Unconventional superconductivity and anomalous response in hole-doped transition metal dichalcogenides
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Unconventional superconductivity and anomalous response in hole-doped transitionmetal dichalcogenides
Evan Sosenko, ∗ Junhua Zhang, † and Vivek Aji ‡ Department of Physics, University of California, Riverside, Riverside, California 92521, USA (Dated: July 16, 2018)Two dimensional transition metal dichalcogenides entwine interaction, spin-orbit coupling, andtopology. Hole-doped systems lack spin degeneracy: states are indexed with spin and valley speci-ficity. This unique structure offers new possibilities for correlated phases and phenomena. We realizean unconventional superconducting pairing phase which is an equal mixture of a spin singlet andthe m = 0 spin triplet. It is stable against large in-plane magnetic fields, and its topology allowsquasiparticle excitations of net nonzero Berry curvature via pair-breaking circularly polarized light. I. Introduction
The interplay of spin-orbit interaction and electron-electron interaction is a fertile area of research where newphases of matter and novel phenomena have been theo-retically conjectured and experimentally realized [1–7].Single-layer transition metal group-VI dichalcogenides(TMDs), MX ( M = Mo , W and X = S , Se , Te ), are di-rect band gap semiconductors that have all the necessaryingredients to explore these phenomena [8–18]. Whilesharing the hexagonal crystal structure of graphene, theydiffer in three important aspects: (1) gapped valleys asopposed to Dirac nodes; (2) broken inversion symmetryand strong spin-orbit coupling yielding a large splittingof the valence bands; and (3) the bands near the chem-ical potential predominantly have the transition metal d -orbital character [19–24].The inversion symmetry breaking and the strong spin-orbit coupling due to the heavy transition element (Moand W) endow the bands with nontrivial Berry curva-ture. A remarkable consequence is that spin-preservingoptical transitions between valence and conduction bandsare allowed, even though the atomic orbitals involved allhave a d -character. Furthermore, the valley-dependentsign of the Berry curvature leads to selective photoexci-tation: right circular polarization couples to one valley,and left circular polarization to the other. This enablesa number of valleytronic and spintronic applications thathave attracted a lot of attention over the last few years[25–27].We are primarily interested in exploiting the bandstructure and valley-contrasting probe afforded by thenontrivial topology in order to study and manipulate cor-related phenomena in these systems. In particular, wefocus on hole-doped systems, where an experimentallyaccessible window in energy is characterized by two dis-connected pieces of spin non-degenerate Fermi surfaces.One can preferentially excite electrons from either Fermisurface. Since the spins are locked to their valley index, ∗ [email protected]; https://evansosenko.com † [email protected] ‡ [email protected] these excitations have specific s z (where the z -axis is per-pendicular to the two-dimensional crystal). We focus onthe possible superconducting states and their properties.Spin-valley locking and its consequence for supercon-ductivity, dubbed Ising superconductivity, has been pre-viously studied for heavily doped p -type and n -typeTMDs [28–32], where Fermi surfaces of each spin arepresent in each valley. Our focus is the regime of max-imal loss of spin degeneracy where the effects are moststriking [33]. The two valleys in the energy landscapegenerically allow two classes of superconducting phases:intervalley pairing with zero center of mass momentum,and intravalley pairing with finite Cooper pair center ofmass. Since center-of-symmetry is broken and spin de-generacy is lost, classifications of superconducting statesby parity, i.e., singlet vs. triplet, is no longer possible. Inthis paper, we study both extrinsic and intrinsic super-conductivity by projecting the interactions and pairingpotential to the topmost valence band. We identify thepossible phases, and analyze the nature of the optoelec-tronic coupling and the response to magnetic fields. Ourmain conclusions are as follows:(1) For both proximity to an s -wave superconductor,and due to local attractive density-density interactions,the leading instability is due to an intervalley pairedstate, where the Cooper pair is an equal mixture of aspin singlet and the m = 0 spin triplet [34].(2) While the valley selectivity of the optical transi-tion is suppressed, it remains finite. Consequently, thetwo quasiparticles generated by pair-breaking circularlypolarized light are correlated such that one is in the va-lence band of one valley and the conduction band of theother. The valley and bands are determined by the po-larity of incident light.(3) The quasiparticles generated in (2) both have thesame charge and Berry curvature. Thus an anomalousHall effect is anticipated as the two travel in the samedirection transverse to an applied electric field.(4) An in-plane magnetic field tilts the spin, modify-ing the internal structure of the Cooper pair, however, nopair-breaking is induced in the absence of scalar impu-rities. The suppression of the effective interaction leadsto a parametric reduction of the transition temperature.In the presence of scalar impurities, pair-breaking is en- µ E(k)E g − E soc soc τ = − τ = +n = −n = + K−K↓↑ ↓↑↓↑ ↓↑ Figure 1. Energy bands for
WSe as given by equation (2)with at = 3 .
939 eV Å − , E g = 1 .
60 eV , and E soc = 0 .
23 eV .Each valley is centered at ± K relative to the center of theBrillouin zone. The energy for a given band depends only onthe distance k measured from the valley center. abled, but the associated critical magnetic field is large. II. Model
The TMD system is described by the effective tight-biding, low-energy, two-valley Hamiltonian [26], H τ ( k ) = at ( τ k x ˆ σ x + k y ˆ σ y ) + E g σ z − E soc τ ˆ σ z −
12 ˆ s z . (1)where the Pauli matrices ˆ s i operate in the spin spaceand ˆ σ i operate in the orbital space with the two Blochorbital states | v ντs ( k ) i (indexed by ν = + for the in-plane orbital state (cid:12)(cid:12) d x − y (cid:11) + iτ | d xy i and ν = − forthe out-of-plane orbital state | d z i ), s = ± is the spinindex, and τ = ± is the valley index corresponding to the ± K point, respectively. The momentum k = ( k x , k y ) ismeasured from the valley center, a is the lattice constant, t is the hopping parameter, E g represents the energy gapbetween the conduction and valence bands, and E soc is the spin splitting energy in the valence bands due tospin-orbit interaction.The energy spectrum, E nτs ( k ) = τ sE soc + n q (2 atk ) + ( E g − τ sE soc ) . (2)with k = | k | and n = 1 ( n = − ) indexing the conduction(valence) band is shown in figure 1.We focus on doped systems such that the chemical po-tential µ lies in the upper valence bands. Within eachband, the Bloch basis eigenstates are written in terms of the orbital states as elements on the Block sphere, | u nτs ( k, φ ) i = cos θ nτs ( k )2 (cid:12)(cid:12) v + τs ( k, φ ) (cid:11) + e − iτφ sin θ nτs ( k )2 (cid:12)(cid:12) v − τs ( k, φ ) (cid:11) , (3)where k x + iτ k y = ke iτφ and tan θ nτs ( k )2 = atτ kE g − E − nτs ( k ) = atτ kE nτs ( k ) − E − τs (0) . (4)The polar angle on the Bloch sphere of the conductionand valence bands are related by θ − τs ( k ) − θ + τs ( k ) = τ π .The mapping of the energy band to the Bloch sphere,parametrized by ( θ, φ ) , encodes the topological character:as one moves from the node out to infinity, the statessweep either the northern or southern hemisphere with achirality determined by the Berry curvature. III. Superconductivity
We consider two approaches to realizing a supercon-ducting state. First, we assume a proximity induced stateobtained by layering a TMD on an s -wave superconduc-tor. Second, we study an intrinsic correlated phase aris-ing from density-density interactions.We use d ντs ( k ) as the annihilation operator for tight-binding d -orbital states, and c nτs ( k ) for the eigenstatesof the non-interacting Hamiltonian, λ k for the energydispersion for Bogoliubov quasiparticles, and ∆ k for thesuperconducting gap function. A. Induced State
A proximity s -wave superconductor will inject Cooperpairs according to H V = X k ,ν,τ B ∗ ν d ν − τ ↓ ( − k ) d ντ ↑ ( k ) + ε h.c. (5)The coupling constants B ν and the overall constant ε depend on the material interface [35]. Using the abbrevi-ated notation c k α = c − τs ( k ) , with α = ↑↓ for τ = s = ± ,projecting onto the upper valence bands yields, P n = − τ = s (cid:0) H + H V − µN (cid:1) = X k ,α ξ k c † k α c k α − X k (cid:16) ∆ ∗ k c − k ↓ c k ↑ + ∆ k c † k ↑ c †− k ↓ (cid:17) + ε, (6)where ξ k = E − + ↑ ( | k | ) − µ and the effective BCS gap func-tion is ∆ k = 12 ( B + + B − ) + 12 ( B + − B − ) cos θ k , (7)with θ k = θ − + ↑ ( | k | ) . This form is identical to the stan-dard BCS Hamiltonian with an effective spin index α .However, the spin state of the Cooper pair is an equalsuperposition of the singlet and the m = 0 componentof spin triplet. The corresponding quasiparticle eigen-states are γ k α = α cos β k c k α +sin β k c †− k , − α , with energies λ k = ± p ξ k + ∆ k , where cos 2 β k = ξ k /λ k . Note that if B + = B − , then ∆ k is a constant and independent of k .Even when B + and B − are different, the constant termdominates. Before exploring the nature of this state, weanalyze the case of intrinsic superconductivity, and showthat the same state is energetically preferred. B. Intrinsic Phase
For a local attractive density-density interaction(e.g. one mediated by phonons), the potential is V ≃ P R , R ′ v RR ′ : n R n R ′ : , with v RR ′ = v δ RR ′ and n R the total Wannier electron density at lattice vector R .Projecting onto states near the chemical potential gives P n = − τ = s (cid:0) H V (cid:1) = X k , k ′ v ( k ′ − k ) × (cid:16) A kk ′ c † k ′ ↑ c †− k ′ ↑ c − k ↑ c k ↑ + A k ′ k c † k ′ ↓ c †− k ′ ↓ c − k ↓ c k ↓ + 2 | A kk ′ | c † k ′ ↑ c †− k ′ ↓ c − k ↓ c k ↑ (cid:17) , (8)where A kk ′ = e i ( φ k ′ − φ k ) sin θ k ′ θ k θ k ′ θ k . (9)The first two terms in equation (8) lead to intravalleypairing, and the third to intervalley pairing. We analyzethe possible states within mean field theory. The BCSorder parameter is χ = v X k g ∗ k h c − k α ′ c k α i , (10)where the form of g k depends on the particular pairingchannel. The resulting Hamiltonian has the same formas the BCS Hamiltonian in equation (6) but with an ef-fective ∆ k = g k · χ . The intravalley pairing has threesymmetry channels, with the couplings given by g k =1+cos θ k , √ e − iφ k g k = sin θ k and e − iφ k g k = 1 − cos θ k .For these channels, since h c − k α c k α i = − h c k α c − k α i , rela-beling k → − k in the sum gives χ = 0 [36]. The interval-ley pairing also has three symmetry channels: g k = √ , g k = √ θ k , and g k = √ θ k ˆk . Of the three, theconstant valued channel is dominant [37]. This is to beexpected, as the local density-density interaction leads tothe largest pairing for electrons of opposite spins. Sincethe intravalley processes have the same spin, they aredisfavored as compared to the intervalley pairing.The key features of the intrinsic superconductingstate are identical to the proximally induced case whendensity-density interactions dominate. We restrict fur-ther analysis to that case, and turn to the question ofpair-breaking phenomena induced either by optical ormagnetic fields. IV. Optoelectronic coupling
The non-interacting system displays valley selectiveoptical excitations. Light of a particular polarizationonly couples to one valley. Since the superconductingstate is a coherent condensate admixing the two valleys,we address whether pair-breaking displays similar valleyselectivity. In particular, we explore whether or not thetwo quasiparticles generated by circularly polarized light,with total energy larger than E g + ∆ k , occupy oppositevalleys, with one in the conduction band and the otherin the valence band.The optical excitations arise from the Berry cur-vature, which acts as an effective angular momen-tum. The electromagnetic potential A , with polariza-tion vector ǫ , is introduced using minimal coupling, H νν ′ τs ( k ) → H νν ′ τs ( k + e A ) , where, in the dipole approx-imation, A = 2 Re ǫ A e − iωt . This yields a perturbedHamiltonian H → H + H A , where H A = H ′ e − iωt + H ′† e iωt , with H ′ = X k ,τ,s H ′ τ d − τs † ( k ) d + τs ( k ) − X k ,τ,s H ′− τ d + τs † ( k ) d − τs ( k ) , (11)and H ′ τ = ateA ( τ ˆx + i ˆy ) · ǫ . The transition rate is pro-portional to the modulus-squared of the optical matrixelements, P nn ′ τs ( k ) , defined by H A = X k ,τ,sn,n ′ eA m ǫ · P nn ′ τs ( k ) c nτs † ( k ) c n ′ τs ( k ) . (12)For circularly polarized light, in the absence of supercon-ductivity, ǫ ± = ( ˆx ± i ˆy ) / √ and ǫ ± · P + − τs ( k ) = ∓ τ √ atm e ± iφ sin θ ∓ ττs ( k )2 . (13)The transition rate matrix elements for optical exci-tations from the BCS ground state are given by equa-tion (13) multiplied by a coherence factor sin β k . Since θ − τs ( k ) − θ + τs ( k ) = τ π , switching either the valley or po-larization transforms sin → cos in equation (13), givingmatrix elements | P ± | = (cid:12)(cid:12) ǫ ± · P + − ++ ( k ) sin β k (cid:12)(cid:12) correspond-ing to matching ( P + ) or mismatching ( P − ) polarization-valley indexes. For a given valley, a chosen polarizationof light couples more strongly than the other, as is ev-ident comparing | P + | to | P − | and shown in figure 2.For incident light with energy E g + | λ k | , right circularlypolarized light ( + ) has a higher probability of promotinga quasiparticle to the right conduction band, as reflectedin the larger matrix element | P + | ≫| P − | . As depictedin figure 3, the partner of the Cooper pair is in the va-lence band in the opposite valley. The other valley hasthe opposite dependence on polarization.This key new result opens the door for valley control ofexcitations from a coherent ground state. For example,the two quasiparticles have the same charge and Berry | ( c / ℏ ) P + | ( G e V ) MoSe WS WSe −10 −5 0 5 10λ k /∆ k | ( c / ℏ ) P − | ( G e V ) Figure 2. Optical transition rate matrix elements | P ± | in thesuperconducting phase as a function of the ratio of the quasi-particle energy λ k to the superconducting gap ∆ k . Materialparameters for MoSe , WS , and WSe are given in [26] and agap of ∆ k = 7 . is chosen for illustrative purposes. Theorder-of-magnitude contrast between | P + | and | P − | causesthe optical-valley selectivity. BCS CondensateE(k)−Ω z +Ω z ǫ + Figure 3. Pair-breaking by right circularly polarized lightleads to an electron in the conduction band of the right valleyand a partner in the valence band of the left valley. Thevalleys interchange for left circularly polarized light. curvature (see below). In the presence of an electric filed,they both acquire the same transverse anomalous veloc-ity. Thus, in contrast to the response in the normal state,an anomalous Hall effect is anticipated with no accom-panying spin current.
V. Berry curvature
The Berry curvature in the non-interacting crystal forleft and right circularly polarized ( ǫ ± ) optical excitations for a given k is ± ++ ↑ ( k ) , where Ω nτs ( k ) = ˆz · Ω nτs ( k ) , (14a) = − nτ (cid:20) k ∂∂k θ nτs ( k ) (cid:21) sin θ nτs ( k ) , (14b) = − nτ at ) ( E g − τ sE soc ) h (2 atk ) + ( E g − τ sE soc ) i / . (14c)The BCS ground state [38] is | Ω i = Y k csc β k γ k ↑ γ − k ↓ | i , (15a) = Y k (cid:16) cos β k − sin β k c † k ↑ c †− k ↓ (cid:17) | i . (15b)This superconducting state is built up from the quasi-particle eigenstates, | k i = csc β k γ k ↑ γ − k ↓ | i , of the k -dependent Hamiltonian λ k (cid:16) γ † k ↑ γ k ↑ + γ †− k ↓ γ − k ↓ (cid:17) . The z -component of the Berry curvature of the correlated stateis zero, ˆz · i ∇ k × h k | ∇ k | k i = Ω − + ↑ ( k ) + Ω −−↓ ( − k ) = 0 . (16)A single optically excited state in the left valley for agiven k is c ++ ↑† ( k ) c − + ↑ | k i , which has a Berry curvature +2 sin β k Ω ++ ↑ ( k ) . The corresponding excitation in theright valley has a Berry curvature of the same magnitudebut opposite sign. VI. In-plane magnetic field and scalar disorder
In this section we discuss the effects of in-plane mag-netic fields and non-magnetic impurities on the super-conducting state. We consider the lightly hole-dopedmonolayer TMDs in the regime where the Fermi levelcrosses the upper valence bands and is well separatedfrom the lower valence bands. In this regime, the sys-tem is a spin-valley locking system with the spin-oppositeFermi pocket at each valley. Without loss of generality,we adopt a simplified model taking into account the va-lence bands only. In a quasi two-dimensional (2D) sys-tem, an in-plane magnetic field couples to quasiparticlesthrough spin paramagnetism with negligible orbital in-teractions. Applying a uniform in-plane magnetic fieldin the x direction B = ( B, , , the system is describedby the Hamiltonian ( ~ = k B = c = 1 ) H τ ( k ) = − k m − µ + τ E soc ˆ s z + µ B B ˆ s x , (17)which is acting on the valley-spin basis φ τ ( k ) =( c τ ↑ ( k ) , c τ ↓ ( k )) T , where µ is the chemical potential, τ = ± is the valley index, ˆ s i are Pauli matrices operatingin spin space, and µ B is the Bohr magneton. The disper-sion relations of the upper and lower valence bands havebeen approximated by a quadratic form with an effectivemass m = E g / (cid:0) a t (cid:1) , where E g is the large energy gapbetween the conduction and valence bands, E g ≫ E soc , a and t are defined under equation (1), and the momen-tum k = ( k x , k y ) is measured from the correspondingvalley center with k = | k | . Note that in this section weuse k to represent momentum measured from the cor-responding valley center and p to represent momentummeasured from the Brillouin zone (BZ) center. We usethe notation that c † τs ( k ) ( c τs ( k ) ) creates (annihilates) aquasiparticle with momentum k and spin s in the valley τ , and c † s ( p ) ( c s ( p ) ) creates (annihilates) a quasiparticlewith momentum p and spin s .The Hamiltonian H τ ( k ) has the spectrum E τ,u/l ( k ) = − k m − µ ± q E soc + ( µ B B ) , (18)with u for the upper ( + ) and l for the lower ( − )band at each valley, and the eigenstates ϕ τ ( k ) =( c τu ( k ) , c τl ( k )) T , where c τu and c τl correspond to thequasiparticles in the band basis which is related to thespin basis through a field and valley-dependent unitarytransformation U τ ( b ) : ϕ τ ( k ) = U τ ( b ) φ τ ( k ) , where b = µ B B/E soc is the dimensionless magnetic field. Ap-plying a uniform in-plane magnetic field shifts both theupper (lower) valence bands at the two valleys by thesame amount (but the opposite amount between the up-per and lower band at each valley), so that the per-fect nesting condition between the Fermi pockets atthe two valleys remains. Meanwhile, the quasiparticlespin acquires a finite in-plane component, i.e., deviatingfrom ± z direction. Explicitly, we have h τ, u | ˆ s z | τ, u i = − h τ, l | ˆ s z | τ, l i = ( τ / / √ b , and h τ, u | ˆ s x | τ, u i = − h τ, l | ˆ s x | τ, l i = ( b/ / √ b . Therefore, at both val-leys, the quasiparticle spin tilts towards the field direc-tion in the upper valence bands and tilts against thefield direction in the lower valence bands. The changeof quasiparticle spin orientation induced by the in-planefield modifies the internal structure of the Cooper pairand affects the pairing strength as shown below.To evaluate the effect of the magnetic field on thesuperconductivity we follow the procedure used in sec-tion III. A local attractive density-density interactionwith pairing strength v can be written as: H int = − v R d r ρ ( r ) ρ ( r ) with the quasiparticle density ρ ( r ) = P s c † s ( r ) c s ( r ) , where c † s ( r ) ( c s ( r ) ) is the Fourier trans-form of c † s ( p ) ( c s ( p ) ). Transforming to momentum spaceand projecting onto the upper valence bands, the pairingHamiltonian has the form H p = − v ′ ( b ) X k , k ′ c † + ( k ) c †− ( − k ) c − ( − k ′ ) c + ( k ′ ) , (19)where we have ignored the upper-band subscript u in theoperators c † τu and c τu . The effective pairing strengthin the presence of in-plane magnetic field is v ′ ( b ) = v / (cid:0) b (cid:1) . The Hamiltonian H p describes an inter-valley pairing with a pairing strength v ′ suppressed bythe in-plane field. At zero field, v ′ = v , c † + = c † + ↑ , and c †− = c †−↓ , so the pairing occurs between opposite spins.At finite fields, v ′ < v and the quasiparticle at valley + ( − ), represented by c † + ( c †− ), has its up (down) spintilted towards the field direction. As a result, the inter-valley pairing contains equal-spin pairing components inthe presence of in-plane field.The mean-field Hamiltonian, using the Nambu-valleybasis Ψ k = (cid:16) c + ( k ) , c − ( k ) , c †− ( − k ) , − c † + ( − k ) (cid:17) T , takesthe form H MF ( k ) = ξ k ˆ η z − ∆ˆ η x , (20)where ˆ η i are Pauli matrices acting on Nambu (particle-hole) space, ξ k = − k / m − µ + q E soc + ( µ B B ) , andthe mean field ∆ = v ′ ( b ) P k ′ h c − ( − k ′ ) c + ( k ′ ) i describesan intervalley pairing field, which we choose to be real forconvenience.In a conventional 2D superconductor with a spin-degenerate Fermi surface, the application of an in-plane magnetic field induces an energy splitting betweenopposite-spin bands. This energy mismatch between op-posite spins creates a pair-breaking effect in the cleansystem characterized by the pair-breaking equation fortemperature T ≤ T c [39], ln T c T = 12 (cid:20) ψ (cid:18)
12 + iµ B B c πT (cid:19) + ψ (cid:18) − iµ B B c πT (cid:19)(cid:21) − ψ (cid:18) (cid:19) , (21)where T c is the transition temperature at zero field in theclean system and ψ ( z ) is the digamma function. Thisequation determines the critical field B c that destroysthe superconducting state at temperature T ≤ T c fromspin paramagnetism. Furthermore, the scattering fromnon-magnetic impurities does not alter this pair-breakingequation (21) such that the critical field remains the sameregardless of the presence of scalar disorders [39].Unlike the conventional 2D superconductors, in oursystem the two single-spin Fermi pockets at differentvalleys remain perfectly nested without the energy mis-match caused by spin paramagnetism, and the spins atthe two pockets are no longer opposite with equal-spincomponents induced by the field. These two differencesgive rise to new features in the spin-valley locking sys-tem from the effects of in-plane magnetic fields. First,in the clean limit, the presence of in-plane magneticfields does not lead to a pair-breaking effect for the lackof energy mismatch, but mildly suppresses the transi-tion temperature through the weakening of the pairingstrength. The suppressed transition temperature T ′ c isrelated to the zero-field transition temperature T c as T ′ c = T c exp (cid:0) − b /v N F (cid:1) in the mean-field theory, where N F is the density of states at the Fermi level. Second, thesuperconducting state is no longer immune to the scalardisorder, since non-magnetic disorder potential can causeintervalley scattering due to the field-induced parallel-spin components on the two pockets. This interplay be-tween the in-plane magnetic field and the scalar disorderleads to a pair-breaking effect.In the presence of dilute randomly-distributed non-magnetic impurities, the Hamiltonian for short-range im-purity potential is given by H imp = X j Z d r U δ ( r − R j ) ρ ( r ) , (22)where R j is the position of the j th impurity and U isthe disorder strength. Transforming to momentum spaceand projecting onto the upper valence bands, H imp canbe written using the Nambu-valley basis Ψ k as H imp = X k , k X j e i ( k − k ) · R j Ψ † k ˆ U Ψ k , (23)with the disorder scattering vertex ˆ U taking the form ˆ U = U ˆ η z + U b √ b ˆ τ x , (24)where ˆ τ i are Pauli matrices operating in valley space.The first term in ˆ U corresponds to intravalley scatteringand the second term corresponds to intervalley scatter-ing. Note that we have ignored the factors e i ( ± K ) · R j in the intervalley terms because e i (2 K ) · R j and e i ( − K ) · R j will appear in pair and cancel each other in the diagram-matical calculation of self energy.The self energy due to impurity scattering after averag-ing over randomly-distributed impurity configurations, inthe first-order Born approximation, is obtained as [40, 41] ˆΣ ( k , iω n ) = n imp Z d k ′ (2 π ) ˆ U ˆ G ( k ′ , iω n ) ˆ U , (25)where n imp is the impurity concentration and ˆ G is the Green’s function matrix of the clean system, ˆ G ( k ′ , iω n ) = ( iω n − ξ k ′ ˆ η z + ∆ˆ η x ) − , with the Matsub-ara frequencies ω n = (2 n + 1) πT . After integrating out ξ in the self-energy, the disorder renormalized Green’sfunction matrix ˆ G = (cid:16) ˆ G − − ˆΣ (cid:17) − can be parametrizedas ˆ G ( k , iω n ) = h i ˜ ω n − ξ k ˆ η z + ˜∆ˆ η x + iF ( ω n ) ˆ η z ˆ τ x i − , (26)where the quantities ˜ ω n , ˜∆ , and F ( ω n ) have the defini-tions ˜ ω n = ω n + (cid:18) τ + 12 τ (cid:19) ω n p ω n + ∆ , (27) ˜∆ = ∆ + (cid:18) τ − τ (cid:19) ∆ p ω n + ∆ , (28) F ( ω n ) = 12 τ ω n p ω n + ∆ b √ b . (29) Here, /τ and /τ are the collision rates correspondingto the disorder-induced intravalley and intervalley scat-tering, respectively, with the expressions τ = 2 U n imp πN F , τ = 1 τ b b . (30)In the superconducting state, the self-consistencyequation for the order parameter is given by ∆ = 14 v ′ T X n Z d k (2 π ) Tr h ˆ η x ˆ G ( k , iω n ) i , (31)where Tr [ . . . ] is the trace of the argument. Explicitly,from equation (26), it has the form ∆ = v ′ πN F T X n ˜∆ q ˜ ω n + ˜∆ . (32)Linearizing the self-consistency equation (32) near thecritical field B c , we obtain the pair-breaking equationdue to the interplay between the in-plane field and thescalar disorder, ln T ′ c ( b c ) T = ψ (cid:18)
12 + δ c πT (cid:19) − ψ (cid:18) (cid:19) , (33)where T ′ c ( b c ) = T c exp (cid:0) − b c /v N F (cid:1) is the transition tem-perature in the clean system in the presence of the field b c = µ B B c /E soc , and the pair-breaking parameter δ c = 1 τ (cid:12)(cid:12)(cid:12)(cid:12) b c = 1 τ b c b c (34)arises from the valley-flip scattering process. Equa-tion (33) determines the in-plane critical field B c ( T ) attemperature T ≤ T ′ c . For b c ≪ , the pair-breaking pa-rameter takes the simple form δ c ≈ τ − ( µ B B c /E soc ) .As T → , the pair-breaking equation (33) canbe approximated by the asymptotic expansion of thedigamma function, which leads to πT ′ c exp [ ψ (1 / (cid:0) b c /τ (cid:1) / (cid:0) b c (cid:1) . At finite disorder concentration, τ − = 0 , when b c ≪ with T ′ c ≈ T c , the critical field atzero temperature is approximated as µ B B c (cid:12)(cid:12)(cid:12)(cid:12) T → ≈ E soc h πe ψ (1 / k B T c τ / ~ i / , (35)where we have put back the Boltzmann constant k B andthe reduced Planck constant ~ . The large spin-orbit in-teraction E soc ( ∼ –
500 meV ) in monolayer TMDs in-dicates that the in-plane critical field B c is significantlyenhanced, well beyond the Pauli limit. VII. Conclusions
In this letter, we report on the nature of the super-conducting state of hole-doped TMDs. Remarkably, thecorrelated state inherits the valley contrasting phenom-ena of the non-interacting state. While the magnitude issmaller, pair-breaking produces quasiparticles that havethe same Berry curvature, and hence the same anomalousvelocity. Thus one predicts an anomalous Hall responseunlike the valley Hall response observed in MoSe .Spin-valley locking leads to large critical magneticfields. A similar phenomena was recently reported inheavily hole-doped (beyond the spin-split gap) NbSe [29]. In the new regime, where only one band per valleyintersects the chemical potential, no pair-breaking occursfor in-plane fields unless disorder is present.While systematic synthesis and characterization of hole-doped systems is still in its early stages, the fact thatother two-dimensional compounds and their bulk coun-terparts are known to be superconducting [42] providesimpetus to explore this novel phenomena. Acknowledgments
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