Direct time-domain determination of electron-phonon coupling strengths in chromium

We report the results of an ultrafast, direct structural measurement of optically pumped phonons in a Cr thin film using ultrashort x-ray pulses from a free-electron laser. In addition to measuring and confirming the known long-wavelength dispersion relation of Cr along a particular acoustic branch, we are able to determine the relative phase of the phonons as they are generated. The Cr sample exhibits two generation mechanisms for the phonons: the releasing of a preexisting charge density wave at higher frequencies, and the creation of an acoustic strain pulse via laser heating that dominates at lower frequencies. For the latter mechanism, we are able to measure the frequency dependence of the time required to generate the phonons. To explain the observed magnitude and slope of the delays, we perform first-principles simulations in the framework of density functional perturbation theory and ab initio molecular dynamics to fit anharmonic phonon models. These results show that the wave-vector dependence of the electron-phonon coupling is the driving mechanism behind the delay times: Phase-space limitation leads to higher times near the zone center. The absolute magnitudes of the delay times measured are found to be much shorter than the equilibrium electron-phonon coupling times we compute, indicating that the coupling strength is greatly enhanced when the electronic system is out of equilibrium with the lattice, as has been seen in bismuth and other systems.

Figure 1

The axes and black data points are from data taken at APS. The central peak is the [002] Bragg peak for Cr, and the Laue fringes around it have a periodicity corresponding to the 28-nm film thickness. Shown behind this are three separate area detector images taken at LCLS, corresponding to different cuts through these fringes on the Bragg rod, achieved by rocking the sample. On the right, a simple schematic of the LCLS experimental setup is shown.

Figure 2

(a) Slices through the Bragg rod for the low-$Q$ fringes combined in one image, integrated vertically with respect to Fig. 1, normalized by the $t<0$ values, and plotted vs time (Bragg peak not shown). (b) The extracted 1D signals for the fringes shown in (a).

Figure 3

The dispersion curve points for all measured fringes on the low-$Q$ side are shown, along with their amplitudes. The red line shown is a best fit through all the frequency data points, whose slope gives the longitudinal speed of sound in the Cr film normal to the surface, which is found to be 7.45 nm/ps for our sample. The enhancement of fringes 5, 6, and 7 is attributed to residual CDW ordering that persists above the Néel temperature. The amplitude fit (blue line) is described in the text.

Figure 4

(a) Experimental phonon generation lag times: The time by which $t=0$ must be shifted for each fringe to achieve the phase relation we expect. (b) Calculated equilibrium lifetimes for the longitudinal acoustic phonon mode in Cr (AFM GGA solution), for small $\vec{q}$ along the high-symmetry $\mathrm{\Gamma }\text{−}H$ line.