Two-component nematic superconductivity in 4Hb-TaS2

Most superconductors have an isotropic, single component order parameter and are well described by the standard (BCS) theory for superconductivity. Unconventional, multiple-component superconductors are exceptionally rare and are much less understood. Here, we combine scanning tunneling microscopy and angle-resolved macroscopic transport for studying the candidate chiral superconductor, 4Hb-TaS2. We reveal quasi-periodic one-dimensional modulations in the tunneling conductance accompanied by two-fold symmetric superconducting critical field. The strong modulation of the in-plane critical field, Hc2, points to a nematic, unconventional order parameter. However, the imaged vortex core is isotropic at low temperatures. We suggest a model that reconciles this apparent discrepancy and takes into account previously observed spontaneous time-reversal symmetry breaking at low temperatures. The model describes a competition between a dominating chiral superconducting order parameter and a nematic one. The latter emerges close to the normal phase. Our results strongly support the existence of two-component superconductivity in 4Hb-TaS2 and can provide valuable insights into other systems with coexistent charge order and superconductivity.

Recently, mounting evidence points out that 4Hb-TaS 2 hosts a chiral superconducting state [19][20][21].However, clear proof for the two-component superconducting state is lacking, and the superconducting phase diagram is yet to be determined.4Hb-TaS 2 is a naturally occurring van der Waals heterostructure of alternating 1T-and 1H-TaS 2 layers (Fig. 1a).1T-TaS 2 is considered to be a Mott insulator and a quantum spin liquid candidate [22][23][24][25].2H-TaS 2 is a superconductor with a critical temperature of 0.7 K [26].4Hb-TaS 2 undergoes a charge density wave transition at T = 315 K [27], which is clearly observed as an abrupt increase in the resistance (see Fig. 1b) accompanied by a five-fold reduction of the Hall number (see inset of Fig. 1b and Extended Data Fig. 4).Nevertheless, the system remains metallic and undergoes a superconduct-ing transition at T c = 2.7 K, as can be seen in Fig. 1b.
We have performed scanning tunneling microscopy (STM) measurements.Each point in Fig. 1c is the 1T charge density wave (CDW) super-cell, which dominates over the weaker 1H charge density modulations [20].Surprisingly, we found modulations of the tunneling conductance in the form of highly oriented stripes with a separation of about ≈ 20 nm, see Fig. 1c.These stripes break the C 6 symmetry of the 4Hb-TaS 2 crystal.
Remarkably, this symmetry breaking is manifested in the superconducting state.When rotating the magnetic field in the basal plane, parallel to the TaS 2 planes of the sample, we observe significant two-fold modulations of the critical field H c2 (Fig. 1d).We link the minimal critical field with the direction of the stripes as both are parallel to an in-plane crystal axis.The stripes found by our STM data allow for nematic symmetry breaking in 4Hb-TaS 2 , including its superconducting state.Therefore, we interpret the two-fold critical field modulation as the signature of a multiple-component nematic superconducting order parameter, following Refs.[28][29][30][31].Naively, an anisotropic H c2 would result in an anisotropy of the vortex core [32].However, the density of states in a vortex core is perfectly isotropic as seen in Fig. 1e.We reconcile the observed nematic state and the isotropic vortex using a theoretical analysis invoking a crossover from a chiral order parameter, which dominates at low temperatures, to a nematic one, appearing near T c .The calculated zero field phase diagram is presented in Fig. 1f.Our model predicts a smaller anisotropy of H c2 at low temperatures, as observed in the experiment and discussed further below.We now focus on the characterization of the stripes and their possible origin.The stripes are well oriented across the entire sample and define a specific direction over a macroscopic length scale, roughly parallel to a crystal axis, as seen in Fig. 2a.Their direction persists even when crossing a CDW domain wall, as demonstrated in Fig. 1c.We focus on the regions with abundant stripes and observe a quasi-periodic modulation of ≈ 19.6 nm.In some rare regions no or only a single stripe is found, and the period is ill-defined.
We have performed local spectroscopic measurements on and off stripe, both in the normal and in the superconducting state well below T c .We found that, for negative voltage biases, the density of states on a stripe differs from off a stripe (Figures 2b and 2c).This bias dependent difference suggests that the stripes represent a change in the local electronic density of states rather than just a topographic modulation.We further note that the superconducting gap itself, observed at low biases, remains the same on and off the stripes (Inset of Fig. 2c).As mentioned above, when we rotate the magnetic field in the basal plane, a remarkable two-fold symmetry of the critical magnetic field H c2 is detected, as seen in Fig. 3, and as detailed in Extended Data Fig. 5.The minimum of H c2 is observed when the field is applied parallel to a crystal axis, similar to the direction of the stripes.Surprisingly, the anisotropy of the critical field is smaller as the temperature is decrease from 2.5 K to 1.8 K, and is much smaller at T = 0.4 K, hardly resolved from the data (see Fig. S1a).One simple explanation for the anisotropy of H c2 is that it stems from a variation in the Josephson vortex pinning due to the stripes.To check this conjecture We model the stripes using a slow modulation of the interlayer stacking configuration, yielding a variation of the perpendicular mass m z in a Ginzburg Landau (GL) theory along an in-plane coordinate, perpendicular to the observed stripes.Naturally, modulation of m z does not affect the round shape of the Abrikosov vortices in the plane.Solving the linearized GL equations, we generically find that the critical field is maximal parallel to the stripes.This is in contrast to our observations, ruling out this mechanism (see supplementary information for details).
Moreover, we have a unique situation in which closer to the critical temperature 4Hb-TaS 2 exhibits a clear nematic superconductivity, while at lower temperatures the superconducting state is much more isotropic (Fig. S1a), as seen in the round vortices (Fig. 1e) and the smaller variations in the in-plane critical field (Fig. 3a).We also note that 4Hb-TaS 2 is different than most multi-component superconductors, where the nematic behavior is observed throughout the entire temperature range [5][6][7].
Having excluded a conventional scenario, we now propose a competition between nematic and chiral superconductivity within a two-component GL theory [30].Such an explanation is consistent with the order parameter invoked by previous works [19,21,33].
The two-component order parameter can be written as η = η cos α sin α e iγ , where η is the amplitude of the superconducting order parameter, and the angles α and γ parametrize the nematic and chiral phases.In a purely nematic state γ = 0, and α dictates the direction of nematicity and resulting anisotropy.On the other hand, the purely chiral order, γ → ±π/2 and α = π/4, is isotropic.Deep in the superconducting state, the system is described by a chiral order parameter [19], leading to isotropic vortices.
The origin of the emergent stripes is not fully understood.A possible scenario is that stacking two dimensional layers with a small mismatch and different CDW patterns generates a finite uniaxial strain (see supplementary material for more information).Regardless, we include this experimentally observed symmetry breaking via a xx − yy term in the GL theory.Solving the GL equations for different temperatures discussed in the supplementary material, we obtain that γ = π/2, although α varies with T .Fig. 1f shows that as the temperature increases and the order parameter is reduced, the strain term dominates to favor the nematic order near the crit- The critical field has a pronounced two-fold symmetry, whose anisotropy is larger at temperatures closer to the critical temperature.Inset: E-beam design of the eight gold contacts to the flake.(b) Similar measurement on a single crystal sample at T = 2.1K, in which the crystal axes can be easily identified.The minimal critical field is observed when the field is applied parallel to an in-plane crystal axes.Inset: micrograph of the single-crystal sample with contacts attached.(c) Resistance measurements as a function of the applied magnetic field direction in the basal plane of the single crystal sample, for a constant magnetic field, |H| = 3.1T , at T = 2.1K.A peak in the resistance corresponds to a minimum in Hc2 at these angles.The various colors represent four different current directions as depicted at the inset.Clearly, the overall behavior is independent of the current direction.
Following this model, our results can be understood in the following way: at low temperatures, the order parameter is mostly chiral, in line with the isotropic vortex core in Fig. 1e and the almost isotropic H c2 in Fig. S1a.At temperatures closer to T c , the order parameter is mostly nematic, consistent with the observed two-fold angular dependence of the critical field seen in Fig. 3.This theory predicts that vortices gradually become anisotropic at higher temperatures as the order parameter becomes nematic.
To summarize, we find a microscopic stripe pattern in 4Hb-TaS 2 and link it to the macroscopic two-fold symmetry of the superconducting critical field.The large variation in the critical field, of about 20%, strongly supports the existence of a nematic order parameter close to the normal state.To reconcile theses findings with the indications of a chiral superconducting ground state, we offer a theoretical model that captures the crossover between a chiral order parameter to a nematic one.The latter is allowed by the strain field.We exclude a simpler explanation of Josephson-vortex pinning along the stripes that predicts a minimum in H c2 opposite to our findings.Our work indicates that the pairing in 4Hb-TaS 2 evolves from a nematic, time-reversal even state at high temperatures to a time-reversal breaking chiral order at very low temperatures, suggesting a unique superconducting phase diagram.

Sample growth.
High-quality single crystals of 4Hb-TaS 2 were prepared using the chemical vapour transport method.Appropriate amounts of Ta and S were ground and mixed with a small amount of Se (1% of the S amount).The powder was sealed in a quartz ampule, and a small amount of iodine was added as a transport agent.The ampule was placed in a three-zone furnace such that the powder is in the hot zone.After 30 days, single crystals with a typical size of 5.0 mm x 5.0 mm x 0.1 mm grew in the cold zone of the furnace.The fact that the van der Waals stacking in 4Hb-TaS 2 arrives in the form of a single crystal allows for clean, homogeneous and large samples.
Transport Measurements.Ohmic contacts were made by attaching platinum wires to the 4Hb-TaS 2 single crystals using silver epoxy, or by evaporating gold over the flake sample.Measurements up to 14T were taken in a cryogenic station with a mechanical rotator probe.The current, of the order of 1mA was supplied by a Keithley 6221 current source, and the voltage drop was measured using a Keithley 2182A nanovoltmeter.The flake sample was measured using a current of 250nA.High-field scans (Figs.S1a and 6) were performed at the National High Magnetic Field Laboratory (NHMFL).In this case the resistance was measured using a Lakeshore 372 resistance bridge at low frequencies with a current of 0.316 mA.Inplane rotation was performed using a piezo stack, while monitoring the angle with three Hall sensors.The critical field was defined as the the field in which the resistance is half of the normal state resistance.For details about negating spurious wobbling effects and control experiments see the supplementary material section.To determine the Hall signal we averaged two current orientations and anti-symmetrized the results with respect to the applied magnetic field.
STM measurements.The samples were cleaved in STM load lock at ultra-high vacuum at room temperature.The cleaved crystals were then inserted directly to the 4K sample stage of the STM head for spectroscopic measurements.Commercial PtIr tips were treated on a freshly prepared Cu(111) single crystal for the stability of the tips and then used for the measurements.All the spectroscopic data were measured using standard lockin techniques with frequency of 733Hz.In Fig. 1c the measurement was performed at a constant current mode with set value of 200 pA.We interpret the change at the signal as originating from a change in conductance, and not due to a change in topography, as the stripes' relative density of state change with bias (Fig. 2b).In Figs.1e, 2c the voltage was set in the range of 0.1 mV to 10 mV depending on the bias scan range, and the AC modulation varied between 2 mV to 200 mV.FIG. 4. The charge density wave transition changes the carrier density dramatically.A careful measurement of the temperature dependence of the hall number reveals a decrease by a factor of five across the CDW transition.The Hall number is not simply related to the actual carriers density in TaS2 polymorphs [34,35] due to the complicated Fermi surfaces [19].We interpret the dramatic change in the Hall number as a result of a Fermi surface reconstruction at the CDW transition that gaps major parts of Fermi surface.Inset: raw measurements of the Hall resistance as a function of the field at various temperatures.IG. 6. Field sweeps at different out-of-plane angles performed at T = 0.3 K.When the applied magnetic field is rotated out of the basal plane (θ = 90 • ), the critical field is reduced drastically, as expected from the quasi-2D nature of the superconducting state.Remarkably, the normal state magneto-resistance is almost isotropic and non-saturating up to 33T.

EXTENDED DATA FIG. 2: FIELD SWEEPS AT DIFFERENT IN-PLANE ANGLES
intentional 90 • in-plane offset.If the variation of the critical field was only due to a possible wobble in the rotation, we would expect that the critical field will be minimal for the two samples at the same angles.However, we observed that the critical field minimum is rotated by 90 • , meaning that the field variation is related only to the crystal and not to the (common) rotation platform (Fig. S1b).(Here the critical field is defined as the field at which the resistance is 90% of the normal state resistance).

Proposed origin of the stripes
We interpret the stripe pattern as a result of the layer mismatch in 4Hb-TaS 2 .Let us consider the geometrical aspects of the 4Hb-TaS 2 layers.The 1H and 1T layers have different in-plane lattice parameters with about 1.5% mismatch [36] and different symmetries [37].Without any special symmetry considerations, this mismatch will either relax isotropically or lead to a two dimensional Moiré pattern.However, at the charge ordered states the 1T and 1H layers form different charge density waves, leading to a non isotropic mismatch.Specifically, the CDW pattern in the 1H layer is parallel to one of the crystal axes [38].In this special direction the strain is maximal and the charge moves to minimize the CDW-induced mismatch.
This uniaxial contraction may lead to a one dimensional Moiré-like pattern parallel to the CDW pattern, with the expected periodicity of 19 nm due to the 1T and 1H lattice 1.5% mismatch [39].The strong electron-phonon coupling in 4Hb-TaS 2 couples the electronic degrees of freedom to the lattice, resulting in a Moiré-like pattern of the local density of states, as seen in Fig. 1c.We speculate that scarce regions with no stripes are due to some local strain relaxation, as the stripes tend to deform close to a dislocation, as seen in Fig. S2.Anisotropic Ginzburg-Landau theory with an in-plane field Our starting point is the linearized GL equation i=x,y,z where the masses are related to the coherence length, xy |α| , and 2 as the Hamiltonian of a particle in a magnetic field with eigenvalues {E}, the critical field is obtained from the minimal energy solution, |α| = min{E}.For the homogeneous case with a magnetic field in the xy plane we have eigenstates |k , k ⊥ , n (k parallel or perpendicular to the magnetic field B in the plane) with energy where, ω = 2eB √ mxymz so that min{E} = ω 2 , and with Φ 0 = /2e.For B z we have H c,z = Φ0 ξ 2

xy
. Since ξ xy ξ z the in-plane critical field is larger than the perpendicular critical field, 1.

Incorporation of the stripe modulation via mass modulation
Here the anisotropy of the critical field is attributed to a modulation of the perpendicular mass m z in a GL theory.The stripes are incorporated by adding a periodic modulation to m z , Here a is the distance between the stripes, and we set the x direction to run perpendicular to the stripes.We treat this modulation as a small perturbation, δm m z .The Hamiltonian used to obtain the critical field is H = H 0 + V , where Here δm mz 2 cos 2 (2πx/a) and p z is the momentum operator along the z direction.We next compute the correction δE to the ground state energy.The relative correction to the critical field is B z.For perpendicular field the screening currents flow in the xy plane and are unaffected by the periodic modulation in m z .Thus, H c,z is unaffected by δm.Next we consider an in-plane field, B = B(cos(θ)x + sin(θ)ŷ).
B ⊥ stripes (θ = 0).We have set the x direction to run perpendicular to the stripes so that in the present case B x, and the cyclotron motion takes place in the yz plane.The perturbation acts nondiagonally on the momentum |k → |k ± 2π a , increasing the eigenvalue of H 0 by E BZ = 2 2mxy 2π a 2 .We define the ratio where B = eB is the magnetic length.Then, up to second order, using . The two terms inside the parenthesis correspond to a virtual transition to k = ±2π/a and either n = 0 or n = 2, respectively.Here n, n denote eigenvalues of a † a.The last term stems from first order perturbation theory in V (2) .This quadratic dependence on the mass modulation δm is confirmed in Fig. S3(a) via a comparison to a numerical solution.From Eq. ( 6), the correction to the critical field is FIG. S3.Correction to the lowest energy eigenvalue with increasing mass variation δm/mz presented for different W .The points are the results from the numerical calculation whereas the lines are the perturbative results.(a) θ = 0 that is magnetic field perpendicular to the stripe direction.The perturbative calculation predicts the correction to be second order in δm/mz, which is confirmed by the numerical calculations.In this case, the correction is expected to change the sign at W = 1 8 (−5+ √ 57), which is also obtained from the numerics.(b) θ = π/2, i.e., magnetic field parallel to the stripe direction.Here the perturbation calculation predicts a first-order correction to the energy eigenvalue, which is confirmed by the numerical calculations.The first-order correction is expected to vanish for W = 1.
B stripes (θ = π/2).In this case the cyclotron motion lies in the xz plane, and the unperturbed Hamiltonian is Adding the perturbation V , each momentum kick k x → k x ± 2π/a acts as a shift operator e ∓ipz∆z/ on the oscillator, where ∆z = 2 B

2π
a .Notice that in this case H 0 displays the Landau level degeneracy with respect to k x .Performing degenerate first order perturbation theory, we obtain a hopping amplitude t between states differing by ∆k x = ±2π/a and having exclusively n = 0, This is justified in the perturbative regime t ω, so that higher n will appear only in second order as t 2 /( ω).Using the Harmonic oscillator result n|e αa For n = 0, the correction to the lowest energy is negative and given by δE . This linear correction is confirmed via a comparison to a numerical simulation in Fig. S3(b).Thus the correction to the critical field is Thus, this model with a varying mass along z predicts an anisotropic H c2 .To find the angle at which H c2 is maximal  17).(b) Chiral versus nematic parametrization of the order parameter using two angles (α, γ), see Eq. ( 14).
(r η = 0) we have a minimum of f h at α = 0. Minimizing with respect to α gives This acquires a solution only below a temperature T * ≤ T c where cη(−rη) bηλ( xx− yy ) ≥ 1. Accordingly, as shown Fig. S5(a), the order parameter is purely nematic for T * < T < T c , and is mixed nematic and chiral below T * .Deep in the superconducting phase it is maximally chiral.
Next, given α and γ, we allow a spatial dependence of the form η = η(x, y)(cos α, sin αe iγ ) T .Substituting this into the gradients terms, they become κ 1 F κ1 + κ 2 F κ2 , where We can see that in the purely chiral case F κ2 = 0, whereas in the purely nematic case F κ2 is a simple mass anisotropy In the fully chiral case we recover isotropy m y m x = 1, while in the nematic case this becomes m y m x = κ1+κ2 κ1−κ2 .We may now accommodate the experimental results within this model.When T T c the order parameter is primarily chiral, α ∼ = π/4, γ = π/2, so Isotropic vortices will be observed.At higher temperatures, this model predicts an anisotropic critical field , which is dominated by the strain xx − yy .Whether the maximal critical field is parallel or perpendicular to the uniaxial strain depends on the signs of κ 2 and λ.It is expected that details of the microscopic model can yield either sign of the anisotropy of the in-plane H c2 .
Our model predicts that when the order parameter |η| weak, nematicity is preferred.One might claim that at the center of the vortex core the vanishingly small order parameter is therefore nematic and we should expect anisotropic vortices for all temperatures.However, the STM measurement effectively probes a finite distance from the core, into the bulk, where the chiral nature of the order parameters is restored and we still expect that the shape of the vortex will represent the bulk order parameter.
Finally, we comment on the behavior of H c2 at T T c , and the suppression of the anisotropy we found at low temperatures.To obtain the behavior of H c2 at low T it is necessary to solve the Landau level problem for the in-plane magnetic field, and then find the energy minimum by varying α and γ.Instead of this complicated procedure, we just show that the nematic part of the phase diagram is suppressed at low T .To this end, we estimate the instability of the chiral phase as a function of increasing magnetic field strength.In the chiral phase, the gap is isotropic, and at small magnetic field the gradient terms merely renormalize r η , see Eqs. ( 2) and (3).Plugging this back into the condition for the stability of the chiral phase, we find an estimate for the critical field strength The nematic phase exists for H * < H < H c .Thus, the region of nematic superconductivity decreases sharply at low temperatures, where r η is large.

FIG. 1 .
FIG. 1. Nematic and chiral superconductivity in 4Hb-TaS2 and their origin.(a) Crystal structure of 4Hb-TaS2 displaying alternately stacked quasi-2D layers of 1T-TaS2 (1T) and half of the 2H-TaS2 (1H) polymorphs.Inset: schematics of the hexagonal crystal, marking the crystal axes and the two directions of the field rotation in the experiment.(b) Resistance as a function of temperature.At T = 315 K the charge density wave transition can be seen by the abrupt increase in the crystal resistance.The superconducting transition is observed at Tc = 2.7 K. Note the logarithmic temperature scale.Inset: Hall number as a function of temperature.The charge density wave phase transition at T = 315 K results in a strong reduction in Hall number.(c) STM topographic image, where each dot represents a single site in the √ 13 × √ 13 CDW patern.The stripes are clearly seen, and they extend across a CDW domain boundary.(d) The critical field (at T = 2.5 K) has a clear 2-fold symmetry as a function of the in-plane direction of the applied magnetic field.(e) Average STM image of a vortex core.The density of states as a function of location, averaged over several vortices, is clearly isotropic.(f) The suggested phase diagram displays the variation of the order parameter η with temperature.A nematic state is favored near Tc, whereas the order parameter becomes chiral at low temperatures.The nematic-chiral crossover is parametrized by the angle α (see details in the text).

FIG. 2 .
FIG. 2. Quasi-periodic conductance modulations in the form of stripes in 4Hb-TaS2.(a) Distance and angular distributions of the stripes, showing a stripe separation with a median Pc = 19.6nmand a clear orientation, roughly parallel to a specific crystal axis.(b) Differential conductance maps of the same field of view measured at E = 160 meV (top), where the stripes are clearly seen, and at E = −140 meV (bottom), where the stripes are invisible.The absence of stripes at different energies shows that they are not simply a topographic modulation.(c) Spatially averaged dI/dV profiles measured on and off stripe, at 4 K. Inset: Low-bias differential conductance spectra measured at 0.4 K on and off stripe.The superconducting gap is unchanged, meaning the superconductivity is homogeneous throughout the sample.

FIG. 3 .
FIG.3.Nematic superconductivity in 4Hb-TaS2.(a) Critical fields of the flake sample at two temperatures.The critical field has a pronounced two-fold symmetry, whose anisotropy is larger at temperatures closer to the critical temperature.Inset: E-beam design of the eight gold contacts to the flake.(b) Similar measurement on a single crystal sample at T = 2.1K, in which the crystal axes can be easily identified.The minimal critical field is observed when the field is applied parallel to an in-plane crystal axes.Inset: micrograph of the single-crystal sample with contacts attached.(c) Resistance measurements as a function of the applied magnetic field direction in the basal plane of the single crystal sample, for a constant magnetic field, |H| = 3.1T , at T = 2.1K.A peak in the resistance corresponds to a minimum in Hc2 at these angles.The various colors represent four different current directions as depicted at the inset.Clearly, the overall behavior is independent of the current direction.

FIG. 5 .
FIG. 5. To deduce the critical field, magnetic field scans were performed at different in-plane angles while keeping a constant temperature.The critical field is clearly changing as a function of the in-plane angle.We show only a quarter cycle to avoid over-lapping of the measurements.

FIG
FIG. S2.Stripe pattern deformation around a dislocation.Local topography measured by an STM in a region with a dislocation (top right of the image), clearly showing the bending of the stripes next to it.