Phonon confinement effects in ultrathin epitaxial bismuth films on silicon studied by time-resolved electron diffraction

The transient temperature evolution of ultrathin bismuth films, epitaxially grown on a silicon single crystal, upon femtosecond laser excitation is studied by time-resolved electron diffraction. The exponential decay of the film temperature is explained by phonon reflection at the interface, which results in a strongly reduced thermal conduction in the cross plane of the layered system. The thermal boundary conductance is found to be as low as $1273\text{W}/\left(\text{K}{\text{cm}}^2\right)$. Model calculations, including phonon confinement effects, explain the linear relationship between the observed film-temperature decay constant and the film thickness. Even for 2.5 nm thin films the phonon transmission probability across the interface is given by bulk properties. Our simulations show that phonon confinement effects are negligible for bismuth-film thicknesses larger than 1 nm.

Figure 1

(a) NC-AFM image of a 2.5 nm thin Bi film on Si(111). The film was grown at 150 K and annealed to 420 K and subsequently held at 400 K for 10 min. The scanning range of the image is $2.5\mu \text{m}\times 2.5\mu \text{m}$. Steps of the underlying Si(111) substrate are clearly visible. The marked area (dashed square) is displayed in (b) with a magnification factor of 3. The step height between two adjacent terraces is one bilayer, i.e., $3.94\text{Å}$. (c) LEED pattern of the 2.5 nm thin Bi film. As evident from the hexagonal symmetry the Bi film is (111) oriented.

Figure 2

Transient surface temperature of a 2.5 nm thin Bi film. The laser fluence is ${\text{Q}}_{\text{A}}=2.3\text{mJ}/{\text{cm}}^2$. The experimental data are shown as markers. The solid line is a fit of Eq. (6). The fit yields a temperature jump of $\left(157\pm 8\right)\text{K}$, a time constant for the increase of ${\tau }_{\text{inc}}=\left(41\pm 4\right)\text{ps}$, and a decay constant of ${\tau }_{\text{dec}}=\left(243\pm 15\right)\text{ps}$. The inset shows the RHEED pattern at a negative delay, i.e., excitation occurs after scattering of the electron probe pulse. The indicated diffraction spot was used to extract the transient temperature evolution.

Figure 3

(Color) (a) Normalized transient temperature evolutions $\left\{\left[T\left(t\right)-T_0\right]/\left(T_{\text{max}}-T_0\right)\right\}$ for different film thicknesses. The base temperature $T_0=90\text{K}$ was the same for all experiment. $T_{\text{max}}$ is the maximum temperature at zero delay. The pump fluence for all experiments was ${\text{Q}}_{\text{A}}=2.3\text{mJ}/{\text{cm}}^2$. (b) Thickness dependence of the decay constant ${\tau }_{\text{dec}}$. (c) Close-up view of the thickness range between 0 and 11 nm. The dashed line in (b) and (c) is a line fit to the experimental data yielding a slope of $\left(93.5\pm 1.2\right)\text{ps}/\text{nm}$. In (c) the results of model calculations for the decay constant using the AMM (dots) and the DMM (solid line) for the determination of the phonon transmission coefficient are additionally displayed (modified AMM/DMM, see text for details). (d) Thickness dependence of the phonon group velocity for the transversal and longitudinal polarized phonons which is used in the model calculations. The horizontal lines are the respective group velocities for bulk Bi.

Figure 4

(Color) (a) Partial VDOS of the system containing 48 Bi-like (red) and 100 Si-like (blue) layers. The acoustic part of the Si-like slab is additionally displayed with a magnification of 30 (green). (b) Partial VDOS of $N=1,2,4,8,16,32,48$ Bi-like layers on 100 Si-like layers. All VDOS are normalized so that the integral over the energy results in unity. The VDOS of the $N=32$ and $N=48$ Bi-like layers in (b) are identical.