Measurements of nonequilibrium interatomic forces using time-domain x-ray scattering

We demonstrate an experimental approach to determining the excited-state interatomic forces using femtosecond x-ray pulses from an x-ray free-electron laser. We determine experimentally the excited-state interatomic forces that connect photoexcited carriers to the nonequilibrium lattice dynamics in the prototypical Peierls-distorted material, bismuth. The forces are obtained by a constrained least-squares fit of a pairwise interatomic force model to the excited-state phonon dispersion relation as measured by the time- and momentum-resolved x-ray diffuse scattering. We find that photoexcited carriers weaken predominantly the nearest-neighbor forces, which drives the measured softening of the transverse acoustic modes throughout the Brillouin zone as well as the zone-center $A_{1g}$ optical mode. This demonstrates a bond-selective approach to measuring electron-phonon coupling relevant to a broad range of photoinduced phase transitions and transient light-driven states in quantum materials.

Figure 1

(a) Diffuse scattering intensity of bismuth in our experimental geometry. (b) Representative time-resolved diffuse scattering signals at the color-coded points shown in (a) at 2.5 $\mathrm{mJ}/{\mathrm{cm}}^2$. (c) Observed frequencies of phonon mean-square displacements for photoexcited bismuth as a function of incident laser fluence. The data correspond to the wave vectors in (d). Solid lines show the best fits to twice the phonon frequency. The inset in (c) shows the Brillouin zones in the lower right quadrant of reciprocal space subtended by the detector in (a), and the line cut along the detector the points in (c) correspond to. The illustrated path (purple) of the line cut shown in (c) and the Brillouin zone boundaries (blue) are also indicated. (e) Time-dependent change in the Bragg intensity of the $\left(223\right)$ peak, showing oscillations of the $A_{1g}$ mode about its new quasiequilibrium position. (f) Time-dependent relative change in the diffuse scattering intensity for three laser fluences at the wave vector $\mathit{Q}=(1.00,1.61,1.39)$ reciprocal lattice units (r.l.u.) [black dot in the inset to (c) and dashed line in (c) and (d)]. The black lines through the data in (b), (e), and (f) correspond to the best fits as described in the text. (g) Fitted frequencies to the oscillations of the $A_{1g}$ mode shown in (e).

Figure 2

(a) Illustration of the crystal structure of bismuth, with key bonds in the model highlighted. Blue and green bonds are the first- and second-nearest-neighbor bonds used in the model. The ninth-nearest-neighbor bonds connect the second-nearest neighbors in the chain direction. (b) Illustration of the nearly sixfold coordinated bonding cluster around a Bi atom (highlighted), showing the three bonds [first-, second-, and ninth-nearest neighbors (NNs)] used in the fit. The direction of the $A_{1g}$ mode ([111] direction) is shown. (c) Traces of force matrices of the three largest interatomic forces as a function of fluence, for atom pairs illustrated in (b).

Figure 3

(a) Reconstructed phonon dispersion along the high-symmetry directions in bismuth. Black lines show the (extrapolated) ground-state dispersion, and blue lines show the photoexcited dispersion at 5.5 $\mathrm{mJ}/{\mathrm{cm}}^2$. Pink shaded areas indicate systematic uncertainties in the dispersion relation as described in the text. (b) DFPT calculations of the ground-state and excited-state forces for the same excitation level as in (a). v.e., valence excitation.