Experimental Characterization of Defect-Induced Phonon Lifetime Shortening

We present the first direct experimental measurement of defect-induced lifetime shortening of acoustic surface phonons. Defects are found to contribute a temperature-independent component to the linewidths of Rayleigh wave phonons on a Ni(111) surface. We also characterized the increase in phonon scattering with both surface defect density and phonon wave vector. A quantitative estimate of the scattering rate between phonon modes and surface line defects is extracted from the experimental data for the first time.

Figure 1

Schematic representation of a HeSE phonon measurement. The ${}^3\mathrm{He}$ beam is scattered from the Ni(111) sample surface, where dashed lines represent the trajectory of the beam and the surface normal. The total scattering angle, ${\theta }_{\mathrm{tot}}=44.4°$, is fixed; the sample can be rotated to change the incidence angle ${\theta }_i$ to investigate specific phononic states. $k_i$ and $k_f$ are the wave vectors of the incident and scattered ${}^3\mathrm{He}$ atoms, with surface parallel components $K_i=k_i\sin {\theta }_i$ and $K_f=K_i+\mathrm{\Delta }K=k_f\sin ({\theta }_{\mathrm{tot}}-{\theta }_i)$, respectively. The ${}^3\mathrm{He}$ beam is nuclear spin polarized along the $x$ direction before entering the solenoids. In the two solenoids, the atoms undergo spin precession, as illustrated by the helices. The final beam polarization is measured along the two perpendicular directions, giving $P_x$ and $P_y$.

Figure 2

(a) Typical HeSE energy transfer spectrum (green squares) obtained at an incident angle ${\theta }_i=29.2°$. The blue dotted line is a Lorentzian function representing the RW phonon peak, while the purple dashed line is a quartic polynomial representing the background. The black solid line shows the sum of the two. The inset shows the raw spin polarization data as a function of the solenoid current magnitude $(I_1^2+I_2^2{)}^{1/2}$, from which the energy transfer spectrum is obtained by Fourier transforming and scaling. (b) Energy transfer spectra for six different values of ${\theta }_i$ from 25.2° to 30.2°, in steps of 1° and including surface parallel momentum transfer, $\mathrm{\Delta }K$. Black lines represent the fits to the sum of a Lorentzian function and quartic polynomial. (c) Centers of the Lorentzian functions in (b), corresponding to the RW phonon energies, as a function of phonon wave vector or momentum transfer, $q$. (d) RW phonon linewidths, obtained from the FWHM of the Lorentzian functions, as a function of $q$. Error bars in this figure and the subsequent figures represent 95% confidence interval of the fitting process. All data are collected along the $\mathrm{\Gamma }\mathrm{M}$ direction on a Ni(111) surface, with a ${}^3\mathrm{He}$ beam energy of 8.07 meV, a surface temperature of 500 K, and a ${}^3\mathrm{He}$ surface reflectivity of $R=0.31$.

Figure 3

(a)–(f) RW phonon linewidths as a function of ${}^3\mathrm{He}$ surface reflectivity. Each marker type represents a certain phonon state, whose energy and momentum are shown in Table 1 and Fig. 2. Dotted lines represent the best fit of the first six data points to $\gamma =-p_{1}R+p_2$. (g) The fitted parameters $p_1$ as a function of phonon wave vectors. (h) Phonon lifetimes $\tau$ determined from the data in (a)–(f) using $\tau =\hslash /\gamma$. Mean free paths $\lambda$ are evaluated using $\lambda =v\tau$, where $v$ is given in Eq. (4). Both decrease monotonically with increasing $q$. All the data correspond to the $\mathrm{\Gamma }\mathrm{M}$ azimuth of a Ni(111) surface at 500 K.

Figure 4

Temperature dependence of defect-induced linewidth broadening. Linewidths are shown for a particular phononic state ($q=0.47{\AA }^{-1}$ along the direction 6° away from the $\mathrm{\Gamma }\mathrm{K}$ azimuth, and $\hslash \omega =7\mathrm{meV}$) at various temperatures and two different values of reflectivity of 0.31 and 0.45. Gray dashed lines provide a guide to the eye, showing a shifted, but otherwise identical temperature dependence at different reflectivities.