Introduction

Since the discovery of high-temperature superconductors (HTSCs) and angle-resolved photoemission spectroscopy (ARPES) has been playing a pivotal role in the research towards the exact mechanisms of this phenomenon1. These studies are based on the assumption that the ARPES spectra directly represent the one-electron spectral function A(ω, k), essentially neglecting matrix element (ME) effects. However, the latter can dramatically depend on energy and k, distorting the signature of the true A(ω, k). Throughout the immensity of ARPES studies, in particular on cuprate HTSCs, most works interpreted the data in terms of A(ω, k) without paying much attention to the ME effects. Another question is related to the extreme surface sensitivity of ARPES, whose probing depth in the conventional VUV-ARPES range of photon energy hv below ~100 eV hardly exceeds 3–5 Å. A still disputed point is whether the SC gap, observed by VUV-ARPES in this thin surface layer, truly represents the intrinsic superconducting (SC) gap in the bulk. The probing depth of the ARPES experiment can be enhanced by pushing it to the soft-X-ray energy range of few hundreds of eV. In the context of SC materials, soft-X-ray ARPES (SX-ARPES, for a recent review see ref. 2) has previously been used to disentangle bulk from surface states in pnictide superconductors3 and, more recently, to probe the three-dimensional (3D) electronic structure of overdoped La2−xSrxCuO4 (LSCO)4,5 and of Bi2Sr2CuO6+x6. However, no SX-ARPES studies have so far addressed the SC gap.

Here, we resolve these two fundamental questions regarding the ME and depth-sensitivity effects on the manifestations of superconductivity in ARPES for the paradigm HTSC cuprate Bi2Sr2CaCu2O8+x (Bi2212). First, we investigate the ME effects by measuring the SC gap through a dense grid of k-space locations, naturally created by the incommensurate lattice superstructure (LSS) in Bi2212. Second, we assess the surface vs bulk dependence of the SC gap from its measurements at different emission angles. Third, we analyze the excitation-energy dependence of the coherent spectral intensity in terms of its connection to electron–phonon interaction or the LSS coherence. Our SX-ARPES experiments used the paradigm HTSC system Bi2212 because of the relatively large magnitude of its SC gap compared to the experimental energy resolution our hv range. Our work puts forward a perspective and methodology for similar verification experiments on other HTSC materials, clearing the way to understand the true mechanisms of their high-Tc superconductivity without overinterpretation of the ARPES data.

Results

Matrix element effects

We start our analysis of the ME effects in Bi2212 from a recap of its crystallographic structure, whose unit cell is shown in Fig. 1a. A peculiarity of Bi2212 is the setting up of the incommensurate LSS along the a-axis with a period λ = 26 Å ~4.8 a0 (refs 7,8) as schematized in Fig. 1b. The corresponding cation displacements, projected onto the ab-plane, form replicas of the fundamental reciprocal vectors along one-dimensional lines in k-space (Fig. 1d).

Fig. 1: Sketch of Bi2212’s lattice structure.
figure 1

a The upper half of the Bi2212 unit cell. b A sketch of the cation displacement projected onto the ac-plane. c The cation displacements projected onto the ab-plane. d The superstructure in k-space. The conventional basis vectors are shown. The red line indicates the measurement direction of our ARPES experiments.

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This LSS manifests itself in the ARPES spectra as multiple replicas of the fundamental band structure and Fermi surface (FS) displaced by QLSS ~ (1/5,1/5)9,10. First of all, we will investigate the intensity of these replicas as a function of hv. Figure 2a–c shows the experimental in-plane FS measured through the first Brillouin zone (BZ) at hv = 205, 314, and 404 eV. We observe that the prominence of the replicas relative to the fundamental FS contours is much enhanced compared to the VUV-ARPES spectra10, and further increases as a function of hv in our soft-X-ray energy range. As the probing depth of the ARPES experiment gradually increases with hv, this observation manifests an increase of the LSS-induced atomic displacements away from the sample surface (see the “Discussion” later). Interestingly, our data show only the main bands and their replicas at QLSS without any noticeable signatures of the shadow bands displaced from the main ones by Q = (1/2,1/2) (ref. 10).

Fig. 2: Experimental in-plane FS in the first BZ.
figure 2

a–c The FS measured at hv = 205, 314, and 404 eV, respectively. The blue dashed contours indicate the fundamental FS. d–f Their corresponding ARPES intensity along XΓX cut (nodal line). The blue arrows indicate the fundamental band dispersions. The replica contours form a dense grid in k-space. Their intensity scales up compared to the fundamental contours, manifesting an increase of the LSS-induced atomic displacements into the sample bulk.

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The LSS provides a natural grid for dense sampling of k-space in the ARPES experiment. Indeed, within the one-step photoemission theory (see, for example, refs 11,12) the ARPES intensity is found as \(I_{\mathbf{K}}^{\mathrm{PE}} \propto {\left\langle f \right|} {\mathbf{A}\cdot\mathbf{P}} {\mathrm{|}i}\rangle|^2\), where 〈f| and |i〉 are the final- and initial-state wavefunctions coupled through the vector potential A of incident electromagnetic field and the momentum operator p. High excitation energies allow us to use the free-electron final state \(\left\langle {\left. f \right| = e^{i{\mathbf{Kr}}}} \right\rangle\), where K is the photoelectron momentum with the in-plane component K||. Neglecting for brevity the experimental geometry and X-ray polarization effects contained in A⋅p, the ARPES intensity for given K appears as the Fourier transform \(I_{\mathbf{K}}^{\mathrm{PE}} \propto \left| {\left\langle {{\mathrm{e}}^{i{\mathbf{Kr}}}{\mathrm{|}}i} \right\rangle } \right|^2\). Expanding |i〉 over the LSS’ in-plane reciprocal vectors g as \(|i\rangle = \sum_{\rm {g}} C_{{\mathbf{k}}_{\Vert} + {\mathbf{g}}}\exp [i({\mathbf{k}}_{\Vert} + {\mathbf{g}}){\mathbf{r}}]\), where k|| is the |i〉‘s in-plane momentum, we obtain \(I_{\mathbf{K}}^{\mathrm{PE}} \propto \left| {\mathop {\sum}\nolimits_{\mathbf{g}} {C_{{\mathbf{k}}_\parallel + {\mathbf{g}}}\left\langle {{\mathrm{e}}^{i{\mathbf{K}}_\parallel {\mathbf{r}}}{\mathrm{|e}}^{{\mathrm{i}}\left( {{\mathbf{k}}_\parallel + {\mathbf{g}}} \right){\mathbf{r}}}} \right\rangle } } \right|^2\). This expression shows, first, that the orthogonality relation \(\left\langle {{\mathrm{e}}^{i{\mathbf{K}}_\parallel {\mathbf{r}}}{\mathrm{|e}}^{i\left( {{\mathbf{k}}_\parallel + {\mathbf{g}}} \right){\mathbf{r}}}} \right\rangle = {\updelta}\left( {{\mathbf{K}}_\parallel ,{\mathbf{k}}_\parallel + {\mathbf{g}}} \right)\) leaves nonzero \(I_{\mathbf{K}}^{\mathrm{PE}}\) only when K|| matches a k|| + g harmonics of |i〉 and, second, that the corresponding ME is determined by the coefficient \(C_{{\mathbf{k}}_\parallel + {\mathbf{g}}}\) of this harmonic varying with g for a detailed evaluation including the out-of-plane dependence of 〈f| and |i〉 (see refs 13,14). In our case, this means that the ME effects will vary across the k-space grid defined by the LSS. Furthermore, in our case this analysis is facilitated by the use of higher hv engaging larger LSS-induced atomic displacements in the bulk of Bi2212 which boost intensity of the LSS-replicas. We will now leverage this idea to verify whether the ME effects, varying along these replicas, affect the SC gap apparent in the ARPES experiment.

Our SC gap analysis used the ARPES data acquired at 205 eV, where the coherent spectral component (see below) was the largest. Figure 3a shows the experimental band structure along the k-space cut #1, indicated in the FS map in Fig. 2a as #1. These k-points have been chosen to avoid the regions of hybridization between the main and replica bands, where the dispersions are distorted15, and keep the gap large enough in comparison with our experimental resolution. Figure 3b represents this band structure symmetrized relative to Fermi level (EF) as IPE(E, k) = IPE(−E, k). Whereas the strict hole–particle symmetry, implying a simultaneous inversion of k as A(ω, k−kF) = A(−ω, −k−kF), would in our case show confusing discontinuous between the regions centered at kF of each replica band, our simplistic symmetrization, albeit not physically rigorous, well serves visualization of the SC gaps through all fundamental and replica bands. These gaps all close below 105 K, as shown in the corresponding Fig. 3c, d. This fact confirms that the observed gaps are the SC ones rather than the pseudogaps16. Moreover, the band dispersions at low temperature show a clear bending, characteristic of the formation of the SC gap. Furthermore, Fig. 3e shows the symmetrized energy-distribution curves (EDCs) in the SC state, extracted from Fig. 3b at k1 through k10. The corresponding values of the SC gap were found by their fitting with the SC spectral function convolved with the energy resolution based on the simplified BCS self-energy \({\sum} {\left( {{\mathbf{k}},\omega } \right) = - i{\it{\Gamma }}_1 + \Delta ^2/\left[ {\omega + \varepsilon \left( {\mathbf{k}} \right) + i{\it{\Gamma }}_0} \right]}\), where Δ is the SC gap width, Γ1 is a single-particle scattering rate, ε(k) then dispersion (with ε(kF) = 0), and iΓ0 is viewed as the inverse pair lifetime, which should be zero in the SC state17. Through the whole grid of our k-space locations, we find ΔSC = 21 ± 3 meV and Γ1 = 34 ± 5 meV (Fig. 3g).

Fig. 3: ARPES data measured at hv = 205 eV.
figure 3

a ARPES intensity in the SC state along the cut #1 in Fig. 2b. b. The corresponding image symmetrized relative to EF, manifesting the SC gap. The dashed lines k1 through k10 indicate EF-crossings for the fundamental bands and their LSS-replicas. c, d The same measurements at 105 K where the SC gap is absent. e Symmetrized EDCs (red) for the SC state, extracted from panel b at the k-points k1 through k10 and their fitting (blue) with the SC spectral function. f The comparison of the EDCs at k1 through k10. g The magnitude of the SC gap (red) and Γ1 (blue) derived by fitting (error bars indicated). These SC gap parameters are the same within the experimental accuracy. The error bars are due to the experimental energy resolution.

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These data show that the SC gap parameters, observed through all k-space locations sampled by the LSS, are the same within the experimental accuracy regarding both the gap magnitude and single-particle scattering rate. Furthermore, our experimental gap magnitude is fully consistent with previous VUV-ARPES results16. As the ME effects must be different between different k-space locations, the observed constant SC gap gives us the sough-for proof of the absence of any significant ME-induced artefacts in its experimental determination by ARPES. We note that STM experiments for Bi2212 have found that the SC gap magnitude slightly (by ~9% of the mean value) differs between different atomic sites8. However, this observation does not contradict to our results where the SC gap is measured in k-space. Analysis of k-dependence of the high-energy dispersion anomalies—so-called waterfalls16,18,19,20—was in our case hindered by overlapping of their spectral signatures coming from different replicas. For the materials without the LSS, the idea to investigate the ME effects by measurements of the SC gap in equivalent k-points can be realized, albeit on a coarser k-grid, by going through different fundamental-lattice BZs. To the best of our knowledge, however, this idea has not yet been applied to any HTSC system.

Depth dependence of the SC gap

We will now address the question whether the SC gap magnitude is depth-dependent. If so, the surface-sensitive ARPES measurements might yield its values deviating from the bulk transport properties. We have accessed this effect by comparing the above ARPES data measured near the normal emission (NE) with those measured at more grazing emission angles.

The experimental off-NE data for the SC state, measured at hv = 205 eV around an emission angle of ~47° and corresponding to the (2, −2) BZ, are presented in Fig. 4. Similarly to the normal-emission data, the in-plane FS (Fig. 4a) shows the fundamental and replica contours. Correspondingly, the experimental band structure along the k-space cut, marked on the map as #2, shows the EF-crossings marked k1 through k6. The symmetrized EDCs at these points (Fig. 4c) and the results of their fitting (Fig. 4d) show that the SC gap decreases from ~32 meV at k1 to ~20 meV at k6. Following the fit with the (simplified) d-wave gap function21 Δk = Δmax cos (2θ), this decrease manifests the gap anisotropy along our inclined path in k-space. Indeed, the previous VUV-ARPES study16 has shown that in the near-optimal doped Bi2212 the SC gap decreases from its maximal value 35–40 meV, characterizing the pseudogap at the antinodal point, to zero at the nodal point. Our values of the SC gap, going from near the antinodal point (k1) to the middle between the nodal and antinodal point (k6), are consistent with this dependence measured at smaller probing depth. Specifically, the gap found at the k6-point is equal, within the experimental uncertainty, to that found in the equivalent k-points k1–k10 in our near-NE data in Fig. 3. This fact gives us the sought for evidence that the SC gap in Bi2212 is independent of the probing depth. This pattern may however change for other HTSCs, for example, for more 3D compounds from the LSCO4,5 and YBCO22 families.

Fig. 4: ARPES data measured at off-normal-emission angles around 47°, corresponding to the (2, −2) BZ.
figure 4

a Experimental in-plane FS measured at hv = 205 eV, with the blue dashed curve indicating the fundamental FS. b ARPES intensity along the cut marked as #2 in panel a. k1 through k6 indicate the EF-crossings. c Symmetrized EDCs at k1 through k6 (red) and their fitting with the SC spectral function (blue). d The magnitude of the SC gap derived by fitting with the error bars indicated. The SC gap decreases from 32 meV at k1 to 20 meV at k6, in accordance with the gap anisotropy. The experimental points are fitted with the d-wave gap function (black line). The arrow indicates k6 which is equivalent to k of the cut #1 through the first BZ (Fig. 3). The same SC gap magnitudes indicate the absence of its depth dependence. The error bars are due to the experimental energy resolution.

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Photon-energy-dependent ARPES response

Finally, we will analyze full hv-dependence of the ARPES spectral structure, presented in Fig. 5a as the out-of-plane FS map as a function of k// (along the direction marked on the in-plane FS map in Fig. 2) and hv increasing from 200 to 480 eV. The narrow-curved contours around 285 and 345 eV (marked in Fig. 5a) are due to the core level electrons excited by high-order radiation from the beamline. Intensity modulations as a function of hv may be connected with the photoemission cross section variations with the final state kz similar to the known intensity variations between the bonding and antibonding states23. The experimental out-of-plane FS map elucidates several important aspects of physics of Bi2212.

Fig. 5: Photon-energy-dependent ARPES response.
figure 5

a The out-of-plane FS map (k// along the ГX direction in the BZ). The FS contours are straight as a function of kz, confirming the 2D character of Bi2212. b Intensity ratio of the replica to fundamental FS contours as a function of hv. Its increase manifests an increase of the LSS-induced atomic \({\int} {I^{\mathrm{PE}}\left( {E,k_\parallel } \right)dk_\parallel }\) displacements into the sample depth. c Coherent spectral weight. Its fast decrease with hv and nearly vanishing above 460 eV suggests mechanisms beyond the conventional DW-picture. The inset illustrates the spectral intensity \({\int} {I^{\mathrm{PE}}\left( {E,k_\parallel } \right)dk_\parallel }\) (yellow) integrated over the area Ω (dashed yellow rectangle), the EDC (blue) at the blue dashed line and its corresponding Icoh component (red).

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First of all, we note that the probing depth of our experiment in the soft-X-ray energy range, which is of the order of 1 nm (ref. 24), is significantly higher compared to the previous VUV-ARPES studies. However, despite the proportional improvement of kz-resolution25, we observe only kz-independent FS contours. This observation extends not only to the ГX (nodal) direction in Fig. 5 but also to the ГM (antinodal) one, where the kz-dispersion might be more pronounced (not shown here for brevity). This fact confirms the purely two-dimensional (2D) character of Bi2212, contrasting it to other HTSC cuprates such as the LSCO4,5 and YBCO22 compounds.

Our second point extends the above analysis of the relative replica to fundamental spectral intensities. Their intensity ratio as a continuous function of hv, derived from the data in Fig. 5a, is presented in Fig. 5b. It shows a gradual increase with hv, which is consistent with previous works in the VUV region where the replicas appear in general weaker than the main band not only for Bi221210,23 but also for other cuprates such as Bi2Si2−xLaxCuO26. This phenomenon should be connected with the increase of photoelectron escape depth with energy. Indeed, the universal curve24 suggests that the escape depth increases from 3–5 Å in the VUV region to 5–10 Å towards 200 eV and further to about 15 Å towards 500 eV. Then the increase of the relative intensity of the replicas identifies an increase of the LSS-induced atomic displacements with the depth into the sample bulk, at least in the CuO-plane second from the surface. This observation is in agreement with the standing-wave photoemission experiments on Bi2212 (ref. 6). Whereas the absence of reliable data on the exact escape depth in Bi2212 has left as with merely a qualitative conclusion on the depth dependence of the atomic displacements, their quantitative evaluation is calling for grazing-incidence X-ray diffraction experiments.

The third and most intriguing point is the energy dependence of the coherent k-dependent spectral weight Icoh(E,k). It was evaluated for every ARPES image, constituting the map in Fig. 5a, in an energy window of 200 meV below EF as \(I^{{\mathrm{coh}}}\left( {E,k_\parallel } \right) = I\left( {E,k_\parallel } \right) - \alpha {\int} {I^{\mathrm{PE}}\left( {E,k_\parallel } \right)dk_\parallel }\), where the k||-integration window Ω around the Г-point is shown in the inset of Fig. 5c, and α is its maximum value leaving Icoh(E,k) > 0 throughout Ω (for robustness against the noise, within 95% in the histogram). The k||-integrated values of Icoh as a function of hv are presented in Fig. 5c. The oscillation of this dependence around 260 eV are attributed to the photoemission cross section variation discussed above. Most importantly, the experimental Icoh shows an unexpectedly fast decay of the coherent spectral weight with hv. This fact is actually seen already in Fig. 2 from the sequence of in-plane FS maps Fig. 2a–c measured with increasing hv. To the best of our knowledge, the observed decay of Icoh is the fastest among all materials investigated so far by SX-ARPES (see the recent review2 and the references therein) including the HTSC cuprates of the LSCO4,5 and YBCO22 families. We can rule out any significant influence of the sample aging on the observed decay as a function of hv, because it was the same when measured in a sequence going from low to high and from high to low hv, where the spectra stayed clear and sharp through the whole experimental round. Moreover, our preliminary measurements on overdoped Bi2212 samples have shown the same fast decay of the coherent spectral weight with hv.

In general, the decay of Icoh is determined by the relation of the de Broglie wavelength of the photoelectrons λB = h/p, where p is expressed through their mass m and kinetic energy Ek as p = (2mEk)1/2, to the thermal atomic-displacement amplitudes. When upon increase of hv or T these two become comparable, the dipole selection rules relax and the coherent Icoh decays2,27. This effect is conventionally described within the Debye–Waller (DW) model as \(I^{{\mathrm{coh}}} = W\left( T \right)I_{T = 0}^{{\mathrm{coh}}}\), where \(W\left( T \right) = {\mathrm{exp}}\left( { - {\mathrm{\Delta }}K^2\left\langle {u^2\left( T \right)} \right\rangle } \right)\) is the DW-factor with the momentum transfer \({\mathrm{\Delta }}K \propto \sqrt {h\nu }\) and mean-squared atomic displacements 〈u2〉, roughly proportional to T. In our data, however, Icoh decays with hv much faster compared to the DW-model with any reasonable <u2> at our working temperature T ~ 14 K. Even more intriguing, at hv = 205 eV Icoh well survives at least up to 105 K (Fig. 3c, d) despite the increase of hv to 460 eV and T to 105 K delivers comparable values of the DW-factor.

One of the sources of such dramatic deviations from the DW-model may be recoil effects28. Physically, the recoil can be viewed as an emission of phonons back into the crystal lattice by the escaping photoelectrons29, where the associated atomic displacements will erode Icoh similarly to the thermally excited phonons. This effect will progressively increase with hv but hardly depend on T, consistent with our experimental results. The prominence of such recoil effects might be related to the known strong electron–phonon interaction in Bi2212, albeit for the valence electrons30. Another scenario9 proposes to relate the anomalous decay of Icoh to the LSS’ incommensurability. We argue, however, that this phenomenon may be attributed rather to a small coherence length of the LSS. Indeed, STM studies8,9 have found them irregular, with the spatial phase slipping and meandering on a scale of few hundreds of Å. These atomic displacements will cut down Icoh as soon as they become comparable with λB. Indeed, whereas the former can be estimated as ±0.4Å 8,9, the quenching of Icoh at hv ~ 460 eV corresponds to a comparable value of ~0.6Å. The quenching is accelerated by the gradual increase of the SX-ARPES probing depth with energy, invoking larger LSS-induced atomic displacements in the deeper atomic layers of Bi2212.

Discussion

We have used SX-ARPES to investigate the SC gap magnitude in Bi2212 as apparent in the photoemission spectra under variation of the ME effects and of the probing depth. The former was varied by measurements of the SC gap over a dense grid of equivalent k-points formed by the LSS-replicas of the FS, taking into advantage strengthening of the LSS-induced atomic displacements into the bulk of Bi2212. The latter was varied by changing from a bulk-sensitive experimental geometry near the NE to a more surface-sensitive one at grazing emission angles. Our SX-ARPES experiments have demonstrated that the experimental SC gap is neither affected by the ME effects nor depends on the depth into the sample, and stays fully consistent with the previous VUV-ARPES results. These results clear up the still-existing doubts on the relevance of ARPES as a tool to explore the intrinsic mechanisms of high-Tc superconductivity in Bi2212. In this context, the observed k-dependence of the dispersion waterfalls16,18,19,20 in Bi2212 is a sign of their (at least partially) extrinsic origin related to the ME effects.

The applicability of ARPES to other HTSC compounds can be verified using the same methodological path. If the LSS is absent, the ME effects can be varied by measurements of the SC gap through different BZs. Furthermore, these effects can be reduced by measurements in the soft-X-ray energy range2 where free-electron character of the photoemission final states reduces the MEs essentially to atomic photoionization cross sections smoothly varying with energy. High-energy resolution in this energy range requires, however, most advanced synchrotron radiation beamlines. The depth dependence of the SC gap, whose absence for Bi2212 might be connected with the purely 2D character of this compound, can come into play for other more 3D HTSCs such as the cuprates from the LSCO4,5 and YBCO22 families. This point also needs verification before using ARPES data to draw valid conclusions about the bulk superconductivity in these materials. Our work puts forward a perspective and methodology for similar verification experiments, clearing the way to understanding the true mechanisms of their high-Tc superconductivity without overinterpretation of the ARPES data.

Our last and equally intriguing point is Bi2212’s decay of the coherent spectral weight with hv, which is much faster even compared to other cuprates. It remains to be explored whether such dramatic deviations from the DW-model hint at recoil effects, invoked by strong electron–phonon interaction in Bi2212, or at small coherence length of the incommensurate LSS.

Methods

High-quality near-optimal doped Bi2212 (Tc ~ 91 K) single crystals were grown by the floating zone method as described elsewhere9. SX-ARPES experiments were performed in the hv range 200–500 eV at the ADRESS beamline of the Swiss Light Source, Paul Scherrer Institute, Switzerland. The photoelectrons were detected with the analyzer PHOIBOS-150 from SPECS GmbH at an angular resolution of ~0.1o. The combined energy resolution was set to 30–40 meV. The EF position of the samples was referred to that of a gold foil in electrical contact with the sample. The FS maps were integrated within EF ± 20 meV. The experiment was performed in vacuum < 9 × 10–11 mbar at a base sample temperature of ~14 K. The samples were cleaved in situ after cooling down.