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Femtosecond electron-phonon lock-in by photoemission and x-ray free-electron laser

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Science  07 Jul 2017:
Vol. 357, Issue 6346, pp. 71-75
DOI: 10.1126/science.aak9946

A deeper look into iron selenide

In the past 10 years, iron-based superconductors have created more puzzles than they have helped resolve. Some of the most fundamental outstanding questions are how strong the interactions are and what the electron pairing mechanism is. Now two groups have made contributions toward resolving these questions in the intriguing compound iron selenide (FeSe) (see the Perspective by Lee). Gerber et al. used photoemission spectroscopy coupled with x-ray diffraction to find that FeSe has a very sizable electron-phonon interaction. Quasiparticle interference imaging helped Sprau et al. determine the shape of the superconducting gap and find that the electron pairing in FeSe is orbital-selective.

Science, this issue p. 71, p. 75; see also p. 32

Abstract

The interactions that lead to the emergence of superconductivity in iron-based materials remain a subject of debate. It has been suggested that electron-electron correlations enhance electron-phonon coupling in iron selenide (FeSe) and related pnictides, but direct experimental verification has been lacking. Here we show that the electron-phonon coupling strength in FeSe can be quantified by combining two time-domain experiments into a “coherent lock-in” measurement in the terahertz regime. X-ray diffraction tracks the light-induced femtosecond coherent lattice motion at a single phonon frequency, and photoemission monitors the subsequent coherent changes in the electronic band structure. Comparison with theory reveals a strong enhancement of the coupling strength in FeSe owing to correlation effects. Given that the electron-phonon coupling affects superconductivity exponentially, this enhancement highlights the importance of the cooperative interplay between electron-electron and electron-phonon interactions.

Many of the properties of complex materials, such as the iron-based pnictides and chalcogenides (1), arise from a coupling of charge, orbital, spin, and lattice degrees of freedom. For example, the cooperative interplay of electron-phonon (EP) and electron-electron interactions has been suspected to play an important role in unconventional superconductors (25), even though the physics is beyond the canonical Bardeen-Cooper-Schrieffer theory (6). Although the EP coupling strength can be inferred from various spectroscopies (7, 8), these techniques rely on nontrivial assumptions and modeling. Ultrafast techniques, especially femtosecond time- and angle-resolved photoemission spectroscopy (trARPES) (9) and time-resolved x-ray diffraction (trXRD) (10), bolstered by the advent of x-ray free-electron lasers (11), now open a window of opportunity for direct measurements of the EP coupling strength with sufficient precision to quantitatively test theories. We combined these two techniques to link electronic and lattice degrees of freedom and determine the EP coupling strength directly and purely from experiments.

The EP coupling strength can be quantified by the deformation potential, defined as the ratio of a band energy shift to the corresponding atomic displacement (12). Experimentally, atoms can be displaced by initiating a coherent phonon mode through photoexcitation of the electrons (13). The light-induced coherent dynamics of the crystal lattice and the electronic band energy can be directly measured by trXRD and trARPES, respectively. Focusing on the coherent lattice vibrations enables extraction of phonon mode–specific information with high precision. Conceptually, this is analogous to electronic lock-in measurements, where a weak electronic signal is extracted by locking-in to a reference signal at the same frequency. Similarly, the light-induced coherent phonon mode provides an internal reference for measuring the EP coupling. Locking-in on the phonon frequency avoids low-frequency contributions from other dynamical processes such as acoustic mode coupling or heat transport, which inevitably accompany optical excitation. Because optical phonons in complex materials are in the terahertz regime, our approach brings this technique to the natural time scales of atoms and electrons, thus enabling insights into microscopic processes.

This coherent lock-in technique is ideal for studying the role of EP interactions in complex materials. In particular, this applies to FeSe, which features strong correlation effects (14), as manifested by robust spin-fluctuations (1517), a tendency toward nematic order (1719), substantial orbital-dependent renormalization of the electron masses (19, 20), and a nonmonotonic pressure dependence of the superconducting transition temperature (2123). The capability to grow high-quality FeSe thin films by molecular beam epitaxy (24) enables advanced experiments.

We report an orbital-resolved coherent lock-in measurement of FeSe that allows us to quantify the EP coupling and assess the importance of electron correlations. Using a combination of trXRD and trARPES, we detect lattice displacements at the subpicometer level and band shifts at the millielectron volt level. Figure 1A is a schematic of the experiment, in which a bulk-like FeSe film (60 unit cells thick) grown on SrTiO3 is photoexcited by an ultrafast 1.5-eV infrared (IR) pump pulse. For trXRD, an 8.7-keV hard x-ray pulse tracks the photoinduced lattice dynamics at a variable time delay Δt; for trARPES, a 6-eV ultraviolet (UV) pulse records the band energy dynamics (25). Benefiting from the single-mode response in FeSe, we lock-in to the coherent A1g optical phonon (26), which corresponds to a periodic variation of the anion height (Fig. 1B, left inset) that has been shown to sensitively influence the electronic band structure (9, 27), superconductivity (23), and antiferromagnetism (10, 28, 29) of iron-based materials.

Fig. 1 Experimental geometry and coherent lattice dynamics.

(A) FeSe/SrTiO3 films were measured by trXRD and trARPES. Δt denotes the delay of the x-ray (turquoise) and UV (purple) probe pulse with respect to the IR pump pulse (red). (B) X-ray intensity of the (004) Bragg peak as a function of Δt. Photoexcitation (F = 1.83 mJ/cm2) initiates a coherent A1g phonon (left inset), resulting in oscillations of the x-ray signal at the A1g frequency (right inset). FT, Fourier transform; arb. u., arbitrary units. (C) Coherent x-ray signal at T = 20 and 180 K (blue and orange, respectively; raw data are shown in fig. S1) after subtracting a smooth background [dotted black line in (B)]. (D) Dependence of the x-ray signal on the incident fluence at T = 20 K. (E) Corresponding displacement δzSe. Time zero is determined by the exponentially decaying initial intensity drop [dotted black line in (D), convolved with the overall time resolution]. Errors of the frequency in (C) denote 1 standard deviation obtained from the fitting. In (C), (D), and (E), solid black lines indicate fits of the data, and curves are offset for clarity by the amounts indicated in each panel.

Figure 1B shows the (004) Bragg peak intensity of the FeSe film measured by trXRD; upon photoexcitation (Δt > 0), the diffracted intensity is periodically modulated owing to the collective displacement of the selenium atoms. After subtracting a smoothed incoherent background, the coherent signal is well fit with an exponentially decaying cosine (Fig. 1C). These data and the corresponding Fourier transforms (Fig. 1B, right inset) demonstrate the cleanliness of the coherent response and exemplify the precision of our measurement, which allows resolution of a hardening of the A1g mode by 0.014 ± 0.003 THz with decreasing temperature. Figure 1D shows that the coherent oscillation amplitude increases with increasing pump fluence, whereas the A1g frequency does not noticeably change. The intensity change observed in trXRD is directly related to the coherent displacement δzSe(t) of the selenium atoms by a structure factor calculation [(29) and supplementary text] based on the symmetry of the A1g phonon (Fig. 1E). We use exponentially decaying cosine fits with a linear background to extract the peak-to-peak amplitude ΔzSe at time zero.

To measure the impact of the coherent A1g phonon on the electronic band structure, we performed trARPES experiments. Photoemission spectra near the Brillouin zone center contain two prominent spectral features (Fig. 2A): (i) a hole-like band, which disperses between energies EEF = 0 and −100 meV, and (ii) a flat band located at EEF ≈ −200 meV (EF denotes the Fermi level). High-resolution equilibrium ARPES studies (19, 20) have determined the orbital characters of the electronic bands near EF by using photon polarization selection rules. Comparison with these studies shows that the first and second bands are of dominant dxz/yz and Embedded Image orbital character, respectively. Electronic band dispersions calculated by density functional theory (DFT) are overlaid in Fig. 2A with an overall renormalization factor of 3, yielding good agreement with both the dxz/yz and Embedded Image bands. Dashed lines denote bands that do not appear in trARPES, likely because of unfavorable photoemission matrix elements and the limited energy resolution.

Fig. 2 Coherent electron dynamics.

(A) Equilibrium photoemission spectrum at T = 20 K along the Γ-X direction. Electronic band dispersions calculated by DFT are overlaid (renormalized by a factor of 3). Dominant features correspond to one of the dxz/yz bands (orange) and the Embedded Image band (blue). Dashed lines indicate unresolved bands, and solid lines indicate detected bands. (B) Photoinduced shift of the dxz/yz band at four delays, corresponding to extrema of the coherent oscillations [denoted by vertical dashed lines in (C)]. (C and D) Momentum-averaged energy shifts of the dxz/yz and Embedded Image bands, respectively. Solid black lines indicate fits of the data. Traces in (C) and (D) are offset for clarity.

The peak energies of the dxz/yz and Embedded Image bands are extracted by fitting two Gaussians to constant-momentum cuts of the spectra (supplementary text). Figure 2B shows the dxz/yz band dispersions at an incident fluence F = 0.62 mJ/cm2 and four representative delay times. At all momenta, the band energy oscillates with the A1g frequency f = 5.25 ± 0.02 THz. The extracted oscillation amplitudes exhibit a momentum k|| dependence of up to 20% between k|| = −0.22 and −0.13 Å−1, yet this is comparable in magnitude to the overall uncertainties (supplementary text). Therefore, we average the energy dynamics within this momentum range and display the averaged dynamics for the dxz/yz and Embedded Image bands in Fig. 2, C and D, respectively. We use exponentially decaying cosine fits with a quadratic background to extract the peak-to-peak amplitudes ΔE at time zero.

Figure 3A compares the selenium displacement δzSe(t) extracted from trXRD with the energy dynamics of the two electronic bands 〈Exz/yz(t)〉 and Embedded Image from trARPES. Time zero is independently determined in the two experiments, each yielding an uncertainty of ~20 fs. Within this accuracy, the lattice and electronic oscillations are synchronous: Both bands shift toward lower energy as the selenium atoms move away from the Fe planes (Fig. 3B). This correspondence is the same as that derived for the related compound BaFe2As2 (29).

Fig. 3 Coherent lock-in at the A1g frequency.

(A) Oscillations of the selenium displacement δzSe (blue) and the momentum-averaged energy shift Embedded Image (orange and green). trXRD and trARPES data were measured with F = 0.46 and 0.37 mJ/cm2, respectively. Black lines indicate fits of an exponentially decaying cosine with a polynomial background to extract the peak-to-peak oscillation amplitudes ΔzSe and ΔEmbedded Image at time zero. (B) Schematic of the A1g phonon mode (top), which periodically modulates the electronic band energies (bottom). (C and D) Lattice and band oscillation amplitudes as a function of the incident pump fluence. Solid lines indicate linear fits. Error bars in (C) and (D) denote statistical uncertainties, whereas the shaded areas represent systematic fluence uncertainties (supplementary text). The error of the fitted slopes accounts for both statistical and systematic uncertainties.

The A1g deformation potential near the Brillouin zone center is quantified by linear fitting of the fluence-dependent amplitudes ΔzSe and Embedded Image shown in Fig. 3, C and D, respectively. We correct for the amplitude reduction caused by the finite time resolutions (supplementary text). Furthermore, for the calculation of the deformation potentials, we include factors to account for the fluence averaging owing to finite pump and probe beam profiles, as well as for the effective excitation densities per FeSe layer, as determined by the pump and probe penetration depths (supplementary text). The fluence dependences indicate that both the trXRD and trARPES experiments sample a linear response, confirming that the coherent signal remains representative of the ground state. The coherent lock-in approach directly yields deformation potentials of ΔExz/yzzSe = −13.0 ± 2.5 and Embedded Image.

The extracted EP deformation potentials allow a comparison with different theoretical approaches. Table 1 shows that canonical DFT, calculated in a nonmagnetic state, underestimates the selenium height and overestimates the A1g frequency. Moreover, the theoretical dxz/yz deformation potential is one order of magnitude smaller than in the experiment. In contrast, the calculated deformation potential of the Embedded Image band is comparable to the experimental value within a factor of 2. The failure of DFT to reproduce these basic properties of FeSe (5, 15, 30) exemplifies the substantial effect of electron correlations on the EP coupling.

Table 1 Comparison of experiment and theory.

Shown are selenium height zSe, A1g phonon frequency Embedded Image, and A1g deformation potentials ΔExz/yzzSe and Embedded Image obtained from experiments, canonical DFT calculations, and DFT+DMFT by Mandal et al. (5); for ΔExz/yzzSe, the band-average (maximum) values are displayed. The experimental value for zSe is taken from Margadonna et al. (30), whereas the deformation potentials are obtained by combining the data shown in Fig. 3, C and D, and applying corrections for spatial integration over pump and probe profiles and effective energy densities (supplementary text). The error of the deformation potentials includes systematic and statistical uncertainties. DFT deformation potentials account for an empirical renormalization factor of 3, and the errors reflect the uncertainty of the renormalization determined from ARPES. Theoretical values for Embedded Image are deduced from quadratic fits to the relative total energy.

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These properties can be quantitatively reproduced by using self-consistent DFT–dynamical mean field theory (DFT+DMFT) to incorporate electron-electron correlation effects (5, 14). For FeSe, this approach (5) yields the correct band structure and selenium height, leading to a softer A1g mode, consistent with experiments (Table 1). In particular, Mandal et al. (5) reported a band-averaged (maximum) A1g deformation potential of |ΔExz/yzzSe| = 10.3 meV/pm (13.4 meV/pm) for the dxz/yz band—in agreement with our experimental value.

Our results differ from previous work (10) on BaFe2As2, which reported that the A1g deformation potential obtained from DFT agrees adequately with experiments. This discrepancy may be associated with the non–orbital-resolved nature of the comparison (9, 10) and differences in the level of electron correlations in the two compounds (14). In contrast, our orbital-resolved lock-in experiment on FeSe establishes a clear case of substantially enhanced EP coupling in the presence of strong correlations.

The important role of correlation-enhanced EP coupling is not universally accepted; most earlier work on iron-based superconductors was focused on spin fluctuations without acknowledging the role of the EP coupling (1417, 22). Moreover, a direct experimental confirmation of the enhanced EP coupling strength has been lacking; phonon spectroscopies, such as Raman (15) and neutron scattering (16, 17), lack orbital resolution, whereas photoemission (7, 19, 20, 24) and tunneling (8) spectroscopy do not resolve phonons directly. Our method bridges this gap and measures all relevant degrees of freedom directly, allowing us to test theories predicting the effect of correlations on the EP coupling.

In particular, FeSe exhibits a substantial and nontrivial increase in critical temperature Tc from 8 to 37 K by application of pressure (2123), which matches the pressure dependence of the EP deformation potential derived from DFT+DMFT (5), emphasizing that correlation effects have a strong impact on superconductivity in FeSe. Moreover, the observation of a much-enhanced out-of-plane A1g mode coupling, compared with DFT calculations (Table 1), suggests a small-momentum-transfer EP coupling, consistent with theoretical results for local EP interactions (5). Such a forward-scattering EP coupling likely interferes constructively with other electronic pairing channels (24), thereby providing a pathway toward superconducting states in which EP and electron-electron interactions act in concert.

The coherent lock-in approach establishes an experimental paradigm for precision measurements of fundamental physical quantities, such as the EP deformation potential, by only relying on a linear, coherent response. It provides a purely experimental and model-free technique that simultaneously yields complete information in time, space, momentum, and energy with high sensitivity. Beyond the immediate implication that the “renormalized” EP coupling strength in iron-based materials is considerably larger than anticipated by conventional theories, this approach also opens the way for unbiased tests of emergent phenomena in other controversial, correlated materials.

Supplementary Materials

www.sciencemag.org/content/357/6346/71/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S12

Table S1

References (3144)

References and Notes

  1. Materials and methods are available as supplementary materials.
  2. Reflectance of mirrors with protected silver coating (www.thorlabs.com/images/TabImages/Thorlabs_Protected_Silver_Coating.xlsx).
Acknowledgments: Use of the LCLS, SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES) under contract DE-AC02-76SF00515. S.G., S.-L.Y., H.S., J.A.S., S.R., J.J.L., T.J., B.M., C.J., A.G., Y.L., D.L., L.C., W.L., R.G.M., T.P.D., W.-S.L., P.S.K., and Z.-X.S. were supported by the DOE-BES Division of Materials Sciences and Engineering under contract DE-AC02-76SF00515 (to the Stanford Institute for Materials and Energy Sciences). The authors gratefully acknowledge technical assistance at LCLS by D. Stefanescu and R. Fiebich. S.G. and D.L. acknowledge partial support by the Swiss National Science Foundation under fellowships P2EZP2_148737 and P300P2_151328, respectively. S.-L.Y. acknowledges support by the Stanford Graduate Fellowship. H.S. acknowledges support from the Fulbright Scholar Program. A.G. acknowledges support by the National Defense Science and Engineering Graduate Fellowship Program. K.W.K. was supported by the Basic Science Research Program through the National Research Foundation of Korea, funded by the Ministry of Science, ICT and Future Planning (NRF-2015R1A2A1A10056200). Y.-D.C. and Z.H. were supported by the Director, Office of Science, BES, of the U.S. DOE under contract DE-AC02-05CH11231. Raw data from the time-resolved x-ray scattering experiment are kept at the LCLS. Time-resolved photoemission raw data are kept at the Shen Laboratory of Stanford University.
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