Mapping momentum-dependent electron-phonon coupling and nonequilibrium phonon dynamics with ultrafast electron diffuse scattering

Despite their fundamental role in determining material properties, detailed momentum-dependent information on the strength of electron-phonon and phonon-phonon coupling (EPC and PPC, respectively) across the entire Brillouin zone has remained elusive. Here we demonstrate that ultrafast electron diffuse scattering (UEDS) directly provides such information. By exploiting symmetry-based selection rules and time resolution, scattering from different phonon branches can be distinguished even without energy resolution. Using graphite as a model system, we show that UEDS patterns map the relative EPC and PPC strength through their profound sensitivity to photoinduced changes in phonon populations. We measure strong EPC to the $K\text{−}\mathrm{point}$ TO phonon of $A_1^{\prime }$ symmetry $\left(K\text{−}{A}_1^{\prime }\right)$ and along the entire TO branch between $\mathrm{\Gamma }\text{−}K$, not only to the $\mathrm{\Gamma }\text{−}{E}_{2g}$ phonon. We also determine that the subsequent phonon relaxation of these strongly coupled optical phonons involve three stages: decay via several identifiable channels to TA and LA phonons $(1-2$ ps), intraband thermalization of the non-equilibrium TA/LA phonon populations $(30–40$ ps) and interband relaxation of the TA/LA modes (115 ps). Combining UEDS with ultrafast angle-resolved photoelectron spectroscopy will yield a complete picture of the dynamics within and between electron and phonon subsystems, helping to unravel complex phases in which the intertwined nature of these systems has a strong influence on emergent properties.

Figure 1

Electron-phonon coupling and phonon-phonon coupling in graphite. (a) Simplified schematic of the electronic band structure of graphite illustrating the influence of the Dirac cones on the electron-phonon scattering process. In these experiments a pump laser pulse drives vertical electronic transitions $\left(\pi \text{−}{\pi }^{★}\right)$. The resulting hot electrons (see text) may inelastically scatter with the $\mathrm{\Gamma }\text{−}{E}_{2g}$ phonon across a Dirac cone and with the $K\text{−}{A}_1^{\prime }$ phonon between Dirac cones, conserving energy and momentum. (b) Phonon dispersion of graphite with the strongly coupled optical modes indicated. The orange bars indicate half the energy of the $K\text{−}{A}_1^{\prime }$ and $\mathrm{\Gamma }\text{−}{E}_{2g}$ modes. The dominant energy and momentum-conserving decay pathways are indicated with colored arrows. For example, the green arrows labeled $A_1^{\prime }$:TA-LA on the left represent the decay from the strongly coupled $A_1^{\prime }$ mode at $K$ to both TA and LA modes at the midpoint of the $\mathrm{\Gamma }\text{−}K$ line. Dotted arrows should be thought of as going in the opposite momentum direction. The side bar (red) provides a schematic of the nonequilibrium LA/TA phonon distribution produced through the decay of $\mathrm{\Gamma }\text{−}{E}_{2g}$ and $K\text{−}{A}_1^{\prime }$ phonons as determined by the UEDS data described in the text.

Figure 2

Evolution of $\mathrm{\Delta }I(\mathit{q},\tau )$ following photoexcitation of graphite (35 fs, 800 nm, $12\mathrm{mJ}/{\mathrm{cm}}^2)$. The dramatic changes reflect the nonequilibrium phonon populations and their time dependence. (a) Raw diffraction pattern of graphite along the [001] zone axis showing sixfold symmetry of the graphene planes. (b) Differential scattering flat field $\mathrm{\Delta }I(\mathit{q},\tau )$ at a time before optical excitation indicating signal to noise. (c) $\mathrm{\Delta }I(\mathit{q},0.5\text{ps})$ provides a map of the relative strength of the $\mathit{q}$-dependent EPC coupling through the increased occupancy of strongly coupled modes. Peaks in $\mathrm{\Delta }I(\mathit{q},0.5\text{ps})$ at the $K\mathrm{points}$ surrounding $\left\{2\overline{1}0\right\}$ (circled) result from the increase in $K\text{−}{A}_1^{\prime }$ population and outline the hexagonal BZ. Scattering from the $\mathrm{\Gamma }\text{−}{E}_{2g}$ TO phonon is forbidden around $\left\{2\overline{1}0\right\}$, but strong coupling to the entire TO branch is evident in the vicinity of $\left\{200\right\}$ as ridges of intensity radiating from $\mathrm{\Gamma }$ (the Bragg peak) to $K$ points. (d)–(f) Nonequilibrium phonon dynamics: relaxation of the transient population of strongly coupled optical modes. (d) At 1.5 ps the peaks evident at $K\mathrm{points}$ in panel (c) have disappeared and diffuse intensity now appears halfway between $\left\{2\overline{1}0\right\}$ and the BZ edge (inset), but is still absent in the $M$ and ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}$ regions. (e) By 5 ps, the character of $\mathrm{\Delta }I(\mathit{q},\tau )$ has changed dramatically to bands of intensity in the ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}\text{−}M\text{−}{\mathrm{\Gamma }}_{\left\{200\right\}}$ direction approximately orthogonal to $\mathit{q}$, with troughs near ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}$ remaining. (f) At 100 ps the ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}\text{−}M\text{−}{\mathrm{\Gamma }}_{\left\{200\right\}}$ bands have become sharper and the troughs near ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}$ have filled in. Strong halos of diffuse intensity are present around the $\left\{100\right\}$ and $\left\{110\right\}$ families of peaks are evident (inset). These halos are weak, but present at 5 ps [inset, panel (e)].

Figure 3

Ultrafast electron diffuse scattering. (a) Intensity of the $\left\{2\overline{1}0\right\}$ Bragg peak showing profoundly nonexponential Debye-Waller dynamics [26]. (b) $\mathrm{\Delta }I(\mathit{q},\tau )$ at select points in the BZ (inset). The rate of increase in the population (EPC) of the TO $K\text{−}{A}_1^{\prime }$ phonon (red, 280 fs) is faster than that for the TO $\mathrm{\Gamma }\text{−}{E}_{2g}$ (cyan, 430 fs) and matches the fast Bragg peak dynamics. The population $K\text{−}\mathrm{LO}$ phonons (green, 730 fs) rises much slower than both TO $K\text{−}{A}_1^{\prime }$ and TO $\mathrm{\Gamma }\text{−}{E}_{2g}$ phonons. The rise in diffuse intensity at the $M$ point (blue, 2.1 ps) is almost an order of magnitude slower than that associated with the TO $K\text{−}{A}_1^{\prime }$ phonon. The slow time-scale decay evident in the Bragg peak [26] and reported in earlier ARPES measurements [13] does not emerge from the dynamics of any single mode, but is a composite of the decay in population of the strongly coupled optical modes (e.g., red, 1.7 ps) and the increase in population of all other modes. (c) Diffuse intensity dynamics for points along the $K\text{−}\mathrm{\Gamma }$ line, and $\mathrm{\Gamma }\text{−}M$ line (b) (inset, magenta). Rise time (blue) and amplitude (red) from single exponential fits to the early-time dynamics are shown. These dynamics are associated phonon-phonon decay (PPC) from the SCOPs to the acoustic modes. Error bars represent covariance in fit parameters. Points close to $\mathrm{\Gamma }$ are not shown due to the interference of the Bragg peak Debye-Waller dynamics of (a).

Figure 4

Diffuse intensity near ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}$ (gray) and ${\mathrm{\Gamma }}_{\left\{200\right\}}$ (cyan). Early-time diffuse intensity changes demonstrate how time resolution and selection rules can be used to separate the dynamics of phonon branches at the same momentum. The cyan curve displays a quasi-biexponential character while the gray curve is a single exponential rise (1.5 ps). The fast cyan component (430 fs, solid line) has been assigned to EPC to the $\mathrm{\Gamma }\text{−}{E}_{2g}$ mode. This signal is absent from the gray curve due to symmetry-imposed selection rules. The 1.5 ps rise in the gray curve (solid line) is associated with the low-frequency LA/TA populations generated through $K\text{−}{A}_1^{\prime }$ phonon decay. The longer time constant in the cyan curve is a composite time scale associated with the decay of the $\mathrm{\Gamma }\text{−}{E}_{2g}$ population and the increase in the low-frequency LA/TA populations. Inset: Diffuse intensity changes for the entire time range. The 34 ps rise in intensity in the gray curve is associated with the intraband thermalization of the nonequilibrium LA and TA populations generated through the decay of strongly coupled $\mathrm{\Gamma }\text{−}{E}_{2g}$ and $K\text{−}{A}_1^{\prime }$ phonons. The 115 ps decay most evident in the ${\mathrm{\Gamma }}_{\left\{2\overline{1}0\right\}}$ data is associated with interband equilibration of the phonon populations in the LA, TA, and ZA bands. Solid gray line shows triexponential fit to the 1.5 ps rise, 34 ps rise, and 115 ps decay.