Abstract
The superconducting state is formed by the condensation of Cooper pairs and protected by the superconducting gap. The pairing interaction between the two electrons of a Cooper pair determines the gap function. Thus, it is pivotal to detect the gap structure for understanding the mechanism of superconductivity. In cuprate superconductors, it has been well established that the gap may have a dwave function. This gap function has an alternative sign change in the momentum space. It is however hard to visualize this sign change. Here we report the measurements of scanning tunneling spectroscopy in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} and conduct the analysis of phasereferenced quasiparticle interference (QPI). We see the seven basic scattering vectors that connect the octet ends of the bananashaped contour of Fermi surface. The phasereferenced QPI clearly visualizes the sign change of the dwave gap. Our results illustrate an effective way for determining the sign change of unconventional superconductors.
Introduction
In superconductors, the charge carriers are Cooper pairs which carry the elementary charge of 2e. This can be easily concluded from the measurements of the quantized flux Φ_{0} = h/2e = 2.07 × 10^{−15} Wb based on the Ginzburg–Landau theory. The core issue and also the most hard problem for understanding the mechanism of superconductivity is how the two conduction electrons are bound to each other to form a pair, the socalled Cooper pair. In the theory of BardeenCooperSchrieffer (BCS), it was predicted that the Cooper pairs in some superconductors, mainly in elementary metals and alloys, are formed by exchanging the virtue vibrations of the atomic lattice, namely phonons. The two electrons in the original paired state (k, −k) will be scattered to another paired state (k′, −k′). This scattering will lead to the attractive interaction V_{k,k′}, and the electron bound state will be formed with the help of suitable Coulomb screening. Condensation of these electron pairs will lead to superconductivity. This condensate is protected by an energy gap Δ(k) which prevents the breaking of Cooper pairs. Usually the gap is a momentum dependent function which is closely related to the pairing interaction function V_{k,k′}. For example, in above mentioned pair scattering picture, one can easily derive the function
Here, \(E\left( k \right) = \sqrt {\varepsilon ^2 + {\it{\Delta}} ^2(k)}\) with ε the kinetic energy of the quasiparticles counting from the Fermi energy E_{F}. If this pairing process can apply to other unconventional superconductors, the sign of the gap Δ(k) would change if the pairing interaction V_{k,k′} is positive.
For cuprate superconductors, it has been well documented that the gap has a dwave form Δ = Δ_{0}cos2θ. This basic form of the gap was first observed by experiments of angle resolved photoemission spectroscopy (ARPES) without sign signature^{1,2}, and later supported by many other experiments, such as thermal conductivity^{3}, specific heat^{4,5}, scanning tunneling microscopy (STM)^{6,7,8}, neutron scattering^{9,10}, and Raman scattering^{11}, etc. Although some of the techniques mentioned above may involve the sign change of the gap, such as the inelastic neutron scattering and STM measurements, they cannot tell how the gap sign changes explicitly along the Fermi surface in the momentum space.
In the model system of cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212), the low energy excitations have been measured very carefully by STM yielding the seven scattering vectors or spots in the Fourier transformed (FT) quasiparticle interference (QPI) pattern^{12,13}. These seven scattering vectors were explained very well with the Fermi arc picture as evidenced by ARPES measurements^{14}. Combining the magnetic effect on the QPI intensities of the different vectors, Hanaguri et al.^{15} showed the well consistency of a dwave gap, if assuming that the magnetic vortices behave like magnetic scattering centers and that leads to the enhancement (suppression) of the joint QPI intensity of two momenta (k_{1}, k_{2}) with the gaps of the same (opposite) signs. The sign problem of a dwave gap was also resolved by the experimental devices based on the Josephson effect^{16,17,18}. Therefore dwave superconducting gap has been well concluded in cuprate superconductors by using many experimental tools.
In this paper, we report the experiments and analysis based on the defectinduced bound state (DBS) QPI method^{19,20} on Bi2212 by concerning the referenced phases at positive and negative energies. Actually in conventional superconductor 2HNbSe_{2} with magnetic impurities, and antiphase feature of the bound states at positive and negative energies were found^{21}. The authors successfully interpreted this effect as the phase difference of the Shiba function at positive and negative biased energies. Our results illustrate the direct visualization of the sign change of the dwave gap in bulk Bi2212.
Results
Topography and tunneling spectra measured on Bi2212
Figure 1a shows a typical topographic image of the BiO surface of Bi2212 with an atomic resolution after cleavage. The supermodulations in the real space can be clearly witnessed on the surface with a period of about 2.54 nm. We can also recognize a square lattice structure with the lattice constant a_{0} about 3.8 Å, even with the appearance of the supermodulations. In Fig. 1b, we present a sequence of tunneling spectra measured along the arrowed line in Fig. 1a. The spectra show the Vshaped bottoms near zero bias, which reveals the intrinsic feature of the gap nodes in optimally doped Bi2212. One can also clearly see the spatial variation of the coherencepeak positions, being consistent with previous reports^{6,7,22}. We then chose one typical spectrum in Fig. 1b and plot it in Fig. 1c. The finite zerobias differential conductance implies the effect of the impurity scattering^{6,7,23} in the case of a nodal gap. We then use the Dynes model^{24} with a dwave gap function Δ(θ) = Δ_{0} cos2θ to fit the tunneling spectrum, and the fitting result is also shown in Fig. 1c by the red line. The resultant fitting yields the parameters of gap maximum Δ_{0} = 42 meV, and the scattering rate Γ = 4 meV. One can see that the fitting curve with the dwave gap function captures the major characteristics except for the feature of the coherence peak on the negativebias side. It should be noted that the gap maximum equals approximately to the coherencepeak energy in a dwave superconductor when the scattering rate is small. Following this conclusion, we carry a statistical analysis of the gap maximum values from the coherencepeak energies of about 200 spectra measured in our experiments on optimally doped Bi2212 samples, and the related histogram of the gap distribution is presented in Fig. 1d. One can see that the distribution of Δ_{0} behaves as a Gaussian shape with an averaged gap maximum \(\bar{\it{\Delta}} _0 = 43\,{\mathrm{meV}}\) obtained from the fitting.
Identification of scattering wavevectors in FTQPI pattern
Figure 2a shows a topographic image with the dimensions of 56 × 56 nm. In this area, we measured the QPI images g(r, E) at different energies, and then obtained the FTQPI pattern g(q, E) in qspace through the Fourier transform. Figure 2b shows a typical FTQPI pattern at –20 meV. Since we have done the lattice correction process by using the Lawler–Fujita algorithm^{25}, the four Bragg peaks are very sharp near the red crosses marked in Fig. 2b. Based on the octet model^{12}, the octet ends of the bananashaped contour of constant energy (CCE) has relatively higher density of states (DOS). Therefore, the major scattering will occur among these hot spots. As expected, we can clearly distinguish all the primary scattering wave vectors q_{i} (i = 1 to 7) indicated by dashed circles in the FTQPI pattern. Besides, two additional pairs of spots, which are indicated by solidline circles, are originated from the supermodulations^{12}. It should be noted that the q_{7} spots at the horizontal direction are not influenced by the supermodulations. Figure 2c shows a schematic plot of the contours at a particular energy below the superconducting gap maximum in a typical cuprate superconductor like Bi2212, and the DOS at the octet ends are set to be maximum. By assuming a certain width for the Fermi arcs in Fig. 2c, we then apply the selfcorrelation and show the simulated FTQPI result in Fig. 2d. One can see that the simulated patterns are similar to those in the experimental FTQPI pattern, which can help identify different primary scattering wave vectors. In a dwave superconductor, the superconducting gap changes its sign in kspace along the Fermi surfaces, and we use different background colors to show this signchange feature in Fig. 2c. With the aid of light pink and light blue colors for different gap signs, we can divide the scattering wave vectors into two groups, i.e., gapsignreversed scatterings (with scattering wave vectors of q_{2}, q_{3}, q_{6}, and q_{7}) and gapsignpreserved ones (q_{1}, q_{4}, and q_{5}).
Energy evolution for different scattering vectors
It is known that the Bogoliubov quasiparticles will be scattered by impurities and the induced standing waves will interfere with each other, giving rise to the Friedellike oscillations of local DOS (LDOS). Figure 3a–f present the QPI images measured at different energies. We can see clear standing waves in these QPI images, which are due to the scattering by some impurities or defects in the sample such as randomly distributed oxygen deficiencies^{23}. The longitudinal periodic modulations as shown in the topography of Fig. 2a mainly contribute the spots enclosed by the solidline circles along q_{y} in Fig. 2b. While the other spots in Fig.2b should be contributed by the scatterings between the octet ends of CCE. We then do the Fourier transform to QPI images and show them for three selected energies in Fig. 3g–i, and the measured QPI images together with the corresponding FTQPI patterns at other energies are shown in Supplementary Fig. 1. One can clearly recognize the scattering spots from the supermodulations and seven characteristic scattering channels in the FTQPI patterns. It should be noted that the coordinates of supermodulation spots in FTQPI patterns are nondispersive in energy, which can also be taken as a justification for the origin of the supermodulation. However, the qvectors of seven characteristic scattering spots seem to be dispersive, which can be naturally illustrated by the octet model^{12}.
To obtain the information of superconducting gaps, we then analyze the energy dispersions of the characteristic scattering wave vectors. The intensities of the spots corresponding to q_{1} and q_{7} scattering vectors are stronger than the spots corresponding to other scattering vectors, and the central coordinates of these spots can be obtained by fitting to a twodimensional Lorentzian function^{13}. In principle, these two scattering wave vectors can provide enough information to fix the coordinates of the octet ends of CCE at various energies within the superconducting gap maximum. The resultant positions of the ends of CCE are shown in the inset of Fig. 4a, and the data allow us to construct part of the Fermi surface. The solid curve in the inset is a fitting result to the positions of the ends of CCE by a circular arc. The curve, which is intentionally cut off by the dashed line^{26}, represents the contour of the Fermi surface of Bi2212 at this doping level. From the inset of Fig. 4a, we can also define the angle of these ends of CCE to the (0, π) to (π, π) direction. With the combination of the defined angle θ and the measured energies as the absolute values of superconducting gaps at the ends of CCE, we can obtain the angular dependent superconducting gap Δ(θ) along the Fermi surface and show it in Fig. 4a. The obtained Δ(θ) data are fitted by two different dwave gap functions, i.e., Δ_{1}(θ) = 39cos2θ meV and Δ_{2}(θ) = 45.6 (0.9cos2θ + 0.1cos6θ) meV, and the latter shows a better consistence with the experimental data. Therefore, the gap function in optimally doped Bi2212 may have a higher order in addition to the dwave component, which is consistent with previous report^{12}. Looking back to other scattering spots, the intensity of q_{4} scattering spot is very weak, thus the dispersion associated with q_{4} scattering spot is not shown here. The positions of q_{2} and q_{6} scatterings are equivalent with only the exchange of k_{x} and k_{y} coordinates. Thus, we focus only on five sets of characteristic scattering spots with the center coordinates obtained by the fittings to twodimensional Lorentzian functions. The obtained energy dispersions of these characteristic wave vectors are presented in Fig. 4b, being consistent with previous reports^{8,12,22}. The solid curves in Fig. 4b represent the predicted energy dispersions based on the Fermi surface in the inset of Fig. 4a and the gap function Δ_{2}(θ). One can see that the calculated curves agree well with the experimental data, which indicates the validity of the octet model and the dwave superconducting gap.
Theoretical approach of using the DBSQPI method
The PRQPI method was first theoretically proposed by Hirschfeld, Altenfeld, Eremin, and Mazin in a twogap superconductor^{27}, which was successfully applied to prove the gapsignchange in ironbased superconductors for a nonmagnetic impurity^{28,29}. The recently proposed DBSQPI method^{19,20} is designed for judging the gapsign issue in ironbased superconductor LiFeAs around a nonmagnetic impurity, and is also successfully used to confirm the gap signchange in (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se from our recent work^{30}. In this method, the QPI image g(r, E) is measured in an area with a nonmagnetic impurity sitting at the center of the field of view (FOV). The FTQPI data, which comes from the Fourier transform of g(r, E), are complex parameters containing the phase information, namely \(g\left( {{\mathbf{q}},E} \right) = \left {g\left( {{\mathbf{q}},E} \right)} \righte^{i\varphi _g\left( {{\mathbf{q}},E} \right)}\). Then the phasereferenced (PR) QPI signal can be extracted from the phase difference between positive and negative bound state energies, that is defined by
Here, g_{r}(q, +E) should be always positive by the definition, and g_{r}(q, −E) should be negative near the bound state energy for the scattering involving the signreversal gaps at k_{1} and k_{2} (q = k_{1} − k_{2}). That conclusion was drawn by the simulation for a nodeless superconductor when the gap changes its sign for different Fermi pockets in LiFeAs (refs. ^{19,20}). Our studies in (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se reveal that this method can also work for concentric twocircle like Fermi surfaces when the gaps on them have opposite signs^{30}.
In cuprates, the gap value varies continuously and gap nodes appear in the nodal direction (Γ − Υ) on the Fermi surface. The FTQPI patterns have already been calculated in several previous works on cuprate systems^{31,32,33,34,35,36,37}. However, it is very curious to check whether this phasereferenced DBSQPI method is still applicable in a dwave superconductor. Before checking, it seems that there are no bound state peaks on the tunneling spectrum in our present sample, but nevertheless the DBSQPI technique can still be applicable. In cuprates, the intrinsic nanoscale electronic disorders such as oxygen vacancies or crystal defects can act as the scattering centers, which will influence the tunneling spectra^{23}. According to the previous calculation^{38}, if the scattering potential is small, the bound state peaks are absent in a dwave superconductor. To further elaborate this issue, we do the theoretical calculations by a standard Tmatrix method^{31} with the details described in Method part. We use a dwave gap in the calculation, and the calculated angledependent superconducting gap and Fermi surface in Supplementary Fig. 2a are consistent with the experimental data. Supplementary Fig. 2b shows the tunneling spectra at an impurityfree area and on site of the nonmagnetic impurity with scattering potential V_{s} = 20 meV. One can see that there are no obvious ingap bound state peaks near zerobias; the nonmagnetic impurities only induce resonance state and produce a continuous change of DOS within the superconducting gap when the scattering potential is small. The further calculated spectra are shown in Supplementary Figs. 3 and 4 with different scattering potentials. One can see that the clear resonance peaks can be observed only when V_{s} is much larger than 100 meV. Otherwise, there are only asymmetric intensity change or some small humps within the gap. The simulation results following the DBSQPI method for a single impurity are shown in Supplementary Figs. 2–4. One can see that the PRQPI signal for the gapsignpreserved scatterings (q_{1}, q_{4}, and q_{5}) are all positive, while the signal for the gapsignreversed ones (q_{2}, q_{3}, q_{6}, and q_{7}) are all negative, which gives a sharp contrast. Concerning the gap sign change issue, the simulated results in a dwave superconductor are consistent with the situation in the s^{±} superconductor LiFeAs (refs. ^{19,20}), and this validates our calculation method.
MultiDBSQPI method applied on experimental data in Bi2212
As shown in the linescan spectra in Fig. 1b, it seems there are no sharp bound state peaks at energies below 30 meV, however, we do have observed the QPI images reflecting the seven characteristic spots. This tells that the standing waves on the QPI images are obviously induced by the widely distributed nonmagnetic impurities which lead to resonance states instead of sharp bound state peaks in Bi2212. For better illustrating this point, we have conducted several other linescan measurements in the sample and presented the results in Supplementary Fig. 5. One can see that, in all three linescan spectroscopies, although the tunneling spectra are relatively homogenous at low energies, small kinks or humps are however observed on many of the spectra at the energies from 10 to 30 meV. In addition, all the spectra show sizable magnitude of DOS near Fermi energy. This is certainly induced by pair breaking effect from impurity scattering. Combining the theoretical calculations and our experimental results, we believe that the resonance states do exist on the spectra and are induced by widely distributed nonmagnetic impurities with relatively weak scattering potentials. Thus, it is difficult to find the exact impurity locations. Given the absence of the bound state peaks, we still try to analyze the measured QPI data to obtain the information of gap sign by using the DBSQPI method.
Figure 5a–f show the PRQPI patterns at different negative energies. The PRQPI patterns at positive energies are not shown here, because all the signals should be positive without any extra phase information according to Eq. (2). At energies from −6 to −12 meV, the g_{r}(q,−E) signals of q_{1} and q_{7} are clear and easily recognized. When the energy is lowered down to below −15 meV, all the seven characteristic scattering spots become even more clear and can also be easily recognized. We then calculate the average value to the signals in the areas within the dashedcircles or ellipses in Fig. 5d–f, and the histograms of the average intensities per pixel corresponding to different scattering channels are shown in Fig. 5g–i. The signals corresponding to q_{1}, q_{4} and q_{5} spots are positive, while those corresponding to q_{2}, q_{3}, q_{6} and q_{7} spots are negative. According to the logic possessed by the DBSQPI method, we naturally argue that those k points in the momentum space connected by q_{1}, q_{4} or q_{5} have signpreserved superconducting gaps, and those connected by q_{2}, q_{3}, q_{6} or q_{7} have signreversed ones. This conclusion is consistent very well with our theoretical calculation above by using a dwave gap function. Since the theoretical model used for the Fermi surface is rather rough, it is thus reasonable to see different intensities of the corresponding spots of the PRQPI signal between the experimental data (Fig. 5g–i) and the calculated results (Supplementary Figs. 2–4), which will be further addressed below. In order to show the validity of this conclusion, we have done some control experiments on other two samples in three different areas, and the results are presented in Supplementary Figs. 6 and 7. One can see that the new results are well consistent with those presented in Fig. 5, and they show exactly the expected results for a dwave superconducting gap. It should be noted that the sign difference for corresponding scattering spots can be easily recognized at energies below 25 meV; however, the sign cannot be resolved when the energy is above 25 meV (Supplementary Fig. 6). The reason for this is that the characteristic scattering spots themselves become blurred at the energies above 25 meV, and only a central spot is left when the energy is near the superconducting gap (~40 meV). This seems to be a common feature in Bi2212 samples, which was also reported in previous studies^{8,13,22}. The reason is unclear yet, and it could be induced by the involvement of the antinodal region which is more associated with the pseudogap^{13,39}. This can also explain the spatial variation of the intensity and shape of the coherence peaks in Bi2212. An alternative picture for this spatial variation of coherence peaks would be the combination of disorder and electron–electron interactions as inferred in twodimensional electron gas system Pb/Si(111) monolayer film^{40}. This requires of course further verification.
Concerning the error bars on the PRQPI intensities, in Fig. 6 we illustrate the statistical results measured on three samples in four different areas at ±20 meV and add the error bars to the related figures. For the measurement in one field of view (FOV) as shown in Fig. 5g–i, it has no doubt for the signs of the PRQPI signal for q_{1} and q_{7} spots. Since the octet model is widely accepted in cuprates, our data for q_{1} and q_{7} undoubtedly show the sign change of a dwave model. For other spots with weaker intensities, such as q_{5} and q_{6}, we calculate the averaged intensity of the PRQPI signal within the two circles by gradually increasing the radius of the outer circle. One can see that the averaged value outside the inner circle (shown in Fig. 6c) is rather stable versus the calculated circle size, and we thus take the average value of those data as the error bar for this particular spot. In this way we did calculations for q_{5} and q_{6} spots. One can see that the sign keeps unchanged even considering the error bars. For four different FOVs measured at ±20 meV as shown in Fig. 5 and Supplementary Figs. 6 and 7, we take average of the intensities for each spot. The corresponding error bar is calculated through the standard deviation defined as \({\mathrm{\Delta }}I = \sqrt {\mathop {\sum}\nolimits_{i = 1}^4 {\left( {I_i  \bar I} \right)^2/4} }\), with \(\bar I\) the averaged intensity of one particular spot in four FOVs. The results are presented in Fig. 6d. It is clear that the error bars are all smaller than the intensities of the corresponding spots.
From above analyses we are confident that the obtained sign of the phasereferenced signal will not change for the related spots even considering the error bars. For some spots, such as q_{3} and q_{4}, the phasereferenced intensity are really very weak. The major reason for this is that the scattering intensities themselves are very weak as shown in Fig. 3. Such weak intensity at large qvector may arise from following three reasons. First, the STM tip is not infinitely sharp, then the highq signals with faster oscillations in real space cannot be completely resolved. Second, the spatial variation of the wave function (Wannier function) may further suppress the highq signals^{41}. Third, our measurements are done in areas with finite size, which leads to the weaker intensity for largeq scattering.
We should point out that there are some differences of the PRQPI intensity for different scattering spots between theoretical calculations and experimental data, although the signs are consistent each other. We realize that the theoretical calculation here can only serve as a qualitative interpretation. The shapes of the simulated FTQPI patterns in Supplementary Figs. 2–4 are clearly different from the measured results, for example, the sizes of the scattering spots cannot be well determined from the theoretical calculations since the FTQPI intensity shows some kind of continuing evolution in the momentum space. Actually, the Fermi surface in cuprates is very complex, and the theoretical model based on a single tightbinding band structure can only capture some features of the QPI results. Furthermore, the cuprate systems have strong correlation effect, and the Fermi arcs/pockets observed by experiment cannot be well understood by the band theory. Thus, it is reasonable to have discrepancy between experiment and theory concerning the intensity and the size of the FTQPI spots.
Discussion
Although consistency has been found between the experimental data and theoretical calculations by using the DBSQPI technique, however, one may argue that the original phasereferenced QPI method^{19,20} was specially designed for the case of a single impurity. In the case of Bi2212, there may be many randomly distributed weak impurities which prevent us from finding an area with a wellisolated impurity. As a result, we can only measure in a large area with multiple impurities with a complex pattern of standing waves in the conductance mappings as shown in Fig. 3a–f. Our primary concern is whether the phase message of the order parameter can be effectively extracted in the case of multiple impurities.
The initial theoretical work of the DBSQPI technique^{19} has actually provided a treatment for the multiimpurity system by applying the phase correction^{42} to the measured FTQPI pattern for multiple impurities g_{m}(q, E). The FTQPI for a single impurity can be derived as
Here \(C\left( {\mathbf{q}} \right) = \mathop {\sum}\nolimits_j {e^{  i{\mathbf{q}} \cdot {\mathbf{R}}_j}}\) is the correction term with R_{j} the location of the jth impurity. The subscripts s, m, and C represent the cases for single impurity, multiimpurity, and correction term, respectively. The φ_{m}(q, E) and φ_{c}(q) are the phases of g_{m}(q, E) and C(q), respectively. The prerequisites for using this formula are assuming identical scattering potentials for all impurities and no interaction among them. The PRQPI signal for a single impurity can be obtained by applying Eqs. (2) and (3) to the corrected FTQPI g_{m}(q, E), see Eq. (4). From the QPI images shown in Fig. 3, one can clearly see that there should be many weak impurities on the surface and it is very difficult to determine the exact coordinates of these impurities. Considering the fact that the correction factor \(C\left( {\mathbf{q}} \right) = \mathop {\sum}\nolimits_j {e^{  i{\mathbf{q}} \cdot {\mathbf{R}}_j}}\) is energy independent and yields a common phase shift for both positive and negative energy, the phase for single impurity after correction can be written as φ_{s}(q, ±E) = φ_{m}(q, ±E) − φ_{C}(q), and the phase difference φ_{s}(q, −E) − φ_{s}(q, +E) = φ_{m}(q, −E)−φ_{m}(q, +E) remains unchanged before and after correction. It means that the sign of PRQPI signal at the negative energy for multiple impurities will be exactly the same as the one for a single impurity.
For this issue we can also get support from the theoretical calculations. To illustrate that, we calculated the results for 60 impurities randomly distributed on the surface with the same scattering potential V_{s} = 20 meV, and also for 60, 100, and 500 randomly distributed impurities with the random scattering potential values from 0 to 100 meV. The results are shown in Supplementary Fig. 8. We want to emphasize that we have already taken the interference between impurities into account in the calculation, in other words the result of the multiimpurity is not the linear summation of the effect arising from individual impurities. Our simulation results show that the PRQPI signal for each particular scattering spot (Supplementary Fig. 8) under different multiimpurity situations mentioned above has the same sign but different value compared with the situation of the single impurity (Supplementary Figs. 2–4). We have repeated each kind of simulation for 100 times with different distributions of the impurities, and can obtain the same conclusion. Hence, this phasesensitive method for multiimpurities can provide us the same message as for a single impurity.
As presented above, we use the multiDBSQPI method to prove the superconducting gap reversal in optimally doped Bi2212, which is consistent very well with the dwave gap structure. However, with these results, one may argue that perhaps the signpreserved gap also gives rise to such changes of the phase difference between the positive and negative energies. It has been calculated and argued that, the scattering from a nonmagnetic impurity can barely induce an impurity bound state in an isotropicswave superconductor^{43}. If the gap is highly anisotropic, or say its value varies in a wide range, there may be some impurity induced bound states even if the superconducting gap is nodeless. We thus carry out further simulations for the superconductors with different gap functions, for examples, a nodal but signpreserved gap \({\it{\Delta}} \left( {\mathbf{k}} \right) = 23\,{\mathrm{cos}}\,k_x  {\mathrm{cos}}\,k_y \,({\mathrm{meV}})\) and a nodeless gap \({\it{\Delta}}\left( {\mathbf{k}} \right) = 23\left {{\mathrm{cos}}\,k_x  {\mathrm{cos}}\,k_y} \right + 2 \, ({\mathrm{meV}})\). The same scattering scalar potential V_{s} = 20 meV is used for the nonmagnetic impurity. The resultant tunneling spectra for above two gaps are shown in Supplementary Figs. 9a and 10a, respectively. One can see that the nonmagnetic impurities only slightly shift the position of the coherencepeaks, and have negligible influence on the LDOS near zero bias. We also calculate the PRQPI images for the single and multiimpurity situation with different forms of the signpreserved superconducting gaps mentioned above, and the related simulations are shown in Supplementary Figs. 9 and 10. One can see that the results for the multiimpurity situation are always similar to the case for a single impurity. Furthermore, all the PRQPI signals for seven characteristic scattering spots are positive in this case. Therefore, we exclude the possibility of the signpreserved gap in Bi2212.
Another argument concerning our conclusion may be that the impurities could be magnetic ones instead of nonmagnetic ones. The calculation for the magnetic scattering should be carried out and compared with experiment. This is actually not relevant for the optimally doped Bi2212 since, as far as we know, no magnetic impurities with even moderate scattering potentials have been reported in literatures^{44}. We further corroborate this point with two more basic arguments. (1) If the magnetic impurities with certain scattering potential are present, we would have seen some strong resonant state peaks within the gap. This has not been observed in such samples. (2) We have also done the calculations for superconductors with a dwave gap with magnetic impurities, and find that the PRQPI signals for seven characteristic scattering spots are of the same signs when the magnetic scattering potential (V_{m}) is smaller than 200 meV, see Supplementary Fig. 11. By using the DBSQPI method, we clearly prove the signchanging dwave pairing symmetry in optimally doped Bi2212. Our experiments and analyzing technique suggest that this method may also be applicable to other unconventional superconductors if the gap has a sign change. This will provide a more easily accessible way to determine the gap structure of unconventional superconductors.
Methods
Sample synthesis and characterization
Optimally doped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} single crystals were grown by the floatingzone technique^{45}. The quality of the sample has been checked by the DC magnetization measurement before the STM measurements. The critical temperature T_{c} is about 90 K as determined from the DC magnetization measurement.
STM/STS measurements
The STM/STS measurements were done in a scanning tunneling microscope (USM1300, Unisoku Co., Ltd.) with ultrahigh vacuum, low temperature, and high magnetic field. The Bi2212 samples were cleaved at room temperature in an ultrahigh vacuum with a base pressure of about 1 × 10^{−10} torr. The electrochemically etched tungsten tips or the Pt/Ir alloy tips were used for all the STM/STS measurements. A lockin technique was used for measuring tunneling spectrum with an ac modulation of 1.5 mV and 987.5 Hz. All the data were taken at 1.5 K. The QPI images were measured with the resolution of 256 pixels × 256 pixels. The setpoint condition was V_{set} = −100 mV, I_{set} = 100 pA. QPI images were measured individually for each pair of positive and negative energies ±E, and each scanning took about 2000 min. The FTQPI patterns were corrected by using the Lawler–Fujita algorithm^{25} to reduce the distortions from the nonorthogonality in the x/y axes. This correction can raise the signaltonoise ratio of FTQPI patterns and it should not give influence on the final determination of the sign of the PRQPI signal corresponding to each primary scattering spot. The presented FTQPI and PRQPI patterns were mirrorsymmetrized to reduce the noise. When treating the QPI data of sample 2, we have removed an unphysical background signal showing as two vertical lines appearing symmetrically around q_{x} = 0. This will not give influence on the final results of PRQPI.
Theoretical calculations
We have employed a single tightbinding band structure similar to the one proposed in the previous report^{46}, with the energy dispersion given by
The parameters (t_{1}, t_{2}, t_{3}, t_{4}, μ) = (100, 36, 10, 1.5, −155) are used in the calculations with the units of meV. Using a standard Tmatrix method^{31}, we simulate the LDOS around a single impurity and for the case of multiple impurities. In Nambu space, the BCS Hamiltonian is given by
Suppose that we have N impurities located at r_{1}, r_{2}, r_{3}, …, and r_{N} (Set N = 1 reduced to the case of a single impurity). The Green’s function in real space^{47} can be formulated by
where \(G_0\left( {{\mathbf{r}},{\mathbf{r}}\prime ,E} \right) = \frac{1}{M}\mathop {\sum}\nolimits_{\boldsymbol{k}} {G_0\left( {{\boldsymbol{k}},E} \right)e^{i{\boldsymbol{k}} \cdot ({\mathbf{r}}  {\mathbf{r}}\prime )}}\) with M the numbers of unit cells, G_{0}(k, E) the unperturbed Green’s function in reciprocal space and the manyimpurity 2N × 2N Tmatrix is determined by
where
Here, V_{i} = V_{s}τ_{3} + V_{m}τ_{0}, with V_{s} the scalar potential, V_{m} the magnetic scattering potential. V_{m} = 0 for nonmagnetic impurities. Then we can obtain the spinsummed LDOS given by
where τ_{i} is the Pauli matrix spanning Nambu space. Referring to Eqs. (2) and (3), we can get the simulated PRQPI images as shown in Supplementary Figs. 2–4 and 8–11. For the case of multiple impurities, we randomly assign the impurities to the lattice sites in our simulated FOV either with the identical scattering potential or with the randomly distributed scattering potentials within 100 meV.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge useful discussions with Tetsuo Hanaguri and Dunghai Lee. We thank Zhen Ke for the programming of lattice correlation by using the Lawler–Fujita algorithm. The work in China was supported by National Key R&D Program of China (grant number: 2016YFA0300401), National Natural Science Foundation of China (NSFC) with the project numbers 11534005, 11574134. Work at Brookhaven was supported by the Office of Basic Energy Sciences (BES), Division of Materials Science and Engineering, U.S. Department of Energy (DOE), through Contract No. desc0012704. R.D.Z. and J.S.W. were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by BES.
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The lowtemperature STM/STS measurements and analysis were performed by Q.G., S.Y.W., Z.Y.D., H.Y. and H.H.W. The samples were grown by G.G., R.D.Z. and J.S.W. Q.G., H.Y. and H.H.W. contributed to the writing of the paper. The theoretical calculation was done by Q.T., Q.G. and Q.H.W., H.Y. and H.H.W. are responsible for the final text. H.H.W. coordinated the whole work. All authors have discussed the results and the interpretations.
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Gu, Q., Wan, S., Tang, Q. et al. Directly visualizing the sign change of dwave superconducting gap in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} by phasereferenced quasiparticle interference. Nat Commun 10, 1603 (2019). https://doi.org/10.1038/s41467019093405
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DOI: https://doi.org/10.1038/s41467019093405
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