Measurement of strong phonon softening in Cr with and without Fermi-surface nesting by inelastic x-ray scattering

Nesting of the Fermi surface can soften and broaden phonons at the nesting wave vectors. However, huge electron-phonon anomalies unexpected from the experimental topology of the Fermi surface have been reported in copper oxide superconductors and their origin remains enigmatic. Here we present results of inelastic x-ray scattering measurements that uncovered similarly pronounced softening of phonons in chromium that occur far from the nesting wave vectors of the Fermi surface. The softening appears in addition to the previously reported phonon anomalies at the nesting wave vectors. Calculations in the local-density approximation show that the anomalies result from the enhancement of the electron-phonon coupling constant. Our results demonstrate that strong electron-phonon coupling alone can result in strong phonon anomalies limited to a small range of wave vectors. They imply that the phonon anomalies in copper oxide superconductors may be explained by an enhanced electron-phonon coupling strength without invoking novel collective modes or some hidden nesting of the Fermi surface.

Figure 1

(a) Phonon dispersion as calculated by means of the Born-von-Karman model along the high-symmetry directions. Data points represent previous neutron data (Ref. 15). Arrows indicate anomalies as observed by neutron scattering (Ref. 15). Branches of interest here are shown in bold. (b) Raw data from inelastic x-ray scattering as function of energy transfer at $T=320\text{K}$ and $\mathbf{Q}=\left(0.5,3.5,0\right)$. The solid line corresponds to a fit to the data using a Lorentzian.

Figure 2

(Color online) Comparison between experiment and Born-von-Karman calculations. The experiments were performed at $T=300\text{K}$. (a) A schematic of the scans in reciprocal space. Hatched ovals indicate previously observed phonon anomalies; [(b)–(g)] calculated (blue circle) and measured (red square) phonon dispersions along the paths indicated in (a); (h) difference $\left(E_{calc}-E_{exp}\right)$ between the calculated and experimental phonon dispersion along the zone boundary N-H (blue circle) and along the high-symmetry direction $\Gamma$-H (red square).

Figure 3

(Color online) Comparison between phonon dispersions and the LDA calculation. [(a) and (b)] Black open circles represent a factor of two more smearing than the blue open squares. [(c) and (d)] Experiment and calculation away from the high-symmetry directions.

Figure 4

(Color online) Joint density of states for electron-hole excitations, whose “hotspots” (in red) correspond to potential phonon anomalies.

Figure 5

(Color online) Interband and intraband contributions to the phonon linewidth and the electronic joint density of states (see Fig. 3). (a) Calculated (100)/(010) cut of the Fermi surface of Cr. Numbers (1,2,3) index the three bands crossing the Fermi surface that are referenced in the legend in (b). (b) and (c) show the contributions of the six relevant electronic transitions to the phonon linewidth and the imaginary part of the susceptibility, respectively, along the [110] direction. Similarly, (d) and (e) and (f) and (g) show the contributions along $\mathbf{q}=\left(h+0.2,h,0\right)$ and along the [100] direction, respectively. The legend in (b) applies to all graphs [(b)–(g)].