Abstract
Spintriplet topological superconductors should exhibit many unprecedented electronic properties, including fractionalized electronic states relevant to quantum information processing. Although UTe_{2} may embody such bulk topological superconductivity^{1,2,3,4,5,6,7,8,9,10,11}, its superconductive order parameter Δ(k) remains unknown^{12}. Many diverse forms for Δ(k) are physically possible^{12} in such heavy fermion materials^{13}. Moreover, intertwined^{14,15} density waves of spin (SDW), charge (CDW) and pair (PDW) may interpose, with the latter exhibiting spatially modulating^{14,15} superconductive order parameter Δ(r), electronpair density^{16,17,18,19} and pairing energy gap^{17,20,21,22,23}. Hence, the newly discovered CDW state^{24} in UTe_{2} motivates the prospect that a PDW state may exist in this material^{24,25}. To search for it, we visualize the pairing energy gap with μeVscale energy resolution using superconductive scanning tunnelling microscopy (STM) tips^{26,27,28,29,30,31}. We detect three PDWs, each with peaktopeak gap modulations of around 10 μeV and at incommensurate wavevectors P_{i=1,2,3} that are indistinguishable from the wavevectors Q_{i=1,2,3} of the prevenient^{24} CDW. Concurrent visualization of the UTe_{2} superconductive PDWs and the nonsuperconductive CDWs shows that every P_{i}:Q_{i} pair exhibits a relative spatial phase δϕ ≈ π. From these observations, and given UTe_{2} as a spintriplet superconductor^{12}, this PDW state should be a spintriplet PDW^{24,25}. Although such states do exist^{32} in superfluid ^{3}He, for superconductors, they are unprecedented.
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Main
Bulk Cooperpair condensates are definitely topological when their superconductive or superfluid order parameters exhibit odd parity^{33,34} Δ(k) = −Δ(−k) with spintriplet pairing. This situation is epitomized by liquid ^{3}He, the only known bulk topological Cooperpair condensate^{35,36}. Although no bulk superconductor exhibits an unambiguously topological Δ(k), attention has recently focused on the compound UTe_{2} as a promising candidate^{1,2,3,4,5,6,7,8,9,10,11,12}. This material is superconducting below the critical temperature T_{c} = 1.65 K. Its extremely high critical magnetic field and the minimal suppression of the Knight shift^{3} on entering the superconductive state both imply spintriplet superconductivity^{1,2}. Temperature^{4}, magnetic field^{4,5} and angular dependence^{5} of the superconductive quasiparticle thermal conductivity are all indicative of a superconducting energy gap with point nodes^{4,5,6}. In the superconductive phase, evidence for timereversal symmetry breaking is provided by polar Kerr rotation measurements^{7} but is absent in muonspinrotation studies^{8}. Furthermore, the superconductive electronic structure when visualized at opposite mesa edges at the UTe_{2} (0–11) surface breaks chiral symmetry^{9}. Dynamically, UTe_{2} seems to contain both strong ferromagnetic and antiferromagnetic spin fluctuations^{10,11} relevant to superconductivity. Together, these results are consistent with a spintriplet and, thus, oddparity, nodal, timereversal symmetry breaking, chiral superconductor^{12}. Figure 1a shows a schematic of the crystal structure of this material, whereas Fig. 1c is a schematic of the Fermi surface in the (k_{x}, k_{y}) plane at k_{z} = 0 (dashed lines; ref. ^{37}). An exemplary order parameter Δ(k) proposed^{5} for UTe_{2} is also shown schematically in Fig. 1c (solid lines), but numerous others have been proposed^{12}, including that of a PDW state^{24,25}. In theory, this PDW, if generated by timereversal and surfacereflection symmetry breaking, is a spintriplet PDW^{25}. Such a state is unknown for superconductors but occurs in topological superfluid ^{3}He (ref. ^{32}).
PDW visualization
In general, a PDW state is a superconductor but with a spatially modulating superconductive order parameter^{14,15}. Absent flowing currents or magnetic fields, a conventional spinsinglet superconductor has an order parameter
for which ϕ_{S} is the macroscopic quantum phase and Δ_{0} the amplitude of the manybody condensate wavefunction. A unidirectional PDW modulates such an order parameter at wavevector P as
meaning that the electronpairing potential varies spatially. By contrast, a unidirectional CDW modulates the charge density at wavevector Q such that
The simplest interactions between these three orders can be analysed using a Ginzburg–Landau–Wilson freeenergy density functional
representing the lowestorder coupling between superconductive and density wave states.
There are two elementary possibilities: (1) if Δ_{S}(r) and Δ_{P}(r) are the predominant orders, they generate charge modulations of forms \({\rho }_{{\rm{P}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\Delta }_{{\rm{P}}}+{\Delta }_{{\rm{P}}}^{* }{\Delta }_{{\rm{S}}}\) and \({\rho }_{2{\rm{P}}}\left({\bf{r}}\right)\propto {{\Delta }_{{\rm{P}}}^{* }\Delta }_{{\rm{P}}}\), that is, two induced CDWs controlled by the wavevector of the PDW; (2) if Δ_{S}(r) and ρ_{Q}(r) are predominant orders, they generate modulations \({\Delta }_{{\rm{Q}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\rho }_{{\rm{Q}}}\), that is, a PDW induced at the wavevector of the CDW. In either case, the PDW state described by equation (2) subsists.
To explore UTe_{2} for such physics, it is first necessary to simultaneously visualize any coexisting CDW and PDW states. Recent experimental advances have demonstrated two techniques for visualizing a PDW state. In the first^{16,17,18,19}, the condensed electronpair density at location r, n(r), can be visualized by measuring the tipsample Josephson criticalcurrent squared \({I}_{{\rm{J}}}^{2}\left({\bf{r}}\right)\), from which
in which R_{N}(r) is the normalstate junction resistance. In the second PDW visualization technique^{17,20,21,22,23}, the magnitude of the energy gap in the sample, Δ(r), is defined as half the energy separation between the two superconductive coherence peaks in the density of electronic states N(E). These occur in tunnelling conductance at signed energies Δ_{+}(r) and Δ_{−}(r) such that
This can be visualized using either normalinsulatorsuperconductor (NIS) tunnelling^{20,21,22} or superconductorinsulatorsuperconductor (SIS) tunnelling from a superconductive STM tip^{17,23,29} whose superconductive gap energy, Δ_{tip}, is known a priori.
CDW visualization in normalstate UTe_{2}
UTe_{2} crystals typically cleave to show the (0–11) surface^{9,24}, a schematic view of which (Fig. 1b) identifies the key atomic periodicities by vectors a* and b*. At temperature T = 4.2 K, this surface is visualized using STM and a typical topographic image T(r) is shown in Fig. 1d, whereas Fig. 1e shows its power spectral density Fourier transform T(q), with the surface reciprocallattice points identified by dashed orange circles. Pioneering STM studies of UTe_{2} by Aishwarya et al.^{24} have recently discovered a CDW state by visualizing the electronic density of states g(r, E) of such surfaces. As well as the standard maxima at the surface reciprocallattice points in g(q, E), the Fourier transform of g(r, E), Aishwarya et al. detected three new maxima with incommensurate wavevectors Q_{1,2,3}, signifying the existence of a CDW state occurring at temperatures up to at least T = 10 K. To emulate this, we measure g(r, V) for −25 mV < V < 25 mV at T = 4.2 K using a nonsuperconducting tip on the equivalent cleave surface to ref. ^{24}. Figure 2a shows a typical topographic image T(r) of the (0–11) surface measured at 4.2 K. The Fourier transform T(q) features the surface reciprocallattice points labelled by dashed orange circles in Fig. 2a, inset. The simultaneous image g(r, 10 mV) in Fig. 2b exhibits the typical modulations in g(r, V) and its Fourier transform g(q, V) in Fig. 2c shows the three CDW peaks^{24} at Q_{1,2,3} labelled by dashed blue circles. Inverse Fourier filtration of these three maxima only shows the incommensurate CDW state of UTe_{2}. Overall, this state consists predominantly of incommensurate chargedensity modulations at three (0–11) inplane wavevectors Q_{1,2,3} that occur at temperatures up to at least 10 K (ref. ^{24}) and with a characteristic energy scale up to at least ±25 meV (ref. ^{24}; Methods and Extended Data Fig. 1).
Normaltip PDW detection at NIS gap edge
Motivated by the discovery that this CDW exhibits an unusual dependence on magnetic field and by the consequent hypothesis that a PDW may exist in this material^{24,25}, we next consider direct PDW detection in UTe_{2} by visualizing spatial modulations in its energy gap^{17,18,20,21,22,23}. The typical tunnelling conductance signature of the UTe_{2} superconducting energy gap is exemplified in Fig. 3a, showing a densityofstates spectrum \(N(E={\rm{e}}{\rm{V}})\propto {{\rm{d}}I/{\rm{d}}V}_{{\rm{N}}{\rm{I}}{\rm{S}}}(V)\) measured using a nonsuperconducting tip at T = 280 mK and junction resistance of R ≈ 5 MΩ. Under these circumstances, researchers find only a small drop in the tunnelling conductance at energies \( E \le  {\Delta }_{{{\rm{UTe}}}_{2}} \) (ref. ^{9}) and concomitantly weak energy maxima in N(E) at the energygap edges \(E\approx \pm {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\) (Fig. 3a, inset). Hence, it is challenging to accurately determine the precise value of the energy gap \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\) (Methods and Extended Data Fig. 3). Nevertheless, we fit a secondorder polynomial to the two energy maxima in measured N(E, r) surrounding \(E\approx \pm {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\), evaluate the images Δ_{±}(r) of these energies and then derive a gap map for UTe_{2} as \({\Delta }_{{{\rm{U}}{\rm{T}}{\rm{e}}}_{2}}({\bf{r}})\equiv [{\Delta }_{+}({\bf{r}}){\Delta }_{}({\bf{r}})]/2\). Its Fourier transform \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{q}}\right)\) presented in Methods and Extended Data Fig. 3 shows three incommensurate energygap modulations occurring at wavevectors P_{i=1,2,3}, consistent with the wavevectors of the CDW modulations discovered in ref. ^{24}. Although this evidence of three PDW states in UTe_{2} is encouraging, its weak signaltonoise ratio owing to the shallowness of coherence peaks implies that conventional \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{NIS}}}\) spectra are inadequate for precision application of equation (6) in this material.
Superconductivetip PDW detection
We turn to a wellknown technique for improving the resolution of energy maxima in g(r, E) measurements. By using SIS tunnelling from a tip exhibiting high sharp conductance peaks, one can profoundly enhance energy resolution for quasiparticles^{26,27,28,29,30,31}. Most recently, this has been demonstrated in electronic fluid flow visualization^{29} microscopy, with effective energy resolution δE ≈ 10 μeV. The SIS current I from a superconducting tip is given by the convolution
Equation (7) demonstrates that using a superconductive tip with high sharp coherence peaks at E_{±} = ±Δ_{tip} in N_{tip}(E) will, through convolution, strongly enhance the resolution for measuring the energies ±Δ_{sample} at which energy maxima occur in N_{sample}(E); it will also shift the energy of these features to E = ±[Δ_{sample} + Δ_{tip}]. In Fig. 3b, we show the \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) spectrum of a UTe_{2} single crystal using a superconducting Nb tip at T = 280 mK. Because the tunnelling current is given by equation (7), the clear maxima in \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) occur at energies ±(Δ_{tip} + Δ_{sample}). With this technique, the energy maxima can be identified with resolution better than δE ≤ 10 μeV when T < 300 mK (ref. ^{29}). Here we use it to improve the signaltonoise ratio of the UTe_{2} superconductive energygap modulations that are already detectable by conventional techniques (Methods and Extended Data Fig. 3).
The UTe_{2} samples are cooled to T = 280 mK, with T(r, V) of the (0–11) cleave surface as measured by a superconductive Nb tip shown in Fig. 3c. Here we see a powerful enhancement in the amplitude and sharpness of maxima in \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) relative to Fig. 3a. Consequently, to determine the spatial arrangements of the energy of the two maxima E_{+}(r) and E_{−}(r) surrounding 1.6 meV exemplified in Fig. 3b, we make two separate g(r, V) maps in the sample bias voltage ranges −1.68 mV < V < −1.48 mV and 1.5 mV < V < 1.7 mV, and in the identical field of view (FOV). The sharp peak of each \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) is fit to a secondorder polynomial \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}=a{V}^{2}+bV+c\), achieving typical quality of fit R^{2} = 0.99 ± 0.005. The energy of maximum intensity in E_{+}(r) or E_{−}(r) is then identified analytically from the fit parameters (Methods and Extended Data Fig. 4). The fine line across Fig. 3c specifies the trajectory of an exemplary series of \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) spectra, whereas Fig. 3d presents the colour map \({\left.{\rm{d}}I/{\rm{d}}V\right}_{{\rm{SIS}}}\) spectra for both positive and negative energy coherence peaks along this line. Periodic variations in the energies at which pairs of peaks occur are obvious, directly demonstrating that E_{+}(r) and E_{−}(r) are modulating periodically but in energetically opposite directions. Using this g(r, V) measurement and fitting procedure (Methods and Extended Data Fig. 4) yields atomically resolved images of E_{+}(r) and E_{−}(r). The magnitude of both positive and negative superconductive energy gaps of UTe_{2} is then \({\Delta }_{\pm }\left({\bf{r}}\right)\equiv \left{E}_{\pm }\left({\bf{r}}\right)\right{\rm{ }}{\Delta }_{{\rm{tip}}} \), in which Δ_{tip} is constant. These two independently measured gap maps Δ_{+}(r) and Δ_{−}(r) are spatially registered to each other at every location with 27pm precision so that the crosscorrelation coefficient between them is X ≅ 0.92, meaning that the superconducting energygap modulations are entirely particlehole symmetric (Fig. 3e,f, Methods and Extended Data Fig. 5).
From these and equivalent data, the UTe_{2} superconducting energygap structure \({\Delta }_{{{\rm{UTe}}}_{2}}\left({\bf{r}}\right)=\left({\Delta }_{+}\left({\bf{r}}\right)+{\Delta }_{}\left({\bf{r}}\right)\right)/2\) can now be examined for its spatial variations δΔ(r) by using
in which \(\left\langle {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\right\rangle \) is the spatial average over the whole FOV. Figure 4a shows measured δΔ(r) in the same FOV as Fig. 3c. The Fourier transform of δΔ(r), δΔ(q), is presented in Fig. 4b, in which the surface reciprocallattice points are identified by dashed orange circles. The three further peaks labelled by dashed red circles represent energygap modulations with incommensurate wavevectors at P_{1,2,3} of the PDW state in UTe_{2}. Focusing only on these three wavevectors P_{1,2,3}, we perform an inverse Fourier transform to show the spatial structure of the UTe_{2} PDW state in Fig. 4c (Methods). This state seems to consist predominantly of incommensurate superconductive energygap modulations at three (0–11) inplane wavevectors P_{1,2,3} with a characteristic energy scale 10 μeV for peaktopeak modulations.
Energy modulations of Andreev resonances
There is an alternative modality of SIS tunnelling, namely, measuring the effects of Andreev reflections. For two superconductors with very different gap magnitudes, when the sample bias voltage shifts the smaller gap edge (UTe_{2} in this case) to the chemical potential of the other superconductor, the Andreev process of electron (hole) transmission and hole (electron) reflection plus electronpair propagation can produce an energy maximum in dI/dV_{SIS} (ref. ^{38}), an effect well attested by experiment^{39}. Here, by imaging the signed energies of A_{±}(r) of two subgap dI/dV_{SIS} maxima detected throughout our studies and identified by the green arrows in Fig. 3b, an Andreevresonance measure of the UTe_{2} energy gap is conjectured as \({\Delta }_{{\rm{A}}}({\bf{r}})\equiv \left[{A}_{+}({\bf{r}}){A}_{}({\bf{r}})\right]/2\). These data are presented in Methods and Extended Data Fig. 7 and show a Δ_{A}(r) modulating with amplitude approximately 10 μeV at wavevectors P_{1,2} state, further evidencing the UTe_{2} PDW state.
Visualizing the interplay of PDW and CDW
Finally, one may consider the two cases of intertwining outlined earlier: (1) Δ_{S}(r) and Δ_{P}(r) are predominant and generate charge modulations \({\rho }_{{\rm{P}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\Delta }_{{\rm{P}}}+{\Delta }_{{\rm{P}}}^{* }{\Delta }_{{\rm{S}}}\) and \({\rho }_{2{\rm{P}}}\left({\bf{r}}\right)\propto {{\Delta }_{{\rm{P}}}^{* }\Delta }_{{\rm{P}}}\) or (2) Δ_{S}(r) and ρ_{Q}(r) are predominant and generate pair density modulations \({\Delta }_{{\rm{Q}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\rho }_{{\rm{Q}}}\). For case (1) to be correct here, a PDW with magnitude 10 μeV coexisting with a superconductor of gap maximum near 250 μeV must generate a CDW on the energy scale 25 meV and exist up to at least T = 10 K. For case (2) to be valid, a normalstate CDW with eigenstates at energies up to 25 meV coexisting with a superconductor of gap magnitude 250 μeV must generate a PDW at the same wavevector and with amplitude near 10 μeV. Intuitively, the latter case seems the most plausible for UTe_{2}.
To explore this issue further, we visualize the CDW in the nonsuperconductive state at T = 4.2 K, then cool to T = 280 mK and visualize the PDW in precisely the same FOV. Figure 4c,d shows the result of such an experiment in the FOV of Fig. 3c. The CDW and PDW images are registered to the underlying lattice and to each other with 27pm precision. Comparing their coterminous images in Fig. 4c and Fig. 4d shows that the CDW and PDW states of UTe_{2} appear spatially distinct. Yet, they are actually registered to each other in space, being approximate negative images of each other (Fig. 4e) and with a measured relative phase for all three P_{i}:Q_{i} pairs of \( \delta {\phi }_{i} \cong \pi \) (Fig. 4f, Methods and Extended Data Fig. 10). A typical example of this effect is shown in a line cut across Fig. 4c,d along the Te chain direction, with the directly measured values shown in Fig. 4g. The direct and comprehensive knowledge of CDW and PDW characteristics and interactions presented in Fig. 4 now motivates search for a Ginzburg–Landau description capable of capturing this complex intertwined phenomenology and that reported in ref. ^{24}.
Conclusions
Notwithstanding such theoretical challenges, in this study, we have demonstrated that PDWs occur at three incommensurate wavevectors P_{i=1,2,3} on the (0–11) surface of UTe_{2} (Fig. 4b,c). These wavevectors are indistinguishable from the wavevectors Q_{i=1,2,3} of the prevenient normalstate CDW at the equivalent surface (Figs. 2c and 4d). All three PDWs exhibit peaktopeak gap energy modulations in the range near 10 μeV (Fig. 4c,g). When the P_{i=1,2,3} PDW states are visualized at 280 mK in the identical FOV as the Q_{i=1,2,3} CDWs visualized above the superconductive T_{c}, every Q_{i}:P_{i} pair is spatially registered to each other (Fig. 4c,d), but with a relative phase shift of \( \delta {\phi }_{i}\, \cong \pi \) throughout (Fig. 4f). Given the premise that UTe_{2} is a spintriplet superconductor^{12}, the PDW phenomenology detected and described herein (Fig. 4) signifies the entrée to spintriplet PDW physics.
Methods
CDW visualization in nonsuperconductive UTe_{2}
Differential conductance imaging of CDW at T = 4.2 K
At T = 4.2 K and using superconducting tips to study the UTe_{2} (0–11) surface, we measure differential tunnelling conductance spectra g(r, V) to visualize the CDW in the normal state of UTe_{2}. Extended Data Fig. 1a–d shows g(r, V) images V = −7 mV, −15 mV, −23 mV and −29 mV with Fourier transform g(q, V) shown as Extended Data Fig. 1e–h. Three CDW peaks at Q_{1,2,3} occur in all g(q, V), representing incommensurate chargedensity modulations with energy scale up to approximately 30 meV, consistent with ref. ^{24}.
CDW visualization at incommensurate wavevectors Q _{1,2,3}
To calculate the amplitude \({g}_{{{\bf{Q}}}_{i}}({\bf{r}})\) of the CDW modulation represented by the peaks at Q_{i} (i = 1, 2, 3), we apply a twodimensional computational lockin technique. Here g(r) is multiplied by the term \({{\rm{e}}}^{{\rm{i}}{{\bf{Q}}}_{i}\cdot {\bf{r}}}\) and integrated over a Gaussian filter to obtain the lockin signal
in which σ_{r} is the cutoff length in the real space. In q space, this lockin signal is
in which σ_{Q} is the cutoff length in q space. Here \({{\sigma }_{{\bf{r}}}=1/\sigma }_{{\bf{Q}}}\). Next, g_{Q}(r, V), the inverse Fourier transform of the combined Q_{i} (i = 1, 2, 3) CDWs, is presented in Extended Data Fig. 1i–l.
To specify the effect of filter size on the inverse Fourier transform, we show in Extended Data Fig. 1m–t the realspace density of states g(r, 10 mV), its Fourier transform g(q, 10 mV) and the evolution of inverse Fourier transform images as a function of the realspace cutoff length σ_{r}. The differential conductance map g_{Q}(r, 10 mV) is shown at a series of σ_{r}, including 10 Å, 12 Å, 14 Å, 18 Å, 24 Å and 35 Å. The distributions of the CDW domains in the filtered g_{Q}(r, 10 mV) images with cutoff lengths of 10 Å, 12 Å, 14 Å, 18 Å and 24 Å are highly similar. The cutoff length used in Fig. 2d is 14 Å, such that the domains of the CDW modulations are resolved and the irrelevant image distortions are excluded. The same filter size of 14 Å is chosen for all three Q_{i} vectors. Formally, the equivalent inverse Fourier transform analysis is carried out for Fig. 4c,d but with a filter size of 11.4 Å to filter both the CDW and PDW peaks.
Simulated UTe_{2} topography
To identify qspace peaks resulting from the (0–11) cleaveplane structure of UTe_{2}, we simulate the topography of the UTe_{2} cleave plane and Fourier transform. Subsequently, we can distinguish clearly the CDW signal from the structural periodicity of the surface. The simulation is calculated on the basis of the ideal lattice constant of the (0–11) plane of the UTe_{2}, a* = 4.16 Å and interTechain distance b* = 7.62 Å. Extended Data Fig. 2a is a simulated T(r) image in the FOV of 14.5 nm. The simulated topography T(r) is in good agreement with experimental T(r) images presented throughout. The Fourier transform, T(q), of the simulated T(r) in Extended Data Fig. 2b shows six sharp peaks, confirming that they are the primary peaks resulting from the cleaveplane structure. Most notably, the CDW peaks in Fig. 2c are not seen in the simulation. They are therefore not caused by the surface periodicity but instead originate from the electronic structure, as first demonstrated in ref. ^{24}.
Normaltip PDW detection at the NIS gap edge of UTe_{2}
Initial STM searches for a PDW on UTe_{2} were carried out using a normal tip at 280 mK. Extended Data Fig. 3a shows a typical line cut of the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum taken from the FOV in Extended Data Fig. 3b. There is a large residual density of states near the Fermi level. The gap depth H is defined as the difference between the gap bottom in the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum and the coherence peak height, that is, \(H\equiv {\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{ }}}_{{\rm{NIS}}}(V\equiv {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}){\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{ }}}_{{\rm{NIS}}}(V\equiv 0)\). Its modulation is extracted from the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) line cut and presented in Extended Data Fig. 3c; it modulates perpendicular to the Te atom chains.
Conventional, NIS tunnelling discloses superconducting energygap modulations as shown in Extended Data Fig. 3a. The superconducting energy gap is defined as half of the peaktopeak distance in the \({\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{ }}}_{{\rm{NIS}}}\) spectrum (Fig. 3a and Extended Data Fig. 3d). Its magnitude \( {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}} \) is found to lie approximately between 250 µeV and 300 µeV. We measure variations in the coherence peak position from the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum at each location r. The two energy maxima near \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\) of each \({\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{ }}}_{{\rm{NIS}}}\) spectrum are fitted with a secondorder polynomial function (\({R}_{{\rm{RMS}}}^{2}=0.87\)). The energy gap is defined as the maxima of the fit, Δ_{+} for V > 0 and Δ_{−} for V < 0. The total gap map \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\equiv [{\Delta }_{+}({\bf{r}}){\Delta }_{}({\bf{r}})]/2\) is derived from Δ_{+} and Δ_{−} (Extended Data Fig. 3e). The Fourier transform of \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\), \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{q}}\right)\) (Extended Data Fig. 3f), shows three peaks at wavevectors P_{i=1,2,3}. They are the initial signatures of the energygap modulations of the three coexisting PDW states in UTe_{2}.
Superconductivetip PDW visualization at the SIS gap edge of UTe_{2}
Tip preparation
Atomicresolution Nb superconducting tips are prepared by field emission. To determine the tip gap value during our experiments, we measure conductance spectrum on UTe_{2} at 1.5 K (T_{c} = 1.65 K), in which the UTe_{2} superconducting gap is closed. The tip gap \( {\Delta }_{{\rm{tip}}} \cong 1.37\,{\rm{meV}}\) is the energy of the coherence peak (Extended Data Fig. 4a). The measured spectrum is fitted using a Dynes model^{40}. The typical \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\) measured at 280 mK on UTe_{2} (Fig. 3b) shows the total gap value \(E= {\Delta }_{{\rm{tip}}} + {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}} \approx 1.62\,{\rm{meV}}\). Therefore, we estimate \( {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}} \approx 0.25\,{\rm{meV}}\), consistent with a previous report^{9} and the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) shown in Fig. 3a and Extended Data Fig. 3.
SIS tunnelling amplification of energygap measurements
To determine the energy of E_{+}(r) and E_{−}(r) at which the maximum conductance in \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}(V)\) occurs, we fit the peak of the measured \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}(V)\) spectra using a secondorder polynomial fit:
This polynomial closely fits the experimental data. Extended Data Fig. 4b,c shows two typical \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}(V)\) spectra measured at +V and −V along the trajectory indicated in Fig. 3c. The evolution of fits g(V) in Extended Data Fig. 4d,e shows a very clear energygap modulation.
Shear correction and Lawler–Fujita algorithm
To register several images to precisely the same FOV, the Lawler–Fujita algorithm is applied to the experimental data. Then, to recover the arbitrary hexagon of the Te lattice, shear correction is applied to correct any image distortions caused by longrange scanning drift during days of continuous measurement.
To correct against picometrescale distortions of the lattice, we apply the Lawler–Fujita algorithm. Let \(\widetilde{T}(\widetilde{{\bf{r}}})\) represent a topograph of a perfect UTe_{2} lattice without any distortion. Three pairs of Bragg peaks Q_{1}, Q_{2} and Q_{3} can be obtained from Fourier transform of \(\widetilde{T}(\widetilde{{\bf{r}}})\). Hence \(\widetilde{T}(\widetilde{{\bf{r}}})\) is expected to take the form
The experimentally obtained topography T(r) may suffer from a slowly varying positiondependent spatial phase shift θ_{i}(r), which can be given by
To get θ_{i}(r), we use a computational twodimensional lockin technique to the topography
for which σ is chosen to capture the lattice distortions. In the Lawler–Fujita analysis, we use σ_{q} = 3.8 nm^{−1}. Mathematically, the relationship between the distorted and the perfect lattice for each Q_{i} is \({{\bf{Q}}}_{i}\cdot {\bf{r}}{\boldsymbol{+}}{\theta }_{i}({\bf{r}})={{\bf{Q}}}_{i}\cdot \widetilde{{\bf{r}}}+{\theta }_{i}\). We define another globalpositiondependent quantity, the displacement field \({\bf{u}}\left({\bf{r}}\right)={\bf{r}}\widetilde{{\bf{r}}}\), which can be obtained by solving equations
Finally, a driftcorrected topography, \(\widetilde{T}\left(\widetilde{{\bf{r}}}\right)\) is obtained by
By applying the same correction of u(r) to the simultaneously taken differential conductance map g(r), we can get
in which \(\widetilde{g}\left(\widetilde{{\bf{r}}}\right)\) is the driftcorrected differential conductance map.
Lattice registration of UTe_{2} energy gap \({{\boldsymbol{\Delta }}}_{{{\bf{UTe}}}_{{\bf{2}}}}{\boldsymbol{(}}{\bf{r}}{\boldsymbol{)}}\)
We measure two separate \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}({\bf{r}},V)\) maps separated by several days and in two overlapping FOVs, with energy ranges −1.68 meV < E < −1.48 meV and 1.5 meV < E < 1.7 meV. Therefore, we obtain two datasets, T_{+}(r) with the simultaneous \({{{\rm{d}}I/{\rm{d}}V }_{{\rm{SIS}}}}_{+}({\bf{r}},V)\) at positive bias and T_{−}(r) with the simultaneous \({{{\rm{d}}I/{\rm{d}}V }_{{\rm{SIS}}}}_{}({\bf{r}},V)\) at negative bias.
After the shear and Lawler–Fujita corrections are applied, the lattice in the corrected topographs of T_{+}(r) and T_{−}(r) become nearly perfectly periodic. Next, we perform rigid spatial translations to register the two topographs to the exact same FOV with a lateral precision better than 27 pm. Extended Data Fig. 5a,b shows two topographs of registered T_{+}(r) and T_{−}(r). Crosscorrelation (XCORR) of two images I_{1} and I_{2}, X(r, I_{1}, I_{2}) at r is obtained by sliding two images r apart and calculating the convolution,
in which the denominator is a normalization factor such that, when I_{1} and I_{2} are exactly the same image, we can get X(r = 0, I_{1}, I_{2}) = 1 with the maximum centred at (0, 0) crosscorrelation vector. Extended Data Fig. 5c shows that the maximum of XCORR between T_{+}(r) and T_{−}(r) coincides with the (0, 0) crosscorrelation vector. The offset of the two registered images are within one pixel. The multipleimage registration method is better than 0.5 pixels = 27 pm in the whole FOV and the maxima of the crosscorrelation coefficient between the topographs is 0.93. All transformation parameters applied to T_{+}(r) and T_{−}(r) to yield the corrected topographs are subsequently applied to the corresponding \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}({\bf{r}},V)\) maps obtained at positive and negative voltages.
Particlehole symmetry of the superconducting energy gap \({{\boldsymbol{\Delta }}}_{{{\bf{UTe}}}_{{\bf{2}}}}{\boldsymbol{(}}{\bf{r}}{\boldsymbol{)}}\)
The crosscorrelation map in Extended Data Fig. 5f provides a twodimensional measure of agreement between the positive and negative \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left(V\right)\) energymaxima maps in Extended Data Fig. 5d,e. The inset of Extended Data Fig. 5f shows a line cut along the trajectory indicated in Extended Data Fig. 5f. It shows a maximum of 0.92 and coincides with the (0, 0) crosscorrelation vector. Thus, it shows that gap values between positive bias and negative bias are highly correlated.
PDW visualization at incommensurate wavevectors P _{1,2,3}
The inverse Fourier transform analysis for PDW state in Fig. 4c is implemented using the same technique described here in Methods. The filter size chosen to visualize the PDW is 11.4 Å. The inverse Fourier transform of the CDW in Fig. 4d is calculated using an identical filter size of 11.4 Å.
Independent PDW visualization experiments
To confirm that the PDW discovered is present in several FOVs, we show a typical example of the gap modulation Δ_{+}(r) from one different FOV in Extended Data Fig. 6. The \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}({\bf{r}},V)\) map is measured in the voltage region surrounding the positive NbUTe_{2} energy maxima near 1.6 meV. The spectra in this FOV are fitted with a secondorder polynomial and shear corrected as described here in Methods. The resulting gap map, δΔ_{+}(r), is presented in Extended Data Fig. 6b. The Fourier transform of this map, δΔ_{+}(q), is presented in Extended Data Fig. 6c. δΔ_{+}(q) features the same PDW wavevectors (P_{1}, P_{2}, P_{3}) reported in the main text.
Energy modulations of subgap Andreev resonances
Surface Andreev bound states must occur in pwave topological superconductors^{41}. Moreover, based on the phasechanging quasiparticle reflections at the pwave surface, finiteenergy Andreev resonances should also occur in the junction between a pwave and an swave superconductor^{42} and are observed in UTe_{2}. Inside the SIS gap, we measure the \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left({\bf{r}},V\right)\) map in the energy range from −500 µeV to 500 µeV. The map is measured in the FOV in Extended Data Fig. 7a, the same FOV as in Figs. 3 and 4. Three conductance peaks are resolved at approximately −300 µeV, 0 and 300 µeV, annotated with green arrows in the typical subgap spectrum in Extended Data Fig. 7b. The energy maximum of the positive subgap states between 200 µeV to 440 µeV is assigned as A_{+}. The energy maximum of the negative subgap states between −440 µeV to −200 µeV is assigned as A_{−}. The averaged energy of the Andreev subgap states is defined as \({\Delta }_{{\rm{A}}}({\bf{r}})\equiv \left[{A}_{+}({\bf{r}}){A}_{}({\bf{r}})\right]/2\), which ranges from 300 µeV to 335 µeV (Extended Data Fig. 7c). Fourier transform of the subgap energies Δ_{A}(q) exhibit two sharp peaks at the PDW wavevectors P_{1} and P_{2} (Extended Data Fig. 7d).
In the case of two superconductors with very different gap magnitude, when the sample bias voltage shifts the smaller gap edge to the chemical potential, the Andreev process of electron (hole) transmission and hole (electron) reflectional plus electronpair propagation may produce an energy maximum in dI/dV_{SIS} at the voltage of smaller gap energy. Hence, the observations in Extended Data Fig. 7d may be expected if the UTe_{2} superconducting energy gap is modulating at the wavevectors P_{1} and P_{2}. Extended Data Fig. 7e shows that the energy of the Andreev states modulate in space with a peaktopeak amplitude near 10 µeV (see histogram in Extended Data Fig. 7f).
Enhancement of signaltonoise ratio using superconductive tips
Superconducting STM tips provide an effective energy resolution beyond the Fermi–Dirac limit. They have therefore been widely used as a method of enhancing the energy resolution of STM spectra^{26,27,28,29,30,31}.
To better quantify the signaltonoise ratio improvement of the measured energygap modulations, we compare the fitting quality of the superconducting gap maps obtained using a normal tip (Extended Data Fig. 3) and a superconducting tip (Fig. 4). The fitting quality is defined using the coefficient
in which \({\rm{d}}I\,/\,{\rm{d}}V\left(V\right)\) is the measured spectrum, g(r, V) is the fitted spectrum and \(\bar{g}\left({\bf{r}}\right)\) is the averaged fitted spectrum. Extended Data Fig. 8a shows a typical spectrum measured using a superconductive tip, \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\) from the FOV in Fig. 3c. Extended Data Fig. 8d is a typical \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum measured using a normal tip from the FOV in Extended Data Fig. 3. The energymaximum noise level is decisively lower in \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\) spectra than in \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectra and the fitting quality \({R}_{{\rm{SIS}}}^{2}\) is substantially higher than \({R}_{{\rm{NIS}}}^{2}\).
Extended Data Fig. 8b,c shows maps of the fitting parameter R^{2} calculated from fitting the dI/dV_{SIS} energymaxima map obtained using a superconductive tip, that is, the \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\) images presented in Fig. 3e,f. Extended Data Fig. 8e,f shows maps of R^{2} calculated from the coherence peak fitting of dI/dV_{NIS} obtained using a normal tip, that is, the \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\) images presented in Extended Data Fig. 3e. Comparing these R^{2} qualityoffit parameter maps, we find that a much larger fraction of normaltip coherence peak maps have poor correspondence with the fitting procedures used. For superconducting tips, the rootmeansquare values of the fitting parameter, \({R}_{{\rm{RMS}}}^{2}\), are 0.98 and 0.99 for the positive and negative coherence peak fitting, respectively. The normaltip \({R}_{{\rm{RMS}}}^{2}\) values are 0.87 and 0.86 for the positive and negative coherence peak fitting, respectively. The superconducting tip therefore demonstrably achieves a marked signaltonoise ratio enhancement for evaluation of \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\) images.
As the signaltonoise ratio is increased in the SISconvoluted coherence peaks measured using a superconducting tip, it has been possible to resolve the UTe_{2} energygap modulations of order approximately 10 μV. Fundamentally, the energy resolution is associated with the ability of the superconductive tip to resolve the energy at which the dI/dV_{SIS} coherence peak reaches its maximum amplitude. Consequently, we determine our energy resolution to be 10 μV.
Thus, the same superconductor energygap modulations in \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\) of UTe_{2} can be observed using either a superconducting tip or a normal tip. However, the former substantially increases the SIS conductance at \(\leftE\right={\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}+{\Delta }_{{\rm{tip}}}\) and allows for considerably better imaging of these energy maxima and thus \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\).
Interplay of subgap quasiparticles and PDW
Here we show simultaneous normaltipmeasured modulations of the UTe_{2} subgap states and \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\) at T = 280 mK, to study their interplay. Extended Data Fig. 9a shows the integrated differential conductance from −250 μV to 250 μV, \({\sum }_{250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{r}},E\right)\). Inverse Fourier transform of the three wavevectors Q_{1,2,3} from \({\sum }_{250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{r}},E\right)\) and P_{1,2,3} from the simultaneous \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\) in Extended Data Fig. 3e are compared in Extended Data Fig. 9c,d. Clearly, from the highly distinct spatial structure of these images, there is no onetoone correspondence between the subgap densityofstates modulations and the simultaneously measured PDW energygap modulations in UTe_{2}. Overall, there is a very weak anticorrelation, with a crosscorrelation value of −0.23 ± 0.05 that is not inconsistent with coincidence. Hence we demonstrate that there is no deterministic influence of the subgap densityofstates modulations on the PDW energygap modulations in superconducting UTe_{2}.
Visualizing the interplay of PDW and CDW in UTe_{2}
The analysis of phase difference between PDW and CDW at three different wavevectors is shown in Extended Data Fig. 10. The inverse Fourier transforms of each CDW and PDW wavevector demonstrate a clear halfperiod shift between the two density waves (Extended Data Fig. 10a–f). This shift motivates the statistical analysis of the phase difference. The phase map of \({g}_{{{\rm{Q}}}_{1}}({\bf{r}},9\,{\rm{mV}})\), \({\phi }_{1}^{{\rm{C}}}\left({\bf{r}}\right)\), and the phase map of \({\Delta }_{{{\rm{P}}}_{1}}({\bf{r}})\), \({\phi }_{1}^{{\rm{P}}}\left({\bf{r}}\right)\), are calculated. The phase difference between two corresponding maps is defined as \( \delta {\phi }_{1} ={\phi }_{1}^{{\rm{C}}}\left({\bf{r}}\right){\boldsymbol{}}{\phi }_{1}^{{\rm{P}}}\left({\bf{r}}\right)\) for the P_{1}:Q_{1} wavevectors. Identical procedures are carried out for P_{2}:Q_{2} and P_{3}:Q_{3}. The histograms resulting from this procedure show that the statistical distributions of the phase shift \(\delta {\phi }_{i}{\rm{}}\) are centred around π (Extended Data Fig. 10j–l). Although the distribution varies, this π phase shift reinforces the observation of the spatial anticorrelation between CDW and PDW.
As shown in the inset of Extended Data Fig. 10g, the three PDW wavevectors are related by reciprocal lattice vectors: P_{2} = P_{1} − G_{3} and P_{3} = G_{1} − P_{1}. Nevertheless, the three UTe_{2} PDWs seem to be independent states when analysed in terms of the spatial modulations of the amplitude of the P_{1,2,3} peaks from Fig. 4 using equation (16). The amplitude of P_{1,2} has a domain width beyond 10 nm in the real space (Extended Data Fig. 10g,h). The amplitude of P_{3} is shortranged, of which the averaged domain width is approximately 5 nm (Extended Data Fig. 10i). The onepixel shift of P_{3} from the central axis is within the error bar of experimental measurements. The spatial distributions of the three PDWs are negligibly correlated with crosscorrelation values of their amplitude of X(P_{1}, P_{2}) = −0.3, X(P_{1}, P_{3}) = 0.9 and X(P_{2}, P_{3}) = 0.28. The weak crosscorrelation relationships indicate that the three PDWs are independent orders.
Data availability
The data shown in the main figures are available from Zenodo at https://doi.org/10.5281/zenodo.7662516.
Code availability
The code is available to qualified researchers from the corresponding authors on reasonable request.
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Acknowledgements
We are extremely grateful to V. Madhavan for generous and incisive advice and guidance on how to execute this project. We acknowledge and thank D. Agterberg, F. Flicker, E. Fradkin, E.A. Kim, S. Simon, J. van Wezel and K. Zhussupbekov for key discussions and theoretical guidance. Research at the University of Maryland was supported by the Department of Energy Award No. DESC0019154 (sample characterization), the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF9071 (materials synthesis), NIST and the Maryland Quantum Materials Center. Q.G., X.L., J.P.C. and J.C.S.D. acknowledge support from the Moore Foundation’s EPiQS Initiative through grant GBMF9457. J.C.S.D. acknowledges support from the Royal Society under award R64897. J.P.C. and J.C.S.D. acknowledge support from Science Foundation Ireland under award SFI 17/RP/5445. S.W. and J.C.S.D. acknowledge support from the European Research Council (ERC) under award DLV788932.
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X.L. and J.C.S.D. conceived the project. S.R., C.B., H.S., S.R.S., N.P.B. and J.P. developed, synthesized and characterized materials. Q.G., J.P.C., S.W. and X.L. carried out the experiments. S.W., J.P.C. and Q.G. developed and implemented analyses. X.L. and J.C.S.D. supervised the project. J.C.S.D. wrote the paper, with key contributions from J.P.C., Q.G., X.L. and S.W. The paper reflects contributions and ideas of all authors.
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Extended data figures and tables
Extended Data Fig. 1 CDW at different voltages in UTe_{2}.
a–d, Measured g(r, V) images of UTe_{2} at T = 4.2 K and at four representative negative sample voltages, −7 mV, −15 mV, −23 mV and −29 mV, in the same 12 nm × 12 nm FOV. e–h, Fourier transform of the g(r, V) images, g(q, V), at different sample voltages, showing the presence of the three wavevectors corresponding to the CDW order (in dashed blue circles). i–l, Inverse Fourier transform of the CDW peaks (Q_{1}, Q_{2}, Q_{3}) at different sample voltages. The CDW pattern is independent from the sample voltages for −29 mV < V < −7 mV. A dashed white circle indicates σ of the Gaussian filter used to isolate CDW peaks in real space. m–t, Cutoff dependence of inverse Fourier transform. n–p,r–t, Inverse Fourier transform of CDW peaks g_{Q}(r, 10 mV) from g(r, 10 mV) in m and g(q, 10 mV) in q. The images of g_{Q}(r, 10 mV) are filtered at different cutoff lengths, 10 Å, 12 Å, 14 Å, 18 Å, 24 Å and 35 Å. The filter size is in the bottomright corner. σ_{r} chosen for Fig. 2d is 14 Å.
Extended Data Fig. 2 Simulated topography of UTe_{2} and its Fourier transform.
a, Simulated topograph, T_{S}(r) of the (0–11) cleave surface of UTe_{2}. b, Fourier transform of simulated topograph, T_{S}(q). The six primary peaks occur at the reciprocallattice wavevectors and are observed in the experimental STM data.
Extended Data Fig. 3 PDW detection using a normal tip.
a, A typical line cut of \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectra obtained at 280 mK along the trajectory shown in b (I_{s} = 1 nA, V_{s} = −5 mV). b, Topograph T(r) obtained using a normal tip. c, Gap depth H distribution along the trajectory in b. d, \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum showing the superconducting gap Δ_{+} and Δ_{−}. e, Image of half the energy difference between superconducting coherence peaks, that is, the superconducting energy gap of \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\), obtained in the same FOV as b, using conventional normaltip imaging. f, \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{q}})\), the Fourier transform of \({\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\). Three peaks are seen at the same wavevector as the normalstate CDW and indicate the existence of three superconducting PDW states (I_{s} = 1 nA, V_{s} = −5 mV).
Extended Data Fig. 4 Determination of the tip gap Δ_{tip} and evolution of dI/dV_{SIS} spectra with parabolic fitting.
a, A typical spectrum measured on UTe_{2} using a superconducting Nb tip at 1.5 K (I_{s} = 100 pA, V_{s} = 4 mV). At this temperature, the UTe_{2} gap is closed, thus the coherence peak value shows the pure Nb tip gap of 1.37 meV. The spectrum is clearly well fitted using the Dynes model. The fitting parameters of the Dynes model are Γ = 0.01 meV, Δ = 1.37 meV. b,c, Line cuts of \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left(V\right)\) spectra measured at both negative bias and positive bias along the trajectory shown in Fig. 3c. d,e, The evolution of \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left(V\right)\) spectra (blue points) from the same data shown in b and c and their parabolic fits g(V).
Extended Data Fig. 5 Spatial registration of topographs and gap maps.
a,b, 12 nm × 12 nm topographs after registration. These topographs were obtained concomitantly as \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left({\bf{r}},V\right)\) maps recording positive and negative coherence peaks, respectively. c, XCORR map of the registered topographs. The correlation coefficient is 0.93, indicating that the two topographs are almost identical. The maxima of the XCORR map is a single pixel wide, which suggests a registration precision of 0.5 pixels, equivalent to registration precision of 27 pm. d, Positive coherence peak map E_{+}(r) from a. e, Negative coherence peak map E_{−}(r) from b. f, XCORR map providing a twodimensional measure of correlation between the positive gap map E_{+}(r) and negative gap map E_{−}(r). Inset, a line cut along the trajectory indicated in f. It shows that the maximum is 0.92 and coincides with the (0, 0) crosscorrelation vector. The strong correlation demonstrates the particlehole symmetry in superconductive UTe_{2}.
Extended Data Fig. 6 PDW repeatability analysis.
a, A topograph recorded in a new FOV away from that seen in Fig. 3c. The image size is 15 nm × 15 nm (V_{s} = 3 mV, I_{s} = 2.5 nA). b, δΔ_{+}(r) map prepared using the same procedure outlined in Methods shows the same gap modulations as Fig. 4a. c, The Fourier transform of δΔ_{+}(r) map, δΔ_{+}(q). (P_{1}, P_{2}, P_{3}) PDW peaks are highlighted with dashed red circles and reciprocal lattice vectors are highlighted with dashed orange circles.
Extended Data Fig. 7 Imaging of subgap Andreev resonances.
a, Topography of the subgap states imaging in the same FOV as Figs. 3 and 4. b, A representative \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\left(V\right)\) spectrum of the subgap states annotated by the green arrows. c, Map of the energy scale of the subgap states modulations Δ_{A}(r). d, Fourier transform of the subgap states modulations Δ_{A}(q). P_{1,2,3} PDW peaks are highlighted with dashed red circles. e, Inverse Fourier transform Δ_{A,P}(r) of PDW peaks P_{1,2,3}. f, Histogram of Δ_{A,P}(r) shows that the PDW modulates within 10 µeV.
Extended Data Fig. 8 Estimation of signaltonoise ratio using fitting quality of spectra measured with superconductive tips and normal tips.
a, Parabolic fit of a typical \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\) spectrum measured using superconductive tips. b,c, Measured R^{2} maps used to estimate the fitting quality of \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{SIS}}}\) spectra for positive energy (b) and negative energy (c). The R^{2} image is from the FOV of Fig. 3c. d, Parabolic fit of a typical \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectrum taken using normal tips. e,f, Measured R^{2} maps used to estimate the fitting quality of \({\rm{d}}I\,/\,{\rm{d}}V{ }_{{\rm{NIS}}}\) spectra for positive energy (e) and negative energy (f). The R^{2} image is from the FOV in Extended Data Fig. 3b.
Extended Data Fig. 9 Modulations of subgap states measured using normal tips.
a, Sum of all subgap states \({\sum }_{250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{r}},E\right)\), measured at T = 280 mK. b, Fourier transform of subgap states \({\sum }_{250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{q}},E\right)\), in which all three wavevectors P_{1,2,3} are present. c, Inverse Fourier transform of P_{1,2,3} from a. d, Inverse Fourier transform of P_{1,2,3} from Extended Data Fig. 3e,f. The filter size is indicated as a dashed white circle.
Extended Data Fig. 10 Phase shift between CDW and PDW.
a–c, Inverse Fourier transforms of the three CDW wavevectors identified \({g}_{{{\bf{Q}}}_{i=1,2,3}}({\bf{r}},\,9\,{\rm{mV}})\) in the same 12 nm × 12 nm FOV as Fig. 3c. d–f, Inverse Fourier transforms of the three PDW \(\delta {\Delta }_{{{\bf{P}}}_{i=1,2,3}}\) wavevectors in the same FOV as Fig. 3c. g–i, Amplitude for all three PDW wavevectors P_{i=1,2,3}. Inset of g is the Fourier transform of the energy gap map, in which the reciprocal lattice points G_{i=1,2,3} are labelled. j–l, Distributions of the relative spatial phase difference \(\delta {\phi }_{i}({\bf{r}})\) between \({\phi }_{i}^{{\rm{C}}}\left({\bf{r}}\right)\) and \({\phi }_{i}^{{\rm{P}}}\left({\bf{r}}\right)\) from three individual wavevectors. Each histogram is centred around π, reinforcing the observation of a general phase difference \( \delta {\phi }_{i} \cong \pi \) between the CDW and the PDW.
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Gu, Q., Carroll, J.P., Wang, S. et al. Detection of a pair density wave state in UTe_{2}. Nature 618, 921–927 (2023). https://doi.org/10.1038/s41586023059197
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DOI: https://doi.org/10.1038/s41586023059197
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