Detection of a Cooper-pair density wave in Bi₂Sr₂CaCu₂O_8+x

Abstract

The quantum condensate of Cooper pairs forming a superconductor was originally conceived as being translationally invariant. In theory, however, pairs can exist with finite momentum Q, thus generating a state with a spatially modulated Cooper-pair density^1,2. Such a state has been created in ultracold ⁶Li gas³ but never observed directly in any superconductor. It is now widely hypothesized that the pseudogap phase⁴ of the copper oxide superconductors contains such a ‘pair density wave’ state^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. Here we report the use of nanometre-resolution scanned Josephson tunnelling microscopy^22,23,24 to image Cooper pair tunnelling from a d-wave superconducting microscope tip to the condensate of the superconductor Bi₂Sr₂CaCu₂O_8+x. We demonstrate condensate visualization capabilities directly by using the Cooper-pair density variations surrounding zinc impurity atoms²⁵ and at the Bi₂Sr₂CaCu₂O_8+x crystal supermodulation²⁶. Then, by using Fourier analysis of scanned Josephson tunnelling images, we discover the direct signature of a Cooper-pair density modulation at wavevectors Q_P ≈ (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀ in Bi₂Sr₂CaCu₂O_8+x. The amplitude of these modulations is about five per cent of the background condensate density and their form factor exhibits primarily s or s′ symmetry. This phenomenology is consistent with Ginzburg–Landau theory^5,13,14 when a charge density wave^5,27 with d-symmetry form factor^28,29,30 and wavevector Q_C = Q_P coexists with a d-symmetry superconductor; it is also predicted by several contemporary microscopic theories for the pseudogap phase^18,19,20,21.

Main

When hole doping suppresses the cuprate antiferromagnetic insulator state the pseudogap regime emerges⁴. Both the high-temperature superconducting state and a charge density wave (CDW) state²⁷ exist therein. However, a ‘pair density wave’ (PDW) state, in which the Cooper-pair density modulates spatially at wavevector Q , is also widely hypothesized^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21} to exist in the pseudogap regime. There are compelling theoretical motivations for a cuprate PDW state. First, microscopic theories for local electronic structure of the hole-doped antiferromagnet predict a cuprate PDW state linked to combined modulations of hole density and antiferromagnetic spin density^6,7,8,9,10. Second, c-axis superconductivity does not appear in La_2−xBa_xCuO₄ until temperatures far below the point at which the CuO₂ planes superconduct, a situation that could be explained by the existence of an orthogonal PDW state in each CuO₂ plane^5,11,13. Third, the existence of a PDW state could explain some of the unusual characteristics observed in the single-particle excitations^15,16,17. Finally, contemporary theories for the cuprate pseudogap phase hypothesize a composite state in which the CDW is directly linked to a PDW of the same wavevector^18,19,20,21.

The definitive signature^1,2,5 of a PDW would be a periodic modulation of the Cooper-pair condensate density with wavevector Q . In principle, this could be visualized using two techniques. First, scanning tunnelling microscopy (STM) using conventional single-electron tunnelling may detect modulations at Q in the locally defined energy gap:

Here Δ₀ is the homogeneous superconductor energy gap and is that of the PDW (refs 1 and 2). A second possibility is scanned Josephson tunnelling microscopy (SJTM) of Cooper pairs^22,23,24 to detect the modulations of Cooper-pair density directly. Again, in the presence of the homogeneous condensate, this should result in a modulating Josephson critical current I_J:

where the first term represents Cooper-pair tunnelling to a homogeneous condensate and the second to the PDW. Imaging the spatial arrangements of the superfluid density ρ_S( r ) would then be possible, since (see Methods section ‘Modelling phase diffusion Josephson dynamics for SJTM’).

The phenomena in equation (1) have not been detected in a cuprate (or any other) superconductor. One challenge is the extreme variation in the differential tunnelling conductance spectra (Fig. 1b) typically detected at the Bi₂Sr₂CaCu₂O_8+x surface T( r ) (Fig. 1a). Visualizing the maximum energy gap (double-headed black arrow in Fig. 1b) yields a heterogeneous ‘gapmap’ Δ( r ) (Fig. 1c), whose Fourier transform (Fig. 1d) exhibits only one sharp finite- q peak (blue arrow). This Δ( r ) modulation occurs at the wavevector Q_SM of a quasi-periodic distortion of the Bi₂Sr₂CaCu₂O_8+x crystal unit cell along the CuO₂ (1,1) direction²⁶ that is referred to as the ‘supermodulation’ (Fig. 1e). Because no other periodic modulations of the maximum energy gap Δ (or any other energy gap) have been detected in cuprate gapmap studies, searches for a cuprate PDW via equation (1) yield a null result.

Figure 1: Spatial variations and modulations in cuprate energy gaps.

a, Typical 35 nm × 35 nm topographic image T( r ) of the Bi₂Sr₂CaCu₂O_8+x surface (crystal ‘supermodulation’ runs vertically). b, Typical differential conductance spectra of Bi₂Sr₂CaCu₂O_8+x. The maximum energy gap Δ is half the distance between peaks. V_B is the bias voltage. c, Spatial arrangement of Δ( r ) (the gapmap) for p ≈ 17% Bi₂Sr₂CaCu₂O_8+x in the same field of view as a. d, The magnitude of the Fourier transform of c, (blue crosses are at q = (π/a₀, 0); (0, π/a₀)), showing a single peak due to the supermodulation²⁶ (blue arrow). H, high; L, low. e, The magnitude of the Fourier transform of topograph (blue crosses are at q = (π/a₀, 0); (0, π/a₀)). The supermodulation is a quasi-periodic modulation along the (1,1) direction with wavevector Q_SM indicated by the blue arrow. f, Simultaneously measured and from d and e along the (1,1) direction. Their primary peaks coincide. FT, Fourier transform; normalization is to DC component.

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The use of superconducting STM tips to measure the spatial variation of Cooper-pair tunnelling (SJTM) therefore remains the most promising approach with which to search for a PDW state (equation (2)). For two superconducting electrodes with temperature dependent energy gaps Δ(T) and phases ϕ₁ and ϕ₂, the Josephson current is given by where ϕ = ϕ₂ − ϕ₁. The Josephson critical current I_J is given by where R_N is the normal-state junction resistance, k is the Boltzmann constant, and 2e is the Cooper-pair charge. Time independence of ϕ would require the Josephson energy , where ϕ₀ is the magnetic flux quantum, to exceed the thermal fluctuation energy kT. For a nanometre-sized Josephson junction with sub-nanoampere I_J, this regime remains out of reach at millikelvin temperatures. However, imaging of phase-diffusion-dominated Josephson tunnelling is possible by using superconducting-tip STM to measure an I(V) whose maximum current is (refs 22, 23, 24 and Methods section ‘Modelling phase diffusion Josephson dynamics for SJTM’).

To search for a cuprate PDW near Q ≈ (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀ where a₀ ≈ 3.8 Å also requires visualizing with nanometre resolution. Moreover, high Δ, low R_N and millikelvin operating temperatures would all be highly advantageous. Motivated in this way, we use a dilution-refrigerator-based STM operated below 50 mK, and achieve a high tip-energy gap Δ_T by picking up a nanometre-sized flake of Bi₂Sr₂CaCu₂O_8+x on the end of our tungsten tip (see Fig. 2a and Methods section ‘Characterizing Bi₂Sr₂CaCuO_8+x nano-flake STM tips’). This immediately converts our measured normal–insulator–superconductor (NIS) tunnelling spectrum (dashed line in Fig. 2b) to a superconductor–insulator–superconductor (SIS) spectrum (solid line in Fig. 2b) exhibiting an energy separation of about 2(Δ_S + Δ_T) between conductance peaks (Methods section ‘Characterizing Bi₂Sr₂CaCuO_8+x nano-flake STM tips’). Figure 2b demonstrates that our d-wave high-temperature superconducting (HTS) tip has Δ_T ≈ 25 meV (double-headed red arrows), and its spatial resolution is evaluated to be 1 nm using a topographic image of the BiO surface of Bi₂Sr₂CaCu₂O_8+x (Fig. 2c) that was measured using this tip in the single-particle tunnelling regime (Methods section ‘Characterizing Bi₂Sr₂CaCuO_8+x nanoflake STM tips’). Next, we measure the I(V) characteristic of this tip as a function of decreasing tip distance from the bulk crystal surface, and thus of decreasing R_N. At T = 45 mK and fixed r , as the superconducting tip is moved forward in ~10-picometre steps the evolution of I(V) characteristics is as shown in Fig. 2d. As the Josephson I_J increases with diminishing distance, the maximum observable current increases as expected (see Methods section ‘Modelling phase diffusion Josephson dynamics for SJTM’ and ‘Characterizing Bi₂Sr₂CaCuO_8+x nanoflake STM tips’).

Figure 2: d-wave HTS tip fabrication for SJTM.

a, Schematic STM tip with nanometre-sized Bi₂Sr₂CaCu₂O_8+x flake adhering. Inset, combined spectroscopic-imaging STM and Josephson circuitry used (R_B = 10 MΩ). b, Conversion of single-particle NIS (dashed) to single-particle SIS (solid) spectra when Bi₂Sr₂CaCu₂O_8+x nano-flake adheres to tungsten tip. The tip gap Δ_T ≈ 25 meV is the difference between the NIS and SIS peaks (red double-headed arrows). c, Topographic image T( r ) (76 nm × 76 nm) using single-particle SIS tunnelling with the same tip as in a and b. Spatial resolution is ~1 nm. Absence of moiré patterns in its Fourier transform (see Methods section ‘Characterizing Bi₂Sr₂CaCu₂O_8+x nano-flake STM tips’) indicates that the nanoflake’s crystal axes (superconducting order parameter) are aligned with the bulk crystal axes (superconducting order parameter). d, Measured evolution of the SJTM I(V) with diminishing tip–sample distance (diminishing R_N) at T = 45 mK (junction formation conditions given in blue text). The maximum current I_c for a typical I(V) is indicated by a dashed green arrow.

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To test the Cooper-pair condensate visualization capabilities of this I_c( r ) imaging technique directly, we use two approaches. First, when Zn impurity atoms are substituted at the Cu sites, muon spin rotation studies show that the Cooper-pair condensate is completely suppressed in a radius of a few nanometres around each Zn atom, in a “Swiss cheese” configuration²⁵. Figure 3a shows a 76 nm × 76 nm image of SIS single-particle tunnelling conductance g( r , −20 meV) using the same tip. One can locate each Zn impurity atom by its scattering resonance peak, which is shifted by convolution with the superconducting tip spectrum from its resonance energy Ω = −1.5 meV to |E| ≈ |Ω| + |Δ_T| ≈ 25 meV (vertical red line in the inset to Fig. 3a). Thus, the Zn impurity atoms are at the dark spots in Fig. 3a. In Fig. 3b we show I_c( r ) measured at ~50 mK in the same field of view as Fig. 3a. This reveals how I_c( r ), and thus the Cooper-pair condensate, is suppressed within a radius R ≈ 3 nm of each Zn atom site, consistent with muon spin rotation studies²⁵. A second test is possible because of the crystal ‘supermodulation’ and the associated modulating superconductivity at wavevector Q_SM (ref. 26) (Fig. 1f). In Fig. 3c we plot the magnitude of the Fourier transform in grey and in orange, both measured simultaneously along the (1,1) direction through Q_SM. This directly demonstrates the capability to visualize the Cooper-pair density modulations that are induced by the Bi₂Sr₂CaCu₂O_8+x crystal supermodulation. Overall, these tests show that nanometre-resolution Cooper-pair condensate visualization is achievable using this d-wave high-gap tip, combined with our millikelvin-operating-temperature SJTM approach.

Figure 3: Cooper-pair condensate visualization using SJTM.

a, The convolution of Ω = −1.5 meV Zn impurity resonance with the d-wave spectrum of the tip shifts the resonance to |E| ≈ |Ω| + |Δ_T|. The measured SIS g( r , E = −20 meV) 76 nm × 76 nm image near this energy (vertical red line) identifies the site of each Zn. b, A 76 nm × 76 nm I_c( r ) image measured at 45 mK in the same field of view as a. A deep minimum (~95% suppression) occurs in I_c( r ) surrounding each Zn site where the Cooper-pair condensate is suppressed²⁵. c, Simultaneously measured magnitude of and along the CuO₂ (1,1) direction. The primary peaks in and from the supermodulation coincide, demonstrating the ability to visualize Cooper-pair density modulations.

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Next, we apply this technique to search for a cuprate PDW in the same Bi₂Sr₂CaCu₂O_8+x samples with T_c = 88 K and hole density p = 17% and at T < 50 mK (see Methods section ‘Defining, measuring and imaging I_c( r )’). The two states already reported to coexist at this p value are the high-temperature superconductor and a CDW with Q = (0.22 ± 0.02, 0)2π/a₀ and (0, 0.22 ± 0.02)2π/a₀ (ref. 27). Figure 4a shows our measured I_c( r ) in the 35 nm × 35 nm field of view (blue dashed box in Fig. 3b). Clear I_c( r ) modulations are immediately observable and, given that from equation (2), they provide the magnitude and wavevector of the component modulations. The magnitude of the Fourier transform of I_c( r ),  , is shown in Fig. 4b, revealing that modulation in I_c( r ) occurs at the wavevectors Q_P = (0.25 ± 0.02, 0)2π/a₀ and (0, 0.25 ± 0.02)2π/a₀ (dashed red circles in Fig. 4b), while nothing is observable in the second harmonic . This situation seems to occur because the modulations are superposed on a much stronger spatially constant critical current (equation (2)). These data provide strong a priori evidence for the existence of a PDW coexisting with a robust Cooper-pair condensate in Bi₂Sr₂CaCu₂O_8+x. In Fig. 4c we show (as blue dots) the measured value of I_c( r ) along the blue line in Fig. 4a, while we show the amplitude and wavelength of the global I_c( r ) modulations (determined from the magnitude and central Q value of the peaks in Fig. 4b) as a red line. From these data, and in general from the magnitude of the peaks at Q_x and Q_y in Fig. 4b, we conclude that the modulation amplitude in Cooper-pair density is ~5% of its average value. Therefore the cuprate PDW at p = 17% is subdominant to the d-wave superconducting state below 50 mK.

Figure 4: Visualizing the Cooper-pair density wave in Bi₂Sr₂CaCu₂O_8+x.

a, A 35 nm × 35 nm field of view I_c( r ) image (dashed blue box in Fig. 3b) showing the I_c( r ) modulations parallel to the CuO₂ x and y axes (supermodulation-induced I_c( r ) modulations are removed). b, , the Fourier transform of I_c( r ) in a (yellow crosses at q = (π/a₀, 0) and (0, π/a₀)). Maxima from I_c( r ) modulations (dashed red circles) occur at Q_P = (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀, and no significant modulations occur in (see Methods section ‘Visualizing supermodulation-induced PDW and R_N( r )’). c, Measured I_c( r ) along the dashed blue line in a (blue dotted line); statistical error bars are the variance of I_c transverse to the dashed line. The red line shows the global amplitude and Q_P of the I_c( r ) modulations (from the circled maxima in b). d, Magnitude of (yellow crosses at q = (π/a₀, 0);(0, π/a₀)) revealing the d-symmetry form factor density wave modulations (dashed red circles) at Q_C = (0.22, 0)2π/a₀; (0, 0.22)2π/a₀ (same hole density as in b).

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Finally, in Fig. 4d we study , the oxygen-sublattice-phase-resolved image of d-symmetry form factor density modulations^28,29,30, from a Bi₂Sr₂CaCu₂O_8+x sample with the same hole density (see Methods section ‘Imaging d-symmetry CDW and avoiding the CDW setup effect in I_c( r )’). The conventional locations of d-symmetry form factor CDW peaks in (dashed red circles) occur at Q_C = (0.22 ± 0.04, 0)2π/a₀ and (0, 0.22 ± 0.04)2π/a₀. We note that in the unprocessed data the modulations in electronic structure and topography occur based upon the sum of both oxygen sublattices () and at wavevector Q = Q_Bragg ± Q_C (ref. 28). No response of the tip to these modulations could produce a spurious I_c( r ) modulation at the PDW wavevector Q_P = (0.25 ± 0.02, 0)2π/a₀ and (0, 0.25 ± 0.02)2π/a₀ (see Methods section ‘Imaging d-symmetry CDW and avoiding the CDW setup effect in I_c( r )’). Obviously, Fig. 4b and Fig. 4d are not identical, with the PDW exhibiting much narrower peaks and thus more spatial coherence. Nevertheless, this PDW wavevector (Fig. 4b, c) is not inconsistent (within joint error bars) with a conventionally defined wavevector of the cuprate CDW state at the same hole density²⁷. However, the observed PDW also exhibits a primarily s/s′-symmetry form factor because Fig. 4b is based on the conventional sum over sublattices, while the CDW state exhibits a primarily d-symmetry form factor, as detected in the difference of sublattice-phase-resolved images²⁸ (Fig. 4d).

The visualization of a Cooper-pair density wave, a long-term challenge in physics^1,2, has now been achieved using SJTM^22,23,24. The subdominant PDW observed in Bi₂Sr₂CaCu₂O_8+x at approximately the same wavevector as a CDW ( Q_P ≈ Q_C) and with s/s′ form factor (Fig. 4), is consistent with Ginzburg–Landau theory^5,11,13,14 in the case (among others) where a d-symmetry superconductor coexists with a d-symmetry form factor CDW (see Methods section ‘Analysing PDW within Ginzburg–Landau theory’). Microscopic theories for a cuprate PDW^{5,6,7,8,9,13,16} requiring Q_P = Q_C/2 cannot be fully tested here because of the observed disorder in at low q (Fig. 4b). However, microscopic models for the pseudogap phase involving the interplay of d-symmetry Cooper pairing and a d-symmetry form factor CDW do yield a PDW with Q_P ≈ Q_C (refs 18, 19, 20, 21) as in Fig. 4, although typically with a d-symmetry form factor. Moreover, the apparently lattice-commensurate value of Q_P, which would arise naturally in strong-coupling theories for density modulations in a doped Mott insulator, motivates further study. Finally, our HTS-tip millikelvin-SJTM approach to nanometre-resolution Cooper-pair condensate imaging also reveals a direct route to visualization of FFLO^1,2 or PDW^{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21} states in other cuprates, iron pnictides and unconventional superconductors.

Methods

Modelling phase diffusion Josephson dynamics for SJTM

The Josephson effect³¹ is first described by

the relationship of the Cooper-pair current I through the junction to the phase difference ϕ between the two electrodes, with I_J the Josephson critical current. Second,

relates the time evolution of that phase difference to the voltage across the junction. If the Josephson coupling energy  ≫ kT so that effects of thermal fluctuations on ϕ can be ignored, there exists a steady-state solution to equations (3) and (4) where V = 0 for I < I_J. At 50 mK, the condition corresponds to a Josephson critical current I_J ≈ 2 nA.

Because of our operating junction resistance R_N ≈ 10 MΩ, the maximum supercurrent used in the SJTM experiments described here was approximately 20 pA. Thus, our experimental junctions all lie in a regime where ϕ exhibits steady-state diffusion through the Josephson potential owing to thermal fluctuations^{32,33,34,35,36,37,38}. This results in a voltage dropped across the junction while the Cooper-pair current is flowing. We analyse the Josephson I(V) characteristics between our d-wave HTS tip and the Bi₂Sr₂CaCu₂O_8+x (BSCCO) surface using this classic phase-diffusion model for small junctions^32,33 within which the I(V) characteristic of the junction is given by

V_JJ is the voltage across the junction and with Z the impedance in series with the voltage source at the high frequencies relevant to repeated re-trapping of the diffusing phase. Importantly, I(V_JJ) is maximal at

where T^* is an effective temperature parameterizing how dissipative the phase evolution is. Spatially resolved measurements of I_c( r ) thus allow one to measure spatial modulations in (refs 22, 23, 24).

Extended Data Fig. 1a shows a diagram of the hybrid spectroscopic-imaging STM/SJTM measurement circuit used. A load resistor R_B = 10 MΩ is voltage-biased in series with the Josephson junction formed between the STM tip and the sample. Extended Data Fig. 1b shows the zero-frequency solutions to an electronic circuit model for this setup. The red and blue lines show the trajectories that the forwards and backwards sweeps take through the I(V) plane. The nonlinearity of the circuit introduces discontinuities in the measured I(V); a discontinuity occurs at the point where I = I_c and thus makes this current experimentally identifiable by the sharp I(V) feature associated with it. Extended Data Fig. 1c shows the same trajectories but as a function of the voltage across the junction V_JJ. This circuit’s nonlinearity also generates hysteresis in the I(V) characteristic even if the dynamics of the Josephson junction itself are over-damped at all frequencies. This hysteresis is indeed observed throughout our SJTM studies of BSCCO (Fig. 2d), and systematic errors due to the hysteresis are avoided during our I_c( r ) imaging experiments by sweeping the applied voltage always in the same direction once the Josephson junction has been formed at each location r .

Characterizing Bi₂Sr₂CaCu₂O_8+x nanoflake STM tips

To detect the gap modulations (equation (1)) requires single electron tunnelling from a non-superconducting electrode (NIS), while to observe the Josephson current modulations (equation (2)) requires Cooper-pair tunnelling from a superconducting electrode (SIS). For our SJTM studies, the high energy gap at the STM tip Δ_T is achieved by picking up a nanometre-sized flake of BSCCO onto the conventional tungsten microscope tip. Extended Data Fig. 1d shows the measured dI/dV spectrum of the tungsten tip before that process as a dashed line; it is precisely as expected for hole density p ≈ 17% BSCCO samples. Simulation of the expected dI/dV for tunnelling between a BSCCO nanoflake tip and sample (solid line) is in good agreement with the typical SIS spectrum observed.

In Extended Data Fig. 2 we compare the performance of two completely different BSCCO nanoflake tips. Extended Data Fig. 2a and b shows their measured dI/dV spectra; they are obviously very similar. Extended Data Fig. 2c and d demonstrates the ~1-nm spatial resolution achieved in topographic images using these two tips. Extended Data Fig. 2e and f shows the Fourier transform of these two topographic images, demonstrating that the supermodulation is easily detected in both cases and that no other unusual characteristics, such as additional q -space peaks or moiré patterns, are observed.

To check the relative orientation of the BSCCO nanoflake crystal axes to the axes of the sample, we evaluate simulations to see what might be expected of such a tip. Extended Data Fig. 3a and b shows a measured NIS topograph T( r ) of BSCCO taken with a conventional tungsten tip, and its Fourier transform. To simulate what would be observed if the crystal axes of a BSCCO nanoflake are oriented (or not) to those of the sample crystal axes we extract a typical ~2-nm-diameter patch from this image and then convolute the patch image with T( r ) at each location r . This results in the simulated topograph that is expected when using a BSCCO nanoflake tip with (or without) axes aligned to the crystal axes. If the axes of the patch are aligned (Extended Data Fig. 3c) to the crystal, the result is as in Extended Data Fig. 3e (see the Fourier transform in the inset). If the axes are misaligned (Extended Data Fig. 3d) with the crystal the result is as in Extended Data Fig. 3f (see the Fourier transform in the inset). As the vivid moiré effects in the Fourier transform of the latter are not observed in the actual experiments (Extended Data Fig. 2e and f), we conclude that the axes of the nanoflake are aligned with those of the BSCCO bulk crystal.

Specifications for HTS-tip, millikelvin SJTM for condensate visualization

The technical challenge of large field-of-view nanometre-resolution SJTM imaging is to measure, in a reasonable time (days), an array of I_c( r ) values with ~1% precision using typically 256 × 256 pixels. Four basic elements are required: (1) an ultralow-vibration dilution-refrigerator-based STM that is engineered for sub-picometre stability in taking spectroscopic data while out of feedback, and typically at T = 50 mK; (2) a high-temperature superconducting tip creation scheme which involves removing a nanometre-sized flake from the BSCCO surface and adhering it to the tip; (3) modification to the electronic circuitry to form a hybrid spectroscopic-imaging STM/SJTM operating condition (Fig. 2a) that allows conventional operation in the SIS configuration for topographic imaging and motion control, and bias voltage sweeps in the microvolt range when in Josephson mode; and (4) control of the capability to reliably and repeatedly carry out spectroscopic maps using a junction resistance of ~10 MΩ at every r and with V spanning tens of microvolts.

Defining, measuring and imaging I_c(r)

Extended Data Fig. 4a shows the histogram of I_c values measured in a typical field of view during these studies. By averaging over the individual measured I(V) from all the locations r contained within the bins indicated by the coloured arrows, we obtain the three representative Josephson I(V) measurements shown in Extended Data Fig. 4b. They span the range of I(V) characteristics measured at different locations. The coloured arrows in Extended Data Fig. 4b give examples where I_c is identified in a particular curve. In general we functionally define the quantity I_c( r ) reported in the main text to be the maximum magnitude of current achieved on sweeping away from V = 0. By comparison with Extended Data Fig. 1b, this current is the maximum Cooper-pair current, I_c, described by the phase diffusion model (see Methods section ‘Modelling phase diffusion Josephson dynamics for SJTM’) that at constant temperature and Z is directly proportional to .

Extended Data Fig. 5a, b and c shows typical examples of repeated I_c( r ) measurement experiments (with slightly different operating parameters but the same BSCCO nano-flake tip). The functionality and repeatability of this technique to achieve I_c( r ) imaging with nanometer resolution is clear.

The typical evolution of measured I(V) during SJTM imaging of I_c( r ) at 45 mK at a specific sequence of locations (numbered 1–9 in Extended Data Fig. 5b) is shown in Extended Data Fig. 6. The almost complete (~95%) suppression of I_c as this sequence passes through the site of a Zn atom is seen in panel 5 of Extended Data Fig. 6. These typical data demonstrate directly the unprocessed signal-to-noise ratio for measurement of I_c and how spatial evolution of unprocessed I(V) spectra yields the high fidelity seen in the I_c( r ) images.

In Extended Data Fig. 7a and b we show the topographic images taken with the same BSCCO nanoflake tip before and after a typical I_c( r ) mapping experiment. Comparison of the topographs shows them to be virtually identical and demonstrates that no significant changes occur in the geometry of the tip during the map. The implication is that the nanoflake at the end of the tip never once touches the surface, and that all such I_c( r ) studies reported are in the true SIS tunnelling limit and are very far from making any point contact.

Finally, our model is that the BSCCO tip has maximum gap Δ_T and negligible modulating component, while that of the sample Δ_S has two components (background component) and (PDW component) and with . The Josephson coupling then consists of a primary coupling that occurs between the uniform components, and a secondary coupling that is between the uniform component of the tip and the modulated component of the sample (as in equation (2)) with the empirical result of convolution presented in Fig. 4. To leading order I_c( r ) is then linear in the PDW component of both the sample gap and superfluid density.

Visualizing supermodulation-induced PDW and R_N(r)

In principle, SJTM allows direct measurement of modulations in the superconducting order parameter amplitude through the product I_JR_N. Here, I_J is the intrinsic, T = 0, Josephson critical current of the junction and R_N is a characteristic resistance of the single-particle tunnelling channel. In taking this product, variations that affect the matrix elements for tunnelling of Cooper pairs and the single-particle tunnelling conductance (1/R_N) in the same way cancel out.

We use dI/dV at a bias voltage V_B far higher than the SIS gap edge as an estimate of the ungapped, ‘normal state’ junction conductance G_N. Simultaneous measurement of this value and I_c( r ) were not possible because the former requires V_B in tens of millivolts while the latter requires V_B of a few tens of microvolts. Moreover, in the picometre range tip–sample separations used in measuring I_c( r ), the single-particle tunnelling current can become destructively large to our BSCCO nanoflake tip. Thus the following procedure was adopted:

• 1 Measure I_c( r ) with setpoint current I_s = I_s1 and bias voltage V_s = V_s1, as shown in Extended Data Fig. 8a.

• 2 In the same field of view measure g₁( r , E), defined as dI/dV for bias voltages V_s1 to −V_s1 with setpoint condition I_s = I_s1, V_s = V_s1 .

• 3 In the same field of view measure g₂( r , E) defined as dI/dV with setpoint condition I_s = I_s2, V_s = V_s2 for bias voltages V_s2 to −V_s2, where V_s2 ≫ V_s1 and V_s2 > Δ_SIS.

• 4 Over the range V = [−V_s1, V_s1] determine the coefficients α( r ) and β( r ), which scale g₂( r ) onto g₁( r ) via g₁( r , E) = α( r )g₂( r , E) + β( r ). In doing the scaling a small region of voltages near V = 0 where the Cooper-pair tunnelling (Josephson) signal exhibits is excluded.

• 5 R_N( r ) for junctions used to measure is then determined by 1/R_N( r ) = G_N( r ) = α( r )/2(g₂( r , +V_s2) + g₂( r , − V_s2)) + β( r ), as shown in Extended Data Fig. 8b.

Now we consider the case of the cuprate supermodulation PDW where the bulk quasi-periodic distortion of the Bi₂Sr₂CaCu₂O_8+x crystal at wavevector Q_SM leads to modulations of the measured I_c( r ) at the same wavevector superposed on the larger background.

where we allow for an arbitrary phase difference φ between modulation in I_J and R_N at Q_SM. For any value of φ, one may neglect the third term if

In the experiments presented in the main text we measure the quantity

as shown in Extended Data Fig. 8a. In Extended Data Fig. 8b we show R_N( r ) derived as above. By comparing the amplitude of and at q = Q_SM normalized to their q = 0 value one may directly determine as in Extended Data Fig. 9c; we find that this ratio is far less than 1 and hence modulations in at Q_SM are faithfully represented by those measured in I_c( r ).

Inset to Extended Data Fig. 8c is , showing directly that variations in the near and equivalent are negligible, so that measurement of I_c( r ) should and does yield a faithful image of the condensate.

Imaging d-symmetry CDW and avoiding the CDW setup effect in I_c(r)

The d-symmetry form factor density wave in underdoped cuprates (refs 39, 40, 41) has been directly detected by spectroscopic-imaging STM by analysing the differential conductance images in terms of which extracts the d-symmetry form factor density wave signature at the correct wavevector. At dopings near optimal, it is known from several experimental techniques that the intensity of the d-symmetry form factor density wave diminishes rapidly⁴². Therefore, to improve the visualization of the d-symmetry form factor density wave signature at such high dopings as used here, we use topographic information instead. In constant-current topographic imaging, the STM feedback system adjusts the tip–sample separation, T, as it scans over the sample surface to maintain a setpoint current, I_s, at a constant applied tip–sample bias V_s. The topographic image T( r ) is related to the tunnelling current

and assuming that the function , which represents the effect of corrugation, work function, and tunnelling matrix elements, takes the form

where κ is the tunnel-barrier-height factor and depends on the work functions of the sample and tip, and LDOS is the local density of electronic states. The recorded value of the relative tip–sample separation is then

Thus high signal-to-noise topographic images obtained by constant-current STM imaging reveal contributions from both the surface structure and variations in the .

While a single cuprate topographic image will contain the signature of the d-symmetry form factor density wave, subtracting one topographic image from another taken at opposite bias polarity can further enhance the cuprate CDW signal. This is because the cuprate d-symmetry form factor density wave modulates in anti-phase⁴¹ between empty and filled states and hence a subtraction of topographic images measured near the pseudogap energy (|V_s| ≈ 100 meV in equation (12)) formed with biases +V_s and −V_s should amplify the CDW contrast.

In Extended Data Fig. 9a and b we demonstrate this procedure using two high-resolution, high-signal-to-noise ratio topographs T( r ) taken at opposite bias polarities (100 mV and −100 mV) with the same conventional tungsten tip; they are spatially registered to each other with picometre precision⁴⁰. Extended Data Fig. 9c is the difference of these topographs, from which the two oxygen-sublattice-sampled images are then extracted, and calculated. An image of reveals a primarily d-symmetry form factor density wave because of the existence of four maxima at the CDW wavevector Q_C = (±0.22, 0)2π/a₀ and (0, ±0.22)2π/a₀ as in Fig. 4d (refs 39 and 41).

One important implication of the fact that the topographic/CDW modulations have d-symmetry form factor is for avoidance of the setup effect when measuring I_c( r ). Equation (12) shows that T( r ) is affected logarithmically by the term . This means the tip–surface distance will be modulated at the same Q as modulations in . One concern might be that, because the CDW modulations produce T( r ) (that is, the tip–sample distance) modulations at this Q , this will generate a spurious I_c( r ) modulation at the identical Q . This would be the infamous ‘setup effect’, but now affecting SJTM.

However, because the CDW in cuprates has a predominant d-symmetry form factor, any conventional image T( r ) and its Fourier transform that are formed by adding the three sublattice images exhibit the actual modulations at wavevector Q = Q_Bragg ± Q_C = (1 ± 0.22, 0)2π/a₀ and (0, 1 ± 0.22)2π/a₀. These only become detectable at the correct CDW wavevector Q_C when one uses a measure of d-symmetry form factor: . Therefore CDW modulations in the unprocessed topograph T( r ); occur at Q ≈ (0.78, 0)2π/a₀ and (0, 0.78)2π/a₀ (dashed red circles in the Extended Data Fig. 10a). Because they are at a completely different wavevector, it is impossible for such modulations to produce, through a setup effect, spurious I_c( r ) modulations at Q ≈ (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀. Therefore the PDW wavevector observed directly by SJTM, and shown in the Extended Data Fig. 10b and in Fig. 4b, is not generated spuriously by a systematic setup effect due to the coexisting CDW.

Analysing PDW within Ginzburg–Landau theory

PDW modulations may be induced by the coupling between a translationally invariant superconducting order parameter and a CDW. We focus on the case of a tetragonal system with q = 0 superconducting order of symmetry, a CDW order parameter with a purely d-symmetry form factor⁴³ and an induced secondary PDW order parameter with s/s′ symmetry form factor. The pairing amplitude is expanded in terms of superconducting order parameters via

where , and are the d- and s′-form factors that either do or do not change sign under 90° rotations. This expansion then defines five complex order parameters that enter the free energy: , , , and . and are required to be independent to implement the U(1) spatial phase degree of freedom because itself is complex. Q and are along symmetry-equivalent orthogonal directions. Under a 90° rotation the B_1g superconducting and s/s′ form factor PDW order parameters transform as and , , , .

In a similar way, the CDW order parameter is

This will contribute two independent complex order parameters to the free energy expansion. Since the CDW order parameter must be real . For purely d-symmetry form factor CDW, the order parameter will have the following property under a 90° rotation:, , , . Now consider the symmetry-allowed free energy terms that will couple , the values and the values. The leading term is simply:

Because the PDW order parameters appear to the power of unity in this term, the free energy can always be lowered by making non-zero, if both and are non-zero. Thus the co-existence of superconductivity and a d-symmetry form factor CDW at wave-vector Q will induce a PDW at wavevector Q with an s/s′ symmetry form factor.

We note that such a coupling is not unique. There exists a similar symmetry-allowed term in the free energy admitting a d-symmetry form factor PDW where the values now transform under 90 degree rotation, like the values. The associated free energy term will have a relative negative sign between the first two terms and the second two terms of equation (15). As the experimentally detected PDW has s/s′ form factor we do not discuss this further here.

Code availability

The data files for the results presented here and the codes used in their analysis are available at https://doi.org/10.17630/83957a84-2186-4c14-8e55-f961a19ec9a9.

Extended data figures and tables

Extended Data Figure 1 SJTM circuit model and phase diffusion dynamics.

a, Circuit diagram of the hybrid spectroscopic-imaging STM/SJTM setup used in these experiments. The voltage source V is from the usual STM bias controller; the load resistor R_B = 10 MΩ; the single-particle tunnelling resistance of the Josephson junction formed between the tip and sample is R_N, and the voltage actually developed across the junction is V_JJ. b, The dynamics of this circuit produces two predominant effects. First there is a sudden change in the I(V) characteristic measured with the external ammeter shown, when the current reaches a value where I_J is the zero-temperature Josephson critical current and T* is an effective temperature parameterizing the dissipative environment of the junction. The second predicted effect is strong hysteresis depending on which direction the external voltage is swept; this is shown as the difference between the solid-red and dashed-blue lines. Both effects are seen very clearly and universally in the measured I(V) throughout our studies reported here. c, The relationship of the current as in b but in terms of the voltage across the Josephson junction V_JJ. d, The dashed curve represents a typical dI/dV spectrum on as-grown BSCCO sample, measured using a normal metallic tungsten tip. The solid line is a simulation for an expected dI/dV spectrum when using a BSCCO superconducting tip; we use the standard equation for tunnelling between two superconductors, each with the density of states identical to the dashed line. The result, in very good agreement with the typical measured SIS spectrum, is shown in Fig. 2b.

Extended Data Figure 2 Spectroscopic/topographic data from two distinct BSCCO tips.

Typical dI/dV spectrum, topography and magnitude of its Fourier transform measured with two completely distinct BSCCO tips on two different BSCCO samples: tip 1 (a, c and e) and tip 2 (b, d and f).

Extended Data Figure 3 Simulation of topography with BSCCO nanoflake tip.

a, Surface topography T( r ) of BSCCO sample; image obtained with a conventional metallic tungsten tip. b, Magnitude of Fourier transform (FT) of a. c, Model for BSCCO nanoflake tip that is aligned with the BSCCO surface in a. d, Model for BSCCO nanoflake tip that is misaligned with the BSCCO surface in a. e, Convolution of BSCCO surface image in a and aligned BSCCO nanoflake tip in c. Inset shows resultant Fourier Transform with Bragg peaks in the corners and supermodulation peaks. f, Convolution of BSCCO surface image in a and misaligned BSCCO nanoflake tip in d. Inset shows resulting Fourier transform with many additional broad peaks caused by moiré pattern effects. Comparison to e demonstrates that the tip used in the studies reported here was well aligned with the sample.

Extended Data Figure 4 Spatial variation of HTS tip–sample Josephson junction I(V).

a, A histogram of all I_c values measured at different locations in the field of view shown in Fig. 3b. The characteristic I_c values associated with the three spectra in b are indicated by the coloured arrows. b, Three I(V) formed by averaging the constituent I(V) from the bins indicated by the coloured arrows in a. They demonstrate the variation in the measured I(V) characteristic that occurs at different locations in the field of view shown in Fig. 3b. The three I_c values are indicated by coloured arrows.

Extended Data Figure 5 Repeatable utility of BSCCO nanoflake tips for I_c( r ) mapping.

Three I_c( r ) images a, b and c measured with the same BSCCO nanoflake tip, at different times (separated by many days) and using different Josephson junction parameters, but closely related in fields of view. Repeatability and fidelity of our I_c( r ) imaging by SJTM is evident.

Extended Data Figure 6 Sequence of measured I_c( r ) along line in Extended Data Fig. 5b.

Nine individual I(V) spectra measured at the locations indicated by numbers 1–9 in Extended Data Fig. 5b. In each case the transition from the SIS resistive branch to the phase diffusion Josephson tunnelling branch is evident. Moreover, as the sequence passes through the site of a Zn atom, the value of I_c diminishes by ~95% from its maximum, as expected from muon spin rotation experiments. Bias, V_B.

Extended Data Figure 7 Before and after topographic images bracketing I_c( r ) map.

a, Topograph taken with BSCCO tip before typical I_c( r ) SJTM map. b, Topograph taken with same BSCCO tip after the same I_c( r ) SJTM map as a. Comparison of a and b shows that tip and surface are very well preserved in our SJTM protocol.

Extended Data Figure 8 Comparison of modulations in I_c( r ) and R_N( r ).

a, A typical measured I_c( r ) image of Bi₂Sr₂CaCu₂O_8+x with the crystal supermodulation effect retained and apparent as strong spatial modulations in I_c along the vertical axis. b, Measured R_N( r ) image simultaneous with a, with the crystal supermodulation effect retained. The spatial modulations in R_N( r ) along the vertical axis are greatly diminished in relative amplitude compared to I_c( r ) modulations in a. c, Inset shows , the magnitude of the Fourier transform of R_N( r ) from b. Plotting the simultaneously measured Fourier amplitudes of and along the (1, 1) direction passing through the wavevector of the supermodulation Q_SM shows that modulations in are negligible. Therefore the predominant effect in the I_c( r )R_N( r ) studied through this work is caused by the I_c( r ) variations, coming from the superfluid density variations of the condensate in the sample.

Extended Data Figure 9 d-symmetry density wave from topography.

a, High-resolution topographic image of typical BiO surface at the same hole-density as the I_c( r ) studies, measured at V = 100 meV. b, High-resolution topographic image of identical (registered for every atom within about 10 picometres) BiO surface as a measured at V = −100 meV. c, Difference between a and b; a CDW exhibits its signature logarithmically in such an image and therefore it can be used to detect the d-symmetry form factor density wave, as in Fig. 4d of the main text.

Extended Data Figure 10 Absence CDW setup effect in I_c( r ).

a, , the Fourier transform magnitude of the sublattice-resolved image O_x( r ) + O_y( r ) derived from δT( r ), the difference between the two unprocessed topographic images T( r, ±100 meV) in Extended Data Fig. 9. We see directly that the actual modulations in topography due to the density of states modulations from the CDW occur at wavevectors (1 ± 0.22, 0)2π/a₀ and (0, 1 ± 0.22)2π/a₀ (dashed circles), as has been reported extensively in the past. These only become detectable at the actual CDW wavevector Q_C = (0.22, 0)2π/a₀ and (0, 0.22)2π/a₀ when one uses a measure of d-symmetry form factor modulations: , as shown in Fig. 4d. Because the physically real modulations in topography and conductance imaging therefore occur at Q = (0.78, 0)2π/a₀ and (0, 0.78)2π/a₀ (dashed circles), it is impossible for them to produce, through a ‘setup effect’, spurious I_c( r ) modulations at the PDW wavevector Q_P ≈ (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀, as indicated by dashed circles in b. b, The measured q -space structure (which samples all sublattices in the conventional form. The PDW maxima occur at Q_P ≈ (0.25, 0)2π/a₀ and (0, 0.25)2π/a₀.

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