Zero bias conductance peak related to spin triplet states in noncollinear magnetized graphene superconducting junctions

Abstract

We investigated the zero-bias conductance peak in a graphene-based ferromagnet/ferromagnet/barrier/d-wave superconductor (F/F/B/d-wave SC) heterojunction. Our research indicates that the spin-triplet pairing states induced by non-collinear magnetizations do not lead to the splitting of the zero-bias conductance peak (ZBCP), and the anomalous Andreev reflection makes a significant contribution to the ZBCP. In the case of half-metal, the triplet bound states appear at zero incident energy due to Klein tunneling, which is coincide with the singlet bound states, resulting in the ZBCP arises solely due to spin-triplet pairing states. The ZBCP can be modulated by the exchange field strength, Fermi level and magnetizations angle. These findings offer deeper understanding of the influence of non-collinear magnetizations on anomalous Andreev reflection in graphene-based F/F/B/d-wave SC heterojunctions and hold promise for the development of graphene-based superconducting spintronic devices.

Introduction

In d-wave superconductors, the pair potential of electron and hole quasiparticles have opposite signs at the Fermi level, giving rise to zero-energy bound states (Andreev bound states)^1,2,3,4,5. For instance, in an N/d-wave superconductor-silicene junction, a zero-bias conductance peak (ZBCP) associated with the spin-singlet pairing states is observed via scanning tunneling spectroscopy and can be utilized to probe d-wave superconducting properties^4,6,7. In ferromagnet (F)/d-wave superconductor traditional-material-based junctions, as the exchange field strength in the F region increases, the ZBCP value decreases and becomes a zero-bias conductance dip (ZBCD) when the F region is fully polarized^8,9,10. Additionally, the spin-triplet pairing states near the ferromagnet/ferromagnet/d-wave superconductor interface can emerge due to non-collinear magnetizations. Niu et al. reported that in a ferromagnet (F)/ferromagnet (F)/d-wave superconductor heterojunction with CuO$_2$, the ZBCP splits into two peaks at zero incident energy when the F region is highly polarized¹¹.

Over the past two decades, since its initial isolation by Andre Geim and Kostya Novoselov in 2004, numerous researchers have devoted considerable efforts to understanding and studying graphene, a single layer of carbon atoms with a honeycomb lattice structure^{12,13,14,15,16}. Its dispersion relation is linear at low energies, and its Fermi level can be manipulated by a gate voltage. The proximity-induced d-wave pairing symmetry can be achieved experimentally by depositing superconducting contacts on a graphene sheet^{16,17,18,19,20,21}, leading to extensive investigations of the transport properties of graphene-based d-wave superconducting junctions. For example, the effect of exchange field strength (h) on the ZBCP has been theoretically investigated in graphene-based F/I/d-wave superconductor (SC) junctions, showing that as the exchange field strength h increases, the height of the ZBCP can be reduced when h is less than the Fermi level ($E_F$ )^23,24,25. It has also been demonstrated that the ZBCP exhibits periodic changes with variations in barrier strength in graphene-based d-wave superconducting heterojunctions²⁶. Moreover, the spin-triplet pairing states can cause splitting of the ZBCP in graphene-based F/Rashba spin-orbit coupling (RSOC)/I/d-wave SC hybrid structures²⁷. The ZBCP has also been studied in ferromagnet (F)/insulator (I)/p-wave superconductor (p-wave SC) hybrid structures²⁸.

The spin-triplet pair correlations possess long-range properties and can propagate extensively in materials with strong scattering resources, which facilitates the design of graphene-based spintronics devices^29,30,31,32. In addition to the RSOC, the non-collinear magnetizations can also induce the spin-triplet pairing states in a silicene-based superconducting junction³³. The spin-triplet pairing states induced by the non-collinear magnetizations has been discussed in the graphene-based F/F/SC junction, where the superconductor (SC) refers to s-wave pairing symmetry²². In contrast to s-wave superconductor, the d-wave superconductor is anisotropic, which indicates the formation of the surface bound states on the d-wave superconductors and the existence of ZBCP^34,35,36. Therefore, the impact of non-collinear magnetizations on the ZBCP is extremely significant research interest. To explore the impact of non-collinear magnetizations on the behavior of ZBCP in a graphene-based d-wave SC junction, we consider a graphene-based F/F/B/d-wave superconductor junction. In this paper, we mainly focuses on normalized charge conductance spectra in the presence of spin-triplet paring states induced by the non-collinear magnetizations.

In this paper, we study the transport properties of graphene - based F/F/B/d-wave superconducting heterostructures. It is shown that non-collinear magnetizations can cause spin-triplet states, but the spin-triplet states will not lead to the splitting of ZBCP due to the relativistic fermions in graphene. In the case of half-metal, the ZBCP is only regulated by the spin-triplet states, resulting in the spin-triplet ZBCP. We can also find that the ZBCP can be controlled by the Fermi energy level, barrier strength, and magnetization angle. In particular, the barrier strength exhibits periodic oscillation modulation, and the magnetization angle can periodically modulate the ZBCP. These results can directly prove both the non-collinear magnetization characteristics and d-wave pairing symmetry associated with graphene relativistic fermions. The paper is organized as follows. Firstly, we express the model and basic formalism in “Model and basic formalism”. Then, results and discussions are given in “Results and discussions” in five subsections: in “Band structure and reflection probabilities” we study the band structure and various reflection probabilities, in “The effect of exchange field on ZBCP” we explored the effects of strength of exchange field on the ZBCP, in “The effect of tunable Fermi level on ZBCP” the effect of Fermi level on the ZBCP is discussed, we also discussed the effect of strength of barrier on normalized charge conductance in “The effect of the barrier on ZBCP”, and we study the effect of magnetization angle on ZBCP in “The effect of magnetization angle on ZBCP”. Finally, the main conclusions of this work are expressed in “Conclusions”.

Model and basic formalism

Fig. 1

Schematic illustration of the graphene-based F/F/B/d-wave SC heterojunction. The left F region is semi-infinite at range of $x\le -L_1$, the range of the middle F region is $-L_1\le x \le 0$, the B region has a thickness extending from $x=0$ to $x=L_2$, and the d-wave superconductor is semi-infinite at range of $x\ge L_2$. $\varphi _n$ ($\varphi _m$) and $\theta _n$($\theta _m$) are the azimuthal angles and polar angles of $h_n$ ($h_m$) in left F region (middle F region), respectively. The B region could be created by an external gate voltage $V_0$, and the $\alpha _s$ is the angle formed by the crystallographic a axis of the d-wave SC with the x axis.

We consider a graphene-based F/F/B/d-wave SC heterojunction residing in the xy plane, and the schematic potential of the model for the relativistic spin-polarized electrons is shown in Fig.  1. The exchange field could be achieved by means of atomic adsorption in the normal graphene and the direction of the exchange field can be controlled by manipulating the type and distribution of adsorbed atoms^37,38. In B region, the barrier can be implemented by either using chemical doping or the electric field effect^39,40. In the d-wave SC region, $YBa_2Cu_3O_{7-\delta }$ thin films grown on an $SrTiO_3$ substrate, and then the d-wave superconductivity in graphene could be realized by the proximity effect with the $YBa_2Cu_3O_{7-\delta }$^41,42. Owing to the conical dispersion relation exhibited by monolayer graphene at low energies, electrons and holes within this system can be regarded as massless Dirac fermions¹⁴. Thus, to describe the low-energy excitations of the F/F/B/d-wave SC junction shown in Fig. 1, one can combine the Dirac Hamiltonian with the Bogoilubov-de Gennes equation to obtain a Dirac-Bogoliubov-de Gennes (DBdG) equation in the presence of non-collinear magnetizations fields and d-wave superconductor as follows^13,43,44,45:

$$\begin{aligned} \begin{pmatrix}H-\mu _j& \Delta \\ \Delta ^*& \mu _j-THT^{-1}\end{pmatrix}{\Psi _{\tau }}={E}{\Psi _{\tau }}, \end{aligned}$$

(1)

where $\mu _j$ ($j=n,m,b,s$ for different regions) is the Fermi level, E stands for the quasiparticles$^{'}$ energy, T is the time reversal operator¹³. $\Psi _{\tau }=(\psi _{A\tau }^\uparrow ,\psi _{B\tau }^\uparrow ,\psi _{A\tau }^\downarrow ,\psi _{B\tau }^\downarrow ,-\psi _{A\bar{\tau }}^{\downarrow *},\psi _{B\bar{\tau }}^{\downarrow *},\psi _{A\bar{\tau }}^{\uparrow *},-\psi _{B\bar{\tau }}^{\uparrow *})^{Tr}$, in which $\Psi _{\tau }$ is the 8 component wave functions for the electron and hole spinors, Tr is the transpose operator. The A and B denote the two inequivalent sites in the hexagonal lattice of graphene^22,46, $\tau$ ( $\bar{\tau }$) indicates two valleys of K ($K^{'}$) and $\tau =\pm 1$ ($\bar{\tau }=-{\tau }$), the arrow index ($\uparrow ,\downarrow$) stands for real spin. The total Hamiltonian is the incorporation of a two dimensional massless Dirac Hamiltonian and the Hamiltonian in different regions ($H=H_0+H_i$). A two dimensional massless Dirac Hamiltonian which describes low energy excitations in one valley of graphene is

$$\begin{aligned} H_0=-i\hbar \nu _Fs_0\otimes (\hat{\sigma }_x\partial _x+\tau \hat{\sigma }_y\partial _y), \end{aligned}$$

(2)

Single particle Hamiltonian in four different regions can be expressed by

$$\begin{aligned} H_{i(x)}={\left\{ \begin{array}{ll}-\overrightarrow{h_n}{\cdot }\overrightarrow{s}\otimes {\sigma _0,\quad x\le -L_1},\\ -\overrightarrow{h_m}{\cdot }\overrightarrow{s}\otimes {\sigma _0,\quad -L_1\le x\le 0},\\ V_0{s_0}\otimes {\sigma _0},\quad 0\le x\le L_2,\\ -U_0{s_0}\otimes {\sigma _0},\quad x\ge L_2, & \end{array}\right. } \end{aligned}$$

(3)

where $\nu _F\approx 10^6ms^{-1}$ is the Fermi velocity. ${\sigma }_{x,y}$ and $\textbf{s}_{x,y,z}$ are Pauli matrices, acting on pseuedospin and real spin spaces of graphene, respectively. $\sigma _{0}$ and $s_{0}$ are the corresponding identity matrices. The exchange field $\overrightarrow{h_j}$ ($j=n,m$) is given by $h(\sin \theta _j\cos \varphi _j,\sin \theta _j\sin \varphi _j,\cos \theta _j)$ with the polar angle $\theta _j$ and the azimuthal angle $\varphi _j$. $V_0$ is the strength in barrier region. $U_0$ indicates the electrostatic potential in the superconducting region. $\Delta$ is the d-wave superconducting energy gap which can be given by

$$\begin{aligned} \Delta _\pm =\Delta _{(t)}\cos (2\theta _s\mp 2\alpha _s)e^{i\phi _{0}}{\Theta }\big (x-L_2\big )\varvec{,}\end{aligned}$$

(4)

where the $+$ (−) stands the electron-like (hole-like) quasiparticles$^{'}$ pairing potential sign, respectively. The superconduction pairing potential depends on the temperature, $\Delta _{(t)}=\Delta _{0}\text {tanh}{\left( 1.74\sqrt{\left( t_{c}/t\right) -1}\right) }$, and we set $t=0.01t_c$ throughout our entire calculation process in which $\Delta _0$ is the superconductor gap at zero temperature and $t_c$ is the critical temperature of superconductor. $\phi _0$ is the macroscopic phase of the SC, $\theta _s$ is the propagation angle for the quasiparticles in the S region, $\alpha _s$ stands for the angle between the a axis of the crystal and the normal of interface, ${\Theta }$ indicates Heaviside step function.

Due to the valley degeneracy in the F/F/B/d-wave SC heterojunction, we consider only the K valley in Eq. (1) ($\tau =1$). Here, we set $\hbar \nu _{F}=1$ for simplicity in calculation. To clearly understand the transport properties of carriers in F/F/B/d-wave SC heterojunction, we assume that a spin-up electron at incident angle $\alpha$ with energy E incident on the junction from left F lead. The total wave function in different regions can be written as

$$\begin{aligned} \begin{aligned}&\Psi _{(x)}^{F_n} =\Psi _{e,\uparrow }^{F_n,+}+r_{N\uparrow \uparrow }\Psi _{e,\uparrow }^{F_n,-}+r_{N\uparrow \downarrow }\Psi _{e,\downarrow }^{F_n,-}+r_{A\uparrow \uparrow }\Psi _{h,\uparrow }^{F_n,-}+r_{A\uparrow \downarrow }\Psi _{h,\downarrow }^{F_n,-}, \\&\Psi _{(x)}^{F_m} =a_1\Psi _{e,\uparrow }^{{F_m},+}+a_2\Psi _{e,\uparrow }^{{F_m},-}+a_3\Psi _{e,\downarrow }^{{F_m},+}+a_4\Psi _{e,\downarrow }^{{F_m},-}+a_5\Psi _{h,\uparrow }^{{F_m},+}+a_6\Psi _{h,\uparrow }^{{F_m},-}+a_7\Psi _{h,\downarrow }^{{F_m},+}+a_8\Psi _{h,\downarrow }^{{F_m},-}, \\&\Psi _{(x)}^B =b_{1}\Psi _{e,\uparrow }^{B,+}+b_{2}\Psi _{e,\uparrow }^{B,-}+b_{3}\Psi _{e,\downarrow }^{B,+}+b_{4}\Psi _{e,\downarrow }^{B,-}+b_{5}\Psi _{h,\uparrow }^{B,+}+b_{6}\Psi _{h,\uparrow }^{B,-}+b_{7}\Psi _{h,\downarrow }^{B,+}+b_{8}\Psi _{h,\downarrow }^{B,-}, \\&\Psi _{(x)}^{S} =t_1\Psi _{S1}^{e+}+t_2\Psi _{S2}^{e+}+t_3\Psi _{S1}^{h-}+t_4\Psi _{S2}^{h-}, \end{aligned}\end{aligned}$$

(5)

where $r_{N\uparrow \uparrow (\downarrow )}$, $r_{A\uparrow \uparrow (\downarrow )}$ are the amplitudes of the normal electron (spin-flipped) reflection, the anomalous (conventional) Andreev reflection, respectively. $a_{1,2};b_{1,2}$ ($a_{3,4};b_{3,4}$) are the amplitudes of spin-up (spin-down) electrons, and $a_{5,6};b_{5,6}$ ($a_{7,8};b_{7,8}$) are the amplitudes of spin-up (spin-down) holes in middle F region and barrier region. $t_{1}$ ($t_{3}$) are respectively corresponding to the amplitudes of spin-up electron (hole) quasiparticles and $t_{2}$ ($t_{4}$) are respectively corresponding to the amplitudes of spin-down electron (hole) quasiparticles in the S region. ± denotes the wave functions traveling along the $\pm {x}$ direction in the heterojunction. We assume that the junction width W is enough large so that the y component of the wave vector q is a conserved quantity upon the scattering processes. By solving the Eq. (1), we can obtain the wave functions associated with the dispersion relation in the different regions which can be expressed by

In F region:

$$\begin{aligned} \begin{aligned} \Psi _{F_j,\uparrow }^{e\pm }&= \begin{pmatrix} \cos \frac{\theta _j}{2}e^{-i\varphi _j}&,\pm \cos \frac{\theta _j}{2}e^{-i\varphi _j}e^{\pm i\alpha _{j,\uparrow }^e}&,\sin \frac{\theta _j}{2}&,\pm \sin \frac{\theta _j}{2}e^{\pm i\alpha _{j,\uparrow }^e}&,0&,0&,0&,0\end{pmatrix}^{Tr}e^{\pm i k_{F_j,\uparrow }^{e}x},\\ \Psi _{F_j,\downarrow }^{e\pm }&= \begin{pmatrix}-\sin \frac{\theta _j}{2}e^{-i\varphi _j}&,\mp \sin \frac{\theta _j}{2}e^{-i\varphi _j}e^{{\pm i\alpha _{j,\downarrow }^{e}}}&,\cos \frac{\theta _j}{2}&,\pm \cos \frac{\theta _j}{2}e^{{\pm i\alpha _{j,\downarrow }^{e}}}&,0&,0&,0&,0\end{pmatrix}^{Tr}e^{\pm i k_{F_j,\downarrow }^{e}x},\\ \Psi _{F_j,\uparrow }^{h\pm }&= \begin{pmatrix}0&,0&,0&,0&,-\sin \frac{\theta _j}{2}e^{-i\varphi _j}&,\pm \sin \frac{\theta _j}{2}e^{-i\varphi _j}e^{\pm i\alpha _{j,\uparrow }^h}&,\cos \frac{\theta _j}{2}&,\mp \cos \frac{\theta _j}{2}e^{\pm i\alpha _{j,\uparrow }^h}\end{pmatrix}^{Tr}e^{\pm i k_{F_j,\uparrow }^{h}x}, \\ \Psi _{F_j,\downarrow }^{h\pm }&= \begin{pmatrix}0&,0&,0&,0&,\cos \frac{\theta _j}{2}e^{-i\varphi _j}&,\mp \cos \frac{\theta _j}{2}e^{-i\varphi _j}e^{\pm i\alpha _{j,\downarrow }^h}&,\sin \frac{\theta _j}{2}&,\mp \sin \frac{\theta _j}{2}e^{\pm i\alpha _{j,\downarrow }^h}\end{pmatrix}^{Tr}e^{\pm i k_{F_j,\downarrow }^{h}x}, \\ \end{aligned}\end{aligned}$$

(6)

where $j=n,m$. The propagation angles are given by $\alpha _{{j},{\uparrow (\downarrow )}}^{e}=\arcsin [q/(E+\mu _j+(-) h_j)]$ and $\alpha _{{j},{\uparrow (\downarrow )}}^{h}=\arcsin [q/(E-\mu _j-(+) h_j)]$. Tr is the transpose operator and the x component of wave vectors during the scattering processes are given by $k_{F_j,\uparrow }^e =({E}+\mu _j+{h_j}){\cos {\alpha _{{j},\uparrow ^e}}}$, $k_{F_j,\downarrow }^e =({E}+\mu _j-{h_j}){\cos {\alpha _{{j},\downarrow ^e}}}$, $k_{F_j,\uparrow }^h =({E}-\mu _j-{h_j}){\cos {\alpha _{{j},\uparrow ^h}}}$, $k_{F_j,\downarrow }^h =({E}-\mu _j+{h_j}){\cos {\alpha _{{j},\downarrow ^h}}}$, respectively.

In B region:

$$\begin{aligned} \begin{aligned} \psi _{B,1}^{e\pm }&=(1,\pm e^{\pm i\gamma _B^e},0,0,0,0,0,0)^{Tr}e^{\pm ik_B^{e}x}, \\ \psi _{B,2}^{e\pm }&=(0,0,1,\pm e^{\pm i\gamma _B^e},0,0,0,0)^{Tr}e^{\pm ik_B^{e}x}, \\ \psi _{B,1}^{h\pm }&=(0,0,0,0,1,\mp e^{\pm i\gamma _B^h},0,0)^{Tr}e^{\pm ik_B^{h}x}, \\ \psi _{B,2}^{h\pm }&=(0,0,0,0,0,0,1,\mp e^{\pm i\gamma _B^h})^{Tr}e^{\pm ik_B^{h}x}, \\ \end{aligned}\end{aligned}$$

(7)

where the propagation angles are given by $\gamma _B^{e(h)}=\arcsin [q/[(E+(-)(\mu _j-V_0)]$ ($j=b$)and the x component of wave vectors during the scattering processes are $k_B^{e(h)}=[E+(-)(\mu _j-V_0)]\textrm{cos}\gamma _B^{e(h)}$.

In S region:

$$\begin{aligned} \begin{aligned}&\Psi _{S1}^{e\pm }=\begin{pmatrix}u_{(\eta ^+)}&,\pm u_{(\eta ^+)} e^{\pm i \eta _1}&,0&,0&,v_{(\eta ^+)}&,\pm v_{(\eta ^+)} e^{\pm i \eta _1}&,0&,0\end{pmatrix}^{Tr}e^{\pm ik_ex},\\&\Psi _{S2}^{e\pm }=\begin{pmatrix}0&,0&,u_{(\eta ^+)}&,\pm u_{(\eta ^+)} e^{\pm i \eta _1},0&,0&,v_{(\eta ^+)}&,\pm v_{(\eta ^+)} e^{\pm i \eta _1}\end{pmatrix}^{Tr}e^{\pm ik_ex},\\&\Psi _{S1}^{h\pm }=\begin{pmatrix}v_{(\eta ^-)}&,\mp v_{(\eta ^-)} e^{\mp i \eta _2}&,0&,0&,u_{(\eta ^-)}&,\mp u_{(\eta ^-)} e^{\mp i \eta _2}&,0&,0\end{pmatrix}^{Tr}e^{\pm ik_hx},\\&\Psi _{S2}^{h\pm }=\begin{pmatrix}0&,0&,v_{(\eta ^-)}&,\mp v_{(\eta ^-)} e^{\mp i \eta _2}&,0&,0&,u_{(\eta ^-)}&,\mp u_{(\eta ^-)} e^{\mp i \eta _2}\end{pmatrix}^{Tr}e^{\pm ik_hx}, \end{aligned} \end{aligned}$$

(8)

$u(\eta )={\frac{\sqrt{E+\Omega (\eta )}}{{2E}}}$ and $v(\eta )=\sqrt{\frac{E-\Omega (\eta )}{2E}}$ are the coherence factors where $\Omega (\eta )=\sqrt{E^2-|{\Delta _0(\eta )}|^2}$, respectively. The propagation angles are given by $\eta _{1(2)}=\arcsin [q/(\mu _j+U_0+(-) \Omega ]$ ($j=s$) ( $\eta ^+=\eta _1$ and $\eta ^-=\pi -\eta _2$) and the x component of wave vectors during the scattering processes are given by $k_{e(h)} =[\mu _j+U_0+(-)\Omega ]\cos {\eta _{1(2)}}$. In S region, we considered that the S region is in a heavily doped regime ($U_0>>E,\Delta _0$), which satisfies the requirement of the mean-field.

All the amplitudes in Eq.( 5) can be obtain by matching the boundary conditions

$$\begin{aligned} \begin{aligned} \Psi _{(-L_1)}^{F_n}=\Psi _{(-L_1)}^{F_m}, \Psi _{(0)}^{F_m}=\Psi _{(0)}^B, \Psi _{(L_2)}^B=\Psi _{(L_2)}^S, \end{aligned} \end{aligned}$$

(9)

After getting all the scattering coefficients for the incident electron with spin-up, the normalized charge conductance can be calculated by means of the theory of generalized Blonder-Tinkham-Klapwijk formula ^47,48,49

$$\begin{aligned} G_e=\int _{0}^{\pi /2}\sum _{\sigma ,\sigma ^{\prime }=\uparrow ,\downarrow }G_\sigma \left( 1-\left| r_{N\uparrow \uparrow }\right| ^2\frac{\cos \alpha _{{\sigma }(\sigma ^{\prime })}^e}{\cos \alpha _{\sigma (\sigma ^{\prime })}^e}-\left| r_{N\uparrow \downarrow }\right| ^2\frac{\cos \alpha _{{\sigma ^{\prime }}(\sigma )}^e}{\cos \alpha _{\sigma (\sigma ^{\prime })}^e}+\left| r_{A\uparrow \uparrow }\right| ^2\frac{\cos \alpha _{{\sigma }(\sigma ^{\prime })}^h}{\cos \alpha _{\sigma (\sigma ^{\prime })}^e}+\left| r_{A\uparrow \downarrow }\right| ^2\frac{\cos \alpha _{{\sigma ^{\prime }}(\sigma )}^h}{\cos \alpha _{\sigma (\sigma ^{\prime })}^e}\right) {\cos \alpha }d{\alpha },\end{aligned}$$

(10)

where the $\sigma$ is $\uparrow$($\downarrow$). $G_\sigma =2e^2N_\sigma (E)/h_n$ is spin-dependent normal conductance, $\begin{aligned}N_\sigma (E)=\big |E+\mu _{j}+{\sigma }h_n\big |W\big /\big ({\pi }\hbar \nu _{F}\big )\end{aligned}$ ($j=n$)is the density of states, the junction width is W. The sum can be replaced by an integral ($\sum _n\rightarrow \int dq$) when the junction width is wide enough. Using Eq.(10), the differential conductance for the present superconducting heterojunction can be obtained easily by the numerical calculations.

Extracting the conductance contributed by conventional Andreev reflection ($R_{A\uparrow \downarrow }$) and the conductance contributed by anomalous Andreev reflection ($R_{A\uparrow \uparrow }$) from the total conductance ($G_e$) are extremely essential for us to study in detail how the spin-singlet paring states and the spin-triplet paring states affect the features of normalized charge conductance with varied values of exchange field. The conductance of the conventional Andreev reflection can be expressed by

$$\begin{aligned} G_{AR}=2\int _{0}^{\pi /2}\left( G_{\uparrow }\left| r_{A\uparrow \downarrow }\right| ^2\frac{\cos \alpha _{{\downarrow }}^h}{\cos \alpha _{\uparrow }^e}+G_{\downarrow }\left| r_{A\downarrow \uparrow }\right| ^2\frac{\cos \alpha _{{\uparrow }}^h}{\cos \alpha _{\downarrow }^e}\right) {\cos \alpha }d{\alpha },\end{aligned}$$

(11)

The conductance of the anomalous Andreev reflection can be expressed by

$$\begin{aligned} G_{NAR}=2\int _{0}^{\pi /2}\left( G_{\uparrow }\left| r_{A\uparrow \uparrow }\right| ^2\frac{\cos \alpha _{{\uparrow }}^h}{\cos \alpha _{\uparrow }^e}+G_{\downarrow }\left| r_{A\downarrow \downarrow }\right| ^2\frac{\cos \alpha _{{\downarrow }}^h}{\cos \alpha _{\downarrow }^e}\right) {\cos \alpha }d{\alpha },\end{aligned}$$

(12)

where the $G_{{\uparrow }(\downarrow )}$ is the spin-up (spin-down) normal conductance.

Results and discussions

In this section, we elucidate the pivotal findings of our research. The parameter $\alpha _s$ is meticulously set to $\pi /4$, and the corresponding energy gap functions for electron and hole quasiparticles are defined as $\Delta _+=\Delta _0\sin 2\theta _s$ and $\Delta _-=-\Delta _0\sin 2\theta _s$, respectively. Notably, a zero-energy bound state can achieve at the interface of the d-wave superconducting junction, attributed to the inverse signs of the pairing potential functions and a precise phase difference of $\pi$. Consequently, a zero-bias conductance peak (ZBCP) emerges as a distinctive feature in the majority of high-temperature d-wave superconducting junctions, allowing for experimental differentiation between d-wave and isotropic s-wave superconductors ^45,46,47. It is imperative to underscore that, in numerical computations, the units for E, $E_F$, $h_n$, $h_m$, $V_0$ and $U_0$ are benchmarked against $\Delta _0$. Furthermore, We also normalize lengths using the superconducting coherence length ${\xi }_s=\varvec{\hbar }{\nu }_F/\Delta _0$. For the sake of uniformity, we stipulate $\mu =\mu _j$ when $\mu _n=\mu _m=\mu _b=\mu _s$.

Band structure and reflection probabilities

Fig. 2

Energy band structure of each region in the F/F/B/d-wave SC heterojunction. Here, the red solid (dotted) line represents the spin-up conduction band (valence band), respectively. The blue solid (dotted) line denotes the spin-down conduction band (valence band), respectively. In left F region, the solid circles stand for the electrons while the circles stand for the holes. The middle panel is the band structure of RSOC.

Firstly, we begin to investigate the energy bands of the graphene-based junction with the non-collinear magnetizations condition (i.e. $\theta _n=\varphi _n=0$, $\theta _m=\varphi _m=0.5\pi$). By diagonalizing the DBdG Hamiltonian Eq. (1), we can obtain the dispersion relations and corresponding spinors for four different regions. For a comparison, we also plotted the energy band diagrams with Rashba spin-orbit coupling for comparison with the energy bands in the middle F region. They can be represented as:

$$\begin{aligned} \begin{aligned}&{E_{F_n}}=\pm \mu _n\pm \left( k_{x}^{2}+q^{2}\right) ^{1/2}\pm h_n,\\&{E_{F_m}}=\pm \mu _m\pm \left( k_{x}^{2}+q^{2}\right) ^{1/2}\pm h_m, \\&{E_{R}}=\pm \mu _R\pm \left( k_{x}^{2}+q^{2}+\lambda ^2\right) ^{1/2}\mp \lambda , \\&{E_{B}}=\pm \mu _b\pm \left( k_{x}^{2}+q^{2}\right) ^{1/2}\pm V_0, \\&{E_{S}}=\Bigg [\left| \Delta _\pm \right| ^2+\Bigg (U_0+\mu _s\pm \sqrt{k_x^2+q^2}\Bigg )^2\Bigg ]^{1/2},\end{aligned}\end{aligned}$$

(13)

where the $\lambda$ stands for the strength of the Rashba spin-orbit coupling. Note that the x component of wave vectors ($k_x$) in the different F regions, RSOC region, B region and S region are $k_{F_{n(m)},\uparrow {/}\downarrow }^{e/h}$, $k_{{RSOC},\uparrow {/}\downarrow }^{e/h}$, $k_{B}^{e/h}$, $k_{S}^{e/h}$, respectively. In our calculations of the reflection probabilities, we take into account a case that an electron with spin-up enters the F/F interface.

Figure 2 illustrates that the presence of an exchange field can induce z-direction exchange splitting in the left F region, resulting in the spin band subgap. In the middle F region, the presence of an exchange field can induce y-direction exchange splitting. There are two sub-bands within the energy band, which are linear, indicating that the carriers are massless. Due to the non-collinear magnetizations in two F regions, the middle F region can generate the effect of spin mixer that influences carriers transport ²². The energy bands corresponding to Rashba spin-orbit coupling also have two sub-bands, but they are parabolic, suggesting that the carriers are massive, yet still serves as a spin mixer in the carriers transport mechanism ⁵⁰. In B region, the spin-up and spin-down energy bands are degenerate, allowing for the possibility of Klein tunneling to occur. The d-wave superconductor is highly doped, and the low-energy excitation energy bands is parabolic. Here, we only consider the case where the incident energy of a spin-up electron is less than the sum of the strength of exchange field and the Fermi level, which is satisfied ($E\le \mu _n+h_n$), and it is located at a (as seen in Fig. 2). In the F/F/B/d-wave SC heterojunction, four types of backscattering possibilities may occur due to the spin mixing that can occur in the middle F region, including the backscattering of $a\rightarrow b$ is the normal electron reflection ($R_{N\uparrow \uparrow }$), $a\rightarrow c$ is the normal spin-flipped reflection ($R_{N\uparrow \downarrow }$), $a\rightarrow d$ is the anomalous Andreev reflection ($R_{A\uparrow \uparrow }$), $a\rightarrow e$ is the conventional Andreev reflection ($R_{A\uparrow \downarrow }$). Therefore, from the analysis of energy bands, we can find that the anomalous Andreev reflection may be realized in the F/F/B/d-wave SC junction.

Fig. 3

Amplitudes of the four types of reflections ($R_{N\uparrow \uparrow }$, $R_{N\uparrow \downarrow }$, $R_{A\uparrow \uparrow }$, $R_{A\uparrow \downarrow }$) as functions of the incident electron energy E and the incident angle $\alpha$. The parameters used in the calculation are $L_1=0.1\xi$, $L_2=0$, $\mu =\mu _j=100\Delta _0$, $U_0=0$, $\varphi _n=\theta _n=0$, $\varphi _m=\theta _m=0.5\pi$, (a-d) $h_n=0.5\mu$, (e-h) $h_n=\mu$.

In Fig. 3, we discussed the four distinct types of reflection spectra for the case where $h_n$ and $h_m$ are non-collinear (i.e. $\theta _n=\varphi _n=0$, $\theta _m=\varphi _m=0.5\pi$). We can find that the probabilities of normal electron reflection $R_{N\uparrow \uparrow }$, normal spin-flipped reflection $R_{N\uparrow \downarrow }$, anomalous Andreev reflection$R_{A\uparrow \uparrow }$, and conventional Andreev reflection$R_{A\uparrow \downarrow }$ are all exist for $h_n=0.5\mu$, as shown in Fig. 3a–d. Then, we can also find that the normal spin-flipped reflection and conventional Andreev reflection are both occur within $0^{\circ }<\alpha <20^{\circ }$, which is consistent with the critical angles $\alpha _{e\uparrow \downarrow }^c$ and $\alpha _{h\uparrow \downarrow }^c$ ($\alpha _{e\uparrow \downarrow }^c=\arcsin \frac{|E+\mu -h_n|}{|E+\mu +h_n|}$ and $\alpha _{h\uparrow \downarrow }^c =\arcsin \frac{|E-\mu +h_n|}{|E+\mu +h_n|}$). The normal electron reflection occurs within the range of $20^{\circ }<\alpha <90^{\circ }$ and the anomalous Andreev reflection occurs the range of $0^{\circ }<\alpha <40^{\circ }$ due to probability density conservation, which is consistent with the theoretically calculated critical angles $\alpha _{e\uparrow \uparrow }^c$ and $\alpha _{h\uparrow \uparrow }^c$ ($\alpha _{e\uparrow \uparrow }^c=\arcsin \frac{|E+\mu +h_n|}{|E+\mu +h_n|}$ and $\alpha _{h\uparrow \uparrow }^c=\arcsin \frac{|E-\mu -h_n|}{|E+\mu +h_n|}$) at $90^{\circ }$. When $h_n=\mu$, as shown in Fig. 3e, f, the left F region behaves as a half-metal case, we can find that the normal spin-flipped reflection and the conventional Andreev reflection are absent (as seen in Fig. 3f, h). The anomalous Andreev reflection is dominant in the spectra within small angles and it can even approach to unity (as seen in Fig. 3g). These indicates that the pure spin-triplet pairing states can be realized in this case. Such an anomalous Andreev reflection may be have an important role in the ZBCP.

The effect of exchange field on ZBCP

Fig. 4

(a–c) The normalized charge conductance ($G_e$) as a function of E for different values of $h_n$ in graphene-based F/F/B/d-wave SC junction. (b–d) The conductance $G_{AR}$ ($G_{NAR}$) from conventional (anomalous) Andreev reflection contribution as a function of E for different values of $h_n$. (a,b) $\theta _m=\varphi _m=0$, (c,d) $\theta _m=\varphi _m=0.5 \pi$. We set $h_m=5\Delta _0$, $\mu =\mu _n=\mu _m=100\Delta _0$. Here, the black curve for $h_m=h_n=0$, and the parameters are chosen to be the same as in Fig. 3.

For a comparison, we investigated the conductance spectra of the F/F/B/d-wave SC junction when the magnetizations transitions are collinear configuration and non-collinear configuration. For the case of collinear magnetizations, i.e. $\theta _n=\varphi _n=\theta _m=\varphi _m=0$ (as seen in Fig. 4a), we can find that the normalized charge conductance always appears a shape of zero bias conductance peak (ZBCP) with the different strengths of exchange field. Then, with enhancement of the strength of exchange field in the left F region, the height of the ZBCP is notably decreased, and when $h_n=\mu$, the conductance spectrum evolves into a shape of zero bias conductance dip. This behavior is consistent with the findings presented in Refs. ^23,27, which also investigated the impact of spin-singlet paring states on ZBCP in graphene-based superconducting heterostructures. To gain a more comprehensive understanding of the relationship between the ZBCP and spin-singlet pairing states, we have extracted the conductance ($G_{AR}$) which contributed by the conventional Andreev reflection ($R_{A\uparrow \downarrow }$), as depicted in Fig. 4b. Due to the anomalous Andreev reflection ($R_{A\uparrow \uparrow }$) is absent, causing the conductance of the anomalous Andreev reflection $G_{NAR}$ is always zero, so it is not shown in Fig. 4b. We can find that the height of the ZBCP completely depends on the magnitude of $G_{AR}$, and it is typically suppressed by the strength of the exchange field $h_n$. This arises from that stronger of the exchange field can create an additional momentum difference of $2h_n/\nu _F$ between the incident electron and the conventional Andreev reflected hole, which in turn suppresses the spin-singlet pairing process ²⁷. As the left F region transforms into a spin half-metal case at $h_n=\mu$, the spin-singlet pairing process cannot occur within the subgap energy range. It comes from there is no corresponding Fermi surface for the hole state with down-spin, resulting in the probability of conventional Andreev reflection ($R_{A\uparrow \downarrow }$) is zero. Therefore, the conductance spectrum evolves into shape of zero bias conductance dip (as seen in Fig. 2a). These characteristics indicate that the emergence of ZBCP is attributed to spin-singlet pairing states in the collinear magnetic configuration.

Fig. 5

(a–c) The normalized charge conductance as a function of E for different strengths of exchange field in middle F region, (b–d) and the conductance $G_{NAR}$ ($G_{AR}$) from anomalous (conventional) Andreev reflection contribution as a function of E for different strengths of exchange field in middle F region. (a,b) $h_n=0.5\mu _n$, (c,d) $h_n=\mu _n$. We set $\mu =\mu _n=\mu _m=100\Delta _0$, $\theta _n=\varphi _n=0$, $\theta _m=\varphi _m=0.5\pi$.

Next, we turn our attention to the normalized charge conductance spectra for the case of non-collinear magnetizations by setting $\theta _n=\phi _n=0$, and $\theta _m=\phi _m=0.5\pi$, where the anomalous Andreev reflection is occurred (as seen in Fig. 3). We notice that the subgap conductance still exhibits as a zero bias conductance peak and without any split regardless of the change in the ratio of $h_n/\mu$, as shown in Fig. 4c. Particularly, the height of the ZBCP is substantial in magnitude even though when the left F region transforms into a spin half-metal case at $h_n=\mu$. This characteristic is entirely different from the results in d-wave superconducting heterostructures with Rashba spin-orbit coupling (RSOC), where studies have shown that spin-triplet pairing leads to the splitting of the ZBCP into two sub-peaks, and the value of the ZBCP is always zero ²⁷. Although both non-collinear magnetizations and RSOC can generate spin mixed states in graphene, from the perspective of energy bands, the energy bands of non-collinear magnetizations and RSOC are different (as seen in Fig. 2): the energy bands of non-collinear magnetization are always linear and the bandgap is not opened, which allowing the graphene Dirac fermions being massless, resulting in ZBCP not splitting. In contrast,, the energy bands corresponding to RSOC exhibits a certain spin bandgap, which makes the graphene Dirac fermions massive and leads to the splitting of ZBCP. More importantly, it can be observed that the ZBCP under non-collinear magnetizations is always higher than that under collinear magnetizations when the strength of the left exchange field is not zero, as seen in Fig. 4a, c. Such a magnitude difference arises from that the value of ZBCP for non-collinear magnetizations case not only derived from the spin-singlet pairing states related conductance $G_{AR}$, but also stemmed from that spin-triplet paring states related conductance $G_{NAR}$. This can be proven by the subgraph in Fig. 4b, d. When $h_n= 0.4\mu$, the magnitude of ZBCP is 1.365, the magnitude of $G_{AR}$ is 0.258, and the magnitude of $G_{NAR}$ is 1.107. It can be found that the magnitude of ZBCP is equal to the sum of $G_{AR}$ and $G_{NAR}$ values at zero incident energy ($E=0$), as shown in Fig. 4c, d. Therefore, the ZBCP arises due to both the spin-singlet and spin-triplet paring states.

Further, we also examine how the strength of the middle exchange field ($h_m$) affects the zero-bias conductance peak under non-collinear magnetizations condition, as shown in Fig. 5. We firstly plot the normalized charge conductance spectrum at $h_n=0.5\mu$, as shown in Fig. 5a, the normalized charge conductance always exhibits a zero-bias conductance peak (ZBCP) as the strength of middle exchange middle increases, and the value of ZBCP is notably enhanced when the strength of the middle exchange field increases. To clearly know the contribution of spin-triplet paring states and spin-singlet paring states to ZBCP. We also plot the conductance curves of $G_{NAR}$ and $G_{AR}$ (subfigure in Fig. 5b), as depicted in Fig. 5b. We can find that the value of ZBCP is equal to $G_{AR}$ at $E=0$ when the strength of middle exchange field is zero. Particularly, the value of the ZBCP is the sum of both $G_{NAR}$ and $G_{AR}$ at $E=0$ when the strength of middle exchange field is nonzero, as depicted in Fig. 5a, b. Additionally, with increasing of exchange field, the value of $G_{NAR}$ at $E=0$ is effective increased (as seen in Fig. 5b). This phenomenon can be explained by the energy bands of the middle exchange field (as seen in Fig. 2): the presence of the middle exchange field causes the energy bands to split, and the splitting of the energy bands becomes more pronounced when the strength of the middle exchange field increases, making the spin-triplet pairing states more easily to occur. From Fig.  5a, b, we can see that when the strength of the middle exchange field is zero ($h_m=0$), only the spin singlet pairing state contributes to ZBCP at $h_n=0.5\mu$. But when the strength of the middle exchange field is non-zero ($h_m\ne 0$), the emergence of ZBCP is attributed to both spin-triplet and spin-singlet pairing states.

Then, we set the left F region as a half-metal case ($h_n=\mu$). We plot the normalized charge conductance graph at a half-metal case, as shown in Fig. 5c. It is evident that the height of the ZBCP notably increases with the enhancement of the middle exchange field strength. Notably, when the strength of the middle exchange field is equal to zero, the normalized charge conductance graph exhibits a zero-bias conductance dip. To gain a clearer comprehension of the normalized charge conductance behavior, we also plot the conductance curves of $G_{NAR}$ at various strengths of the middle exchange field, as shown in Fig. 5d. Particularly, the value of $G_{NAR}$ is also zero because the energy bands of the middle exchange field are degenerate when the strength of middle exchange field is equal to zero ($h_m=0$), which prevents the anomalous Andreev reflection occurring. Then, we can find that the strength of middle exchange field can effectively enhance the value of $G_{NAR}$, and moreover, the value of ZBCP is consistently equal to the value of $G_{NAR}$ at $E=0$, as shown in Fig. 5c, d. Consequently, the emergence of ZBCP is only attributed to spin-triplet pairing states for various strengths of the middle exchange field when $h_n=\mu$.

The effect of tunable Fermi level on ZBCP

Fig. 6

(a–c) The normalized charge conductance as a function of E for different Fermi level in middle F region, (b–d) and the conductance $G_{NAR}$ ($G_{AR}$) from anomalous (conventional) Andreev reflection contribution as a function of E for different Fermi level in middle F region. (a,b) $h_n=0.5\mu _n$, (c,d) $h_n=\mu _n$. We set $\mu =\mu _n=100\Delta _0$, $h_m=5\Delta _0$, $\theta _n=\varphi _n=0$, $\theta _m=\varphi _m=0.5\pi$. Other parameters are same as Fig. 4.

Graphene, compared to traditional materials, possesses a tunable chemical potential or Fermi level, which can be easily adjusted through gate voltage or doping. Therefore, exploring the impact of the Fermi level on the spin-singlet pairing states and spin-triplet pairing states in a graphene-based F/F/B/d-wave SC junction is significant. Figure 6a, c present the conductance for different Fermi levels $\mu _m=25\Delta _0$, $50\Delta _0$, $70\Delta _0$, and $100\Delta _0$ in the middle F region, under non-collinear magnetizations condition. To understand the influence of both of spin-singlet pairing states and spin-triplet pairing states on the ZBCP, we firstly consider the conductance graph at $h_n=0.5\mu$, as shown in Fig. 6a. The normalized charge conductance consistently exhibits a zero bias conductance peak (ZBCP), which does not split with increasing Fermi level. In addition, as the Fermi level increases, the ZBCP significantly increases, which is in sharp contrast to the findings discussed in reference  ²⁷. Its results indicate that the characteristics of split ZBCP with zero bias conductance cannot be altered by the mismatch strength of the Fermi wave vector. We also plot the conductance curves of $G_{NAR}$ and $G_{AR}$ (subfigure in Fig. 6b) as function of E for varying Fermi levels. We can find that the values of both $G_{AR}$ and $G_{NAR}$ increase with the Fermi level increased at zero incident energy ($E=0$). We can also observe that the value of ZBCP is the sum of $G_{AR}$ and $G_{NAR}$ at zero incident energy ($E=0$). Therefore, both the spin-singlet paring states and the spin-triplet pairing states are the origin of the ZBCP at $h_n=0.5\mu$.

Next, we consider the influence of Fermi level on ZBCP at the half-metal case ($h_n=\mu$), where the anomalous Andreev reflection ($R_{A\uparrow \uparrow }$) occurs, but there is no conventional Andreev reflection ($R_{A\uparrow \downarrow }$) (as seen in Fig. 3e–f. We can find that the height of the ZBCP also increases with increasing the Fermi level. We can also find that the Fermi level can effectively improve the the conductance of the anomalous Andreev reflection ($G_{NAR}$), moreover, the value of ZBCP is always equal to the value of $G_{NAR}$ at $E=0$, as shown in Fig. 6c, d, which further affirms that only the spin-triplet pairing states effects the ZBCP under the half-metal case ($h_n=\mu$).

The effect of the barrier on ZBCP

Fig. 7

(a,b) The normalized charge conductance as a function of E for different barrier strengths $\kappa$. (c,d) The normalized charge conductance as a function of values of barrier strength $\kappa$ at $E=0.5$. (a–c) $\theta _n=\varphi _n=\theta _m=\varphi _m=0$, (b–d) $\theta _n=\varphi _n=0$ and $\theta _m=\varphi _m=0.5\pi$. We set $\mu =\mu _j=100\Delta _0$, $h_n=\mu$, $h_m=10\Delta _0$.

In what follows, we will proceed to discuss the impact of strength of barrier on the normalized charge conductance under the collinear magnetizations and non-collinear magnetizations conditions. The relatively thin barrier strength in region B is described by a finite dimensionless parameter $\kappa =V_0 L_2$, i.e. $V_0\rightarrow \infty$ and $L_2\rightarrow 0$. Note that in the limit of the thin barrier, we can attain $\gamma _B^{e}, \gamma _B^{h}\rightarrow 0$ and $-k_B^{e} L_2, k_B^{h} L_2\rightarrow \kappa$. In Fig. 7a,b, we depict the conductance graph of normalized charge conductance as a function of E for various barrier strengths values ($\kappa$) in the case of half-metal. In condition of collinear magnetizations, it is observed that the normalized charge conductance consistently presents a shape of ZBCD as the barrier strength ($\kappa$) increases, as depicted in Fig. 7a. However, the non-zero bias conductance shows a monotonically decreasing with increasing barrier strength. In order to better understand the regulatory effect of barriers on non-zero bias charge conductance, we plotted the normalized charge conductance under the situation of $E=0.5\Delta _0$ as a function of barrier strength ($\kappa$), as depicted in Fig. 7c. We can find that the conductance perfect oscillates with a $\pi$-periodicity as a function of the barrier strength; within one period, the conductance first decreases monotonically and then increases monotonically. This periodic variation is attributed to the Klein tunneling effect caused by relativistic low-energy Dirac fermions in graphene ²⁷.

In the following discussion, we focus on the normalized charge conductance for instances of non-collinear magnetizations with $h_n=\mu$. Therefore, the normalized charge conductance only depends on the anomalous Andreev reflection ($R_{A\uparrow \uparrow }$). We can find that the value of the ZBCP remains constant with different barriers, as depicted in Fig. 7b. This indicates the emergence of a spin-triplet bound state at $E=0$, where the ZBCP can be referred to as the spin-triplet zero-bias conductance peak. Additionally, the non-zero bias conductance exhibits a monotonically decreasing with increasing barrier strength. We also plotted the normalized charge conductance at $E=0.5$ as a function of $\kappa$, as depicted in Fig. 7d. It is clear seen that the $\pi$ periodic oscillating behavior is also exhibited, and the maximum of the conductance for $E=0.5$ is located at $\kappa =n\pi$ (i.e., $n = 0, 1, 2...$) and the minimum of $G_e$ is located at near $\kappa =(0.5 + n)\pi$. More importantly, the oscillating behaviors are almost symmetric with the $\kappa =n\pi$. These characteristics exhibit substantial differences compared to that in the graphene-based FM/RSOC/I/d-wave SC junction ²⁷, where the anomalous Andreev reflection is also exist. Therefore, due to the existence of zero energy state and spin-triplet bound state, the zero-bias conductance peak (ZBCP) cannot be modulated by the barrier strength. However, the normalized charge conductance at $E\ne 0$ can be periodically modulated by the barrier strength.

The effect of magnetization angle on ZBCP

Fig. 8

(a–c) The spectrum of normalized charge conductance as a function of E and $\theta _m$. (d–i) The spectrum of the conductance $G_{NAR}$ $(G_{AR})$ from anomalous (conventional) Andreev reflection contribution as a function of E and $\theta _m$ for different strengths of exchange field in left F region. (a,d,g) $h_n=0.4\mu$, (b,e,h) $h_n=0.7\mu$, (c, f, i) $h_n=\mu$. We set $\theta _n=\varphi _n=0$, $\varphi _m=0.5\pi$, $\mu =\mu _j=100\Delta _0$. The other parameters are the same as in Fig. 4.

Furthermore, we discussed the magnetization angle effects on charge conductance, as depicted in Fig. 8. We consider three types of exchange fields in the left F region, which are $h_n=0.4\mu$, $0.7\mu$ and $\mu$. Figure  8 corresponds to normalized charge conductance ($G_e$), the anomalous Andreev reflection ($G_{NAR}$) and the conventional Andreev reflection ($G_{AR}$), respectively. In Fig. 8, we can observe that $G_e$, $G_{NAR}$ and $G_{AR}$ all exhibit $\pi$ periodicity. Then in Fig. 8a–c, it is seen that the normalized charge conductance shows a gradual enhancement when the magnetization angle increases $\theta _m$ from $\theta _m=n\pi$ to $\theta _m=(0.5+n)\pi$ (where n represents non-negative integers), and one can also find that the maximum conductance always situates at $E=0$ with the magnitization angle $\theta _m=(0.5+n)\pi$, which is referred to the maximum value of ZBCP. From Fig.  8d–f, we can find that $G_{NAR}$ also gradually increases from zero when the angle changes from $\theta _m=n\pi$ to $\theta _m=(0.5+n)\pi$ and reaches its maximum at $\theta _m=(0.5+n)\pi$. However, the $G_{AR}$ reaches its peak at $\theta _m=n\pi$ and gradually decreases to near disappearance with the angle of $\theta _m=(0.5+n)\pi$, as shown in Fig. 8g,h. Especially special is the $G_{AR}$ are almost zero with all magnetization angle when $h_n=\mu$, as shown in Fig. 8i, which leads a zero-bias conductance dip can always be achieved when $\theta _m=n\pi$ and $E=0$. Therefore, these indicate that the normalized charge conductance is primarily attributed to the presence of spin-triplet pairing states in non-collinear configurations and the spin-triplet pairing states can be modulated by the magnetization angle.

Conclusions

In conclusion, through the numerical calculation of the spin-dependent DBdG equation, we have theoretically investigated the differential conductance of a graphene-based Ferromagnet/Ferromagnet/Barrier/d-wave Superconductor (F/F/B/d-wave SC) heterojunction. The research findings indicate that in the case of non-collinear magnetizations, both anomalous Andreev reflection and conventional Andreev reflection can occur when $h_n\ne \mu$, whereas only the anomalous Andreev reflection takes place when when $h_n=\mu$, and the anomalous Andreev reflection can be separated from the conventional Andreev reflection. The spin-triplet pairing states induced by non-collinear magnetizations do not cause the splitting of the zero-bias conductance peak (ZBCP). We can also find that the triplet bound states emerge at $E=0$, and the ZBCP is only governed by the spin-triplet pairing states when $h_n=\mu$, resulting in the spin-triplet ZBCP. In the case of collinear magnetizations, a zero-bias conductance dip (ZBCD) appears when $h_n=\mu$, due to the absence of both conventional and anomalous Andreev reflection. The insulator barrier strength periodically modulates the normalized charge conductance at when $E\ne 0$. In the graphene-based F/F/B/d-wave superconducting junction, the Fermi level can effectively tune the spin-triplet states, achieving a significant enhancement of the ZBCP. Additionally, the non-collinear magnetization angle can periodically modulate the ZBCP. These novel features can realize a pure spin-triplet ZBCP and confirm the d-wave pairing symmetry induced by proximity effects in graphene. Our findings bring great promise for the large-scale realization of graphene-based superconducting spintronic devices in the future.