Abstract
The interactions between electrons and lattice vibrations are fundamental to materials behaviour. In the case of group IV–VI, V and related materials, these interactions are strong, and the materials exist near electronic and structural phase transitions. The prototypical example is PbTe whose incipient ferroelectric behaviour has been recently associated with large phonon anharmonicity and thermoelectricity. Here we show that it is primarily electronphonon coupling involving electron states near the band edges that leads to the ferroelectric instability in PbTe. Using a combination of nonequilibrium lattice dynamics measurements and first principles calculations, we find that photoexcitation reduces the Peierlslike electronic instability and reinforces the paraelectric state. This weakens the longrange forces along the cubic direction tied to resonant bonding and low lattice thermal conductivity. Our results demonstrate how freeelectronlaserbased ultrafast Xray scattering can be utilized to shed light on the microscopic mechanisms that determine materials properties.
Introduction
The narrow gap IV–VI semiconductors and group V semimetals are characterized by coupled electronic and lattice instabilities^{1,2,3,4,5}. The general behaviour has been understood at least qualitatively in terms of electronphonon interactions associated with the unsaturated plike bonding in these materials. In the case of PbTe, a ferroelectric instability develops with decreasing temperature, but the material remains in the paraelectric, rocksalt phase to the lowest temperatures measured^{6,7}.
Recent measurements^{8,9,10,11} on PbTe reveal anomalous lattice dynamics with increasing temperature, especially in the soft transverse optical (TO) phonon branch associated with the ferroelectric instability. This has been interpreted in terms of both giant anharmonicity^{8,12,13,14,15,16} and local symmetry breaking due to offcentring of the Pb ions^{9,10}. The observed anomalies have prompted renewed theoretical and computational interest. Much of this work has concentrated on the role of lattice anharmonicity and phononphonon interactions^{12,13,14,15,16,17,18,19}. On the other hand, a few theoretical studies have focused on the importance of electronphonon coupling, for example, in the link between the mixed ioniccovalent character of Pb–Te bonds and the soft mode behaviour^{5,16} or the large polarizability of the Pb ion’s lone 6s^{2} pair^{20,21}. The instability has also been discussed in terms of the large polarizability of resonant valence plike bonds^{3,4}, which has recently been associated with longrange interactions, the low lattice thermal conductivity, and the high thermoelectric performance^{22}. Thus, the origin of the soft phonon branch and the associated anharmonicity in PbTe remains controversial, partially due to the difficulty of separating the coupled electronic and lattice interactions experimentally.
The incipient ferroelectric behaviour of IV–VI compounds is strongly sensitive to carrier concentration^{23}, which suggests the importance of the valence and conduction ptype bands very near the band edges in determining the equilibrium structure^{1}. Ultrafast photoexcitation provides a means to affect the electronphonon coupling by redistributing the carrier population near the band edges at constant volume and without introducing defects associated with doping. Our motivation for the present work is to use a combination of ultrafast Xray scattering and first principles calculations to isolate the role of electronphonon interactions in photoexcited PbTe.
In particular, differential changes in the momentumdependent phononphonon displacement correlations can be accessed via Fouriertransform inelastic Xray scattering (FTIXS)^{24,25}. In this technique, an ultrafast pump pulse is used to suddenly photoexcite carriers on a time scale that is fast compared with all the relevant phonon periods. In turn, the modification of the bandpopulation renormalizes the phonon frequencies and, in principle, the eigenvectors as well, mixing the acoustic and optical branches. The projection of the initially thermally occupied vibrational modes onto these new modes produces a nonequilibrium state where the displacement correlations evolve in time. This method thus allows us to extract explicit information about the coupling of the electronic states near the energy gap with lattice vibrations. In previous experiments on photoexcited Ge^{24,25}, temporal oscillations in the Xray diffuse scattering were observed, corresponding to correlations between phonons of equal and opposite momenta belonging to the same branch and oscillating at twice the phonon frequency.
Here we measure oscillations in momentumdependent phonon–phonon displacement correlations for PbTe excited across its direct band gap. In addition to oscillations at twice the transverse acoustic phonon frequency, we measure the sum and difference frequencies for phonons of different branches. These combination modes result from a sudden inhomogeneous change in the interatomic forces and the dynamical matrices on photoexcitation. Our simulations, based on ab initio calculations, reproduce the experimentally observed combination modes due to a weakening of the longrange interactions along the bonding directions. This inhomogeneous change in forces is a direct consequence of the redistribution of the population of electronic states near the direct bandgap at the Lpoint that also results in a stabilization of the paraelectric phase. Thus, we demonstrate that the particular coupling of electronic states near the band edges via the zone centre TO mode is the primary mechanism that induces the TO mode softening that drives the incipient ferroelectric behaviour. Our results are consistent with the resonant bonding picture of the soft mode behaviour, verifying that it is primarily the electronic states near the energy gap that are responsible for this effect. These findings open new opportunities for the development of more efficient thermoelectric materials (for example, by controlling the identified mechanism for incipient ferroelectricity via alloying) and contribute to the general understanding of mechanisms for ferroelectric phase transitions.
Results
Identification of combination modes in FTIXS spectra
The experiments were performed at room temperature using 50 fs, 8.7 keV Xrays from the Xray pump probe instrument at the Linac Coherent Light Source (LCLS)^{26} freeelectron laser. We excite ∼2 × 10^{20} cm^{−3} carriers per Lvalley (∼0.5% valence excitation) across the direct band gap of lowcarrierdensity (4 × 10^{17} cm^{−3}) ntype PbTe with 60 fs infrared pump pulses with central wavelength of 2 μm (see Methods section). Subsequently, we track the Xray diffuse scattering as a function of time delay τ between the infrared pump and Xray probe pulses, referencing differential changes in the scattering intensity with and without the photoexcitation. Absorption of the pump pulse produces correlated pairs of phonons with equal and opposite momenta that are probed by femtosecond Xray diffuse scattering. The intensity of this scattering is proportional to snapshots of the equaltime phononphonon displacement correlations as a function of momentum transfer^{24}. The amplitudes of the temporal oscillatory components are directly related to the strength of the interaction of the phonons with the photoexcited electron density, through changes in the dynamical matrices^{27}.
Figure 1 presents twodimensional Xray scattering images from PbTe for a fixed crystal configuration. Each pixel corresponds to a different momentum transfer Q. Figure 1a shows the average diffuse scattering intensity I_{0}(Q), from the unexcited state, that is, when the pump arrives after the Xray probe (τ<0). The overlaid black lines represent the boundary between different Brillouin zones (BZ) for the fcc Bravais lattice. Two BZs of interest, G=(1 3) and G=(0 4), are labelled in reciprocal lattice units (rlu) of 2π/a_{0}, where a_{0} is the length of a side of the conventional cubic cell. The white arrows point along the reduced wave vectors q=Q−G, where in this case q≈(0 q_{y} 0), that is, it is approximately along the highsymmetry Δ (Γ to X) direction in each zone. Figure 1b–d are differential scattering intensity patterns δI(τ;Q)=I(τ;Q)−I_{0}(Q), in which the unexcited state frame (Fig. 1a) is subtracted from the pattern at each time delay. The time delays shown here are τ=0, 0.5 and 2.5 ps. Increases in the scattering intensity relative to the reference frame are marked in red while decreases in the intensity are shown in blue. While at the moment of excitation (τ=0 ps) there are not yet significant changes in scattering intensity, within 0.5 ps, a strong decrease in intensity is seen along the Δ direction in the G=(1 3) BZ while an increase in intensity is seen along Δ in the G=(0 4) BZ. By 2.5 ps, the diffuse scattering shows a general increase and the prominent decrease along the Δ line in the (1 3) BZ has largely subsided.
Figure 2 shows the relative differential changes δI(τ;Q)/I_{0}(Q) along the two Δ lines indicated in Fig. 1, in the time domain (Fig. 2a,c) and the frequency domain (Fig. 2b,d). The precise trajectory for the reduced wave vectors (q_{x} q_{y} q_{z}) is shown in the upper portion of Fig. 2b,d. In our experimental geometry, the diffuse scattering along Δ is sensitive primarily to transverse phonons polarized along (0 0 1). From top to bottom the data in Fig. 2a span from near zonecentre to zone edge in the G=(1 3) BZ and in Fig. 2c from near zone centre to approximately half the distance to zone edge in the G=(0 4) BZ. The data in the () zone show a rapid decrease in scattering followed by damped oscillatory behaviour consisting of multiple frequencies that disperse with increasing q_{y} on top of a several picosecondscale increase. Meanwhile in the (0 4) zone, the data show an increase in scattering with damped oscillations.
In Fig. 2b,d, we plot the Fourier transform of the timedomain data for the (1 3) and (0 4) zones as a function of q_{y}. Several dispersive features are immediately apparent in the spectra on top of a broad and relatively featureless lowfrequency background originating from the slow variations in the diffuse scattering over time. The dispersive features in Fig. 2b match well with the frequencies expected for the first transverse acoustic overtone (2TA) and TO and acoustic combination modes (TO±TA) as extracted from the INS data reported by Cochran et al.^{28} (overlaid in red). The overtone is due to phononphonon correlations between modes at +q and −q within the same branch^{24}, while the combination modes originate from correlations between different branches. These mixed terms were not observed in previous FTIXS measurements^{24,25}. Photoexcitation can lead to combination modes only through inhomogeneous changes in the various elements of the dynamical matrix, mixing the eigenvectors. The predominance of the combination modes indicates that photoexcitation couples the TO and TA eigenvectors in the nonequilibrium state and does not just renormalize their frequencies as would occur through an overall scaling of the dynamical matrix. This mixing of the branches is consistent with a picture where photoexcitation primarily affects the longrange interactions that have been linked to the ferroelectric instability through resonant bonding^{22}. As shown below, our first principles calculations support this picture while also pinpointing the associated electronic states that are responsible for instability.
In the (0 4) BZ, the sensitivity to optical phonons is low and we therefore only expect to see overtones of the acoustic modes, consistent with the results shown in Fig. 2d. Here we attribute the dispersion to the 2TA overtone. The discrepancy with the neutron results (red line) is likely due to systematic deviations from the Δline for this zone in our geometry (finite q_{x} and q_{z}; see coordinate trajectory in the figure and Methods section).
First principles simulation of photoexcited PbTe
To simulate the experimental conditions, we calculate the time evolution of the displacement correlations between phonon modes due to a sudden promotion of valence electrons to the conduction band^{27}. The initial values of these displacement correlations for unexcited PbTe were calculated using density functional theory (DFT, see Methods section). We calculate the interatomic force constants of photoexcited PbTe using constrained density functional theory (CDFT)^{29}, taking that 0.5% of valence electrons are promoted into the conduction band, as estimated in the experiments (see Methods section). In the CDFT calculation, we assume that electron and hole populations thermalize rapidly within their respective excitedstate bands according to FermiDirac distributions, which is reasonable since the typical electronelectron scattering times are on the order of tens of femtoseconds. This scattering time and the excitation pulse itself are much shorter than the shortest phonon period. Thus, for the simulations we consider the photoexcitation as sudden.
A sudden change of interatomic force constants before and after photoexcitation leads to the time evolution of the displacement correlations between phonon modes, which was derived starting from the equations of motion of damped harmonic oscillators for each phonon mode^{27}. The timedependent Xray diffuse scattering signal is proportional to those displacement correlations^{24,27} and its Fourier transform gives the simulated FTIXS spectrum along Δ in the (1 3) BZ shown in Fig. 3a. The FTIXS spectrum calculation makes use of the excitedstate harmonic and anharmonic (3^{rd} order) force constants computed using CDFT, as well as the groundstate harmonic force constants computed using DFT that characterize the initial thermal state. Here the anharmonic force constants are only used to introduce damping. To better compare the simulation and experiments, we filter the experimental data to remove the slowly varying background before taking the Fourier transform, as shown in Fig. 3b. Both combination modes (TO±TA) and an overtone mode (2TA) appear in the calculations and are labelled. The simulation matches the experimental data, capturing many of its general features and thus reflecting the closeness of the model in depicting the effects of ultrafast photoexcitation on the material.
Figure 3c shows the experimental (black lines, unfiltered) and calculated (orange lines) timedependent diffuse scattering at select coordinates along Δ in the (1 3) BZ. Note that the model calculations do not account for relaxation of the photoexcited carriers or the heating of the lattice, which could explain the picosecond rise seen in the data but not the model. Importantly, the strong initial dip and phase of the oscillation observed in the experiment is well reproduced by the model. In the calculations, this coincides with substantial hardening of the TO phonon branch of up to ∼28% at Γ (Fig. 4a,b), accompanied with a distinctive weakening of longrange interactions involving both 4^{th} and 8^{th} nearest neighbours (Fig. 5). These particular force constants correspond to interactions with the second and third neighbours along the <100> bonding direction in the cubic crystal structure of PbTe (see Fig. 6). The calculations reproduce the anomalously strong longrange interactions that are expected along this direction that have been attributed to resonant bonding involving pband valence electrons^{22}. It is further observed that the experimental lifetimes seen in Fig. 3c appear shorter than those predicted in the simulations.
It is especially important to note here that the appearance of combination modes is a direct indication of the nonequilibrium state following photoexcitation and could only appear on a sudden inhomogeneous change in the interatomic forces. To illustrate the degree of mixing we show in Fig. 4c the projection of the calculated TA eigenvectors of the excited state onto those of the equilibrium TO state, , along the Δ line. Here the projection represents the degree of equilibrium TO character that has been mixed into the TA modes on photoexcitation. Such mode mixing directly leads to the experimentally observed combination modes seen in Figs 2 and 3.
The general agreement between the measured and simulated twophonon FTIXS spectra indicates that photoexcitation of PbTe stabilizes the paraelectric phase, as described below. The specific observation of combination modes along the Δ line is consistent with differential changes in the realspace force constants, as obtained in the calculations. Furthermore, the initial rise of the oscillation after τ=0 in the (0 4) BZ (Fig. 2c) is consistent with a sudden softening of the TA branch frequency on photoexcitation. This conclusion can be drawn independently of the calculations since the square of the phonon displacements, and thus the diffuse scattering, is inversely proportional to the phonon frequency^{24}. In contrast, the initial dip of the response in the (1 3) BZ (Fig. 2a) likely originates from a hardening of the TO branch near zone centre after photoexcitation, as predicted by calculations (see Fig. 4b). Furthermore, we note that the absence of a clear signal at 2TO in the (1 3) BZ in the experiment is consistent with the simulation results, in which the square of the phonon displacement dampens at twice the rate of the already strongly anharmonic TO mode.
Discussion
The origin of the photoinduced stabilization of the paraelectric phase of PbTe can be understood from the following arguments. Figure 7 shows the highest valence and lowest conduction bands of photoexcited PbTe, calculated with and without displacement of the atoms corresponding to the soft TO phonon at zone centre (Γpoint) (Fig. 7a) and at the Xpoint (Fig. 7b). The solid line shows the bands for the equilibrium atomic positions and the dashed lines show the bands with the TO phonon mode frozen in. The colour shading of the bands indicates the change of occupation of electron and hole states following photoexcitation (see Methods section). In Fig. 7a, we see that the zone centre TO(Γ) phonon displacement lowers the energy of states near the valence band maximum and increases the energy of states near the conduction band minimum, particularly along the L–W line. This is in accordance with the JonesPeierls distortion mechanism, which stabilizes the rhombohedral structure of the group V elements and the ferroelectricallydistorted IV–VI compounds by coupling the lowlying conduction bands with the higher lying valence bands in the (111) face of the fcc Brillouin zone^{2}. The underlying effect of electronphonon coupling here is the same as that of the Kohn anomaly in threedimensional metals and the Peierls instability of onedimensional metals. This explains the strong photoinduced softening of the zone centre A_{1g} mode in the group V semimetal bismuth^{30,31}, where photoexcitation reduces the Peierls distortion and drives the system closer to the symmetric phase. Similarly, in the case of PbTe, a relatively small change of the occupation of these bands (filling the lower conduction bands and vacating the highest valence bands) reduces the effectiveness of the TO displacement in lowering the total electronic energy. The result is an increase in the softTO mode frequency at Γ, as observed in calculations for PbTe (Fig. 4b). In contrast, the TO mode at X (Fig. 7b) strongly couples the valence bands causing them to repel along L–W, shifting the highest energy band upwards under phonon displacements. Consequently, in PbTe photoexcitation gives rise to a softening of the TO frequency at X (Fig. 4b) at the same time that it reduces the softmode behaviour with the near Γ hardening.
The interatomic forces, particularly the aforementioned longrange interatomic terms with 4th and 8th neighbours, can be understood in terms of these strong phonon wavevectordependent frequency shifts and their coupling with the conduction and valence bands near the energy gap. Via Fourier transform, this momentum dependent phonon frequency renormalization directly translates to the prominent longrange realspace coupling noted in ref. 22. By promoting the carriers from valence to conduction band, the wavevectordependent renormalization of phonon frequencies due to the electronphonon coupling becomes weaker^{32}, resulting in shorterrange interatomic forces (see Fig. 5). Our measurements and calculations of photoexcited PbTe demonstrate that the electronic states very near the direct gap dominate this mechanism and that the effect on mode frequencies depends strongly on the phonon wavevector.
We note that the strong electronTO(Γ) phonon coupling occurs between bands that originate from halffilled degenerate states in the absence of ionicity^{2}. This is compatible with the resonant bonding picture^{22,33}, which argues that the zone centre opticalphonon displacement tends to lower the energies of the halfsaturated pbonds on one side of the atom relative to the other^{5}, leading to its strong coupling with the electronic bands. However, the band picture provides more detailed insights into the phononmomentumdependence of the electronphonon contribution to the phonon frequencies, which is absent from the resonant bonding viewpoint, indicating the strong coupling of the electronic states along L–W line with the soft TO modes.
In summary, we show that the impulsive redistribution of the electronic states near the band edges is consistent with a weakening of the longrange interactions along the <100> cubic direction. This stems from photoexcitation of delocalized carriers from plike resonant bonding states to plike antibonding states, thus reducing the propensity for a ferroelectric distortion^{5} and stabilizing the paraelectric state. We show that this effect softens the TA branch while hardening the TO branch near zone centre. These frequency shifts arise in the first principles calculation of photoexcited harmonic force constants, independent of anharmonic effects, indicating that the soft mode behaviour is primarily a consequence of the strong electronphonon coupling. Our results reconcile the real and reciprocal space pictures of bonding and confirm the importance of electronic states near the band edges in determining the equilibrium structure in resonantly bonded structures^{2,3,4,5}.
Methods
Experimental details
The reported measurements were performed at the Xray pump probe instrument of the LCLS Xray freeelectron laser (FEL) using infrared pump pulses (60 fs, 350 μJ, 0.6 eV) generated from an optical parametric amplifier laser and Xray probe pulses (50 fs, 8.7 keV) with energies selected using a diamond (111) monochromator^{34} leading to a bandwidth of ∼0.5 eV. Both beams approached the sample at a grazing incidence angle (<5°) and the entire sample space was contained within a Hefilled chamber to minimize background air scattering. A large area, 2.3 Mpixel CornellSLAC Pixel Array Detector with 110 × 110 μm^{2} pixels and a 120 Hz readout rate captured the resulting Xray diffuse scattering over a wide region of reciprocal space. Preliminary measurements were also taken at the Stanford Synchrotron Radiation Light source (Xray only) and the Spring8 Angstrom Compact Free Electron Laser (SACLA) (timeresolved, 1.5 eV optical pump/9 keV Xray probe).
Diffuse scattering patterns were recorded at a repetition rate of 120 Hz and filtered on a shotbyshot basis according to fluctuations in both the electron beam energy and Xray intensity. Images passing the filter constraints were subsequently sorted into 100 fswide time delay bins according to the pershot output of a timing diagnostic tool accurately tracking the relative arrival time between pump and probe^{35}. The resulting images in each bin are summed and normalized by the sum of Xray intensities of the shots corresponding to those images. The reported results stem from the analysis of an average of ∼10,000 shots per time delay bin, which accounting for unused, filteredout shots, overhead in mechanical movement speeds, and freeelectron laser machine downtime, is collected in a matter of several hours in realtime.
Sets of room temperature measurements were taken at two infrared pump fluences, differing by a factor of approximately two. Considering uncertainties in the incidence angle and size of the focused beam, the estimated photoexcited carrier density of the higher pump fluence (the reported data) is 2 × 10^{20} cm^{−3}. Other than differences in the amplitude of the differential intensity timedomain responses linear with the pump fluence, no notable variations are seen in a comparison of data from the two fluences. Furthermore, the damage limit of the PbTe samples, from exposure to the combined fluence of both infrared and Xray laser beams, was assessed at both SACLA and LCLS. The chosen aggregate pump and probe fluence for the measurements was thus kept below this limit. No evidence of sample damage was observed during the experiment.
Sample growth
Single crystals of PbTe were grown by a modified Bridgman technique with {100} crystallographic orientation at Oak Ridge National Laboratory with ntype carrier concentration of 4 × 10^{17} cm^{−3}.
Geometry and data extraction
We chose the geometric configuration of the experimental setup to simultaneously capture diffuse scattering corresponding to a highsymmetry Δ (Γ to X) line for both an alleven (0 4) and an allodd (1 3) Brillouin zone (BZ), with select sensitivity to transverse phonon polarization scattering contributions (q parallel to the ydirection). The geometry was chosen such that the Ewald sphere misses the zone centre (Γ) by ∼1% for the (1 3) BZ to prevent damage to the detector by the Bragg diffracted beam.
To enhance the signaltonoise contrast, multiple neighbouring pixels are binned and averaged together to represent specific coordinates along the subsequent reduced wavevector directions. The Δ direction of the (1 3) BZ navigates from q∼(0 −0.1 0) to q∼(0 −1 0) in (0 −0.05 0) steps, with each coordinate represented by the closest (in reciprocal space) 50 neighbouring pixels, while the Δ direction of the (0 4) BZ moves from q∼(0 −0.05 0) to q∼(0 −0.425 0) in (0 −0.025 0) steps, with each coordinate represented by the closest 10 neighbouring pixels. It should be noted that q_{x} and q_{z} coordinate values collected near the Δ line of the (1 3) BZ remain close to zero throughout the entire reduced wavevector, the same coordinate values deviate slightly from the Δ line of the (0 4) BZ. This is due to finite curvature of the Ewald sphere. The deviation from zero for q_{x} is ∼−0.003 rlu for each q coordinate (segment of binned pixels) moving away from Γ and for q_{z} is ∼−0.002 rlu. This divergence explains the mismatch with the overlaid dispersion along Δ extracted from two times the frequency of the INS results of ref. 28 in Fig. 2d.
To produce the FTIXS, twophonon dispersion in Fig. 2b,d, we perform a FTbased analysis to translate the timedomain data into the frequencydomain. Specifically, we apply a Fast Fourier transform algorithm to the timedomain data extracted at each q coordinate of a reduced wavevector under investigation. Before employing the algorithm, additional points are padded to each timedomain trace, effectively extending the time delay range to artificially increase the frequencydomain resolution for visual clarity. The padded points in each trace take on the value of the last recorded point in the experimental data. For the reported data, 175 padded points are used for the away from zone centre Δ direction traces. The magnitudes of the Fourier coefficients are illustrated in false colour on a base10 logarithmic scale.
First principles calculations
The harmonic (second order) and anharmonic (third order) force constants for photoexcited PbTe were calculated using the CDFT approach described in Murray et al.^{29}. The electronic temperature used in the calculation was 2,000 K, and carriers were assumed to (separately) thermalize within the valence and conduction bands, keeping the total carrier density in the conduction band fixed at 0.5% of the valence electrons. Thus, carrier recombination between conduction and valence bands was assumed to be negligible in the timescale of the experiments. The electronic temperature and the photoexcited carrier concentration values were chosen to give a qualitative picture of the effect of photoexcitation in the experiment. The computed interatomic force constants do not depend strongly on the electronic temperature, and they depend linearly on the photoexcited carrier concentration. We thus interpolated 3rd order force constants for 0.5% carriers using those calculated for 0 and 1%. The second order force constants for nonexcited PbTe were calculated using density functional theory (DFT). All the second and third order force constants were calculated from atomic forces using a realspace finite difference supercell approach^{36} and the Phono3py code^{37}. All CDFT and DFT calculations were carried out using the ABINIT code^{38} and employing the local density approximation and Hartwigsen–Goedecker–Hutter normconserving pseudopotentials. We used the theoretical lattice constant for the local density approximation functional and Hartwigsen–Goedecker–Hutter pseudopotentials for the nonexcited material. The lattice constant with the photoexcitation was kept the same since it relaxes on much longer timescales than that at which we track the phonon response. Forces were computed on 216 atom supercells, using an energy cutoff of 15 Ha and 4shifted 2 × 2 × 2 reciprocal space grids for electronic states. Phonon frequencies, mode eigenvectors, and phonon decay rates were calculated for reduced wave vectors throughout the Brillouin zone as described in He et al^{39}.
The Xray diffuse scattering signal arising from phonon squeezing for reduced wave vectors q was calculated by taking the overlap of the change, , in the inverse dynamical matrix before and after photoexcitation with mode eigenvectors, u_{q,λ} and of the photoexcited system. This determines the crosscorrelation between modes λ and λ′ immediately following photoexcitation, which then gives rise to damped oscillations at the sum and difference frequencies, , the damping rate of the signal being the average of the energy damping rates of the two modes involved. These damped oscillations of the crosscorrelation contribute to the Xray diffuse scattering signal for momentum transfer, Q=q+G, scaled by the polarization factor Q·u_{q,λ} of each mode. Details of this method are in Fahy et al.^{27}.
Data availability
The data that support the findings of this study are available from the corresponding authors on request.
Additional information
How to cite this article: Jiang, M. P. et al. The origin of incipient ferroelectricity in lead telluride. Nat. Commun. 7:12291 doi: 10.1038/ncomms12291 (2016).
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Acknowledgements
This work is supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DEAC0276SF00515. I.S. acknowledges support by Science Foundation Ireland and Marie Curie Action COFUND under Starting Investigator Research Grant 11/SIRG/E2113. S.F. and É.D.M. acknowledge support by Science Foundation Ireland under Grant No. 12/1A/1601. O.D. acknowledges support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, through the Office of Science Early Career Research Program. A.F.M. and B.C.S. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. J.C. acknowledges financial support from the Volkswagen Foundation. Portions of this research were carried out at the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory. LCLS is an Office of Science User Facility operated for the U.S. Department of Energy Office of Science by Stanford University. Preliminary experiments were performed at SACLA with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2013A8038) and at the Stanford Synchrotron Radiation Light source, SLAC National Accelerator Laboratory, which like the LCLS is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE AC0276SF00515.
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D.A.R. conceived this project. M.P.J. and M.T. led the experiment, supported by C.B., M.C., J.C., O.D., J.M.G., T.H., M.C.H., M.K., A.M.L., P.Z., D.Z. and D.A.R. M.P.J. analysed the data. I.S, É.D.M. and S.F., performed the DFT and CDFT calculations and contributed to the interpretation along with D.A.R., M.P.J., M.T., O.D. and R.M. The sample was grown and characterized by A.F.M., B.C.S. and O.D. T.S. helped with preliminary Xray experiments. The manuscript was written by M.P.J., M.T., I.S., S.F. and D.A.R., with feedback from all coauthors.
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Jiang, M., Trigo, M., Savić, I. et al. The origin of incipient ferroelectricity in lead telluride. Nat Commun 7, 12291 (2016). https://doi.org/10.1038/ncomms12291
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