A schematic diagram showing two subsystems of electrons and phonons with relevant processes in a different time scale, starting with initial laser heating of electrons near a substrate surface, diffusion of the hot surface electrons to bulk, heat transfer from electrons to phonons, and finally thermal relaxation of the laser-excited area via heat diffusion to dark substrate area or sample holders or via blackbody radiation.
(a) Time dependence for the electron temperature. (b) Time dependence for the phonon temperature. (c) Dynamic showing oscillations with a period equal to a round-trip time for sound to travel across the slab. The simulations with were done using TTM with laser fluence of and a pulse width (FWHM) of . All relevant TTM parameters are listed in Table II. and .
(a) Time dependence for the electron temperature. (b) Time dependence for the phonon temperature. (c) Temporal behavior for the dynamic of a gold substrate with . All other parameters are similar as in Fig. 2. represents collective effects from retarded sound waves originated from each atom in the substrate and reflections from the boundaries. The oscillations for thinner film disappear in this example of a thick film where the round-trip time is longer than the frictional damping time.
(a) Time dependence of dynamic for a gold substrate with . (b) The dependence of the interatomic distance change at various locations of the gold substrate, showing the greatest distance increment occurs at the middle of a slab. All other parameters are similar as Fig. 2. (c) Time dependence for the change of the linewidth for the Bragg peak for a gold substrate with , showing a similar pattern as in (a). The linewidth is defined as the root mean square of the second moment. (d) Time dependence for the change of the integrated intensity of the Bragg peak.
(a) Half oscillation period vs slab thickness for a thin gold film with and 200. The slope equals to the sound velocity showing the time needed for sound waves to travel across the slab. (b) The time delay for the first major peak in and its dependence on slab thickness, in the presence or absence of the electronic Grüneisen parameter . For thinner films the inverse of the initial slope of is slightly greater than the sound velocity of . The slope is slightly greater than the sound speed, indicating the effective length is less than the slab thickness. At a greater thickness the initial linear dependence becomes levelled off, i.e., those atoms much deeper inside a thick slab are not influenced significantly by surface laser heating. A laser fluence of was used here.
The dependence of the maximum for dynamic on the slab thickness, but with a different laser fluence [subplot (a)], frictional damping rate [subplot (b)], and penetration depth of the electron beam [subplot (c)], showing approximately a linear dependence of the maximum on the slab thickness. According to the static thermal expansion mechanism, the change in the interatomic spacing reflects the temperature change at that particular site, and is a local effect independent of other sites or the thickness of the film. However, the nonthermal dynamic expansion presented in this work is a nonlocal effect of the interatomic spacing changes on the surface layers. It represents the retarded wave propagation from all stressed atoms in the slab with each atom acting as a sound wave source caused by thermally induced impulsive stress acting on the atom. A laser fluence of was used here.
Comparison of dynamic for a gold film with [subplot (a)] and [subplot (b)] in the presence (with a ratio of electron and phonon Grüneisen parameters ) or absence of the electron-induced stress . Some differences are present at very short times when the electron temperature is highly elevated, but those differences decrease when both electron and phonon temperatures become equilibrated. Given the same fluence, the electron temperature is higher for a thinner slab; therefore, the electronic contribution to stress is greater. The peak delay time is shortened in the presence of the electron Grüneisen parameter. A laser fluence of was used here.
(a) Two-dimensional plot of the atomic displacement vs time and site index , assuming a gold film with and . The displacement is defined as the position change of the atom at a given time with respect to the center of mass . The inset window on the top shows the displacement vs the site number at a given time indicated by the cross-wire lines. The oscillation patterns illustrate “breathing” motion like an open-end standing wave of a fundamental mode with the slab thickness equal to half wavelength. The inset window to the right is the time dependence of the atomic displacement. The oscillation period is the round-trip traveling time for the sound wave. (b) Two-dimensional plot of the relative interatomic displacement vs time and site number . (c) Two-dimensional plot of the atomic displacement vs time in picoseconds and site number with . For a thicker slab, the time for sound wave to travel across the slab can become substantially longer than the damping time due to friction, and the periodic oscillations are damped out. (d) Two-dimensional plot of the relative interatomic displacement vs time and the site index .
The dependence of , based on the dynamic mechanism, on the square of the slab thickness, but with a different laser fluence [subplot (a)] and frictional damping rate [subplot (b)]. is the ratio between the slab thickness increment and the total slab thickness. It shows a dependence of on from the dynamic expansion mechanism. In contrast, depends only on the temperature jump but not on for the ordinary static linear thermal expansion mechanism. A laser fluence of was used here.
(a) The overall for a gold film of , including the dynamic mechanism from the thermally induced impulsive stress and the static linear thermal expansion mechanism. The dash line indicates the peak delay time of . (b) Dynamic for a gold film while omitting the contribution from static linear expansion. The damping parameter due to friction was arbitrarily set at , which corresponds to a time constant of about . A laser with fluence of , a pulse width of , and were used. (c) The overall for a gold film of , including the dynamic mechanism from the thermally induced impulsive force and the static linear expansion mechanism. (d) Dynamic for a gold film without the contribution from the static linear expansion.
(a) for a gold film of with overall contributions from both dynamic and static mechanisms. (b) Dynamic for a gold film of without the contribution from the static linear expansion. All other parameters are similar to Fig. 10, except that a smaller laser fluence of was used.
(a) Time dependence of , with both dynamic and static contributions, assuming a different skin depth (absorption length) of the laser light than the actual value given in Table II. The peak delay time for is also slightly influenced by a change in . A smaller laser fluence of was used here with a pulse width (FWHM) of . (b) Magnitude of the maximum for vs . A smaller magnitude for is obtained because of a smaller force or temperature gradient with a larger .
(a) Comparison of the experimental UEC data and model simulations for an aluminum film with . With for Al–Al interatomic equilibrium distance corresponds to thickness of . A laser fluence of , a laser pulse width of , a pulse delay of , the optical reflectivity at , and for the penetration depth of transmitted electron beam were used. The damping parameter , with the time constant is about . The ratio between the electronic Grüneisen parameter and the phonon Grüneisen parameter is set at 0.74. (b) for an aluminum film with a different electron-phonon coupling .
(a) The dependence of the half oscillation period for on the thickness of a thin aluminum film. The inverse of the slope corresponds to the sound velocity of . The full oscillation period represents the round-trip traveling time for sound. (b) The delay time for the first major Bragg peak vs aluminum film thickness in the presence or absence of the electron Grüneisen parameter . Because of the varying fast rise of the electron temperature, the short-lived impulsive force due to the electronic stress shortens the delay time for the first major peak in and causes the inverse slope to appear slightly greater than the sound velocity. In the absence of the inverse slope is much closer to the actual sound speed in aluminum. The oscillation period of the subplot (a), however, was not significantly affected even if was set to 0.
Nomenclature for symbols.
The parameters (Ref. 22) for gold in the TTM. ( reduced specific heat for electrons, thermal conductivity for electrons, the specific heat for phonons, the electron-phonon coupling, the laser skin depth, the optical reflectivity at , and the linear thermal expansion coefficient. A force constant of and Au–Au distance of were used, corresponding to sound velocity of at room temperature.)
The parameters (Refs. 25 and 33) for aluminum in the TTM. [ the optical reflectivity at . A force constant of and Al–Al distance of were used, corresponding to sound velocity of (Ref. 16). The electron-phonon coupling was reported by Tas and Maris (Ref. 33)].
The comparison between the dynamic expansion/contration mechanism and the linear thermal expansion mechanism.
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