Abstract
The interactions between electrons and antiferromagnetic magnons (AFMMs) are important for a large class of correlated materials. For example, they are the most plausible pairing glues in high-temperature superconductors, such as cuprates and iron-based superconductors. However, unlike electron-phonon interactions (EPIs), clear-cut observations regarding how electron-AFMM interactions (EAIs) affect the band structure are still lacking. Consequently, critical information on the EAIs, such as its strength and doping dependence, remains elusive. Here we directly observe that EAIs induce a kink structure in the band dispersion of Ba1−xKxMn2As2, and subsequently unveil several key characteristics of EAIs. We found that the coupling constant of EAIs can be as large as 5.4, and it shows strong doping dependence and temperature dependence, all in stark contrast to the behaviors of EPIs. The colossal renormalization of electron bands by EAIs enhances the density of states at Fermi energy, which is likely driving the emergent ferromagnetic state in Ba1−xKxMn2As2 through a Stoner-like mechanism with mixed itinerant-local character. Our results expand the current knowledge of EAIs, which may facilitate the further understanding of many correlated materials where EAIs play a critical role.
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Introduction
Electron–boson interactions belong to the most fundamental microscopic processes in solids, which are responsible for various fascinating properties. For example, electron–phonon interactions (EPIs) could result in conventional superconductivity or charge density waves1,2, whereas the high-temperature superconductivity in cuprate and iron-based superconductors is proposed to be related to the interactions between electrons and antiferromagnetic (AFM) spin fluctuations, i.e., AFM magnons (AFMM)3,4,5. The electron–boson interactions "dress" the electrons up, and convert the electrons into quasiparticles. Theoretically, the effects of electron–boson interaction can be described by a complex self-energy of the electronic structure6,7, which is determined by both the electron–boson matrix element g (i.e., interaction potential) and the material-specific bosonic structure8. Experimentally, the band structure is renormalized, and sometimes an abrupt distortion, i.e., a kink, in the otherwise smooth band dispersion can be detected by angle resolved photoemission spectroscopy (ARPES)6,7. Based on the ARPES data, one can extract the electron self-energy and the electron–boson coupling constant λ, which characterizes the total renormalization strength and directly gives Tc in BCS theory.
The kinks induced by electron–phonon interaction have been detected in various materials, while the coupling constant λ is found to be less than 1 in most metals9,10,11. In contrast, the contribution of AFMM to the electron self-energy has not been experimentally identified, although electron-AFMM interactions (EAIs) holds the key to the general understanding of many correlated phenomena like unconventional superconductivity. Although kink features have been widely observed in the dispersion of many cuprate superconductors12,13,14,15,16,17,18,19, the origin has been controversial, as the energy scales of kinks are found in both the magnetic resonances20 and the phonon modes9,21,22. The presence of kinks in the heavily overdoped regime without many AFM fluctuations9,10,23,24 disfavors a magnon origin. Moreover, the energy scale of AFM spin fluctuations is comparable to that of the electronic dispersion near Fermi energy (EF) in cuprates or iron-based superconductors25,26,27, which would lead to a renormalization of the entire band rather than a kink near EF28,29. Therefore, one cannot isolate or identify the EAIs in the retrieved total self-energy.
To reveal the characteristics of the EAIs, one requires a compound that has both robust AFMM excitations and relatively large electron bandwidth. This dilemma condition is usually difficult to fulfil, while Ba1−xKxMn2As2, an isostructure material for 122-type Fe-based superconductors30, could be an ideal system. On one hand, BaMn2As2 is an AFM insulator (AFI) with a Néel temperature TN = 625 K. The magnetic moments of Mn2+ ions in BaMn2As2 point along the c axis, forming a G-type AFM order [Fig. 1b]. K doping turns Ba1−xKxMn2As2 into a metal [Fig. 1a], while the TN and magnetic moments persists31,32. Inelastic neutron scattering studies find strong AFMM excitations in heavily doped samples33. On the other hand, the electron bandwidth could be large in Ba1−xKxMn2As2 as observed in the parent compound BaMn2As234. At x > 0.19, it shows an emergent ferromagnetic (FM) ground state with the FM moments in the ab-plane contributed by As-4p orbital rather than the canting of Mn AFM moments32,35,36, while the underlying mechanism of the emergent FM is unknown. Here, by conducting ARPES studies on Ba1−xKxMn2As2, we directly observe a kink feature of clean EAIs origin. We subsequently unveil its doping-dependent and temperature dependent behaviors. Its renormalization on the electronic structure is unusually strong, which drives the emergent ferromagnetic state through a Stoner-like mechanism.
Results and discussions
Electronic structure of Ba1−xKxMn2As2
Figure 1d–f shows ARPES results measured using vacuum ultra-violet (VUV) photons, which resolve the Fermi surface structure of Ba0.7K0.3Mn2As2 in its three-dimensional Brillouin zone. In the Γ-X-M plane measured with 78 eV photons [Fig. 1e], the photoemission intensity mapping shows a large pocket α in rounded square shape centered at Γ. The α pocket is absent in the ZRA plane [Fig. 1f] and highly dispersive along kz, forming a drum-shaped Fermi surface [Fig. 1d]. The spectral intensity at zone center shows two-dimensional character and persists along kz [Fig. 1d–f], which is contributed by bands β and β’ whose band top locates near EF. Except for some variation in the spectral weight due to the photoemission matrix element, the Fermi surfaces measured using VUV photons are consistent with the soft X-ray ARPES results (Section 1 of Supplemental Materials), confirming the bulk nature of the measured bands. By calculating the Fermi surface volume in the three-dimensional Brillouin zone, we estimated the hole doping based on Luttinger theorem. The estimated hole doping agrees well with the K concentration at various doping levels (Section 2 of Supplemental Materials).
Along Γ-X, the hole-like bands α and β are resolved [Figs. 1g–i]. The \(\beta ^{\prime}\) band is relatively weaker but observable in the second-derivative spectra [Fig. 1h], which is also reproduced in calculations34. Both the Fermi surfaces and band dispersions of Ba0.7K0.3Mn2As2 roughly agree with the pristine BaMn2As2 after a chemical potential shift31,34. The bandwidth is in the energy scale of 1 eV, which is one order of magnitude larger than that in most Fe-based superconductors, such as the isostructural Ba1−xKxFe2As237. The large bandwidth of Ba1−xKxMn2As2 provide a clean playground for observing the kinks of electron–boson interactions and retrieving the self-energies.
Strong electron–magnon interactions in Ba1−xKxMn2As2
The α band shows a kink around the binding energy EB = 50 meV, as shown by the photoemission image and MDCs [Fig. 2a–c, and Section 4 of Supplementary Materials]. The bands β and \(\beta ^{\prime}\) also show similar features (Section 5 of Supplemental Materials). The kink in electronic dispersion is a general signature of electron–boson interactions7. At the kink feature of band α, the Fermi velocity vF is renormalized from the velocity of large-scale dispersion \({v}_{F}^{0}\) by a factor of ~6.1 [Fig. 2b]. The corresponding electron–boson coupling constant reaches \(\lambda={v}_{F}^{0}/{v}_{F}-1\simeq 5.1\). The coupling constant of the kink in Ba0.7K0.3Mn2As2 is colossal and larger than those of most of the reported kinks [Fig. 2f]11,38,39,40,41,42,43,44,45,46,47, except those near the anti-nodal region of cuprates that are enhanced by superconductivity18,19.
To retrieve the self-energy of band α, we conducted numerical fittings on the MDCs of Fig. 2a using Lorentzian peaks and a constant background (see Section 6 of Supplemental Materials for details). The momentum location of Fig. 2a is set slightly off the zone center to avoid the influence from \(\beta ^{\prime}\) band to the fittings. The real part of the self-energy contributed by the kink-related bosonic modes is obtained by Re Σb(E, k) = E(k) − E0(k), where E0(k) is a parabolic estimation of the "bare band" dispersion. The imaginary part of the self-energy is obtained by \(|{{{{{\rm{Im}}}}}}\,{\Sigma }^{{{{{{\rm{raw}}}}}}}(E)|=|{v}_{F}^{0}|\times {{{{{\rm{FWHM}}}}}}/2\), where FWHM is the full-width at half-maximum from fitting the MDCs. Corresponding to the kink energy of Ba0.7K0.3Mn2As2, the Re Σb peaks at ~50 meV [Fig. 2d], where the imaginary part of the self-energy Im Σraw also shows a step [Fig. 2e], indicating the energy scale of the kink-related bosonic modes. After subtracting Im Σraw by a background (Im Σothers) that accounts for extrinsic broadening and other effects of electron correlations (Section 6 of Supplemental Materials,48,49), we obtain the Im Σb that is contributed by the kink-related bosonic modes. The Im Σb matches well with the Kramers-Kronig (KK) transformation of Re Σb [Im \({\Sigma }_{KK}^{b}\) in Fig. 2d]. Vice versa, the KK transformation of Im Σb [Re \({\Sigma }_{KK}^{b}\) in Fig. 2c] also matches well with Re Σb. The KK conjugation between the Re Σb from dispersion and Im Σb from FWHM helps to confine the fitting processes and supports the self-consistency of the extracted self-energies.
The conventional Landau quasiparticle picture requires Im Σ < < EB near EF, while in Ba0.7K0.3Mn2As2 the Im Σb rises quickly below EF and exceeds EB [Fig. 2e]. The large magnitude of self-energy comes from the strength of the electron–boson interactions and indicates a deviation from conventional quasiparticle picture. Besides, although the simulated ARPES spectra based on the quasiparticle spectral function reproduces the kink feature (Section 7 of Supplemental Materials), it does not give the high-energy spectral weight at EB > 100 meV. The high-energy spectral weight increases at higher EB [Fig. 2a, b], which is opposite to the expectation for the high-energy increasing of Im Σraw in a conventional quasiparticle picture. The deviations from conventional quasiparticle picture also exist in the ARPES spectra of cuprates50 and manganites51. Similar to the method for cuprates with large coupling constant18, the self-energy analysis here is a semi-conventional quasiparticle-like approach. The high-energy spectral weight that deviates from conventional quasiparticle picture could relate with a polaronic metallic state induced by strong correlations with bosons like phonons or magnetic excitations51,52.
The obtained Im Σb and Re Σb consistently indicate a bosonic energy scale that extends up to ~100 meV and peaks at ~50 meV [Fig. 2d, e]. Based on the extracted self-energy, we can roughly estimate the Eliashberg function α2F(ω) of the kink by \({\alpha }^{2}F(\omega )=\frac{\partial {{{{{\rm{Im}}}}}}\Sigma (E)}{\pi \partial E}{|}_{E=\omega }\) (see Section 6 of Supplemental Materials for details), which characterizes the corresponding bosonic density of states (DOS) weighted by their effective interactions with electrons7. As shown in Fig. 2f, the energy scale of α2F(ω) is distinct from the phonon energies below 25 meV according to the inelastic neutron scattering (INS) on Ba1−xKxMn2As2 (x = 0.25)33, and the phonon spectra calculated for Ba1−xKxMn2As2 (x = 0 and 0.5) (Section 11 of Supplemental Materials,53,54,55,56). EPI could exist below 25 meV, while it is not relevant to the kink at 50 meV. On the other hand, the energy scale of α2F(ω) matches that of magnon spectra of Ba0.75K0.25Mn2As2 [Fig. 2f], which has been resolved by the INS magnetic intensity and reproduced by Heisenberg-model-based simulations (Section 12 of Supplemental Materials,33,57). After considering the momentum restriction of single-magnon absorption/emission processes, the simulated α2F(ω) also exhibits the same energy scale (Section 13 of Supplemental Materials,33). The consistent energy scale provides compelling evidence that the kink feature in Ba1−xKxMn2As2 is due to the strong interactions between itinerant electrons and AFMMs.
Kinks are present at band α for dopings x = 0.1, 0.2, and 0.3 of Ba1−xKxMn2As2 [Fig. 3a–d]. Since there is no ferromagnetic transition at doping levels lower than x = 0.19, it further supports that the kink originates from AFM magnons rather than FM magnons. The large-scale band dispersion of α is hole-like near EF with respect to the zone center and becomes relatively steep at ~EB = 0.3–0.7 eV [Fig. 3e]. When the doping increases, the dispersion of band α shifts with chemical potential in a rigid band manner (Section 9 of Supplemental Materials). Though the steep dispersion below the kink of x = 0.3 [Fig. 3c] is reminiscent of the "waterfall" feature in cuprates58,59,60, its doping-dependent energy scale in Ba1−xKxMn2As2 is distinct from the fixed energy scale of "waterfall" features at various dopings59,60,61. As the chemical potential lowers across the hole-like dispersion [Fig. 3f], the "bare band" velocity \({v}_{F}^{0}\) below the kink increases from 2.68 ± 0.11 eV ⋅ Å at x = 0.1 to 7.41 ± 0.43 eV ⋅ Å at x = 0.3. The coupling constant is obtained for each doping based on \(\lambda={v}_{F}^{0}/{v}_{F}-1\), which increases with doping by approximately three times from x = 0.1–0.3 [Fig. 3g]. The Fermi velocity vF is nearly doping independent [Fig. 3a–d], which is likely a coincidence due to that both the \({v}_{F}^{0}\) and λ increase with doping. The energy of kink feature gradually shifts towards lower energy with increasing doping [Fig. 3a–c and g], which is consistent with the doping-dependent energy shift of magnons as revealed by both the INS magnetic intensity and the simulated magnon DOS (Section 13 of Supplemental Materials,33,57).
The emergent ferromagnetic ground state
As the K doping increases, a ferromagnetic state emerges in Ba1−xKxMn2As2 at dopings x > 0.19 [Fig. 1a and Refs. 32,35], which is proposed to be itinerant from the spin polarization of As-4p holes35,36. When Ba0.7K0.3Mn2As2 enters ferromagnetic phase with decreasing temperature, the quasiparticle-like weight of band α increases and becomes sharper [Fig. 4a]. The enhancing of quasiparticle-like weight across the Curie temperature is reported in many ferromagnetic systems of transition metal compounds62,63,64. On top of a temperature-independent incoherent weight that extends to near EF [inset of Fig. 4e], the quasiparticle-like weight increases continuously with decreasing temperature [Fig. 4e and g]. The major enhancement takes place at the ferromagnetic transition.
The kink feature persists above the Curie temperature in the spectra of Ba0.7K0.3Mn2As2 [Fig. 4a, b]. From 30 K to 120 K, the Fermi velocity vF increases from ~0.55 eV ⋅ Å to ~0.83 eV ⋅ Å, while the \({v}_{F}^{0}\) is almost fixed around 3.4 eV ⋅ Å, indicating larger coupling constant λ at lower temperatures. Based on \(\lambda={v}_{F}^{0}/{v}_{F}-1\), the EAIs coupling constant λ increases by ~70% from 120 to 30 K [Fig. 4f], while a prominent increase occurs between 100 and 65 K across the ferromagnetic transition. As summarized in Fig. 4f, g, both the quasiparticle-like weight and the EAI coupling constant λ follow the emergence of ferromagnetic moment with decreasing temperature. Meanwhile, the emergence of ferromagnetic state with doping is also accompanied by an enhancement of EAI coupling constant [Fig. 3g]. These results indicate an intimate relationship between the ferromagnetic state and the EAIs.
Across the Curie temperature, our ARPES data resolve no detectable exchange splitting in Ba0.7K0.3Mn2As2 [Fig. 4a]. Even if there is a finite exchange splitting obscured by extrinsic momentum broadening, the estimated upper limit of itinerant magnetic moment is less than 40% of that from magnetic susceptibility measurements (see Section 16 of Supplemental Materials for details). Furthermore, our DFT+U calculation, without considering the EAIs, gives diminishing exchange splitting at As-4p bands either (Section 15 of Supplemental Materials). As doping increases from x = 0.1 to x = 0.3, the Fermi surface volume evolves as a normal metal rather than a half-metal (Section 2 of Supplemental Materials). These results indicates that the ferromagnetism in Ba1−xKxMn2As2 is in mixed itinerant and localized character.
The itinerant holes are likely playing critical roles in the ferromagnetic transition, considering the coincident enhancement of EAIs and ferromagnetic transition with either doping or cooling. The EAIs in Ba1−xKxMn2As2 could influence the ferromagnetic transition through enhancing the DOS at EF, noted as N(EF). Here we estimate the N(EF) directly by its definition \(N({E}_{{{{{{\rm{F}}}}}}})=\frac{\pi }{4}\int \frac{dS}{4{\pi }^{3}}\frac{1}{{v}_{F}}\), where the integration goes over the Fermi surfaces (Section 10 of Supplemental Materials). Note that this estimation only involves the size of Fermi surfaces and the Fermi velocities without considering the experimental photoemission spectral weight. Based on the v\({}_{F}^{0}\) without the renormalization by EAIs, the calculated "bare" DOS N0(EF) is ~0.3 spin−1eV−1As−1. This is similar to previous DFT results that does not meet the Stoner criteria65. However, after considering the renormalization by EAIs, the N(EF) estimated from renormalized vF is greatly enhanced with respect to N0(EF). The N(EF) further increases with doping or cooling [Fig. 5], following the doping dependence and temperature dependence of the coupling constant λ. It becomes most enhanced in the ferromagnetic regime, reaching 1.52 ± 0.32 spin−1eV−1As−1 at x = 0.3, which is nearly three times of that estimated by DFT calculations at an even higher doping x = 0.465. As summarized in Fig. 5, whenever N(EF) is increased roughly to ~1.2 spin−1eV−1As−1 either by doping or cooling, the broad ferromagnetic transition takes place. These results suggest that a Stoner-like mechanism is at work for driving the ferromagnetic ground state through As-4p holes. The As-4p states could obtain a localized character through the Mn-As hybridization that projects the the As-4p orbital to the flat bands at high binding energies34, while the itinerant As-4p holes near EF are strongly renormalized by the EAIs to reach the DOS of Stoner criteria and give rise to the ferromagnetism in Ba1−xKxMn2As2 with mixed itinerant and localized character.
The EAI is one of the fundamental interactions in condensed matter. Our ARPES study on Ba1−xKxMn2As2 provides a clean observation of EAI-induced kink. Our findings illustrate quantitatively how the EAIs can renormalize the electron band and help to induce ferromagnetism in Ba1−xKxMn2As2, which reveals a unique pathway to realizing emergent ground states like ferromagnetism by the strong interaction between electrons and AFM order. It also offers an unprecedented opportunity to directly study the self-energy and Eliasgberg function α2F(ω) of EAIs, and our quantitative analysis reveals various important facts about the EAIs.
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The coupling constant λ of EAIs in Ba1−xKxMn2As2 (x = 0.3) can reach as high as 5.4, which is much higher than most other known electron–boson interactions in metals. The large coupling constant demonstrates that the effect of the EAIs can be overwhelmingly strong and could induce emergent ground states. The high spin S = 5/2 of Mn and relatively small bandwidth of magnons give rise to high magnon DOS, which could be responsible for the large coupling constant. Nevertheless, the underlying behavior of EAI should be universal and could be applied to other systems like cuprates and iron-based superconductors.
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The EAIs vary strongly with both temperature and doping, even if the AFM order is relatively robust. The EPIs usually decrease at higher carrier concentrations due to screening. In contrast, the EAIs in Ba1−xKxMn2As2 is enhanced by three times from x = 0.1–0.3 with increasing hole concentration. Based on simulations (Section 13 of Supplemental Materials), the increased scattering phase space at larger Fermi surfaces could explain the enhanced coupling constant at higher hole dopings.
These features of EAIs put constrains on the theories. The extracted electron self-energy and Eliashberg function α2F(ω) are critical for revealing the electron–magnon matrix element g and other microscopic characteristics of EAIs at a more quantitative level, which encourages future inelastic scattering studies on the magnon spectra with absolute value. As these characteristics of the EAIs could be general in other materials with AFM excitations, such as high-temperature superconductivity in cuprates and iron-based compounds, our results would facilitate further understanding of the related emergent phenomena.
Methods
The Ba1−xKxMn2As2 single crystals used in this study were grown by the flux method as described elsewhere32,66. The chemical composition and K doping level were determined by electron probe microanalysis (EPMA). The magnetic susceptibility was measured by Quantum Design Dynalcool system. VUV-ARPES experiments were performed at beamlines BL5-2 of the Stanford Synchrotron Radiation Light Source (SSRL), I05 of the Diamond Light Source, and BL4.0.3 of the Advanced Light Source (ALS). The energy and angular resolutions were set at 15 meV and 0.1°, respectively. Soft X-Ray ARPES experiments were performed at ADRESS of the Swiss Light Source67. The energy and angular resolutions were set at 80 meV and 0.1°, respectively. All samples were cleaved in situ and measured under a vacuum better than 8 × 10−11 mbar. The homogeneity of the cleaved surfaces was confirmed by ARPES, STM, STS, and EPMA (Sections 1–3 of Supplemental Materials and refs. 34,68,69). Each group of data for doping dependence or temperature dependence study was taken in the same beamline facility. Furthermore, the data taken at different beamlines shows consistent electronic structure and spectral features. For doping x = 0.3 and temperature T = 30 K, the data taken at SSRL, Diamond, and ALS give identical values of kF and vF within the experimental resolution. Further analysis gives the similar EAI coupling constant λ and density-of-staes N(EF).
Data availability
The relevant data supporting our key findings are available within the article and the Supplementary Information file. All raw data generated during our current study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We gratefully acknowledge the valuable discussion with Prof. J. Zhao, and experimental support of Dr. D. H. Lu, Dr. M. Hashimoto, Dr. Y. B. Huang, Dr. Z. Sun, Dr Z. T. Liu, and Dr. D. W. Shen. We thank the Diamond Light Source for time on beamline I05, the Advanced Light Source (U.S. DOE contract no. DE-AC02-05CH11231) for access to beamline 4.0.3, the Stanford Synchrotron Radiation Light Source for the access to beamline 5-2, and the Swiss Light Source for time on SX-ARPES endstation of beamline ADRESS. Some preliminary data were taken at National Synchrotron Radiation Laboratory (NSRL, China) and BL03U at Shanghai Synchrotron Radiation Facility. This work is supported in part by the National Natural Science Foundation of China (Grants No. 11888101, 12074074, and 11790312), the National Key R&D Program of the MOST of China (Grants No. 2017YFA0303004 and 2016YFA0300200), Project supported by Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01), and Shanghai Rising-Star Program (20QA1401400).
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T.Y., M.X., C.W., Q.Y., X.L., R.P., and H.X. collected the ARPES data. T.Y., H.X., R.P., and D.F. analyzed the ARPES data. W.Y. and T.Z. measured the STM. Y.S. measured the EPMA. W.L. and X.W. conducted the phonon and electronic band calculations. H.X. performed the simulations. J.B. and G.C. provided the single-crystal samples. P.D., J.D., and V.S. assisted the experiments at beamlines. T.Y., H.X., R.P., and D.F. wrote the paper. H.X., R.P., and D.F. are responsible for the infrastructure, project direction, and planning. All authors have discussed the results and the interpretation.
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Yu, T.L., Xu, M., Yang, W.T. et al. Strong band renormalization and emergent ferromagnetism induced by electron-antiferromagnetic-magnon coupling. Nat Commun 13, 6560 (2022). https://doi.org/10.1038/s41467-022-34254-0
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DOI: https://doi.org/10.1038/s41467-022-34254-0
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