Observation of photo-induced plasmon–phonon coupling in PbTe via ultrafast x-ray scattering

In polar semiconductors, Fröhlich electron–phonon interactions lead to strong coupling between collective electronic and longitudinal lattice excitations. The coupling can be pronounced in the group IV–VI compounds due to the combination of high polarizability and large longitudinal optical (LO)/soft transverse optical (TO) phonon splitting near zone center. This leads to rapidly dispersing LO phonon–plasmon coupled (LOPC) modes^1–4 that can affect nonequilibrium properties, such as carrier relaxation⁵ and transport,⁶ and have implications for thermoelectric transport,⁷ ferroelectricity,⁸ and superconductivity at high carrier densities.⁹ Inelastic neutron scattering (INS) measurements^1,2,10–12 on several doped group IV–VI semiconductors, including PbTe, show an anomalous dip in the low-wavevector (long-wavelength) dispersion of the LO phonon branch, due to screening from the free carrier concentrations.

Ultrafast photoexcitation can be used to transiently control material properties, for example, through the excitation of large amplitude vibrational motion or the generation of large carrier densities. Such excitation can, furthermore, lead to a nonequilibrium state with dramatically different properties from the ground state. In the case of PbTe, where the LO–TO splitting is large, we may, therefore, expect that photoexcitation could be an effective control parameter for nonequilibrium properties. For example, the spectrum of THz emission from ultrafast laser excited PbTe was recently shown to be widely tunable depending on the photocarrier density.¹³ Thus, it is important to better understand the coupling between collective electronic and longitudinal lattice excitations. Photoexcited LO-phonon–plasmon coupled (LOPC) modes have been observed in all-optical experiments in the III–V compound GaAs^14–16 as well as PbTe¹⁷ and PbTe_0.95S_0.5.¹⁸ Ultrafast, time-resolved x-ray diffuse scattering has the advantage that it allows us to directly observe the dispersion of the LOPC modes and their renormalization due to photo-excitation. Here, we present femtosecond Fourier-transform inelastic x-ray scattering (FT-IXS)^19–22 measurements of near zone center excitations in photoexcited PbTe. Using this method, we have previously shown that near bandgap photoexcitation in PbTe couples the TO and transverse acoustic (TA) modes at high wavevector along the bonding direction, reducing the ferroelectric instability and stabilizing the paraelectric state.²² In this case, photoexcitation is also expected to strongly affect the LO phonon through coupling to the photoexcited plasma. Indeed, in the current work, we observe a heavily damped mode that strongly disperses with the increasing wavevector from near the zone center TO frequency to the LO frequency. We attribute the time- and wavevector-dependent signal to squeezed oscillations of the LOPC mode, likely due to coupling to longitudinal acoustic (LA) modes that drive shifts in the electronic bands via the acoustic deformation potentials.

The experiment was performed at the x-ray pump probe (XPP) instrument²³ of the Linac Coherent Light Source (LCLS) x-ray free-electron laser. Details of the experimental setup can be found in Ref. 22. Briefly, infrared (IR) pulses of light (60 fs, 350 μJ, 0.6 eV) generated from an optical parametric amplifier laser were used as the pump source and hard x-ray pulses (50 fs, 8.7 keV) as the probing mechanism. The energy of the pump source was chosen to just exceed the direct bandgap of PbTe (∼0.31 eV at room temperature²⁴). A large area Cornell-SLAC Pixel Array Detector (CSPAD) captured the resulting x-ray diffuse scattering over a wide region of reciprocal space. Diffuse scattering patterns were recorded at room temperature tracked as a function of time delay τ between the IR pump and x-ray probe pulses at binned step sizes of 100 fs.

We chose a fixed sample and detector configuration such that we capture scattering with a momentum transfer near the $(1¯13)$ Brillouin zone. Brillouin zones with odd h + k + l are sensitive to both optical and acoustic phonons for the rock salt structure. The sample was detuned ∼1% from the Bragg condition for reciprocal lattice vector $G=(1¯13)$ in reciprocal lattice units (rlu) to prevent the full intensity of the Bragg reflection from hitting the detector. In this geometry, two high-symmetry reduced wavevector directions are captured simultaneously (approximately): Γ to X with $q∼(0 qy 0)$ and Γ to W with $q∼(qx 0 qz=2qx)$⁠. Notably, in our measurement scheme, the two directions have varying sensitivity to phonon polarization due to the $(Q·e)2$ dependence in the scattering intensity, where e is the phonon polarization and $Q=G+q$ is the momentum transfer (divided by $ℏ$⁠). Thus, the diffuse scattering along Γ to X is primarily sensitive to phonons of transverse polarization [along the (001), z-direction]. Conversely, the diffuse scattering along Γ to W is largely sensitive to phonons of longitudinal polarization.

The differential scattering intensity $δI(τ;Q)=I(τ;Q)−I0(Q)$ is collected as a function of time-delay, τ. $I0(Q)$ is the unpumped signal, collected for time delays where the x-ray probe arrives prior to the pump pulse (⁠$τ<0$⁠). The behavior along the two high-symmetry directions described above is shown in the time-domain in Fig. 1. Figures 1(a) and 1(b) show $δI(τ;Q)/I0(Q)$ along the respective Γ to X and Γ to W wavevectors. Each trace represents the time-dependent differential changes in a pixel on the detector and, thus, a unique Q along one of the labeled high-symmetry wavevectors. The topmost traces depict data nearest to zone center. The traces extracted along the Γ to X wavevector reveal a strong decrease in scattering intensity immediately following the arrival of the pump pulse and weak modulations with long periods. The activity along Γ to X across the entire BZ has already been described in a prior report.²² Along the Γ to W wavevector, on the other hand, substantial coherent oscillations with much shorter periods are present, damping on a sub-picosecond timescale. Moreover, the oscillation period shortens with increasing q from Γ, as can easily be seen in the positive dispersion shown in the Fourier transform [Fig. 1(d)].

A sudden change in the interatomic force constants before and after photoexcitation leads to the time evolution of the displacement correlations between phonon modes. The time-dependent diffuse x-ray signal is proportional to those displacement correlations and its Fourier transform gives the FT-IXS spectrum.^19,25 We plot the FT-IXS spectra^19–21 of the collected time-domain data in Figs. 1(c) and 1(d) for a range of q from near Γ toward X and W, respectively. The amplitudes are displayed in false color on a logarithmic (base-10) scale as a function of $|qy|$ in Fig. 1(c) and $|qz|$ for 1(d). The complete trajectories of q coordinate values for both wavevectors are displayed in rlu above the spectra.

We identify some of the features in the FT-IXS spectra with the help of first principles calculations^22,25–27 to obtain the displacement correlations between phonon modes due to a sudden promotion of valence electrons to the conduction band. The full details of the calculations can be found in Ref. 22. In Fig. 1(c), an overtone transverse acoustic mode (2TA) is identified as shown by the black trace. The appearance of this mode near Γ is consistent with the results discussed in Ref. 22 in which the same mode is identified along Γ to X, all the way out to zone edge.

Similarly, along Γ to W, the calculated excited-state dispersion for the overtone TA mode is overlaid as a black trace in the spectrum of Fig. 1(d) and agrees well with the low-frequency feature near ∼0.25 THz. Note that although this specific wavevector direction is primarily sensitive to longitudinal phonon polarization, the overtone TA branch still appears due to a non-negligible residual sensitivity to transverse polarization. However, these calculations do not predict the intense and broad highly dispersive feature appearing at higher frequencies. As described below, we attribute this feature as likely due to the displacement correlations between the photoexcited LOPC mode and LA modes that also couple to the photoexcited plasma.

The interaction between plasmons and the LO phonon mode in polar materials^1–4 screens the macroscopic electric field associated with the LO branch, reducing its strength by a factor equaling the low frequency dielectric constant $ε(q,0)=ε0(1+ks2/q2)$⁠, where k_s is the Thomas–Fermi screening wavevector.²⁸ In the long-wavelength limit, the coupling is largest when the zone center plasma frequency $ωp(q=0)$ equals the LO frequency $ωLO(q=0)$⁠. Here, the plasma frequency depends on the concentration of free carriers n, their charge and effective mass e and $m*$⁠, the vacuum permittivity $ε0$⁠, and the high-frequency dielectric constant $ε∞$ as

For group IV–VI semiconductors, $ωp(q=0)$ ∼ $ωLO(q=0)$ for carrier concentrations as small as ∼10¹⁷ cm⁻³⁴.

At higher carrier densities, when $ωp(q=0)$ exceeds $ωLO(q=0)$⁠, the dielectric function can be approximated in the quasistatic limit (⁠$ω→$ 0) as $ε(q,0)$⁠. For $q≪ks$⁠, the electric field of the LO mode is dramatically screened leading to dispersion in the lower frequency coupled plasmon–LO phonon,¹ 

where $ωTO(q=0)$ is the zone center TO phonon frequency (∼0.95 THz for PbTe at room temperature).¹¹ 

For large enough carrier density, such as in dense photoexcitation, the Fermi energy exceeds the thermal energy and we can approximate the carriers as a degenerate Fermi gas. In this case, we approximate the Fermi-wavevector within a single L-valley as $kF=(3π2(n/4))1/3$⁠, such that the Thomas–Fermi screening wavevector

Thus, in our experiments, strong plasmon–phonon coupling is expected from the initially low-carrier-density (4 × 10¹⁷ cm⁻³) n-type PbTe upon photoexcitation of $∼2×1020 cm−3$ carriers per L-valley above the bandgap (⁠$∼0.5$ % valence excitation). In this excitation regime, the frequency of the photoexcited plasma $ωp(q=0)$ (∼73 THz) far exceeds $ωLO(q=0)$ (∼3.42 THz for PbTe).¹¹ Here, $kF≈0.186$ rlu and $ks≈0.083$ rlu, respectively. The use of a degenerate Fermi gas model to compute these parameters is reasonable since the approximate Fermi energy (∼1.3 eV) is larger that the expected thermal energy of the photoexcited carriers (⁠$≈0.5$ eV). Moreover, the Debye screening wavevector for this temperature and density is very similar to the Thomas–Fermi screening wave vector.²⁹ The particular details of this model do not have a significant impact on the forthcoming interpretation of our experimental results due to the weak $n1/6$ dependence for the screening wavevector seen in Eq. (3).

The calculated dispersion in $ω−$ for this simple model at both equilibrium and photoexcited carrier density of $n=4×1017cm−3$ and $2×1020cm−3$ is shown in Fig. 2, purple and green traces, respectively. In the photoexcited case, k_s is $∼2.7$ times larger than that for the lower density even though the change in carrier density is more than three orders of magnitude, reflecting the $n1/6$ dependence in Eq. (3). Nonetheless, a dramatic shift in the calculated equilibrium dispersion appears at low q. Due to the low carrier concentration of our samples, the expected equilibrium dispersion deviates appreciably from the LO frequency only in a small region near q = 0. This is reflective of the relatively weak screening of the LO phonon mode at such low carrier densities, and the screened portion of the dispersion would be difficult to resolve in most measurements. On the other hand, for the dense photoexcited case, the screened region of the dispersion of the LOPC mode extends considerably further from the zone center, reflective of the decreased screening length (increased k_s).

The data in Fig. 1 extend to approximately the k_s calculated above. Thus, we expect that the LOPC mode $ω−(q)$ should appear prominently in this region, given the strong sensitivity to longitudinal phonon polarization along this wavevector direction. The computed low-q screened LO phonon dispersion [Eq. (2)] along Γ to W is overlaid in blue. A close agreement with the highly dispersive spectral feature observed in the FT-IXS experiment is achieved for the estimated photoexcited carrier density. Thus, we attribute the broad high frequency dispersive feature to the photo-induced LOPC mode.

In the FT-IXS measurement, the LOPC mode would be expected to appear at either twice its frequency or in combination with low energy acoustic modes.²² 

The photoexcited electron–hole plasma gives rise to a coupling between the LO and LA modes. This arises because the shifts of the conduction and valence bands caused by strain act like an electric field on the carriers, driving them as the LA mode oscillates. The electric field of the charge carriers then interacts with the polar LO phonon. The details of this plasma-mediated interaction between the LO and LA modes are given in  Appendix A. This LO–LA interaction, arising only in the presence of the photoexcited plasma, gives rise to a phonon squeezing signal at the sum and difference frequencies of the LO and LA modes, as observed in the experiment. The theory predicts that the signal goes to zero as $q→0$ and peaks at a wave vector, $q=ksε∞/εst$⁠, where $ε∞$ is the high-frequency dielectric constant of PbTe (in the absence of the carriers) and $εst$ is the static dielectric constant of PbTe. This agrees well with observed range of wave vectors at which a large squeezing signal is detected. In  Appendix B, we show that IXS oscillations at twice the LO mode frequency would not occur in the classical coupled phonon–plasma system, suggesting that such a signal should be weak in the current experiment. In principle, coupling of the LO mode to single-particle carried excitations could also give rise to IXS oscillations near the LO mode frequency.

The observed photo-induced plasmon–phonon state and the resulting screening of the LO phonon branch are consistent with early INS data on PbTe at degenerate carrier concentrations.^1,10,11 The difference is that in the current experiment, we measure the screened LO mode dispersion in a material with much lower equilibrium carrier concentration (⁠$4×1017$ cm⁻³ prior to photo-excitation) that is suddenly excited to very high concentration (10²⁰–10²¹ cm⁻³). We note that from a plasma and screening perspective, our photoexcited density is in a similar regime as the highly doped crystals measured with INS; however, we emphasize that the current measurements do not reflect the spontaneous scattering from single LOPC and other excitation, but instead they represent the time-dependent scattering from the nonthermal and nonstationary state produced by the sudden excitation. This results in the measurement of correlated phonon pairs, for example, in the TA overtone and TA ± TO combination modes seen in Ref. 22 to span the entire Brillouin zone. Here, we have shown further that a high photoexcited carrier density in PbTe substantially affects the LO modes by sudden modification of the screening, giving rise to the generation of nonthermal LOPC modes of PbTe.