Abstract
Theoretically, it has been known that breaking spin degeneracy and effectively realizing spinless fermions is a promising path to topological superconductors. Yet, topological superconductors are rare to date. Here we propose to realize spinless fermions by splitting the spin degeneracy in momentum space. Specifically, we identify monolayer holedoped transition metal dichalcogenide (TMD)s as candidates for topological superconductors out of such momentumspacesplit spinless fermions. Although electrondoped TMDs have recently been found superconducting, the observed superconductivity is unlikely topological because of the near spin degeneracy. Meanwhile, holedoped TMDs with momentumspacesplit spinless fermions remain unexplored. Employing a renormalization group analysis, we propose that the unusual spinvalley locking in holedoped TMDs together with repulsive interactions selectively favours two topological superconducting states: interpocket paired state with Chern number 2 and intrapocket paired state with finite pair momentum. A confirmation of our predictions will open up possibilities for manipulating topological superconductors on the devicefriendly platform of monolayer TMDs.
Introduction
The quest for material realizations of topological chiral superconductors with nontrivial Chern numbers^{1,2,3,4} is fuelled by predictions of exotic signatures, such as Majorana zero modes and quantized Hall effects. Unfortunately, natural occurrence of bulk topological superconductors are rare with the best candidates being superfluid ^{3}He (ref. 5) and Sr_{2}RuO_{4} (ref. 6). Instead, much recent experimental progress relied on proximityinducing pairing to a spin–orbitcoupled band structure building on the proposal of Fu and Kane^{7}. Their key insight was that a paired state of spinless fermions is bound to be topological and that the surface states of topological insulators are spinless in that the spin degeneracy is split in position space (rspace): the two degenerate Dirac surface states with opposite spin textures are spatially separated. Nevertheless, despite much experimental progress along this direction^{8,9,10,11,12,13,14}, the confinement of the helical paired state to the interface of the topological insulator and a superconductor limits experimental access to its potentially exotic properties.
Another type of exotic paired states that desires material realization is the finitemomentumpaired states, which has long been pursued since the first proposals by Fulde and Ferrell^{15} and by Larkin and Ovchinnikov^{16}. Most efforts towards realization of such modulated superconductors^{17,18}, however, relied on generating finitemomentum pairing using spinimbalance under a (effective) magnetic field in close keeping with the original proposals. Exceptions to such a spinimbalancebased approach are refs 19, 20 that made use of spinless Fermi surfaces with shifted centres. More recently, there have been proposals suggesting modulated paired states in cuprate highT_{c} superconductors^{21,22,23}. However, unambiguous experimental detection of a purely modulated paired state in a solidstate system is lacking.
We note an alternative strategy that could lead to pairing possibilities for both topological and modulated superconductivity: to split the spin degeneracy of fermions in momentum space (kspace). This approach is essentially dual to the proposal of Fu and Kane, and it can be realized in a timereversalinvariant noncentrosymmetric system when a pair of Fermi surfaces centred at opposite momenta ±k_{0} consist of oppositely spinpolarized electrons (see Fig. 1a). When such a spinvalleylocked band structure is endowed with repulsive interactions, conventional pairing will be suppressed. Instead, there will be two distinct pairing possibilities: interpocket and intrapocket pairings, where the latter will be spatially modulated with pairs carrying finite centreofmass momentum ±2k_{0}.
What is critical to the success of this strategy is the materialization of such kspacesplit spinless fermions. A new opportunity has arisen with the discovery of a family of superconducting twodimensional (2D) materials, monolayer groupVI transition metal dichalcogenides (TMDs) MX_{2} (M=Mo, W, X=S, Se)^{24,25,26,27}. Although the transition metal atom M and the chalcogen atom X form a 2D hexagonal lattice within a layer as in graphene, monolayer TMDs differ from graphene in two important ways. First, TMD monolayers are noncentrosymmetric, that is, inversion symmetry is broken (see Fig. 1b,c). As a result, monolayer TMDs are directgap semiconductors^{28} with a type of Dresselhaus spin–orbit coupling^{29,30} referred to as Ising spin–orbit coupling^{31}. This spin–orbitcoupled band structure leads to the valley Hall effect^{30,32}, which has established TMDs as experimental platforms for pursuing valleytronics applications^{30,32,33,34,35,36}. Our focus, however, is the fact that there is a sizable range of chemical potential in the valence band that could materialize the kspace spinsplit band structure we desire (see Fig. 1d). Second, the carriers in TMDs have strong dorbital character and, hence, correlation effects are expected to be important. Interestingly, both intrinsic and pressureinduced superconductivity have been reported in electrondoped (ndoped) TMDs^{24,25,26,27} with the debate regarding the nature of the observed superconducting states still ongoing^{37,38,39,40,41}.
Here we propose to obtain kspacesplit spinless fermions by lightly holedoping (pdoping) monolayer TMDs such that the chemical potential lies between the two spinsplit valence bands. We investigate the possible paired states that can be driven by repulsive interactions^{42} in such lightly pdoped TMDs using a perturbative renormalization group (RG) analysis going beyond meanfield theory^{38,43}. We find two distinct topological paired states to be the dominant pairing channels: an interpocket chiral (p/d)wave paired state with Chern number C=2 and an intrapocket chiral pwave paired state with a spatial modulation in phase. The degeneracy can be split by the trigonal warping or Zeeman effect.
Results
Spinvalleylocked fermions in lightly pdoped monolayer TMDs
The generic electronic structure of group IV monolayer TMDs is shown in Fig. 1d. The system lacks inversion symmetry (see Fig. 1b,c), which leads to a gapped spectrum and a S_{z}preserving spin–orbit coupling. Such Ising spin–orbit coupling^{31} acts as opposite Zeeman fields on the two valleys that preserve timereversal symmetry. Furthermore the spin–orbit coupling is orbitalselective^{44} and selectively affects the valence band with a large spinsplit^{29}.
By lightly pdoping the TMDs with the chemical potential μ between the spinsplit valence bands, spinvalleylocked fermions can be achieved near the two valleys (see Fig. 2a,b). Assuming negligible trigonal warping at low doping, we can use a single label τ=↑,↓ to denote the valley and the spin. Denoting the momentum measured from appropriate valley centres ±K by q, the kinetic part of the Hamiltonian density is
where μ is the chemical potential, m is the effective mass of the valence band and c_{q,↑}≡ψ_{K+q,↑} and c_{q,↓}≡ψ_{−K+q,↓} each annihilates a spinup electron with momentum q relative to the valley centre K or a spindown electron with momentum q relative to the valley centre−K (see Fig. 2a). Hence, the spinvalleylocked twovalley problem is now mapped to a problem with a single spindegenerate Fermi pocket. Nonetheless, the possible paired states with total spin τ_{z}=±1 and τ_{z}=0 in fact represent the novel possibilities of intrapocket modulated pairings with total τ_{z}=±1 and interpocket pairing with total τ_{z}=0, respectively (see Fig. 2a,b).
Pairing possibilities
To discuss the pairing symmetries of the two pairing possibilities, it is convenient to define the partialwave channels with respect to the twovalley centres ±K. Since a total spin τ_{z}=±1 intrapocket pair consists of two electrons with equal spin, Pauli principle dictates such pairing to be in a state with odd partialwave . Stepping back to microscopics, such pairs carry finite centreofmass momentum ±2K and form two copies of phasemodulated superconductor^{15}. This case may or may not break timereversal symmetry due to the absence of locking between the s of the two pockets τ=↑,↓. For the total τ_{z}=0 interpocket pairing, the allowed symmetries of a superconducting state is further restricted by the underlying C_{3v} symmetry of the lattice. In particular, the absence of an inversion centre allows the pairing wavefunction in each irreducible representation to be a mixture between parityeven and odd functions with respect to the Γ point^{45}. Specifically, swave mixes with fwave and dwave mixes with pwave (see Fig. 2c,d). Among the irreducible representations of C_{3v}, two fully gapped possibilities are the trivial A_{1} representation, which amounts to (s/f)wave pairing () and a chiral superposition of the 2D E representation, which amounts to a mixture of p±ip and d∓id pairing (). The mixing implies that the nontopological fwave channel that is typically dominant in trigonal systems as a way of avoiding repulsive interaction will be blocked together with swave by the repulsive interaction in the pdoped TMDs. Hence, it is clear that the pairing instability in channel is all one needs for topological pairing in the pdoped TMDs.
Two distinct topological paired states
To investigate the effects of the repulsive interactions between transition metal dorbitals, we take the microscopic interaction to be the Hubbard interaction, which is the most widely studied pardignamtic model of strongly correlated electronic systems
where W is the ultraviolet energy scale, U>0 and n_{i,s} is the density of electrons with spin s on site i. By now, it is well established that the interaction that is purely repulsive at the microscopic level can be attractive in anisotropic channels for lowenergy degrees of freedom, that is, fermions near Fermi surface. The perturbative RG approach has been widely used to demonstrate this principle on various correlated superconductors. For the model of pdoped TMDs defined by Equations (1) and (2), the symmetryallowed effective interactions at an intermediate energy scale close to the Fermi level in the Cooper channel (see Supplementary Note 1) would be:
where q and q′ are the incoming and outgoing momenta. Now, the remaining task is to derive the effective inter and intrapocket interactions g_{↑,↓}(q,q′) and g_{↑,↑}(q,q′) perturbatively in the microscopic repulsion U and check to see whether attraction occur in the channel (see Methods and Supplementary Note 2).
Before going into the details of calculation, it is important to note that isotropic pairing with is forbidden by Pauli principle in the total τ_{z}=±1 channel and blocked by the bare repulsive interaction in the total τ_{z}=0 channel. Hence, we need to look for attraction in the anisotropic channel, which is given by the momentumdependent part of g_{ττ′}^{(0)}. With our assumption of isotropic dispersion at lowdoping, one needs to go to the twoloop order to find momentum dependence in the effective interaction. Fortunately, it has been known for the model of Equations (1) and (2) that effective attraction is indeed found in anisotropic channels at the twoloop order^{46}. Here we carry out the calculation explicitly (see Methods and Supplementary Note 2) and find the effective interactions in the channel to be attractive, that is,
for τ,τ′=↑,↓, where is the angle associated with the momentum transfer, and is the normalized angularmomentumone eigenstate in 2D.
In the lowenergy limit, the effective attractions in the channel at the intermediate energy scale Λ_{0} in Equation (4) will lead to the following two degenerate topological paired states (see Methods): the interpocket (p/d)wave pairing, which is expected to be chiral (see Fig. 3a) and the modulated intrapocket pairing (see Fig. 3b). The degeneracy is expected for the model of Equations (1) and (2) with its rotational symmetry in the pseudo spin τ. There are two ways this degeneracy can be lifted. First, the trigonal warping will suppress intrapocket pairing as the two points on the same pocket with opposing momenta will not be both on the Fermi surface any more (see Fig. 3c). On the other hand, a ferromagnetic substrate will introduce an imbalance between the two pockets, which promotes intrapocket pairing^{47} (see Fig. 3d).
Discussion
The distinct topological properties of the two predicted exotic superconducting states lead to unusual signatures. The interpocket paired state (see Fig. 3a) is topological with Chern number C=2 because of the two pockets (see Methods). The Chern number dictates for two chiral edge modes, which in this case are Majorana chiral edge modes each carrying central charge (1)/(2) (refs 1, 48). This is in contrast to d+id paired state on a single spindegenerate pocket, which is another chiral superconducting state^{49,50,51,52,53,54} with four chiral Majorana edge modes. An unambiguous signature of two Majorana edge modes in the interpocket chiral paired state will be a quantized thermal Hall conductivity^{1} of
at temperature T, where c=1 is the total central charge. In addition, signatures of the chiral nature of such state could be revealed by a detection of timereversal symmetry breaking in polar Kerr effect and muon spin relaxation measurements. Finally, a sharp signature of anisotropy of the pairing will be the maximization of the critical current in a direct current superconducting quantum interference device (dc SQUID) interferometry setup of Fig. 4a at some finite flux Φ_{max}≠0.
The intrapocket paired state (see Fig. 3b) is not only topological, but also its phase of the gap is spatially modulated with e^{i2K˙r} and e^{−i2K˙r} for spinup and down pairs, respectively, where r is the spatial coordinate of the centreofmass of the pair (see Supplementary Note 3). Since the gaps on the two pockets are not tied to each other in principle, the system may be either helical respecting timereversal symmetry (C=0) or chiral (C=2). Either way, there will be a Majorana zero mode of each spin species at a vortex core so long as τ_{z} is preserved. What makes the intrapocket paired state distinct from existing candidate materials for topological superconductivity, however, is its spatial modulation. Smoking gun signature of the modulation in phase would be the halved period (hc)/(4e) of the oscillating voltage across the dc SQUID setup in Fig. 4b in flux Φ due to the difference between the pair momenta on the two sides of the junction. Another signature of the intrapocket paired state will be the spatial profile of the modulated phase directly detected with an atomic resolution scanning Josephson tunnelling microscopy^{23,55}.
In summary, we propose the kspace spin splitting as a new strategy for topological superconductivity. Specifically, we predict lightly pdoped monolayer TMDs with their spinvalleylocked band structure and correlations to exhibit topological superconductivity. Of the monolayer TMDs, WSe_{2} may be the most promising as its large spinsplitting energy scale^{56} allows for substantial carrier density within the spinvalleylocked range of doping^{57}. The rationale for the proposed route is to use a lower symmetry to restrict the pairing channel. The merit of this approach is clear when we contrast the proposed setting to the situation of typical spindegenerate trigonal systems. With a higher symmetry, trigonal systems typically deals with the need for anisotropic pairing due to the repulsive interaction by turning to the topologically trivial fwave channel^{2,49}. The ndoped TMDs whose lowenergy band structure is approximately spindegenerate fall into this category. Hence, experimentally realized superconductivity in ndoped systems would likely be topologically trivial even if the superconductivity is driven by the same repulsive interaction we consider here. The predicted topological paired states in pdoped TMDs are a direct consequence of the spinvalley locking, which breaks the spin degeneracy in kspace and creates two species of spinless fermions. Experimental confirmation of the predicted topological superconductivity in pdoped TMDs will open unprecedented opportunities in these highly tunable systems.
Methods
Perturbative RG calculation
For the RG calculation, we follow the perturbative twostep RG procedure in ref. 49, which has been used to study superconductivity in systems such as Sr_{2}RuO_{4} (ref. 58) and generic hexagonal lattices with spin degeneracy^{49}. Taking the Hubbard onsite repulsion in Equation (2) as the microscopic interaction, the first step is to integrate out higherenergy modes and obtain g_{τ,τ′}^{(0)} in Equation (3), the lowenergy effective interactions in the Cooper channel at an intermediate energy close to the Fermi level. The second step is to study the evolution of these effective interactions as the energy flows from Λ_{0} to 0, which is governed by the RG equations.
In the first step, we calculate the inter and intrapocket effective interactions g_{inter}^{(0)}(q,q′)≡(q,q′) and g_{intra}^{(0)}(q,q′)≡g_{τ,τ}^{(0)}(q,q′) in terms of the incoming and outgoing momenta q and q′ order by order in U until we obtain attraction in one of them in certain partialwave channel . Following ref. 46, we find the effective interactions to be (see Supplementary Note 2)
and
where p=q±q′ is the momentum transfer, C>0 and C′<0 are momentumindependent constants coming from tree level and oneloop order, and the momentumdependent terms come solely from twoloop order.
Each partialwave component is given by the projection of g_{inter/intra}^{(0)}(q,q′) on to the normalized angular momentum eigenstate in 2D, , where is the angle associated with the momentum transfer p. We find
where H_{n} is the n^{th} harmonic number and α≡(U^{3}m^{2})/(2π^{3}) and β≡(U^{3}m^{2})/(64π^{3}) are postive constants related to density of states and interaction strength. Here terms with α and β come from contributions with one particle–particle and one particle–hole bubble, and two particle–hole bubbles, respectively (see Supplementary Note 2). The α term in acquires an extra minus sign on top of from the closed fermion loops in Supplementary Fig.1 (3g) and (3h). Meanwhile, the α term in contains an implicit factor because of the fact that the outgoing external momenta in Supplementary Fig. 1 (3a) and (3b) are exchanged, which is equivalent to setting .
Note that with even s is forbidden since intrapocket pairs have equal spin, and that for odd s since they correspond to the spintriplet states with τ_{z}=0 and ±1, respectively. While λ_{inter}^{(0),0}>0 as expected from the bare repulsion, the most negative values are .
In the second step, we derive and solve the RG equations to study the evolutions of the effective interactions as the energy E lowers from Λ_{0} to 0. Using in Equation (8) as the initial values for the RG flows, the channel with the most relevant attraction in the lowenergy limit E→0 is the dominant pairing channel. Under the assumption that the energy contours for 0<E<Λ_{0} are isotropic, different partialwave components do not mix while the inter and intrapocket interactions with the same can in principle mix. By a procedure similar to that in refs 50, 59, we find the RG equations up to oneloop order to be
and
where the inverse energy scale is the RG running parameter, d_{1}(y)≡(δΠ_{pp}^{ss}(±2K))/(δ y), d_{2}(y)≡, and d_{3}(y)≡(δΠ_{ph}^{ss}(0))/(δ y). Here Π_{pp/ph}^{ss′}(k) is the noninteracting static susceptibility at momentum k in the particle–particle or particle–hole channel defined in Supplementary Note 1. Since the lowenergy band structure is well nested at ±2K in the particle–particle channel, the Cooper logarithmic divergence appears not only at k=0 but also ±2K (see Supplementary Note 1). Thus, d_{1}(y)=1. On the other hand, since the lowenergy band structure is poorly nested at any k in the particle–hole channel and is far from van Hove singularity, the particle–hole susceptibilities do not diverge in the lowenergy limit (see Supplementary Note 1). Thus, in the lowenergy limit y→∞. Therefore, with logarithmic accuracy, the inter and intrapocket interactions renormalize independently with the wellknown RG equation in the Cooper channel
with i=inter, intra. The RG flow , which solves the RG equation, shows that the pairing interaction in channel becomes a marginally relevant attraction only if the initial value . Since we concluded that the most negative initial values occur in the channels for both inter and intrapocket interactions in the first step of the RG procedure, we expect degenerate inter and intrapocket pairings in the lowenergy limit.
The Chern number of interpocket paired state
The interpocket chiral paired state becomes just a spinful p+ip paired state with total spin τ_{z}=0 when we map the spinvalleylocked twopocket problem to a spindegenerate singlepocket problem. The spinful p+ip pairing comprises two copies of ‘spinless’ p+ip pairings as the Bogoliubovde Gennes Hamiltonian of the former can be written as
where the lowenergy dispersion ɛ_{q}=−(q^{2})/(2m)−μ, the gap function and c_{q,±}≡(c_{q,↑} ±c_{q,↓} )/. Since a spinless p+ip paired state has Chern number C=1, where with , the τ_{z}=0 spinful p+ip paired state in the singlepocket system has C=2. Hence, the interpocket chiral pairing in the twopocket system has C=2 as well.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information file.
Additional information
How to cite this article: Hsu, Y.T. et al. Topological superconductivity in monolayer transition metal dichalcogenides. Nat. Commun. 8, 14985 doi: 10.1038/ncomms14985 (2017).
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Acknowledgements
We thank Reza Asgari, Debdeep Jena, Katja Nowak, Grace Xing, K. T. Law and Andrey Chubukov for helpful discussions. Y.T.H. and E.A.K. were supported by the Cornell Center for Materials Research with funding from the NSF MRSEC programme (DMR1120296). E.A.K. was supported in part by by the National Science Foundation (Platform for the Accelerated Realization, Analysis, and Discovery of Interface Materials (PARADIM)) under Cooperative Agreement No. DMR1539918. A.V. was supported by Gordon and Betty Moore Foundation and in part by Bethe postdoctoral fellowship. M.H.F. acknowledges support from the Swiss Society of Friends of the Weizmann Institute.
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Y.T.H. carried out the RG calculations to identify the dominant paired states. Y.T.H. and A.V. analysed the topological properties for the paired states. Y.T.H. and M.H.F. analysed the pairing symmetries for the paired states. E.A.K. supervised the project and wrote the paper with contributions from Y.T.H., A.V. and M.H.F.
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Hsu, YT., Vaezi, A., Fischer, M. et al. Topological superconductivity in monolayer transition metal dichalcogenides. Nat Commun 8, 14985 (2017). https://doi.org/10.1038/ncomms14985
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DOI: https://doi.org/10.1038/ncomms14985
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