Superconductivity emerging from a stripe charge order in IrTe2 nanoflakes

Superconductivity in the vicinity of a competing electronic order often manifests itself with a superconducting dome, centered at a presumed quantum critical point in the phase diagram. This common feature, found in many unconventional superconductors, has supported a prevalent scenario in which fluctuations or partial melting of a parent order are essential for inducing or enhancing superconductivity. Here we present a contrary example, found in IrTe2 nanoflakes of which the superconducting dome is identified well inside the parent stripe charge ordering phase in the thickness-dependent phase diagram. The coexisting stripe charge order in IrTe2 nanoflakes significantly increases the out-of-plane coherence length and the coupling strength of superconductivity, in contrast to the doped bulk IrTe2. These findings clarify that the inherent instabilities of the parent stripe phase are sufficient to induce superconductivity in IrTe2 without its complete or partial melting. Our study highlights the thickness control as an effective means to unveil intrinsic phase diagrams of correlated van der Waals materials.

The thickness of exfoliated IrTe 2 crystals was estimated by atomic force microscopy (AFM). The optical and AFM images for typical IrTe 2 nanoflakes on top of a SiO 2 /Si substrate are shown in Supplementary Fig. 1. The obtained flakes are typically several µm 2 in area and 4-160 nm in thickness, as indicated by a line profile in the insets of Supplementary Fig. 1. We found that the optical contrast, AFM thickness, and Raman spectra of IrTe 2 nanoflakes are nearly unchanged even after a month in ambient condition, as shown in Supplementary Fig. 1. These results confirm that nanometer-thick and stable IrTe 2 flakes can be successfully isolated from a bulk crystal. The stripe charge order in IrTe 2 nanoflakes is characterised using Raman spectroscopy below T s . For several nanoflakes with different thicknesses ranging from 31 nm to 160 nm, we found multiple peak splitting in the Raman spectra taken at 77 K well below T s . These has the well-defined edges along three crystalline directions due to its hexagonal symmetry ( Supplementary Fig. 4a). The atom-resolved STM image and its fast Fourier transform confirm that the topmost hexagonal Te atoms show no charge-ordered phase ( Supplementary   Fig. 4b). The unit cell, Te, and Ir atoms are indicated by black parallelogram, grey and green balls, respectively. Supplementary Fig. 4c describes the local density of states (LDOS) of the IrTe 2 nanoflake, showing its metallic behaviour. Comparing with bulk IrTe 2 , we can conclude that the intrinsic characteristics of the structural and electronic properties on thin v nanoflakes remain the same at room temperature [4].
At low temperature (T STM = 85 K), we found three energetically equivalent chargeordered domains on IrTe 2 nanoflake ( Supplementary Fig. 5). We have rigorously mapped out the nanoflake by taking more than a hundred of STM images resolving charge-ordered phases along all domain boundaries. We could find only three stripe ordered domains as expected from three-fold rotational symmetry, but no hexagonal ordered phases which can be superconducting at lower temperatures. In bulk IrTe 2 crystals, superconducting hexagonal vi ordered phases appear at the specific locations where three striped phases meet [5]. It is noteworthy that bulk crystals always have non-negligible portion of hexagonal domains upon rapid thermal quenching. Thus, we carefully examined the existence of the hexagonal phase after rapid thermal quenching (1 K/sec) especially at A and B indicated in Supplementary with thicknesses ranging from 11 to 20 nm, exhibit the dominant stripe ordering with period of 5a 0 , which is different from that (8a 0 ) of bulk IrTe 2 at the same temperature [6].

Supplementary Note 4: Transport properties
The thermal cycling effect on the IrTe 2 nanoflakes was investigated by measuring the in-plane resistivity. Supplementary Fig. 7 shows the temperature-dependent resistivity ρ(T ) curves, normalised by its room temperature value ρ(300 K), for the 56 nm-thick nanoflake, which were taken at different thermal cycles with different temperature ramping rates of were repeatedly observed at T s,dn = 60-80 K for cooling (Fig. 7a) and at T s,up = 180-210 K for warming ( Supplementary Fig. 7b). The transition temperatures T s,dn and T s,up vary at different cooling and warming runs, consistent with previous reports [7]. The superconducting transitions, however, are almost identical at different cooling runs. As shown in Supplementary Fig. 7c, domains with different T c or H c2 is unlikely to be the origin of the observed multi-steps in MR. Instead, such a behaviour can be induced by the weak links and the associated vortex dynamics in 2D superconductors [8][9][10]. As found in the STM results ( Supplementary   Fig. 6 and Fig. 2), the stripe-charge-ordered domains with different orientations are formed in IrTe 2 flake. The resulting domain boundaries, known to have a high resistance [11], may serve as weak links, which are more fragile to external currents or magnetic fields and produce the multi-steps, particularly for B c. The ix This is distinct from the almost T -linear dependence of B c c2 (T ) for B c [12,13]. We thus conclude that 2D superconductivity is realised in IrTe 2 nanoflakes.
The field orientation dependence of B c2 (θ) further supports the same conclusion. The  Fig. 9l and Fig. 3c). These results indicate that the 2D superconductivity is realised even for relatively thick IrTe 2 flakes.

Supplementary Note 5: Superconducting gap estimation
Superconducting gap of thin films or nanoflakes can be extracted from the temperature dependent self-field critical current [15][16][17]. When the thickness d of the superconducting system is much smaller than the penetration depth λ(0) (d λ), superconductivity is destroyed by excessive self-field, induced by the supercurrent. In this case, the temperature dependent self-field critical current density, J c,sf (T ), is closely related with London pene- xiii are in excellent agreement with those measured using other techniques for a wide range of superconducting thin films of metals, nitrides, oxides, cuprates, pnictides, borocarbides, MgB 2 , and heavy fermions [15][16][17], which is recently extended to various superconducting nanoflakes including NbSe 2 [16], FeSe on SrTiO 3 [16], PdTe 2 on SrTiO 3 [18], and ionic gated MoS 2 [19].
The temperature dependent J c,sf (T ) data for representative IrTe 2 nanoflakes with d = 21, 56, 90, and 140 nm are shown in Supplementary Fig. 10. The penetration depth for doped IrTe 2 is found to be ∼ 150 nm [20], much larger than the thickness of the nanoflakes. This ensures to apply the self-field model to the measured J c,sf (T ) data. In all cases, J c,sf (T ) and the corresponding λ(T ) data are in good agreement with the fits to the model, assuming a single s-wave gap with BCS temperature dependence [21]. From the extracted ∆ SC (0) and the measured T c , we obtained the gap ratio 2∆ SC /k B T c ∼ 5.6, which is much larger than the BCS value of 2∆ SC /k B T c = 3.53. These results show that superconductivity in IrTe 2 nanoflakes is in the strong coupling regime.

Supplementary Note 6: Density functional theory calculations
When a nanoflake is exfoliated and anchored by metal electrodes on top of a substrate, the in-plane strain is expected to be applied at low temperatures because of the different thermal contraction between the nanoflake and the substrate ( Supplementary Fig. 11a).
For IrTe 2 nanoflakes in stripe charged phase, the thermal expansion coefficients are α ∼ +110×10 −6 K −1 along the a-axis and ∼ −21×10 −6 K −1 along the b-axis [22], much larger in magnitude than those of Si (α ∼ +2.56×10 −6 K −1 ) [23] and other 2D materials e.g. [24]. Using these thermal expansion coefficients we estimate the substrate-induced tensile strain up to ∆a/a ∼ 3% and ∆b/b ∼ −0.7% at 20 K. In the realistic case, there are domains of stripe order with different orientations in IrTe 2 nanoflake, which releases the substrate-induced strain. The resulting strain would be moderate, 0.1-0.3% [25], but has a strong impact on the stability of the stripe-charge-order phases as found in a recent study [26] . The strain-induced changes of the periodicity and also the Ir-Ir dimer density significantly modify the charge transfer between Ir and Te states and thus the electronic structures.
To investigate the effect of the substrate-induced strain, we performed the density funcxiv tional theory calculations for estimating electron-phonon coupling and T c . Supplementary   Fig. 11 shows the phonon dispersion curves and electron-phonon coupling (EPC) of the stripe phase without and with the 3.0% in-plane tensile strain. When the in-plane strain becomes larger than a critical value of 3.1%, we observed the imaginary phonon frequency near the Γ point, indicating the structural instability. However, this critical strain, estimated from calculations, is far larger than the experimental one, ∼ 0.1% [26]. The observed extreme sensitivity of the stripe-charge-ordered phases to the strain implies that the 5a 0 stripe phase is much closer to the structural instability than expected in DFT-based calculations. xv in-plane strain Ir-Ir dimer distance structural instability and the resulting enhancement of the electron-phonon coupling with increase of the strain.
The magnitude of electron-phonon coupling constant λ qv is indicated by the colour in the dispersion curves. We computed the superconducting parameters, such as the logarithmic averaged phonon frequency ω log , the electron-phonon coupling (EPC) constant λ ep , and superconducting temperature T c using the Eliashberg EPC and Allen-Dynes formula [27,28], where α 2 F (ω) is the Eliashberg function and µ * is the effective Coulomb repulsion parameter.
The superconducting parameters depending on the in-plane strain are summarised in Table I.
We have found that the enhanced EPC and T c with increasing in-plane strain from 0 to 3.0%. With increasing in-plane strain, the overall phonon bands become softer resulting in increasing λ ep , which follows the equation λ ep ∝ 1 ω 2 where ω 2 is the averaged phononfrequency square [29]. Especially, α 2 F (ω) with 3.0% strain shows the two additional peaks in the low frequency range of 4 meV and 7 meV, which contribute the enhanced λ ep . In contrast, assuming elongation of the c-axis lattice parameter up to ∆c/c ∼ 9%, two orders of magnitude larger than typically found in other nanoflakes [30,31], we found much smaller change in λ ep and T c . Considering the calculated T c smaller than observed in experiments, we cannot rule out the possibility that superconductivity in IrTe 2 is not fully captured by xvi the DFT-based calculations. Nevertheless, the clear trend of λ ep and T c suggests that the tensile strain, rather than interlayer elongation, may play an important role for suppression of the stripe order and appearance of superconductivity in IrTe 2 nanoflakes.