Superconductivity-induced distortions of phonon lineshapes in niobium

Superconductivity-induced changes in phonon lineshapes in niobium have been reinvestigated by high-resolution inelastic neutron scattering. We show that the changes go beyond a simple change in lifetime and frequency when the phonon frequency is close to the superconducting energy gap $2\Delta$. The observed lineshapes in elemental niobium are qualitatively similar to those found previously in borocarbide superconductors and agree very well with those predicted by the theory of Allen et al. [Phys. Rev. B 56, 5552 (1997)]. Our results indicate that the peculiar phonon lineshapes in the superconducting state predicted by the theory of Allen et al. [Phys. Rev. B 56, 5552 (1997)] are a general phenomenon and not restricted to a particular class of compounds.

Figure 1

(a) Dispersion of the transverse-acoustic phonon branch in the (100) direction. The line was calculated from the force-constant model published in Ref. 12. Filled symbols depict our results for $T=10.5\text{K}$, open symbols are taken from Fig. 9 in Ref. 2. (b) Phonon dispersion in the off-symmetry direction $\left(0.16,x,x\right)$. The line was again calculated from the force-constant model, whereas the open circles denote results of DFT calculations from Ref. 13. They were shifted upward by 0.9 meV in order to match the experimental value at (0.16,0,0). The arrow denotes the momentum resolution in the direction vertical to the scattering plane.

Figure 2

(Color online) Energy scans taken at $\mathbf{Q}=\left(0.14,1,1\right)$ at (a) $T=2\text{K}$ and (b) 10.5 K. The thick (red) line in (a) represents a simulation based on the lattice-dynamical model of Ref. 12 and the resolution in $\mathbf{Q}$ and $\omega$ of the spectrometer. The resulting spectrum was fitted by five Gaussians being 0.15 meV apart from each other (thin lines). (b) The thick (red) line was obtained by convoluting each Gaussian in (a) by the same Lorentzian and the same rigid energy shift to fit the normal-state data.

Figure 3

(a) Frequency and (b) linewidth (half width at half maximum) of transverse-acoustic phonons in the (100) direction observed above and far below $T_c$. The solid line in (a) represents DFT results (Ref. 13) multiplied by a factor 1.5 to match the experimental data. The inset in panel (a) shows the observed energy shift $\Delta E$ on entering the superconducting state while the dashed, vertical line indicates the superconducting energy gap $2\Delta =3.2\text{meV}$. The linewidth values were obtained by fitting the experimental spectra with a convolution of a Lorentzian and the resolution function of the spectrometer as indicated in Fig. 2a. Note that deviations from a Lorentzian in the superconducting state as discussed in the text were not taken into account in these fits.

Figure 4

(Color online) Energy scans of a phonon with a frequency well above the superconducting energy gap $2\Delta$ taken (a) in the superconducting state and (b) in the normal state, respectively, at $\mathbf{Q}=\left(0.2,1,1\right)$. The red lines depict fits by convoluting the resolution function of the spectrometer with a Lorentzian. The blue line was calculated from the theory of Allen et al. (Ref. 9).

Figure 5

(Color online) Energy scans taken at (a): $\mathbf{Q}=\left(0.16,1,1\right)$, (b): (0.17,1,1), and (c): (0.18,1,1) at temperatures above (red) and below $T_c$ (blue). The fit curves represent a convolution of a Lorentzian with the resolution function. The theoretical curves were obtained using the theory of Allen et al. (Ref. 9) and a value of the superconducting gap as indicated by the dashed vertical line. The experimental background is indicated by the thin horizontal lines.

Figure 6

(Color online) The dashed (red) lines and the full (black) ones depict the theoretical curves before and after convoluting them with the experimental energy resolution. The value of the superconducting energy gap used in the calculations is $2\Delta =3.2\text{meV}$. For the sake of comparison with Fig. 5, the experimental background indicated by the horizontal dotted line has been added. The occurrence of two peaks in the top panel is an artifact related to the decomposition of the phonon line into a small number of individual components: a decomposition into a larger number of components would lead to densely spaced sharp peaks between $3\text{meV}<E<3.2\text{meV}$. However, the lineshape would barely change after convolution with the experimental energy resolution.