Thickness-dependent electron–lattice equilibration in laser-excited thin bismuth films

For a quantitative analysis the integrated signal of the Bragg-peaks has been determined as a function of pump–probe time delay Δt by fitting each peak separately with a Gaussian function superimposed on a (linear) background. Figure 2(a) shows as an example the result of such an analysis for the same 15 nm Bi film and the same pump fluence of F = 0.69 mJ cm⁻² as in figure 1(c); the integrated diffraction signal has been normalized to the value measured at negative delay times, i.e. before sample excitation. Figure 2(a) also contains (as violet data points) the time dependence of the diffuse scattering signal measured over an intervall ${\rm{\Delta }}q$ = $(3.8\pm 0.2)$Å⁻¹.

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As already mentioned above, the integrated diffraction signal of the Bragg-peaks exhibits an order-dependent decrease within a few ps, while the diffuse scattering increases on a similar time-scale. Within the Debye–Waller model the intensity decrease of l = 0 Bragg-peaks ${I}_{{hk}0}({\rm{\Delta }}t)$ is caused by the increase of the r.m.s. atomic displacement perpendicular to the c-axis⁴ $\;{\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle ({\rm{\Delta }}t)$ and can be expressed as:

$$\begin{eqnarray}&&{I}_{{hk}0}({\rm{\Delta }}t)={I}_{{hk}0}^{0}\cdot {{\rm{e}}}^{-\frac{1}{2}{G}_{{hk}0}^{2}{\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle ({\rm{\Delta }}t)},\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&\iff \qquad -\mathrm{ln}\left(\displaystyle \frac{{I}_{{hk}0}\left({\rm{\Delta }}t\right)}{{I}_{{hk}0}^{0}}\right)=\displaystyle \frac{1}{2}{G}_{{hk}0}^{2}\cdot {\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle ({\rm{\Delta }}t).\end{eqnarray} \tag{ 1b }$$

Herein ${I}_{{hk}0}^{0}$ denotes the scattering signal of the unpumped sample (measured at negative time delays) and G_hk0 the length of the reciprocal lattice vector corresponding to reflection (hk0). Equation (1b) is particularly useful to verify whether the experimental data can be described within the Debye–Waller model: plotting the negative logarithm of the normalized intensity as a function of ${G}_{{hk}0}^{2}$ should result in a linear dependence with a slope determined by the increased r.m.s. displacement ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle .$

Figure 2(b) shows this dependence for the the asymptotic intensity ${{\rm{I}}}_{\infty }$ (signal averaged over the delay range 12 ps $\leqslant \;{\rm{\Delta }}\;{\rm{t}}$ $\leqslant$ 15 ps) as a function of ${G}_{{hk}0}^{2}$ for different fluences (color-coded as labelled in figure 2(c)). The measured data follow indeed the linear behavior expected from the DW-model indicating that the lattice excitation perpendicular to the c-axis is dominated by incoherent, statistical atomic motion. From the slope of linear fitting curves (solid/dashed curves) the corresponding asymptotic r.m.s. displacement ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }$ has been determined, which is shown in the inset of figure 2(b) as a function of pump fluence. The dashed curve represents a quadratic fit to the data indicating a slight non-linearity in the degree of lattice excitation with fluence.

It should be emphasized that the linear functions used to fit the measured data intersect the origin, a direct consequence that diffraction of MeV electrons can be described in the kinematic limit for these sample thicknesses. At non-relativistic energies experiments have frequently shown a finite intercept with the ordinate in dependencies similar to figure 2(b) [40, 41] and/or exhibit direct beam (zero order) attenuation [41, 42].

We verified that the DW-model does not only describe the asymptotic late time behavior but the experimental data for all delay times, giving evidence that coherent contributions to the atomic motion (perpendicular to the c-axis) are negligible even directly after excitation. Such coherent contributions, i.e. an excitation of the ${{\rm{E}}}_{g}$ optical phonon mode, can in principle occur by stimulated Raman-scattering [43] and, for stronger excitation, via anharmonic coupling to the displacively excited ${{\rm{A}}}_{1g}$ mode (motion along the c-axis) [44]. Time-resolved x-ray diffraction measurements [45] have however shown, that the coherent excitation of the ${{\rm{E}}}_{g}$-mode is weak and strongly damped, which has been explained by the very short lifetime of the electronic states driving the mode [46, 47]. Therefore, the diffraction data can be used to determine ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle$ as a function of delay time. As an example figure 2(c) shows ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right)$ measured on the same 15 nm Bi film as before for different fluences.

Frequently r.m.s. displacements measured in time-resolved diffraction experiments are used to derive an induced lattice temperature rise ${\rm{\Delta }}{T}_{{\rm{L}}}$ [14, 40, 41, 48] implying thermal occupation of the phonon modes of the system. For Bi this is a reasonable assumption only for long time delays after full electron-lattice equilibration (see below [27, 35]). Therefore, we only converted ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }$ into a temperature (see left ordinate in the inset of figure 2(b)). This requires knowledge of the temperature dependence of the Debye–Waller-factor at high temperatures. With the Debye–Waller factors reported by Fischer et al [38] for temperatures of 294 K and 516 K, respectively, and assuming a Debye-like linear increase with temperature our measured ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }$ yield unrealistically high temperatures (e.g. ${\rm{\Delta }}{T}_{{\rm{L}}}$ = 400 K at F = 1.1 $\mathrm{mJ}\;{\mathrm{cm}}^{-2}$ well above the melting point). Alternatively, using a Debye temperature of ${{\rm{\Theta }}}_{{\rm{D}}}=120$ K, as derived from x-ray diffraction measurements at room temperature [39] (equivalent to the parametrized temperature dependent Debye–Waller factor reported by Gao et al [49]), more realistic, lower ${\rm{\Delta }}{T}_{{\rm{L}}}$ are obtained, but still the final temperature is in excess of the melting temperature at the highest fluences. In contrast, using transmission electron diffraction a much lower Debye-temperature of ${{\rm{\Theta }}}_{{\rm{D}}}\approx 90$ K has been found for Bi nanoparticles by analyzing the temperature dependent decrease of the (110) Bragg-peak intensity [50]. Applied to our data this would result in a correspondingly smaller laser-induced temperature increase (e.g. ${\rm{\Delta }}{T}_{{\rm{L}}}$ = 180 K at F = 1.1 mJ cm⁻²). The temperature estimates shown in figure 2 are based on the results of first principle calculations by Arnaud and Giret [35] (compare their figure 2(b)) yielding a maximum temperature close to the melting point at F = 1.1 mJ cm⁻², the highest fluence we were able to perform measurements. When the fluence was increased above 1.1 mJ cm⁻² accumulative sample damage was observed as evidenced by a progressive permanent reduction of the Bragg-peak intensities.

Electronic excitation of Bi leads to rather strong changes of the interatomic forces and thus to phonon softening, as has been first observed for the ${{\rm{A}}}_{1g}$ optical mode through time-resolved reflectivity measurements [51, 52]. DFT calculations have shown that phonon softening extends over large parts of the Brillouin zone [22]. Using sub 100 fs laser pulses for excitation it occurs on a time-scale short compared to even the highest frequency modes and induces a squeezed phonon state [27] where the initial room temperature r.m.s. displacement is too small in relation to the weakened interatomic forces. The system relaxes by a rapid increase of the r.m.s. displacement within a few hundred fs [27, 35], as demonstrated by the data depicted in figure 3 showing ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right)$ measured with finer delay stepping on a 15 nm Bi film for an excitation fluence of 1 mJ cm⁻².

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These data very clearly evidence a two-step evolution in the lattice response with a very fast sub-ps component (blue curve; guide to the eye) subsequently followed by the relaxation towards the final equilibrated state (green curve) which is reached on a few-ps time-scale (see figure 2). Following the interpretation of Johnson et al [27] we associate the initial component with the dynamics of the squeezed phonon state while the slower response is due to ordinary electron–phonon coupling and the transfer of electronic excess energy to the lattice.

We would like to note that the generation of squeezed phonon states as a result of electronically induced changes of the interatomic forces [53] is a more general phenomenon and has been used to map phonon dispersion relations [54] and identified as a precursor of non-thermal melting in covalent semiconductors [55]. Bi with its Peierls-distorted equilibrium structure exhibits a pronounced sensitivity to electronic excitation. Therefore, lattice softening and squeezing effects are particularly strong: the initial increase of the r.m.s. displacement amounts to approximately 40% of the total ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }$. Moreover, even after relaxation of the squeezed state ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right)$ cannot be directly converted into changes of the lattice temperature ${{\rm{T}}}_{{\rm{L}}}$ since both, the amount of energy that is transferred from the excited electronic system to the lattice and the inter-atomic forces change with time during electron-lattice equilibration. Within a simple Debye-picture where ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \propto {\rm{\Delta }}{\text{}}{T}_{{\rm{L}}}/{{\rm{\Theta }}}_{{\rm{D}}}^{2}$ (${{\rm{T}}}_{{\rm{L}}}\gg {{\rm{\Theta }}}_{{\rm{D}}}$) the Debye-temperature ${{\rm{\Theta }}}_{{\rm{D}}}$ is not a constant but depends on the degree of electronic excitation (e.g. electronic temperature [35]) and is, therefore, a time-dependent quantity.

Qualitatively similar results as discussed so far for 15 nm films have been obtained also for Bi films with 8 nm, 25 nm and 55 nm thickness. All samples exhibit a two-step temporal evolution of the r.m.s. displacement with a 'fast' (softening/squeezing) and 'slow' (electron-lattice energy exchange) component. While the speed of the initial 'fast' increase is limited by the temporal resolution of the experiment, the "slow" response caused by electron-lattice relaxation exhibits a pronounced thickness dependence. This is demonstrated by the data shown in figure 4.

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Figure 4(a) compares time-dependencies of ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right),$ which lead to the same (within the experimental error) asymptotic r.m.s. displacement ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }=0.029$ Ų, for the four different film thicknesses. To quantify the speed of electron-lattice equilibration by an energy relaxation time$\;{\tau }_{{\rm{E}}}$ these time dependencies have been fitted by an exponential increase (dashed curves). Only data measured at delay times ${\rm{\Delta }}\;t\;\geqslant \;$ 0.8 ps have been taken into account in order to clearly separate energy relaxation from the initial squeezed state dynamics. For the examples shown in figure 4(a) ${\tau }_{{\rm{E}}}$ decreases from 3 ps for the 55 nm thick film to 1.8 ps at 8 nm thickness. While these values are in principal agreement with previously reported results [14, 15, 17] our data give clear indication for an increased electron–lattice coupling with decreasing sample thickness. This conclusion is confirmed by figure 4(b) which shows ${\tau }_{{\rm{E}}}$ as determined from a large number of measured ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right)$ (more than 90 delay scans) for the different sample thicknesses as a function of ${\rm{\Delta }}\langle {u}_{\perp }^{2}{\rangle }_{\infty }$. Despite fluctuations in the data it is evident that electron-lattice equilibration becomes faster at stronger excitation and, in particular, with decreasing film thickness.

Let us first comment on the observed decrease of ${\tau }_{{\rm{E}}}$ with increasing excitation strength. Indications for a fluence dependent energy relaxation in Bi have been reported before [14, 15], but only over a very limited range of fluences. Such behavior can be attributed to electronic density of state effects [56, 57], which are expected to be particularly strong in Bi [35] since the electronic density of states is low at the Fermi-level but increases significantly at higher energies—states that are occupied for the high excitation levels used in the experiments here.

With respect to the observed thickness dependence it has to be stressed that time-resolved x-ray diffraction experiments [25] have revealed an effective energy deposition depth of approximately 26 nm at similar excitation fluences, which is larger than the optical absorption depth of Bi ($1/\alpha \approx 13{\rm{nm}}$ at λ = 800 nm [58]). Furthermore, ambipolar electron-hole-diffusion leads to a distribution of the deposited energy deeper into the sample. Using a diffusivity D = 2.3 cm² s⁻¹[25], we estimate a diffusion length of about 15 nm during the first ps after excitation. Therefore, a homogeneous excitation profile prior to electron-lattice equilibration can be assumed for film thicknesses up to at least 25 nm and, therefore, the thickness dependent energy relaxation can not be attributed to transport effects within the film (which might play a role for the 55 nm film).

To explain the thickness dependence we refer to work on the thermal boundary conductance (Kapitza conductance) of metal–insulator interfaces. For these systems it has been proposed that direct coupling of metal electrons to phonons of the insulating substrate provide an additional channel for energy transport across the interface [59, 60]. Experiments on thin Au-films have provided evidence that this effect might be particularly effective as long as electrons and lattice are not in equilibrium [61, 62]. It will lead to a faster cooling of the hot electrons in the film and thus also to faster equilibration with the lattice. Since it is an interface effect its absolute magnitude does not depend on film thickness, but its relative weight will increase with decreasing thickness which accelerates the equilibration in thinner films, as observed in our experiments.

In a simplified picture we may decompose the total equilibration rate ${{\rm{\Gamma }}}_{{\rm{E}}}=1/{\tau }_{{\rm{E}}}$ into a bulk contribution due to the direct electron–phonon coupling ${{\rm{\Gamma }}}_{{\rm{E}},\mathrm{Bi}}=1/{\tau }_{{\rm{E}},\mathrm{Bi}}$ in the Bi film and a contribution ${{\rm{\Gamma }}}_{{\rm{E}},\mathrm{Int}}=1/{\tau }_{{\rm{E}},\mathrm{Int}}$ from the coupling across the interface to substrate phonons, as schematically shown in figure 5(a). While for given excitation conditions ${\tau }_{{\rm{E}},\mathrm{Bi}}$ is independent of the film thickness d, the interface contribution is thickness dependent as ${\tau }_{{\rm{E}},\mathrm{Int}}\propto d$ [63], which leads to:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\rm{E}}}={{\rm{\Gamma }}}_{{\rm{E}},\mathrm{Bi}}+{{\rm{\Gamma }}}_{{\rm{E}},\mathrm{Int}}=A+B\cdot \displaystyle \frac{1}{d}.\end{eqnarray} \tag{ 2 }$$

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Figure 5(b) shows the total equilibration rate ${{\rm{\Gamma }}}_{{\rm{E}}}$—obtained from exponential fits to the time dependencies of ${\rm{\Delta }}\langle {u}_{\perp }^{2}\rangle \left({\rm{\Delta }}t\right)$ depicted in figure 4(a)—as a function of the inverse film thickness. The measured data exhibit indeed a linear dependence as predicted by (2). This supports the interpretation of an interface-mediated contribution to the relaxation dynamics in thin films and allows to separate both parts. From the parameters of a linear fit (gray dashed curve) the bulk equilibration time can be determined as $1/A={\tau }_{{\rm{E,Bi}}}\approx 3.6\;\mathrm{ps}.$ For the given case the interface contribution ${{\rm{\Gamma }}}_{{\rm{E,Int}}}=B/d$ exceeds ${{\rm{\Gamma }}}_{{\rm{E}},\mathrm{Bi}}$ at film thicknesses below 10 nm highlighting the importance of this relaxation channel at nanoscale system dimensions.