For $m\gt {E}_{F},$ the ZBCP also appears but sharper compared to $m\lt {E}_{F}$ situation. These different spectral line features are attributed to two different natures of order parameters of f-wave superconductivity, which can be used to distinguish between f₁-wave and f₂-wave. As ${\rm{\Gamma }}$ increases, the ZBCP rapidly narrows and decreases until becoming smooth as can be seen in figures 2(b) and (c). This indicates that the influence of quasi-particle lifetime on the conductance spectrum is greater in the TI-based FM/f₂-wave SC tunnel structure than in the f₁-wave SC tunnel structure. However, comparing figures 1 and 2, we find that the influence of m on the conductance spectrum of TI-based FM/f₂-wave superconducting tunnel structure is the same as that of the f₁-wave SC junction. It then follows that the specular Andreev reflection and Andreev retro-reflection in the TI-based FM/f-wave SC tunneling structure can be distinguished by the phenomenon that the tunneling conductance disappears at $m={E}_{F}.$ This criterion is also applicable to the FM/s-wave, ${d}_{{x}^{2}-{y}^{2}}+i{d}_{xy}$ mixed wave and p-wave SC tunneling structure based on TI [22, 23, 31].

The interaction of the magnetic gap and quasi-particle lifetime on the shot noise in the FM/f₁-wave SC junction based on TI is presented in figure 3, where (a), (b), (c) and (d) give the cases of ${\rm{\Gamma }}=0,$ ${\rm{\Gamma }}=0.8{{\rm{\Delta }}}_{0},$ ${\rm{\Gamma }}={{\rm{\Delta }}}_{0}$ and ${\rm{\Gamma }}=1.2{{\rm{\Delta }}}_{0},$ respectively. As shown in figure 3(a), there are two dips obtained at $eV=\pm {{\rm{\Delta }}}_{0}$ in the shot noise spectrum when ${\rm{\Gamma }}=0,$ m = 0, or $m\ll {E}_{F}.$ As m increases, the two dips transforms into two peaks. Comparing figures 3(a)–(d), one can see that the distance between the two dips or peaks gradually decreases with the increase of ${\rm{\Gamma }},$ and then merges into a zero-bias shot noise dip (ZBSD) or ZBSP, finally becomes smooth for $m\lt {E}_{F}.$ In order to see this transformation more clearly, the zero-bias shot noise spectrum for $m=0$ is presented in figure 4. These features of the shot noise spectrum indicate that shortening the lifetime of quasi-Dirac particles will prevent the formation of Andreev resonance peaks and eliminate the dips of FM/f₁-wave SC junctions based on TI. It should be noted that the differences of the effect of m on the conductance, when the amplitude of ${\rm{\Gamma }}$ is small, as m increases from zero, the value of shot noise, especially approximately within the energy range of $(-\sqrt{{{{\rm{\Delta }}}_{0}}^{2}-{{\rm{\Gamma }}}^{2}},\sqrt{{{{\rm{\Delta }}}_{0}}^{2}-{{\rm{\Gamma }}}^{2}}),$ first increases, then decreases to zero when $m/{E}_{F}=1,$ and then increase rapidly for $m\gt {E}_{F}.$ This result shows that the shot noise spectrum of the FM/f₁-wave SC junction based on TI strongly depends on both the magnetic gap effect and the quasi-particle lifetime.

Figure 5 shows the influence of the magnetic gap effect and quasi-particle lifetime on the shot noise spectrum of the FM/f₂-wave SC junction based on TI, where (a), (b) and (c) give the cases of ${\rm{\Gamma }}=0,$ $0.2{{\rm{\Delta }}}_{0}$ and $0.4{{\rm{\Delta }}}_{0},$ respectively. We can see from figure 5(a) that, unlike figure 3(a), a ZBSP is obtained when $m\lt {E}_{F}$ and the quasi-particle lifetime is ignored. There is a split in the ZBSP for $m\gt {E}_{F}.$ These differences stem from two different natures of order parameters of the f-wave SC, which can also be used as a criterion to distinguish between f₁-wave and f₂-wave. One can see from figures 5(b) and (c) that the value of shot noise decreases and the splitting is gradually smoothed out as the quasi-particle lifetime decreases. This indicates that reducing the lifetime of quasi-Dirac particles will both prevent the formation of zero-bias Andreev bound states and suppress proximity effects. Comparing figures 5, 3, 2, and 1, we can see that the shot noise and tunneling conductance have zero values at $m={E}_{F}$ for the entire energy range regardless of whether the quasi-particles lifetime is considered. This can also be easily obtained from equations (5) and (6). At the same time, according to our calculations, the Andreev reflection scattering coefficient ${r}_{h}=0,$ i.e., there is no Andreev reflection for any incident angle. Therefore, $m={E}_{F}$ is the transition point from Andreev reflection being the predominant process to Andreev retro-reflection being the predominant process of the FM/f-wave SC junction based on TI.

In this paper, the interplay between the quasi-particle lifetime and the magnetic gap effect on the conductance spectrum and the shot noise spectrum of the FM/f₁-wave and f₂-wave SC junction based on TI are theoretically investigated. It is found that due to the two different natures of order parameters of the f-wave superconductivity, for f₁-wave and f₂-wave, there are differences in both tunneling conductance spectrum and shot noise spectrum. These different features can be used to distinguish between f₁-wave and f₂-wave. Additionally, we have found that reducing the lifetime of quasi particles will transform the energy-gap peaks into a zero-bias peak in the conductance spectrum, transform the energy-gap dips into a zero-bias dip in the shot noise spectrum, and smooth out the zero-bias conductance peak as well as the zero-bias shot noise dip. An increase in the magnetic gap will suppress the tunnel conductance and shot noise when the conventional Andreev retro-reflection dominates, while it will enhance the conductance and shot noise when the specular Andreev reflection is dominant. The increase of the quasi-particle lifetime will enhance both specular Andreev reflection and Andreev retro-reflection. When Fermi energy equals the magnetic gap, two types of reflections will transform from one dominant to the other dominant.

In summary, the results show that both tunnel conductance and shot noise exhibit different behaviors with changes in the magnetic gap for $m\gt {E}_{F}$ and $m\lt {E}_{F},$ and they also exhibit different behaviors with changes in quasi-particle lifetime for f₁ and f₂ waves. These different features not only help to better understand the specular Andreev reflection in the FM/f–wave SC junction based on TI but also provide insight into experimentally determining the f-wave pairing symmetry. Furthermore, the phenomenon of tunnel conductance and shot noise vanishing at $m={E}_{F}$ can be used to experimentally distinguish specular Andreev reflection from Andreev retro-reflection.