Quasielastic light scattering in rutile, $\mathrm{ZnSe}$, silicon, and ${\mathrm{SrTiO}}_3$

Quasielastic light scattering (QELS) has been investigated in the crystals of ${\mathrm{TiO}}_2$ (rutile), $\mathrm{ZnSe}$, silicon, and ${\mathrm{SrTiO}}_3$. The temperature dependence of the linewidth for the QELS has been measured in detail by backward light scattering interferometry and by impulsive stimulated thermal scattering technique in the temperature range from 5 to $670\mathrm{K}$. The quasielastically scattered spectra observed consist of two components, which can be classified into two types, namely, types I and II, depending on the linewidth. The analyses have shown that the linewidth of the QELS I changes from the well-known $q^2$ to a $q^1$ dependence with either decreasing temperature or increasing $q$, where $q$ is the wave-vector transfer in the scattering experiment. It has been found that the linewidth of the QELS I in arbitrary phonon regimes including “hydrodynamic,” “collisionless,” and “intermediate” can be roughly estimated solely in terms of average sound velocity and the “phonon Knudsen number” $q\overline{l}$, where $\overline{l}$ is the mean free path of thermal phonons. A broad doublet spectrum, which was first reported by Hehlen et al. in Phys. Rev. Lett. 75, 2416 (1995), has been observed in ${\mathrm{SrTiO}}_3$ at low temperatures, and its origin has been also discussed in terms of phonon Knudsen number.

Figure 1

Experimental setup for backscattering $(\theta ={180}^{\circ })$.

Figure 2

Experimental setup for ISTS. Here, AOM represents acousto-optic modulator. The modified Sagnac interferometer, which is enclosed with the dashed lines, enables continuous change of beam separation without causing any time delay between the two separated pulses at the focused position.

Figure 3

(Color online) Backscattered spectrum observed in rutile at $297\mathrm{K}$ with $\mathit{q}\Vert \left[100\right]$. The LAs and TAs represent longitudinal and transverse acoustic Brillouin lines, respectively. The two QELS components are labeled as types I and II. The smooth curve is a Lorentzian fit to the QELS I. Note that the vertical scales are normalized at the peak height of the LA Brillouin component. Also note that the middle and the lower spectra are constructed by combining the spectra obtained at different resolutions.

Figure 4

(Color online) Backscattered spectrum observed in $\mathrm{ZnSe}$ at $297\mathrm{K}$ with $\mathit{q}\Vert \left[110\right]$. The LAs and TAs represent longitudinal and transverse acoustic Brillouin lines, respectively. The two QELS components are labeled as types I and II. The smooth curve is a Lorentzian fit to the type I QELS. Note that the vertical scales are normalized at the peak height of the LA Brillouin component. Also note that the middle and the lower spectra are constructed by combining the spectra obtained at different resolutions.

Figure 5

(Color online) Backscattered spectrum observed in silicon at comparable temperatures of $523\mathrm{K}$ (upper) and $473\mathrm{K}$ (lower) with $\mathit{q}\Vert \left[111\right]$. The LAs and SRs represent the Brillouin scattering due to longitudinal acoustic and surface Rayleigh modes, respectively. The two QELS components are labeled as types I and II. The smooth curve is a Lorentzian fit to the type-I QELS. Note that the vertical scales are normalized at the peak height of the LA Brillouin component. In the lower trace, the QELS II is very weak and cannot be well separated from the components of the type I and the Brillouin.

Figure 6

(Color online) Backscattered spectrum observed in ${\mathrm{SrTiO}}_3$ $297\mathrm{K}$ with $\mathit{q}\Vert \left[001\right]$. The LAs and TAs represent longitudinal and transverse acoustic Brillouin lines, respectively. The two QELS components are labeled as type I and II. The smooth curve is a Lorentzian fit to the type-I QELS. Note that the vertical scales are normalized at the peak height of the LA Brillouin component. Also note that the middle and the lower spectra are constructed by combining the spectra obtained at different resolutions.

Figure 7

(Color online) The temperature dependence of the linewidth for the type I and II QELS’s measured by backscattering in ${\mathrm{TiO}}_2$ (rutile) $(q=7.34\times {10}^7{\mathrm{m}}^{-1})$, $\mathrm{ZnSe}$ $(q=7.49\times {10}^7{\mathrm{m}}^{-1})$, silicon $(q=10.2\times {10}^7{\mathrm{m}}^{-1})$, and ${\mathrm{SrTiO}}_3$ $(q=6.00\times {10}^7{\mathrm{m}}^{-1})$. The circles and the triangles represent ${\Gamma }_{\mathrm{I}}$ and ${\Gamma }_{\mathrm{II}}$, respectively. The solid curves represent the $D_{\mathrm{th}}{q}^2$ values. Note that the ordinate axes is scaled logarithmically and the unit is in angular frequency rather than in units of frequency $(\Gamma ∕2\pi )$.

Figure 8

(Color online) Temperature variation of the type-I QELS in ${\mathrm{SrTiO}}_3$. The dots are the experimental spectra and the smooth curves are least-squares fits. For $72\mathrm{K}$, a double unshifted Lorentzian has been assumed and fitted.

Figure 9

(Color online) The broad doublet spectra observed in ${\mathrm{SrTiO}}_3$: from the upper trace, 297, 200, 153, 110, 90, 76, 60, 50, 40, 30, and $15\mathrm{K}$. The arrows indicate the broad doublet, and the LA’s and TA’s represent longitudinal and transverse acoustic Brillouin lines, respectively. The direction of $\mathit{q}$ was chosen to be almost parallel to crystalline [100] direction in cubic symmetry. The TA Brillouin frequency changes across the structural phase transition temperature of $105\mathrm{K}$.

Figure 10

Temperature dependence of the ISTS signal in ${\mathrm{TiO}}_2$ (rutile) in the temperature range from 6 to $296\mathrm{K}$. The left graphs are traced from 0 to $100\mathrm{\mu }\mathrm{s}$, while the right ones are traced from 0 to $5\mathrm{\mu }\mathrm{s}$. The origin of the spikes appearing at temperatures 296 and $202\mathrm{K}$ is discussed in the text.

Figure 11

Temperature dependence of ${\Gamma }_{\mathrm{I}}$ measured by ISTS in ${\mathrm{TiO}}_2$ (rutile), $\mathrm{ZnSe}$, and ${\mathrm{SrTiO}}_3$. The ordinate axes are logarithmically scaled and normalized by $q^2$ for direct comparison with $D_{\mathrm{th}}$. The values of $q$ are indicated in the inset. The measured values well follow the $D_{\mathrm{th}}$ curves in a wide range of temperature.

Figure 12

(Color online) The linewidths of the QELS I and II in the units of the average Brillouin frequency versus the phonon Knudsen number $q\overline{l}$. Legends for the symbols are shown in the figure. The ISTS data points circled by the oval are considered as artifacts, and they should be shifted to $q\overline{l}\lesssim 1$ according to the discussion made in the text. For ${\Gamma }_{\mathrm{II}}$ in silicon, only the room temperature value is presented here because of the lack of accuracy for other temperatures.

Figure 13

(Color online) The least-squares fit of common logarithm of Eq. (16) to that of the measured ${\gamma }_{\mathrm{I}}$ (solid line). The ordinate axis measures the common logarithm of ${\gamma }_{\mathrm{I}}$. The dashed line is the linear function $(1∕3)q\overline{l}$, which is expected from heat transport theory.