Topological pairings in Janus monolayer TaSSe

TL;DR

This study investigates the topological superconducting states in Janus monolayer H-TaSSe, revealing the emergence of full-gap, topologically nontrivial pairings influenced by strong Rashba effects, with potential for helical or chiral topological superconductivity.

Contribution

It is the first to analyze the pairing symmetry and topological states in Janus monolayer H-TaSSe, highlighting the influence of Rashba effects on topological superconductivity.

Findings

Identified a full-gap, topologically nontrivial s+f+p-wave pairing at higher chemical potentials.

Discovered a time-reversal broken d+p+f pairing with large Chern number at lower chemical potentials.

Suggested H-TaSSe as a candidate for helical or chiral topological superconductor.

Abstract

The Janus monolayer transition metal dichalcogenides[TMDs] MXY[M=Mo,W, etc. and X,Y=S,Se, etc.] has been synthesized recently, and the Rashba spin splitting arises in it owing to the breaking of out-of-plane mirror symmetry[\href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.235404}{Phys. Rev. B 97, 235404 (2018)}]. Here we study the pairing symmetry of superconducting Janus monolayer H-TaSSe by solving the linearized gap equation at the critical temperature $T_c$. We find that the strong Rashba effect in H-TaSSe could produce topological superconducting states which differs from that in its parent monolayer H-TaS$_2$ and H-TaSe$_2$. More specifically, at the chemical potential $\mu>-0.08$eV, we obtain a time-reversal invariant s+f+p-wave mixed pairing state. This pairing state is full-gap and topologically nontrivial, i.e. $\mathbb{Z}_2=1$. However, a time-reversal broken d+p+f pairing belonging to the 2-dimensional irreducible representation $E$ appears at lower chemical potential. It can host a large Chern number $C=-6$ at appropriate pairing strengths. The results suggest the monolayer H-TaSSe to be a candidate helical or chiral topological superconductor.

Tables

Table 1. Table 1: Classification of basis gap functions based on C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} symmetry. They are introduced as: C ​ ( 𝒌 ) = 1 3 ​ ∑ i = 1 3 cos ⁡ k i , C + ​ ( 𝒌 ) = 1 3 ​ ∑ i = 1 3 ω i − 1 ​ cos ⁡ k i , S ​ ( 𝒌 ) = 1 3 ​ ∑ i = 1 3 sin ⁡ k i , S + ​ ( 𝒌 ) = 1 3 ​ ∑ i = 1 3 ω i − 1 ​ sin ⁡ k i , C − ​ ( 𝒌 ) = C + ∗ ​ ( 𝒌 ) , S − ​ ( 𝒌 ) = S + ∗ ​ ( 𝒌 ) formulae-sequence 𝐶 𝒌 1 3 superscript subscript 𝑖 1 3 subscript 𝑘 𝑖 formulae-sequence subscript 𝐶 𝒌 1 3 superscript subscript 𝑖 1 3 superscript 𝜔 𝑖 1 subscript 𝑘 𝑖 formulae-sequence 𝑆 𝒌 1 3 superscript subscript 𝑖 1 3 subscript 𝑘 𝑖 formulae-sequence subscript 𝑆 𝒌 1 3 superscript subscript 𝑖 1 3 superscript 𝜔 𝑖 1 subscript 𝑘 𝑖 formulae-sequence subscript 𝐶 𝒌 superscript subscript 𝐶 𝒌 subscript 𝑆 𝒌 superscript subscript 𝑆 𝒌 C(\bm{k})=\frac{1}{\sqrt{3}}\sum\limits_{i=1}^{3}\cos{k_{i}},C_{+}(\bm{k})=\frac{1}{\sqrt{3}}\sum\limits_{i=1}^{3}\omega^{i-1}\cos{k_{i}},S(\bm{k})=\frac{1}{\sqrt{3}}\sum\limits_{i=1}^{3}\sin{k_{i}},S_{+}(\bm{k})=\frac{1}{\sqrt{3}}\sum\limits_{i=1}^{3}\omega^{i-1}\sin{k_{i}},C_{-}(\bm{k})=C_{+}^{*}(\bm{k}),S_{-}(\bm{k})=S_{+}^{*}(\bm{k}) , with 𝒙 ± = ( 𝒙 ± i ​ 𝒚 ) / 2 subscript 𝒙 plus-or-minus plus-or-minus 𝒙 𝑖 𝒚 2 \bm{x}_{\pm}=(\bm{x}\pm i\bm{y})/2 and the phase factor ω = e − i ​ 2 ​ π 3 𝜔 superscript 𝑒 𝑖 2 𝜋 3 \omega=e^{-i\frac{2\pi}{3}} .

$\Gamma$		Singlet	Triplet
$A_1$		${\Psi }^{A_1,on}=\frac{1}{\sqrt{2}}$ ${\Psi }^{A_1,nn}=C\left(𝒌\right)$	${𝒅}^{A_1,z}=S\left(𝒌\right)𝒛$ ${𝒅}^{A_1,xy}=i[S_{-}\left(𝒌\right){𝒙}_{+}-S_{+}\left(𝒌\right){𝒙}_{-}$]
$A_2$			${𝒅}^{A_2,xy}=S_{-}\left(𝒌\right){𝒙}_{+}+S_{+}\left(𝒌\right){𝒙}_{-}$
E		$\{\begin{array}{cc}{\Psi }_1^{E,nn}=C_{+}\left(𝒌\right)& \\ {\Psi }_2^{E,nn}=C_{-}\left(𝒌\right)& \end{array}$	$\{\begin{array}{cc}{𝒅}_1^{E,z}=S_{+}\left(𝒌\right)𝒛& \\ {𝒅}_2^{E,z}=S_{-}\left(𝒌\right)𝒛& \end{array}$ $\{\begin{array}{cc}{𝒅}_1^{E,xy}=\sqrt{2}S\left(𝒌\right){𝒙}_{+}& \\ {𝒅}_2^{E,xy}=-\sqrt{2}S\left(𝒌\right){𝒙}_{-}& \end{array}$ $\{\begin{array}{cc}{𝒅}_1^{E,\widetilde{xy}}=\sqrt{2}{S}_{-}\left(𝒌\right){𝒙}_{-}& \\ {𝒅}_2^{E,\widetilde{xy}}=-\sqrt{2}{S}_{+}\left(𝒌\right){𝒙}_{+}& \end{array}$

Equations

$$h_0(\mathbf{k})$$

$${\xi }_{\mathbf{k}}$$

$${\mathbf{\beta }}_{\mathbf{k}}$$

$${\mathbf{\alpha }}_{\mathbf{k}}$$

$$H_0(\mathbf{k})={\xi }_{\mathbf{k}}+{\mathbf{g}}_{\mathbf{k}}\cdot \mathbf{\sigma }.$$

$${\Delta }_i^{\Gamma ,\alpha }(\mathbf{k})=\{\begin{array}{ll}{\Psi }_i^{\Gamma ,\alpha }i{\sigma }_y& singletpairing\\ {\mathbf{d}}_i^{\Gamma ,\alpha }\cdot \mathbf{\sigma }i{\sigma }_y& tripletpairing\end{array},$$

$$\Delta (\mathbf{k})/\Delta =\sum _{\alpha }{c}^{\Gamma ,\alpha }{\Delta }^{\Gamma ,\alpha }(\mathbf{k}),$$

$${\Delta }_{s_1{s}_2}(\mathbf{k})$$

$$V_{s_1{s}_2{s}_3{s}_4}(\mathbf{k},{\mathbf{k}}^{\prime })$$

$$=$$

Full text

I Introduction

Topological superconductors(TSCs), which are characterized by a full-gap bulk and robust gapless surface states, have gained much attention for their novel topological properties recentlyFu and Kane (2008); Schnyder et al. (2008); Kitaev (2009); Fu and Berg (2010); Ryu et al. (2010); Qi and Zhang (2011); Das et al. (2012); Fidkowski et al. (2013); Sato and Ando (2017). Several superconductors are considered to be unconventional and even topologically nontrivial, including Sr2RuO4Rice and Sigrist (1995); Baskaran (1996); Nelson et al. (2004) and some heavy fermion compoundsStewart (1984); Varma (1985); De Visser et al. (1987). There are also theoretical proposals for realizing the topological superconducting states artificially by using the s-wave superconductor and spin-orbital coupling(SOC). For instance, through proximity effect, the s-wave superconductor on a strong topological insulator becomes a spinless p+ip-wave superconductorFu and Kane (2008). Besides, by applying a strong perpendicular magnetic field to an s-wave Rashba superconductor, topological superconducting states can emergeSato et al. (2009); Sau et al. (2010). However, there’s no definitive experimental evidence for TSCs until now, so it’s still a challenge to search for them theoretically and experimentally.

Transition metal dichalcogenides(TMDs) are one kind of noncentrosymmetric layered van der Waals materials, which have been studied for decades as they always exhibit superconductivity, CDW and other electronic phenomenaClayman and Frindt (1971); Boaknin et al. (2003); Huang et al. (2007); Berthier et al. (1976); Mutka (1983); Castro Neto (2001). Superconductivity in the bulk TMDs is generally thought to be conventional BCS-type. However, in the ultra-thin films, a strong enhancement of the in-plane critical field indicates possible unconventional superconductivity caused by the large Ising SOC JM et al. (2015); Xi et al. (2015); Saito et al. (2016); Xing et al. (2017); Lu et al. (2018); Barrera et al. (2018). Two kinds of parity-mixed pairings, i.e. s+f-wave and d+p-wave, have been suggested as candidate unconventional superconducting states in the monolayer TMDsYuan et al. (2014); Hsu et al. (2017).

This work is on the basis of newly synthesized Janus TMDs MXY(M=Mo,W,Ta, etc. and X,Y=S,Se,Te, etc.)Lu et al. (2017); Zhang et al. (2017); Li et al. (2018); Hu et al. (2018); Shi and Wang (2018); He and Li (2018); Zhou et al. (2019); Yagmurcukardes et al. (2019). In a Janus TMD molecular layer, M atomic plane is sandwiched by two different atomic planes X and Y, which breaks the out-of-plane mirror symmetry, leading to a Rashba-type spin splitting in the band around $\Gamma$. Previous works on the Janus layered materials mainly focused on the electronic structure, while the superconducting pairing phase of them has not been discussed. We choose H-TaSSe as our present pairing-symmetry study object based on the following facts: First, its parent phases H-TaS2 and H-TaSe2 are intrinsic superconductors. Thus, H-TaSSe is very likely to be a superconductor too. Second, there are $\Gamma$-centered FS sheets in the normal states of H-TaS2 and H-TaSe2Rossnagel et al. (2005); Sanders et al. (2016). Similar FS sheets in H-TaSSe then will be subjected to a large Rashba spin splitting, which could affect the pairing symmetry. Therefore, because of the strong SOC effects, including both the Rashba and Ising type, singlet and triplet pairings would be significantly mixed together, and topological superconducting states differing from that in monolayer H-TaX2 may be achieved in Janus monolayer TaSSe. We simply ignore CDW here for the suppression of it in the 2D TMDsNavarro-Moratalla et al. (2016); Yang et al. (2018).

By solving the linearized gap equation at the critical temperature, we obtain the most favorable mixed pairing phases of Janus monolayer TaSSe at different pairing interactions and chemical potentials. Our results demonstrate that, at relatively high chemical potential, a time-reversal invariant(TRI) s+f+p-wave pairing state of irreducible representation(IR) $A_{1}$ will dominate the nearest-neighbor(NN) pairing channels. Compared to the nodal s+f-wave pairing of monolayer H-TaS2, this s+f+p-wave pairing has a large additional $p\pm ip$-wave component induced by the Rashba SOC, making the superconductor to be full-gap and topologically nontrivial. At a lower $\mu$, however, a time-reversal broken(TRB) d+p+f-wave pairing belonging to 2-dimensional(2D) IR $E$ appears in the phase diagram. It can be a large-Chern-number[$C=-6$] chiral TSC at appropriate pairing strengths. On the other hand, the TRB d+p-wave pairing of monolayer H-TaSe2 always hold a trivial Chern number. This difference could be owing to the Rashba-induced non-unitary p- and f-wave pairing components in the $E$ phase of H-TaSSe.

II Model

The crystal structure of monolayer H-TaSSe is shown in Fig. 1(a). It can be viewed as the monolayer H-TaS2[H-TaSe2] whose bottom[top]-layer S[Se] atoms are replaced by Se[S]. Given the absence of literatures about H-TaSSe, and note that it will preserve robust electronic structure from its parent phasesHu et al. (2018), we construct its normal Hamiltonian based on H-TaS2 and H-TaSe2. Thus a tight-binding model describing both H-TaS2 and H-TaSe2 can be given by

[TABLE]

where $k_{j}=\bm{k}\cdot\bm{R_{j}}$, with bonding vectors $\bm{R}_{1}=a\bm{x}$, $\bm{R}_{2}=a(-\frac{1}{2}\bm{x}+\frac{\sqrt{3}}{2}\bm{y})$, $\bm{R}_{3}=a(-\frac{1}{2}\bm{x}-\frac{\sqrt{3}}{2}\bm{y})$ and $\bm{R}_{4}\equiv\bm{R}_{1}$, $a$ the lattice constant. $t,t_{1}$ denote the NN and next nearest-neighbor(NNN) hopping. $\mu$ is the chemical potential and $\beta_{so}$ the Ising SOC parameter. We fix $(t,t_{1},\beta_{so})=(-0.06,-0.14,0.067)$eV, and set the chemical potential $\mu$ as an adjustable parameter. When $\mu=0$eV, this model can fit the band structure of H-TaS2 given by experiments and DFT calculations very wellNavarro-Moratalla et al. (2016); Sanders et al. (2016); Zhao et al. (2017). On the other hand, a chemical potential $\mu=-0.18$ eV will give a FS topology of H-TaSe2, which has dog-bone-shaped electron Fermi pockets centered at $M$Rossnagel et al. (2005).

The Rashba spin splitting caused by the Mirror asymmetry of monolayer H-TaSSe can be written as

[TABLE]

The parameter $\alpha_{R}$ is taken from the H-WSSeHu et al. (2018), whose Rashba splitting strength is 158 meV$\cdot$Å. This Rashba strength is corresponding to an energy 0.032 eV in our lattice model if we assume the lattice constant $a=3.3$ Å, the same as in monolayer H-TaS2Sanders et al. (2016). The selection of $\alpha_{R}$ is reasonable for the atomic-structure similarity between tantalum and tungsten.

These two kinds of SOC can be combined as $\bm{g_{k}}=\bm{\alpha}_{\bm{k}}+\bm{\beta_{k}}$, then the normal Hamiltonian of monolayer H-TaSSe is

[TABLE]

As a result, the spin splitting around $\Gamma$ is Rashba-type while around K it keeps Ising-typeHu et al. (2018). The chemical potential $\mu$ of H-TaSSe is hard to be determined in our model, and its FS topology can be either similar to that of H-TaS2 or H-TaSe2. In view of this, both two cases are under our consideration and corresponding calculations are also made in section IV. The low-energy spectrum and FS of H-TaSSe at the two specific chemical potentials, $\mu=0$ and -0.18 eV, are simulated and shown in Fig. 1(b)-(c). It can be seen from the figures that the first Lifshitz transition occurs nearly at $\mu=-0.15$ eV. The “Q” labels the position of the corresponding van Hove singularity(vHS) in the Brillouin zone(BZ) .

III Symmetry Analysis and Method

The point-group symmetry of monolayer H-TaSSe is C3v, basis gap functions of which are all presented in Table 1. Only the on-site and nearest-neighbor[NN] pairing channels are under our consideration. According to the spin angular momentum of the pair, i.e. 0[spin-singlet] or 1[spin-triplet], gap functions will take the matrix form as:

[TABLE]

where $\Psi^{\Gamma,\alpha}_{i}$ is the singlet order parameter and $\bm{d}^{\Gamma,\alpha}_{i}$ the $\bm{d}$-vector for the triplet pairing. Thus, the stable pairing of our system can be written as a linearized combination of these basis functions:

[TABLE]

where $\Delta$ is the gap size value and $c^{\Gamma,\alpha}$ denotes the relative amplitude of $\Delta^{\Gamma,\alpha}(\bm{k})$, with the orthogonal relation $\sum_{\Gamma,\alpha}|c^{\Gamma,\alpha}|^{2}=1$.

From Table 1 we can get that $\Psi^{A_{1},nn}=C(\bm{k})$ represents the extended s-wave pairing, and $S(\bm{k})$ the f-wave pairing, while $C_{\pm}(\bm{k})$ and $S_{\pm}(\bm{k})$ denotes the chiral $d{\pm}id$ and $p_{\mp}ip$-wave pairings, respectively. Dispersions of these gap functions in the momentum space are shown in Fig. 2. SOC could destroy the inversion symmetry, giving rise to admixtures between singlet and triplet pairings. In addition, the $\bm{d}$-vector would tend to be paralleled to $\bm{g_{k}}$. So, in monolayer H-TaSSe with both strong in-plane Rashba and out-of-plane Ising SOC, the large admixtures between $\psi$ , $\bm{d}^{xy}$ and $\bm{d}^{z}$ are expected.

We capture the $T_{c}$ and the most stable pairing $\Delta(\bm{k})$ by solving the linearized gap equation:

[TABLE]

where $G(\bm{k}^{\prime},i\omega_{n})$ is the normal-state Matsubara Green’s function, and $V(\bm{k},\bm{k}^{\prime})$ the superconducting interaction. The form of $V(\bm{k},\bm{k}^{\prime})$ can be given by the basis gap functions as follow:

[TABLE]

where $v_{0},v_{1}$ represent the on-site and NN pairing constants and the positive(negative) values of them denote attractive(repulsive) interactions. Index $i$ is used to distinguish two equivalent basis pairing functions in the same IR. For 2D IR $\Gamma=E$, index $i=1,2$, while for 1D IR $\Gamma=A_{1},A_{2}$, it is omitted.

IV Result and discussion

We obtain the $v_{0}$ versus $v_{1}$ pairing phase diagrams of H-TaSSe at specific chemical potential $\mu=0$eV and $-0.18$eV, which are presented in Fig. 1(c)(d). To investigate the strong Rashba effect on the pairing symmetry, the corresponding pairing phase diagrams of monolayer H-TaS2 and H-TaSe2[or can be viewed as a p-doped H-TaS2] are also shown in Fig. 1(a)(b).

As illustrated in Fig. 1(a), a nodal TRI mixed state $\Psi^{A_{1},nn}+\bm{d}^{A_{1},z}$ belonging to IR $A_{1}$, which is denoted as “s+f-wave[$A_{1}$,Nodal]”, dominates the phase diagram of H-TaS2 if the NN pairing interaction is attractive and stronger than the on-site one. The s+f-wave pairing is a 2D Weyl superconductor and holds 12 nodal points on the $\Gamma$-centered FSChen and An (2019). Otherwise, the conventional on-site s-wave has the largest gap amplitude. Border between the two phases is roughly depicted by the gray dash.

In the H-TaSSe[see Fig. 1(c)], the large Rashba SOC induces helical p-wave component $\bm{d}^{A_{1}，xy}$ to the s+f-wave pairing, changing this nodal phase into full-gap. Typical relative amplitudes of these three components are $(c^{A_{1},nn},c^{A_{1},z},c^{A_{1},xy})$=(-0.24,0.91,-0.34). The helical p-wave gap is larger than the extended s-wave gap on the $\Gamma$-centered FS. As a result, this mixed s+f+p-wave state is topologically nontrivial with a $\mathbb{Z}_{2}=1$ indexSato and Fujimoto (2009).

In the p-doped H-TaS2 with chemical potential $\mu=-0.18eV$, a mixed state $\Psi^{E,nn}+\bm{d}^{E,z}$[d+p-wave] of the 2D IR $E$ dominates the NN pairing channels[see Fig. 4(b)]. In principle, there are four components that actually can mix with each other: $\Psi_{1}^{E,nn},\bm{d}_{1}^{E,z},\Psi_{2}^{E,nn}$ and $\bm{d}_{2}^{E,z}$, whose relative amplitudes denoted by ($c_{1}^{E,nn},c_{1}^{E,z},c_{2}^{E,nn},c_{2}^{E,z}$). However, the minimization of the system’s free energy gives the most stable d+p-wave mixed states with either $(c_{1}^{E,nn},c_{1}^{E,z})=0$ or $(c_{2}^{E,nn},c_{2}^{E,z})=0$. In another words, the admixture occurs only between d-id- and p+ip or d+id- and p-ip-wave. These two mixed states are equivalent and both break the time-reversal symmetry, allowing a calculation of Chern numberSchnyder et al. (2008) on them. We adopt the d+p-wave pairing with specific amplitudes $(c_{1}^{E,nn},c_{1}^{E,z})=(-0.47,0.89$), and the calculation gives a zero Chern number. In fact, the whole E phase in Fig. 4(b) is topologically trivial.

In the H-TaSSe with the same chemical potential, non-unitary pairings $\bm{d}^{E,xy}_{i}$ and $\bm{d}^{E,\widetilde{xy}}_{i}$ are induced to the $E$ phase. Based on the free-energy minimization, we choose $i=1$ for our following discussions. $\bm{d}^{E,xy}_{1}$ is an f-wave pairing emerging on the $\uparrow\uparrow$ pairing channel while $\bm{d}^{E,\widetilde{xy}}_{1}$ a p-ip-wave pairing on the $\downarrow\downarrow$ channel. We denote this mixed state $\Psi^{E,nn}+\bm{d}^{E,z}+\bm{d}^{E,xy}+\bm{d}^{E,\widetilde{xy}}$ as the d+p+f-wave. It keeps topologically trivial in the diagram except the regions $v_{1}>0.27eV$, where the nontrivial Chern number is $C=-6$, as shown in Fig. 1(d). Typical relative amplitudes of the basis functions in the trivial and nontrivial cases are ($c_{1}^{E,nn},c_{1}^{E,z},c_{1}^{E,xy},c_{1}^{E,\widetilde{xy}}$)=(0.87,-0.39,-0.12,-0.28) and (0.8,-0.48,-0.14,-0.33), respectively. It is evident that a small modification of the ratios between these basis functions leads to a topological phase transition.

Comparing the phase diagram of H-TaS2 to that of H-TaSSe, we can see that strong Rashba effect could change the topology of the NN pairing phase by inducing additional triplet pairings with vector $\bm{d}\perp\bm{z}$. Therefore, the possible topological pairing phase in H-TaSSe can be either an s+f+p-wave $\mathbb{Z}_{2}$ or a d+p+f-wave chiral mixed state depending on the chemical potential. On the edges, the former hosts robust helical Majorana zero modes(MZMs), while the latter has six channels of chiral ones. The corresponding energy spectrum of them are displayed in Fig. 4. Recent superconducting conductance experiments observed zero bias conductance peak of the thin flakes of superconducting 2H-TaS2 and 2H-TaSe2Galvis et al. (2013, 2014), indicating novel superconductivity in the atomic limit of 2H-TMDs. Theoretical calculations suggest that the novel pairings of thin-layer 2H-TMDs could be a topological s+f-wave or d+p-waveYuan et al. (2014); Hsu et al. (2017), both of which require a NN pairing attraction larger than the on-site coupling. Consequently, assuming the same interaction relation $v_{1}>v_{0}$ here, it is very likely to realize the s+f+p-wave or d+p+f-wave pairing in monolayer H-TaSSe.

To further probe the pairings of monolayer H-TaSSe with respect to the chemical potential, we make a $v_{1}/v_{0}$ versus $\mu/v_{0}$ pairing phase diagram as shown in Fig. 5, with $v_{0}=0.1eV$. Consistent with the above results, once $v_{0}>v_{1}$, the most stable phase is the conventional s-wave. Otherwise, it would be the NN pairings: The TRI s+f+p-wave pairing state dominates the parameter regions where $\mu>-0.08eV$, while the d+p+f-wave pairing is stabilized in other regions. As can be seen, when $\mu$ decreases, the $E$ phase emerges before the onset of the first Lifshitz transition[-0.15 eV]. This emergence of $E$ phase can be qualitatively understood as follows: At a high chemical potential, there are FS sheets centered at K(K’), where $S(\bm{k})$[f-wave], the dominant part of the s+f+p-wave, has the maximum gap value[see Fig. 3(b)]. As $\mu$ falls, the Fermi level will approach the saddle points M and Q, but away from K(K’). Gap functions $C+(\bm{k})$[d-wave] and $S+(\bm{k})$[p-wave], the main components of the $E$ phase, have gap-maximum at M and Q[see Fig.3(c)-(d)]. Generally speaking, the larger gap on the FS, the higher $T_{c}$. Therefore, when $\mu$ decreases to about $-0.08$eV, the $T_{c}$ of $E$ and $A_{1}$ phases become equal and a phase transition emerges. Most of the $E$-phase regions in the diagram are topologically trivial, except two islands with Chern numbers -6. These two islands located nearly on both sides of the vHS[-0.15 eV], indicating that the FS topology of the normal H-TaSSe is not essential for the topology of $E$ phase actually.

V Conclusion

In this paper, the possible pairings of Janus monolayer H-TaSSe at different parameters have been investigated. By analyzing the pairing symmetry and calculating the linearized gap equation, we’ve identified a $\mathbb{Z}_{2}$ topological s+f+p-wave pairing state at the relatively high chemical potential, and a chiral d+p+f-wave pairing state with a large Chern number $C=-6$ at the lower one. The results show that Janus monolayer TaSSe can be a promising intrinsic helical or chiral TSC. Although here we merely focus on H-TaSSe, the conclusion can be applied to other superconducting Janus monolayer TMDs with a strong Rashba splitting. Our work will help to find the TSCs and realize the MZMs in the future.

VI ACKNOWLEDGMENTS

We thank X. Xun and H. Ya for useful discussions. This work is supported by NSFC Project NO.111774126 and 973 Projects No.2015CB921202.