Antinodal kink in the band dispersion of electron-doped cuprate La2−xCexCuO4±δ

Angle-resolved photoemission spectroscopy (ARPES) measurements have established the phenomenon of kink in band dispersion of high-Tc cuprate superconductors. However, systematic studies of the kink in electron-doped cuprates are still lacking experimentally. We performed in situ ARPES measurements on La2−xCexCuO4±δ (LCCO) thin films over a wide electron doping (n) range from 0.05 to 0.23. While the nodal kink is nearly invisible, an antinodal kink around 45 meV, surviving above 200 K, is observed for n ~ 0.05–0.19, whose position is roughly independent of doping. The fact that the antinodal kink observed at high temperatures and in the highly overdoped region favors the phonon mechanism with contributions from the Cu–O bond-stretching mode and the out-of-plane oxygen buckling mode.


INTRODUCTION
The kink phenomenon in cuprates, as a sudden change in the slope of band dispersion revealed by angle-resolved photoemission spectroscopy (ARPES), remains unsolved and intriguing [1][2][3] . The kink is also indirectly revealed in inelastic neutron scattering (INS), inelastic X-ray scattering (IXS), Raman, and optical measurements as an anomaly in the density of states or intensities [4][5][6][7] . However, despite decades of extensive studies, the origin of kink remains controversial, electron-boson coupling (EBC), spin fluctuations, and spin-charge separation have been suggested as possible causes 2,[8][9][10] . Among them, EBC is arguably the most preferred mechanism, with phonons, magnetic resonance modes, and polarons serving as potential candidates for the bosonic mode [11][12][13] . Since the electron-phonon coupling is generally regarded as the pairing glue in conventional superconductors, the bosonic mode referred from the kink in the cuprates is widely argued to play a significant role in the high-T c superconductivity.
In the most extensively studied cuprates Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO) and Bi 2 Sr 1.6 La 0.4 CuO 6+δ , ARPES data display a strong kink with the binding energy of~70 meV along the nodal direction and a weaker kink around~40 meV near the antinode 5,14,15 . While the antinodal kink is widely believed to be related to the magnetic resonance mode or the oxygen buckling mode (out-of-plane oxygen O2 moves parallel to the c-axis, B 1g mode) 5,12 , the nodal kink is more complicated 5,10,16,17 . In La 2−x Sr x CuO 4 , benefited from the large sample size, both ARPES and INS experiments can be performed. The nodal kink appears in LSCO at the metal-insulator transition, which is interpreted as the Cu-O bond-stretching mode (also known as the breathing mode, E u mode) or polaron mode 4,13,18 . In electron-doped cuprates Nd 1.85 Ce 0.15 CuO 4±δ (NCCO), Sm 1.85 Ce 0.15 CuO 4±δ (SCCO), and Eu 1.85 Ce 0.15 CuO 4±δ (ECCO), ARPES experiments suggest that the antinodal photo-hole couples to full-breathing phonons (q = [0.5, 0.5, 0], 70 meV), and the nodal photo-hole couples to halfbreathing phonons (q = [0.5, 0, 0], 50 meV) 19 . The significant phonon softening along the [100] direction in many cuprates is discovered by the INS measurements 20 . Although the kink seems universal in the hole-doped and electron-doped cuprates, systematic studies of doping and temperature dependence of the kink in electron-doped cuprates are still lacking. Especially, there are no ARPES results for the kink effect in La 2−x Ce x CuO 4±δ (LCCO), which has the largest rare-earth ion radius and the highest T c among the electron-doped cuprates. Motivated by this, we prepared high-quality LCCO films by the elaborate ozone/vacuum annealing method 21 , and conducted ARPES measurements to systematically study the kink effect in this material.

Antinodal kink
We plot the doping evolution of the band structures of LCCO in Fig. 1. Figure 1a, b displays the band dispersions near the node and antinode with the electron doping level (n) varies from n0 .05 to n~0.23, respectively. The black lines are the band dispersions extracted from the momentum distribution curves (MDCs) by Lorentzian fitting, and the red lines are the assumed linear bare bands. The kink displays itself as a discrepancy between those two lines. It is clear that the kink is indiscernible in the nodal region, but it is ever-existing near the antinode, except for the heavily overdoped sample with n~0.23. While in the holedoped LSCO, the kink is more conspicuous along the nodal direction 8 . To get a better view, we show the second derivative images of the band dispersions at n~0.084 in Fig. 1d. The dashed white lines indicate the antiferromagnetic (AF) zone boundary (AFZB), where the nodal band shows a band-folding behavior with no existence of kink (cut 1). While the AF gap locates at the AFZB at the near hotspot (cut 2), the kink position at the antinode (cut 3) is away from it, which resembles the case in NCCO 22 .
In general, ARPES probes the single-particle spectral function directly, and the many-body interactions exhibiting themselves as renormalization effects in band dispersion are included in the selfenergy. The peak position of the real part of self-energy (Re Σ) corresponds to the characteristic energy of the kink. We extract the antinodal band dispersions as shown in Fig. 2a and plot the Re Fig. 2b, where the curves are offset vertically for clarity. It is clear that except at n~0.23, all the samples indicate a kink at 40-50 meV, with the mean energy at~45 meV, as summarized in Fig. 2c. In the underdoped region below 0.11, the  kink shows energy softening with the increasing electron doping level, which may be related to the Coulomb screening of the increasing free charge carriers. The characteristic energy slightly increases in the overdoped region, possibly due to the less effective screening or the interaction with the O2 branch 20,23 . We shall clarify that the kink energy is the sum of the bosonic coupling energy and the superconducting gap energy, while the influence of the superconducting gap is neglected in our work [24][25][26] . The superconducting gap in the electron-doped cuprates is relatively small, which is below our energy resolution and thus will not cause any meaningful difference to our conclusion [27][28][29] .

Imaginary part
To gain more insight into the kink phenomena, we introduce the imaginary part of self-energy (Im Σ) along the antinodal direction for n~0.06, 0.11, and 0.14 in Fig. 3a. The dotted lines in Fig. 3a show a considerable slope change around 45 meV. Electron-boson coupling, electron-electron interactions, and impurity scattering are all identified mechanisms responsible for the finite lifetime of a photo-hole 30 . The electron-electron interaction is approximately linear above the kink energy, and we subtract the linear background to obtain the boson contribution (solid lines) 19,31 . As shown in Fig. 3b, by the Kramers-Kronig relation, the Re Σ ΚΚ has a peak at~45 meV, where the boson contribution to Im Σ has a steplike feature. The similar lineshape between the Re Σ and Re Σ ΚΚ further supports the existence of the antinodal kink. The rest of the dataset is shown in Supplementary Fig. 1.
There are several ways to estimate the EBC strength. For n~0.11, the band velocity is renormalized from v F 0 = 1.6 eVÅ −1 to v F = 0.95 eVÅ −1 . Conventionally, the EBC constant is obtained as λ ¼ v 0 F =v F À 1~0.7. One can also estimate the EBC strength from the real part of self-energy to be λ ¼ À dReΣ E ð Þ dE E¼EF~0 .9 at n~0.11 32 . According to the Eliashberg formula, the electron-boson spectral function α 2 F(ω, k) is obtained by the derivative of Im Σ, as seen in Supplementary Fig. 2 for n~0.11. The coupling strength λðϵ; kÞ ¼ 2 R ωmax 0 α 2 Fðω; k; ϵÞ=ωdω is calculated to be λ~0.7 for n~0.11, where ω max is the maximum phonon energy 32,33 . Therefore, we summarized that in the optimally doped LCCO, the EBC strength is about 0.8 ± 0.2. In the optimally doped NCCO, SCCO, and Bi2212 along the nodal direction, the coupling constant is estimated to be λ~0.8 19,34 .

Temperature dependence of the kink
To get more information on this kink effect, we performed temperature-dependent ARPES experiments in the antinodal region of LCCO and plotted Re Σ at different temperatures in Fig. 4a, b. The electron doping levels are estimated to be n~0.084 and 0.175, which lies in the underdoped and the highly overdoped region of the superconducting dome of LCCO, respectively 35 . For n~0.084, the peak positions of Re Σ stay at similar energies, which are represented as the red dots in Fig. 4c, and the kink survives at T = 200 K. Meanwhile, Re Σ shows a twopeak structure at around 30 meV and 70 meV at 120 K, indicating a possibility that the antinodal region may consist of two electronboson couplings. Within the commonly used Eliashberg-type electron-boson coupling model, there is a softening of the kink at high temperatures due to the finite temperature quantum statistics. While the softening is expected in order of a few meV, it is beyond the energy resolution in the current study 2,25,31,34 . The case is the same in the overdoped region at n~0.175. For quantitative analysis, we calculate the spectral weight of Re Σ by integrating over E F to E F −200 meV and show the results in Fig. 4d. This sum area includes all the disparity between the bare band and renormalized band above −200 meV. Extracted using a linear bare band, the sum area of Re Σ is a reasonable way to comparatively estimate the contribution of bosonic modes below 200 meV to the real part of self-energy. For comparison, we also extract the spectral weight of Re Σ in the nodal region of BSCCO and LSCO 8,10 . The result suggests that the antinodal renormalization in LCCO is comparable to the renormalization effect along the nodal direction in the hole-doped cuprates. The magnitude of Re Σ decreases slowly with increasing temperature, and its energy shows little doping dependence, resembling the behavior in the hole-doped cuprates 11,36,37 .

Comparison of the kinks for different cuprates
To advance our understanding of the antinodal kink in LCCO, we plot and compare the energy scales of the kinks in different cuprate systems in Fig. 5. In the hole-doped cuprate BSCCO, the nodal kink stays at~70 meV, and the antinodal region shows a kink feature at~40 meV 14 . It has been interpreted as the parallel and isotropic shifts of multiple kinks 26 . While the antinodal kink is not prominent, combined ARPES and INS data have shown a~70 meV in the nodal region of LSCO 4,18 . In electron-doped cuprates NCCO, SCCO, and ECCO, the 70 meV kink near the antinode is regarded as the Cu-O bond full-breathing mode 19 . Along the nodal direction, the 50 meV kink detected in NCCO and SCCO is related to the Cu-O bond half-breathing mode. In LCCO with the largest La 3+ ion radius among the lanthanide elements, besides the disappearance of the nodal kink, the antinodal kink alters to~45 meV, which is downshifted from the antinodal kink energy in other electron-doped cuprates at~70 meV.

DISCUSSION
We now try to understand the origin of this universal kink in the antinodal region in LCCO. There are several candidates with a possible origins. First of all, the widely debated magnetic resonance mode in the hole-doped cuprates is unlikely to explain the antinodal kink in LCCO. While the magnetic resonance mode in the hole-doped cuprates is around 40 meV 12,38 , it is found to bẽ 10 meV in the electron-doped cuprates by neutron scattering 39,40 , and this energy scale is much smaller than the observed kink energy at~45 meV.
Second, some previous work pointed out that the antinodal kink in the electron-doped cuprates may originate from the opening of the AF gap 3,41,42 . We rule out this assumption for the following reasons: (1) the momentum position where the renormalization happens is away from the AFZB, as seen in Fig.  1c; (2) in the overdoped region n~0.14-0.19 where the AF fluctuations have vanished, the kink effect remains strong. And this kink is robust at n~0.15 at 200 K, suggesting that the proximity to the AF gap is not the mechanism.
Next, we turn to the extensively studied phonon modes. By comparing the energy scales, we found that the typical energy of the B 1g mode is at~40 meV and the Cu-O bond-stretching mode is at~40-70 meV, which is close to the kink we observed  43 . Besides, the B 1g mode is Raman-active and has been confirmed in PCCO, NCCO, SCCO, and Gd 1.85 Ce 0.15 CuO 4±δ (GCCO) by Raman experiments 6 . A mode at 308.5 cm −1 (~38 meV) observed in LCCO is likely ascribed to the B 1g mode 44 . Moreover, the mode energy decreases continuously with increasing rare-earth radius, which is consistent with our data. All these results imply that the kink observed in the antinodal region in LCCO may be attributed to the B 1g and E u phonon modes.
However, there are still some concerns. Firstly, we suggest a phonon origin based on the features of the kink, for its existence in the highly overdoped region and high temperatures. But the specific phonon mode is an assumption considering the similar energy scales. More experiments like Raman, optical, and INS in LCCO are needed to provide confirming information, though the last one may be difficult to perform due to the thin-film form.
Moreover, the kink along the nodal direction is inconspicuous in LCCO and not prominent in NCCO/SCCO compared to that near the antinode. This behavior has a huge difference from the holedoped cuprates, where the nodal kink is strong and holds some anomalous characteristics. Recent works have proposed an idea by assigning the anomalously nodal kink in BSCCO to the spincharge separation 10,45 . It was suggested that quasiparticle excitations maintain coherence within the Fermi pocket, and electrons with higher binding energies than the energy of the kink decay into spinons and holons, causing the abrupt change of the bandwidth. At the antinodal region of LCCO, the decay of the quasiparticles is likely processed by phonon emission that causes the line broadening, while the nodal band dispersion remains sharp. Since the antinodal region seems conventional in LCCO, the nodal region may embody intrinsic physics, like the high-T c superconductivity and the electron-hole symmetry/asymmetry. Is there any relation between the dichotomy of the nodal and antinodal region in the hole-and electron-doped cuprates? Is the kink influenced by the distance to the cross point of AFZB and Fermi surface? What mechanisms may be concealed in the linearlike nodal band dispersion and the much smaller superconducting gap in LCCO? To answer these questions, further studies are needed.
In summary, we have observed a universal kink at~45 meV in the antinodal region of LCCO, the energy scale of which is roughly doping-independent and temperature-independent. We attribute the kink to the Cu-O bond full-breathing mode and the out-ofplane oxygen buckling mode by comparing the kink energy and considering the results in other electron-doped cuprates. It implies that phonon modes are universal in both hole-and electrondoped cuprates.

Samples and ARPES measurements
High-quality LCCO thin films were grown by pulsed laser deposition (PLD) with Ce concentrations x = 0.1, 0.15, and 0.19. The electron doping level, determined by oxygen and Ce stoichiometry, is estimated by fitting the Fermi surface volume using the tight-binding model. By the two-step ozone/vacuum annealing method described previously, we were able to measure ARPES spectra on LCCO films from n~0.05 to 0.23 21 . For n0 .05-0.19, we performed ARPES measurements with a Scienta R4000 analyzer and a VUV helium light source. He-IIα resonant line (40.8 eV) was used and the base pressure of the ARPES system is at~4 × 10 −11 torr. To ensure that the samples at all doping levels are in the normal state, all measurements were carried out at 30 K, unless otherwise specified. For n0 .23, ARPES spectrum was measured at the Dreamline beamline of the Shanghai Synchrotron Radiation Facility (SSRF) with a Scienta Omicron DA30L analyzer at~20 K, with the photon energy of 55 eV. The experiments were performed with an energy resolution of~30 meV.

DATA AVAILABILITY
Data are available from the corresponding author upon reasonable request.
Received: 26 October 2021; Accepted: 28 March 2022; Fig. 5 Comparison of the kink energies. The energy scales of the kinks along the nodal direction (empty triangle symbols) and near the antinodal region (solid square symbols) in the hole-and electron-doped cuprates. The dashed lines at 40 and 70 meV represent the characteristic energies of the kinks. In LCCO, the nodal kink is disappeared, but the antinode has a universal kink at4 5 meV (indicated as the blue square).