Introduction

The field of intense mid-infrared (MIR) laser matter interaction has recently gained wide ranging interests due to novel phenomena like photon acceleration in metasurfaces1 to generation of attosecond pulses reaching carbon K-edge2, and demonstration of megafilamentation in atmosphere3, which may open doors to many exciting new applications4 and help probe novel phenomena5. Intentional MIR femtosecond laser-induced damage (fs-LID) can be used for the sub-surface micromachining of multi-layer materials, such as the fabrication of buried silicon waveguides operating in the telecommunications band6. Motivated by promise of such science and technology advances, there are several major efforts under way for the development of large MIR laser facilities such as the Extreme Light Infrastructure (ELI ALPS)7 in Europe or BESTIA (Brookhaven Experimental Supra-Terawatt Infrared at Brookhaven Accelerator Test Facility)8. The ability to accurately predict MIR fs-LID threshold (fs-LIDT) is crucial to the design and development of new sources9 in this wavelength regime. However, fs-LID and material modification studies performed so far have been mostly limited to near-infrared (NIR) wavelengths near 780–800 nm. Thus, at present, a significant gap remains in theoretical understanding and experimental data and results for fs-LID at MIR wavelengths10,11, which substantially slows down the fundamental and applied research in this area.

In parallel with this gap, our research is motivated by the fundamental fact that the dual nature of light supports two regimes of energy transfer from light to matter. First, light is absorbed by the small discrete energy portions referred to as photons. Second, as an electromagnetic wave, light supplies energy to the electron sub-system by periodically accelerating and decelerating electrons. These periodic electron dynamics are frequently referred to as laser-driven electron oscillations. These oscillations can be characterized by a cycle-averaged energy referred to as the ponderomotive energy. Under the single frequency approximation, the ponderomotive energy of an oscillating conduction-band electron:

$$\begin{array}{c}{W}_{{\rm{P}}}=\frac{{e}^{2}{E}_{0}^{2}}{4{m}_{{\rm{CB}}}{\omega }^{2}}\end{array}$$
(1)

is expressed via effective electron mass mCB; carrier laser frequency ω, and peak electric field E0 of the driving laser pulse. Eq. (1) assumes the amplitude of the laser-driven momentum oscillations is eE0/ω. Due to λ2 scaling in Eq. (1), the ponderomotive energy of high-intensity MIR pulses can exceed the energy of a single photon ħω, introducing new effects into the absorption of laser-pulse energy. In typical semiconductors, ultrafast absorption is coupled to three types of electronic excitations: inter-band transitions from valence to conduction band separated by a band gap of few eV; intra-band transitions within the same energy band; and transitions from the valence band to defect levels or from defect levels to the conduction band12. For interaction of good-quality crystals with sub-200-fs laser pulses, contributions of the defect-to-band and band-to-defect transitions can be neglected since they involve electron-defect interactions that require more time than the pulse duration13,14.

At ultraviolet (UV), visible, and NIR wavelengths, \({W}_{P}\ll \hslash \omega \) even at laser intensity as high as fs-LID and ablation thresholds. Under those conditions, intra-band excitation and free-carrier absorption15 as well as the inter-band transitions are associated with quantized variations of electron energy due to absorption of single or multiple laser photons. The inter-band excitation happens in the multiphoton regime since inter-band tunneling is negligible17,18 according to the Keldysh photoionization model16. This regime can be referred to as photon-driven.

In the opposite limit of \({W}_{P}\gg \hslash \omega \), the photon energy is small enough to provide quasi-continuous variations of electron energy. As a criterion, one can consider photon energy of 0.04 eV corresponding to \(\lambda \ge 25\) μm that is the room-temperature electron energy19. For this field-driven absorption regime, tunneling ionization dominates in inter-band excitation16, and deposition of ponderomotive energy via electron-particle collisions becomes a major mechanism of light absorption. Specific features of tunneling ionization20 suggest that the intra-band absorption of the ponderomotive energy dominates in the overall energy transfer from light to the electrons19.

Therefore, a transition from the quantized photon-driven to continuous field-driven regime of absorption is expected in the wavelength range from 1 to 25 μm (corresponding photon energy is 1 eV – 0.04 eV) for typical semiconductors. Studies of wavelength scaling of fs-LID (permanent modification) or ablation (material removal) thresholds can substantially assist in a search for the fundamental transition between the two absorption regimes.

Experimental data and theoretical results11,17,18 suggest an increasing LIDT with increasing wavelength in the photon-driven regime. An ultimate threshold of the field-driven breakdown is that by DC electric field in a semiconductor21, but it is orders of magnitude lower than peak time-dependent field at fs-LIDT11,18,22,23,24,25,26,27,28. Therefore, it is reasonable to attribute the transition between the photon-driven and field-driven regimes to a maximum of wavelength scaling of fs-LIDT at some wavelength in the range 1 to 25 μm.

In this work, we present a systematic study of irreversible ultrafast MIR laser interactions with the single crystal silicon surface (Fig. 1). Multiple stages of fs-LID and ablation of single-crystal silicon by MIR femtosecond pulses are reported, and scaling of single-pulse thresholds of each individual stage as a function of laser wavelength is measured experimentally, demonstrating a pronounced maximum in the explored MIR part of the optical spectrum. These observations show a significant qualitative difference between the strong-field ultrafast laser-solid interactions at MIR and those at UV, visible, and NIR wavelengths, for which absorption is assumed to be photon-driven11,18,22,23,24,27,29,30,31,32,33. Estimations of the Keldysh parameter demonstrate that a transition from multiphoton to tunneling regime of the photoionization happens at shorter wavelengths than the position of the damage or ablation threshold maximum. Therefore, the maximum of the wavelength scaling of ablation and LID threshold is reached under domination of the tunneling ionization and cannot be attributed to the transition between the two photoionization regimes. Two traditional models of laser-induced damage based on the Gamaly approximation29 and the Keldysh photoionization theory16 fail even to qualitatively reproduce the wavelength scaling of the damage/ablation threshold. Qualitative analysis of involved mechanisms suggests attributing the experimental data to the transition from the photon-driven to field-driven regimes of absorption that is substantially influenced by a competition between different inter-band excitation paths and laser-induced modification of band structure of silicon34.

Figure 1
figure 1

A schematic workflow of the research includes (a) damaging of a silicon surface by a single ultrashort laser pulse with Gaussian space distribution of fluence; detection and characterization of the four regions of a damage spot (see description in the text) by (b) SEM, (c) AFM, (d) cross-sectional TEM, and (e) μ-Raman spectroscopy; (f) measurement of wavelength scaling of the thresholds of formation of each characteristic damage-spot region followed by fitting of the scaling with the Keldysh and Gamaly models; and (g) theoretical analysis of competing direct (vertical blue arrows) and indirect (bent green arrows) inter-band electron transitions between initial (solid red) and laser-disturbed (black dashed) energy bands.

Full size image

Results

Single-pulse MIR fs LID on single-crystal silicon (see Methods) was characterized ex-situ by scanning electron microscopy (SEM), atomic force microscopy (AFM), cross-sectional transmission electron microscopy (TEM), and \(\mu \)-Raman spectroscopy (Figs. 1–4).

Figure 2
figure 2

Side-by-side SEM (a), AFM (b), and AFM lineout (c) of the same damage spot produced at fluence of 0.84 J/cm2 at wavelengths 2.75 μm. Regions of interest 1–4 are described in the main text. SEM overlay of panel (a) shows local fluence. AFM lineout of panel (c) is taken along the blue dotted line shown in the panel (b).

Full size image
Figure 3
figure 3

Cross-sectional TEM images (a–d) of the sub-surface damage below the same site featured in Fig. 2. Four different regions of the same damage site are shown: (a) region 2 (amorphization occurs), (b) between regions 2 and 3, (c) region 3 (recrystallization occurs), and (d) region 4 (crater). The sacrificial platinum layer, protective gold layer, laser-modified surface region, and bulk silicon substrate are indicated. Electron diffraction of the darker contrast areas show a crystalline signal (e) while bright white areas indicate amorphization (f). The local incident laser fluence increases from left to right for each TEM image; i.e. images further towards the right are closer to the central crater (g). TEM was performed at the Center for Electron Microscopy and Analysis (CEMAS) at The Ohio State University.

Full size image
Figure 4
figure 4

μ-Raman spectroscopy (A,C) of a damage spot produced at fluence of 0.87 J/cm2 at wavelength 3.5 μm. Peak intensity of the 520 cm−1 signal (A) shows crystalline phase, while the 480 cm−1 signal (C) shows amorphous phase. An SEM image of the spot (B) is shown for proper identification of different regions of the damage spot. Dotted lines (a–d) indicate the different regions of surface or subsurface LID discussed in the main text. μ-Raman spectroscopy measurements were performed at the US Air Force Research Laboratory in Dayton, Ohio.

Full size image

Figure 2 shows representative SEM and AFM images of a damage spot. Three characteristic regions of surface modification were observed, and Fig. 2a shows the approximate local fluence within each of the regions. The SEM reveals a central ablation spot, surrounded by a distinct crater rim (region 4 in Fig. 2a). Region 3 in Fig. 2a is enclosed by the middle ring and shows a similar contrast to the central crater. The third ring, barely visible, encloses region 2 (Fig. 2a), and exhibits a change in contrast compared to the other two regions. Region 1 in Fig. 2a has no outer boundary, and represents the undamaged silicon substrate with no permanent changes detected via SEM or AFM ex-situ. In the AFM spatially resolved depth profile of the same damage site (Fig. 2b,c), the same 3 rings are all visible, each with the same lateral dimensions as in the SEM images (Fig. 2a). The height profile from the AFM images (Fig. 2c) shows that region 2 is slightly raised over regions 3 and 1, and confirms that area 4 is an ablation crater. Extra SEM images (see Supplementary Materials) suggest that the chaotic nature and nano-roughness within the shorter wavelength damage site is far more pronounced compared that of the longer wavelength, for a fixed fluence.

Figure 3 depicts representative sub-surface structure of the four regions of the damage spot shown in Fig. 2. The topmost layer of the sample shown with a white color has amorphous structure as identified by electron diffraction (Fig. 3f). Deeper layers of darker colors have crystalline structure (Fig. 3e). Within the resolution limit of the imaging device, no polycrystalline areas could be found. Figure 3d presents an image of the sub-surface layers of the innermost ring (region 4 in Fig. 2), where ablation formed the crater. A relatively non-uniform 20–25 nm thick amorphous layer is observed in the middle ring (region 3 in Fig. 2). The region bound by the barely visible outermost ring has a relatively thick (50 nm), surprisingly uniform amorphous layer. In each image of Fig. 3, a very thin (1–2 nm thick) native oxide layer is visible on the sample surface in the form of a slight contrast change. Extra TEM data (see supplementary Materials) suggest that the middle ring and the onset of very thick amorphization do not occur in the same location. The middle ring boundary has a lower fluence threshold than the crystalline/amorphous boundary. The crystalline/amorphous boundary occurs ~2 um closer to the center than the outer ring.

Spatially resolved micro-Raman spectroscopy of a typical damage site (Fig. 4) was performed at the 520 cm−1. Raman peak associated with the crystalline phase (Fig. 4a) and the 480 cm−1. Raman peak attributed to amorphous structure (Fig. 4c). The micro-Raman images match up with a SEM image of the damage spot (Fig. 4b) to properly attribute the crystalline and amorphous contents to the specific parts of the damage spot. Red dotted lines are aligned to certain transitions in either the Raman mapping or the SEM. Line (a) shows the transition from the highly amorphous region to the un-damaged sample Line (b) is aligned to the middle ring of the SEM image within the highly amorphous region. On the Raman mapping, this ring is not distinguished, indicating that there is no significant change in crystallinity below this ring. Line (c) is aligned to the transition between highly amorphous and highly crystalline signals on the Raman mapping. This transition occurs several um closer to the center than the middle ring. Line (d) shows that the regions both inside and outside the rim show a strong crystalline signal. Overall, the Raman mapping is consistent with the TEM images of Fig. 3.

Figure 5a illustrates the approach to the measurement of LID thresholds for the three most probable damage effects under consideration: phase explosion (center of a LID spot), ablation, and ultrafast melting (most outer part of the spot). The diameter of each ring, on each damage site, for each wavelength, was measured from SEM images. The data were then plotted as dependence of diameter squared on logarithm of peak fluence to determine LIDT (Fig. 5a) for each region. Scaling of the damage thresholds with wavelength are plotted in Fig. 5b. Damage thresholds remain constant or rise slightly with increasing wavelength, except for the longest wavelength where a drop in damage thresholds occurs for every regime. A comparison of the experimental wavelength scaling to the predictions of the Gamaly29 and Keldysh16 models is shown in Fig. 5b (see details of the simulations in Materials and Methods and in Supplementary Materials). The Gamaly model29 qualitatively agrees with the strong ablation experimental data and even captures the overall trend for much of the data at shorter wavelengths for indirect band gap, but fails to capture the reduction in LIDT for the longest wavelength. The simulations with the Keldysh photoionization model16 greatly overestimate the experimental LIDTs and do not even qualitatively reproduce the experimentally observed trend.

Figure 5
figure 5

Scaling of area of laser-modified spot on the surface with peak laser fluence at wavelengths of 2.75 μm (a) to illustrate the approach of evaluation of threshold for phase explosion (red), ablation (green), and melting (blue) by extrapolating a linear fit towards zero area. (b) Wavelength scaling of LID threshold as obtained from experimental data (dots with error bars), Gamaly model (dotted line – for indirect band gap; dashed line – for direct band gap), and two-band Keldysh model (solid).

Full size image

Discussion

SEM (Fig. 2), AFM (Fig. 2), and TEM (Fig. 3) images as well as the Raman spectra mapping (Fig. 4) suggest three stages of surface MIR laser damage each characterized by a well-defined threshold (Fig. 5a). Formation of amorphous silicon in the outermost region indicates that the velocity of the cooling front exceeds the critical speed of amorphization in silicon (12–25 m/s)22. This region raises slightly above the sample surface and exhibits significant uniformity of the amorphous-layer thickness attributed to the Gaussian space distribution of fluence. These features, along with the measured threshold, are consistent with those of the ultrafast melting at NIR wavelengths35,36, except the reduced thickness (30 nm for NIR versus 50 nm for MIR) attributed to strong linear absorption of the NIR photons. Therefore, it is reasonable to assume that the ultrafast melting induced at MIR wavelengths is driven by the mechanism established for the NIR wavelengths: crystal-lattice instability by increase of free-carrier density to approximately 10% of the valence band electron density36,37,38,39,40, i.e., about 1022 cm−3. It is believed the energy of the excited electrons does not substantially affect that process38,39.

The region bound by the middle circle demonstrates the features associated with the ultrafast melting and amorphization, but several nm of material are removed through spallative ablation. A similar phenomenon was observed at MIR wavelengths on Ge10 and at NIR wavelengths on Si35. However, at NIR wavelengths, typically only the top oxide layer is removed in this zone35. In contrast, MIR wavelengths show removal of both the oxide layer and some underlying material.

Both the cross-sectional TEM (Fig. 3) and the \(\mu \)-Raman mapping (Fig. 4) show a transition from amorphous to crystalline material several microns inside the middle ring. For comparison, at NIR wavelengths, a strong amorphous phase persists even partially inside the crater rim35. Furthermore, the single-crystal recrystallization revealed by TEM (Fig. 3) also differs from the NIR case, where polycrystalline phase was observed in the form of hillock formation within the crater rim35. The difference between the amorphization by MIR and by NIR pulses35,41,42 is partly justified by the fact that the MIR experiments were performed on a (100) surface while the NIR experiments were reported for a (111) crystalline surface. It has been shown that a (111) crystal surface is amorphized more readily after laser-induced melting than the (100) surface41,42,43, while the transition from amorphization to crystallization occurs at a lower threshold fluence on a (100) surface compared to (111). Moreover, (111) surfaces were found to recrystallize into a polycrystalline phase at higher fluences when compared to (100) surfaces.

The specific band structure of silicon34 is favorable for a variety of laser-stimulated microscopic process that may contribute to the surface modification. The most rigorous approach to interpretation of the reported above data should consider ab-initio simulations of the ultrafast strong-field MIR interactions44. However, available ab initio simulations miss a proper incorporation of electron-particle collisions and can be technically done over 10–20 femtoseconds of the laser-solid interaction45. Other numerical models16,26,29,33,38,39,46,47 are not valid for analysis of the high-intensity MIR laser-silicon interactions because they use the approximations that do not fit the indirect-gap band stricture of silicon. For these reasons, we limit our discussion to qualitative analysis of the following relevant items: A) inter-band electron excitation; B) absorption mechanisms; C) microscopic criteria of ablation threshold, and D) transient modifications of band gaps.

Prior to the analysis of microscopic mechanisms, we note that contribution of the native oxide layer to the ablation can be neglected in spite of its significant absorption around 2.6–2.8 μm and 4.4 μm48,49. A primary ablation of the layer would be expected if it appreciably contributed to the energy deposition. However, structure of the ablated spot (Figs. 2 and 3) suggests that the surface layer was blown up from bottom at each tested wavelength. A reasonable interpretation of that specific ablation morphology should consider a dominant electron excitation and absorption of laser-pulse energy in sub-surface layers of crystalline silicon.

The major processes of energy deposition in crystalline silicon are influenced by the specific structure of indirect-gap energy bands34. Since amplitude of the laser-driven variations of electron momentum eE0/ω is 3–5 times larger than the half-width of the first Brillouin zone at threshold fluence, contributions to the inter-band transitions may simultaneously occur in the vicinity of Γ, X, and L points34 of the silicon band structure. In spite of complicated nature of those processes, some substantial physics of the ultrafast electron excitation and associated absorption can be delivered by simple estimations of characteristic parameters with proper values of energy gaps and effective masses (Fig. 6a)34.

Figure 6
figure 6

(a) A sketch of competing direct (blue vertical arrows) and indirect (bent green arrows) inter-band electron transitions contributing to the nonlinear absorption and electron-hole plasma generation. V1 and V2 depict the two virtual states involved in the indirect transitions. (b) Scaling of the photoionization rate with peak laser intensity at wavelength of 4150 nm to illustrate similarity of contributions of the heavy-hole (HH – black solid), light-hole (LH – red dashed), and split-off (SO – blue dotted) valence bands to the total rate of the inter-band electron excitation (green dash-dotted). (c) Wavelength scaling of critical plasma NP (red squares), epsilon-near-zero NENZ (green dots), and epsilon-near-one NENO (blue diamonds) densities evaluated by equations of section 7 of Supplementary materials for meff = 0.18 of free-electron mass and Drude-model collision time τD = 1 fs. (d) A sketch of laser-induced reduction of the B branch of conduction-band bandwidth along (100) direction due to modification of the initial band (red solid) by ponderomotive energy WP of laser-driven electron oscillations. WP is negative in the center of the Brillouin zine due to negative effective mass, and positive at the zone edge due to positive effective mass near the X valley.

Full size image

A dominating regime of the laser-driven direct inter-band electron excitation (transition paths 1, 2, and 3 in Fig. 6a) can be identified by evaluation of the Keldysh adiabaticity parameter16:

$$\begin{array}{c}\gamma =\frac{\omega \sqrt{m\Delta }}{eE}\end{array}$$
(2)

where m is a proper value of reduced effective electron-hole mass, and Δ is an energy gap between involved valence and conduction bands at the characteristic band-structure points. Estimations for the Γ point (effective electron mass is 0.188 of free-electron mass me0; direct band gap is 3.28 eV34) are summarized in Table 1. At all wavelengths, the values of the Keldysh parameter are well below 1.0 at the Γ and X points and suggest domination of the tunneling regime for the direct inter-band transitions (see Supplementary Materials). Parameters of the L valley support the multiphoton regime of the inter-band transitions, the rate of which is several orders of magnitude lower than the rate of the tunneling transitions (see Supplementary Materials). Therefore, major contributions to the direct inter-band promotion of valence electrons is expected from the Γ and X points of the band structure.

Table 1 Laser wavelength λ; photon energy ħω; peak fluence FABL/FMELT, peak intensity IABL/IMELT, and peak electric field E0ABL/E0MELT of laser pulses at threshold of ablation/melting. For the ablation/melting thresholds, there are shown the Keldysh parameter γABL and γMELT (for the direct transition from the LH valence to the conduction band at the Γ point); the Franz-Keldysh reductions of band gap δΔABL and δΔMELT for LH valence band; ponderomotive energy WpABL and WpMELT of the conduction-band electron oscillations at the Γ point; and ratios WpABL/ħω and WPMELT/ħω of the ponderomotive energy to photon energy.
Full size table

The specific indirect-gap band structure of silicon also favors indirect inter-band transitions (Fig. 6a). They can be considered as multiphoton excitation of a valence electron to a virtual state V1 (for the Γ-to-L transition) or V2 (for the Γ-to-X transition) followed by phonon interaction with the excited electron to promote it to either L or X valley of the conduction band12,19,34. Rates of the indirect transitions around the Γ point (see Supplementary Materials and Methods) can be estimated using the lifetime δti (i = V1 or V2) of the virtual states from the uncertainty relation20:

$$\begin{array}{c}\delta {t}_{i}\cdot \delta {W}_{i}\ge \hslash \end{array}$$
(3)

by assuming the energy uncertainty δWi is as much as the indirect band gaps (i.e., 1.12 eV for the Γ-to-X transition and 2.0 eV for the Γ-to-L transition34). Ratio of the total probabilities P4 and P5 of the indirect transitions (arrows 4 and 5 in Fig. 6a) to the probability P2 of the direct inter-band excitation (arrow 2 in Fig. 6a) can be evaluated via the direct-transition rate wMP2 by the Keldysh formula16, multiphoton-transition rates wV1 and wV2 for the direct excitation of valence electrons to the virtual states, and the time τeph of electron collisions with polar phonons12,19 (see Methods):

$$\begin{array}{c}\frac{{P}_{5}}{{P}_{2}}=\frac{{w}_{{\rm{V}}2}\delta {t}_{{\rm{V}}2}}{{w}_{{\rm{MP}}2}{\tau }_{{\rm{eph}}}},\frac{{P}_{4}}{{P}_{2}}=\frac{{w}_{{\rm{V}}1}\delta {t}_{{\rm{V}}1}}{{w}_{{\rm{MP}}2}{\tau }_{{\rm{eph}}}}.\end{array}$$
(4)

Estimations (see Supplementary Materials) suggest that the ratio of the transition rates of Eq. (4) is at least of the order of 101–102 for V1 and 102–103 for V2 at wavelength 2.75 μm. The smallest reported electron-phonon collision time measured at NIR laser wavelengths is about 30–60 fs50,51. However, that collision time shows a strong dependence on energy of conduction-band electrons52. Small photon energy characteristic of MIR wavelengths is favorable for generation of the conduction-band electrons with energy of the order of 0.1 eV (Table 2). In this case, the electron-phonon collision time is as large as few hundreds of femtoseconds52. Therefore, the ratio of the virtual-level lifetime δtV1 and δtV2 (about 0.5–1.0 fs) to τeph is of the order of 0.001. Equation (4) suggests that the rates of the indirect excitations from Γ to X and L valleys are similar to or higher than the rate of the direct transitions at the Γ-point at 2.75 μm. With increase of wavelength, the ratio of the transition rates reduces by few orders of magnitude because the tunneling regime of photoionization exhibits very weak wavelength scaling16, but the rate of the multiphoton excitation to the virtual levels V1 and V2 substantially decrease with reduction of photon energy (see Supplementary Materials). Therefore, within the MIR regime, a significant contribution of the indirect Γ-to-X transitions to the overall free-carrier generation at shorter wavelengths is replaced by domination of the direct inter-band tunneling excitation at the longer wavelengths.

Table 2 Laser wavelength at spectrum peak λ; photon energy at the laser wavelength ħω; number of laser photons to bridge the indirect band gap for Γ-to- X transitions Nind = < Δind/ħω + 1 > (Δind = 1.12 eV34); excessive energy δind = Nind ħω – Δind of the electrons arriving to the X valley; number of laser photons to bridge the direct band gap at the Γ point Ndir = < Δdir/ħω + 1 > (Δdir = 3.28 eV34); and excessive energy δdir = Ndir ħω – Δdir of the electrons generated at the Γ point.
Full size table

An attempt to evaluate the rate of the Γ-point direct inter-band electron transitions by the Keldysh formula16 applied to three valence bands (Fig. 6b and Supplementary Materials) delivers the total rate about 1017 1/(fs cm3). Therefore, the total density of the laser-generated electron-hole pairs is of the order of 1019 1/cm3 by the end of a laser pulse according to the Keldysh model. This is two to three orders of magnitude below the density required to initiate the ultrafast melting37,38. This difference can result from the contribution of the indirect inter-band transitions that are neglected in the original Keldysh model16, but also from the simplified energy-momentum relation for the central part of the Brillouin zone utilized in the Keldysh formula16.

Since the duration of a single oscillation cycle of conduction-band electrons (9.17 fs at 2750 nm to 13.83 fs at 4150 nm) is almost one order of magnitude smaller than the characteristic time of momentum de-phasing by electron-phonon collisions at the bottom of the conduction band of silicon52, the collisions can be considered as a minor perturbation to the laser-driven oscillatory intra-band electron dynamics. This approach implies that the oscillating electrons and holes possess some ponderomotive energy of the oscillations that is retained in the free-electron sub-system due to the collisions. Estimated ponderomotive energy at the Γ point (Table 1) substantially exceed the photon energy. Moreover, at the ablation threshold, the ponderomotive energy exceeds direct band gaps, bandwidth of the lowest conduction bands of silicon (about 2. 7 eV), and even electron affinity of crystalline silicon (about 4.05 eV34). The latter fact means the conduction-band electrons pumped by the ponderomotive energy can be effectively emitted from the crystal to produce local violation of electric neutrality53,54 that is favorable for a spallation regime of ablation. Moreover, enhancement of the tunneling mechanism of the electron photoemission with reduction of driving laser frequency55 may make an appreciable contribution to reduction of the ablation threshold with increase of laser wavelength. Therefore, the collision-driven deposition of ponderomotive energy can be a more effective ultrafast mechanism of energy transfer from laser pulses to the crystal than absorption of single or multiple photons via the electron-photon-phonon collisions.

Several microscopic criteria have been reported to simulate ablation or damage threshold27,28,29,30,31,32,33,37,56,57 including critical energy, critical free-carrier plasma density NP, free-carrier plasma-resonance density, epsilon-near-zero (ENZ) free-carrier density NENZ, epsilon-near-one (ENO) free-carrier density NENO, phonon-softening plasma density, and crystal-lattice-instability plasma density (see Supplementary Materials, section 7). The Gamaly model29 based on deposition of critical amount of energy was employed to simulate the measured wavelength scaling of ablation and modification thresholds (see Fig. 5), but it did not qualitatively reproduce the scaling trend. It is notable that the critical-energy ablation criterion of the Gamaly model29 is based on a rough estimation of energy balance that neglects some important microscopic processes by assuming instant transitions of virtually all valence electrons to the conduction band and formation of solid-density plasma. In case of high-rate one-photon absorption, this assumption may be reasonable, but it is evidently violated at small rates of nonlinear absorption characteristic of the MIR wavelengths. Therefore, the process of energy transfer from light to the electrons of a solid is extended in time and supports “leakage” of the transferred energy to other process not incorporated into the model.

The microscopic ablation criteria based on critical free-carrier plasma density17,22,31,33,46, ENZ, and ENO free-carrier density36,56,57 employ the critical density of electron-hole plasma that is reached when the plasma frequency equilibrates with laser frequency. Those criteria assume the plasma is quasi-equilibrium and its critical density scales as λ−2 with laser wavelength (Fig. 6c). That scaling predicts reduction of those critical plasma densities by more than one order of magnitude with a transition from NIR to MIR wavelengths (Fig. 6c). It is remarkable that those criteria are based on the plasma critical density (that is about 1019–1020 cm−3 in the MIR wavelength range) that is well below the electron density (about 1022 cm−3) required to initiate the ultrafast melting36,37,38,39,40. Moreover, during the first 100–200 fs in silicon, experiments have shown the reflectivity change remains less than 10% even at fluence 1.5 times the melting threshold58. The weak variations of refractive index during a 200-fs laser pulse10 signals that the laser-generated electron-hole plasma is substantially transparent. This could occur if the plasma is, for example, far from equilibrium59,60. This hypothesis is further supported by estimations of the energy of laser-generated conduction-band electrons (Table 2) that suggest the electron-particle collision time is of the order of few hundreds of femtoseconds52. This special feature of the MIR wavelength range is favorable for generation of the non-equilibrium free-carrier plasma at the bottom of the conduction band. Therefore, the criteria based on the concept of plasma critical density may not be valid at the MIR wavelengths. In contrast, silicon excitation at visible and NIR wavelengths produces conduction-band electrons with average energy about few eV37. At that energy, electron-particle collision time is of the order of 1 fs52 and provides ultrafast equilibration of the free-carrier plasma. Therefore, the critical-plasma density criteria including ENZ and ENO criteria (see supplementary Materials) are more reasonable at shorter wavelengths.

To identify an adequate microscopic criterion of MIR fs-LID and ablation thresholds, we note that ultrafast melting and ablation of a solid should be attributed to weakening or breaking inter-atomic bonds36,37,38,39,40. Those bonds are provided by valence electrons. Therefore, weakening of the bonds should be associated with excitation the valence electrons to the conduction band36,37,38,39,40. Then, a reasonable criterion for the LID and ablation thresholds by ultrashort laser pulses should consider the electron density required to induce crystal-lattice instability via weakening of the inter-atomic bonds10,36,37. This density is obviously an intrinsic property of a crystal and must be a function of material rather than laser-pulse parameters. The λ−2 scaling of the ENZ, ENO, and critical-plasma densities (Fig. 6c) contradicts this conclusion and suggests that those criteria assume reduction of the density of broken or weakened bonds by more than one order of magnitude with the transition from NIR to MIR wavelengths. There is no a reasonable justification of this strong influence of laser wavelength on the intrinsic property of silicon.

The high values of threshold intensity (Table 1) imply significant transient modifications to the band structure of silicon, i.e., distortions of the shape of the bands61 and variations of the band gaps. There are several mechanisms of those modifications. For example, a phenomenon similar to the dynamic Franz-Keldysh effect44,62 reduces band gap. Since the high-intensity limit of that effect44 is like the one for dc electric fields63, the band-gap reduction can be estimated as follows63:

$$\begin{array}{c}\delta \Delta ={[{(e{E}_{0})}^{2}\frac{{\hslash }^{2}}{{m}_{{\rm{RE}}}}]}^{\frac{1}{3}},\end{array}$$
(5)

where mRE is the effective reduced electron-hole mass in the direction parallel to electric field E0. Estimations for <111> direction around the Γ point suggest very substantial reduction of the band gaps of silicon both at the ablation and at the melting thresholds. Moreover, significant wavelength dependence of that effect62 suggests an enhancement of its contribution for the longer-wavelength of the tested MIR range.

The fundamental properties of silicon are favorable for several other effects that also modify the band gaps21. First, the top valence band and the lowest conduction band of silicon originate from the same hybrid atomic state12,19,64, those two energy bands are coupled. That coupling enhanced by electric field of laser radiation64,65,66 results in reduction of the band gaps64,65,66. However, its contribution cannot be estimated in a simple way and requires extended numerical simulations.

Second, band-gap shrinkage results from increase of the conduction-band population N according to the N1/3 law66,67,68. It reduces the band gaps by 1–2 eV at the conduction-band electron density 1021 1/cm3and appreciably contributes to the overall modification of band structure21.

Third, generation of the conduction-band electrons makes another significant contribution to the band-gap modification by the band filling effect66,67,68. That modification results from the dominating filling of the lowest-energy conduction states that favors absorption of one extra photon to promote the newly arriving electrons to higher-energy vacant states of the band21. However, the influence of the ultrafast band-filling effect may be smaller compared to the usual estimations66,67,68 because of effective re-distribution of the electrons over almost entire conduction band due to the laser-driven oscillations21. Also, this effect can be substantially compensated by the band-gap shrinkage effect37.

Finally, the ponderomotive energy substantially contributes to the band-gap modification16,61,69. According to Eq. (1), that energy can make either positive or negative contribution to the band-structure modification depending on sign of effective mass at specific parts of the energy bands. For example (Fig. 6d), the X valley of the conduction band (Fig. 6a) is up-shifted due to positive effective electron mass while the Γ-point segment of the band is down-shifted because of the negative effective mass near the Γ point. Such ponderomotive-energy effects reduce bandwidth of the energy bands, remove degeneracy of the bands, and can produce very prominent modification of the band gaps. The small value of energy of the conduction electrons (Table 2) generated by MIR wavelengths suggests extended electron-particle collision time52 that is favorable for this mechanism of band-gap modification.

The special feature of silicon is that all those effects simultaneously contribute to the band-structure modification by either band-gap reduction (e. g., the analog of the dynamic Franz-Keldysh effect, band-gap shrinkage, and enhancement of band coupling) or band-gap increase (band filling and ponderomotive energy) making overall dynamics of the band-structure modification very complicated. Moreover, the ultrafast dynamics driven by the femtosecond laser pulses can be appreciably influenced by quantum interference between different paths of the laser-driven electron excitations. Also, the effects of the non-parabolic, non-Kane conduction band topology may be very important. Time-dependent ab initio simulations are the most appropriate to attack that complicated problem, but the currently available approaches cannot properly address those effects45.

Conclusions

Single-shot fs-LIDA experiments on silicon followed by cross-sectional TEM, SEM, AFM, and micro-Raman spectroscopy have revealed unique LIDA morphology attributed to multiple stages of surface LIDA. For certain well defined fluences, a “weak” spallative ablation occurs which was not observed for NIR wavelengths under similar conditions. Surface LID thresholds were determined for the three distinct stages of damage at four MIR wavelengths. These studies deliver significant insights into the involved damage mechanisms at the MIR wavelengths.

Ultrafast melting, followed by amorphization, was observed, like in NIR experiments, but it occurs well above the plasma critical density, despite the prediction of the high-collision-rate Drude model. Spallative ablation of a few layers of silicon is the only means by which oxide layer removal was observed with MIR wavelengths. The nanoroughness of the remaining material after ablation decreases with wavelength.

Qualitative analysis of the absorption mechanisms and associated electron excitations by the ultrashort laser pulses suggests attributing the maximum of the wavelength scaling of LID thresholds to simultaneous action of several effects. They include a transition from domination of indirect inter-band excitation to domination of direct inter-band excitation; laser-driven modification of energy bands; enhancement of tunneling electron emission; and the transition from photon-driven absorption regime the field-driven regime. As discussed above, the shorter (e. g., NIR) wavelengths are favorable for dominating contributions of the multiphoton absorption to the inter-band electron promotion, intra-band excitation, and electron emission. In contrast, increase of laser wavelength towards longer-wavelength part of the MIR range leads to reduction of photon energy, increase of ablation thresholds, and corresponding increase of ponderomotive energy of the oscillating conduction electrons. Respectively, the probability of all the multiphoton effects reduces, but the contributions of the field effects increases with increase of laser wavelength. That transition between the absorption regimes is associated with increase of the contribution of ponderomotive energy to the intra-band electron excitation and overall absorption, resulting a decreased damage threshold. Therefore, the wavelength scaling of the melting and ablation thresholds may reflect a transition from the photon-dominated regime of absorption to the field-dominated regime. More detailed justification of this hypothesis requires extended ab initio simulations not available currently.

Methods

LID experiments

The laser source was The Ohio State University (OSU) Extreme Mid-Infrared (EMIR) optical parametric amplifier (OPA)1,70,71. A schematic of the experiment is shown in Fig. S1. 1-on-1 LID experiments were performed in air, with p-polarized pulses (λ = 2.75, 3.15, 3.75, 4.15 μm, \({\tau }_{{\rm{FWHM}}}=200\,{\rm{fs}}\)) and an angle of incidence of 31 degrees. By varying the pulse energy, the peak fluence was varied from 0.25 to 2 Jcm−2. The energy of every pulse was recorded. The beam profile of the focal spot illuminating the surface of the sample was recorded once for each wavelength, and focal spot sizes ranged from \(20-25\) μm FWHM (geometric mean of the horizontal and vertical spot sizes). Reported fluences are calculated after projection of the measured focal spot profile onto the sample surface (surface normal fluence). The sample used was an n-type single crystal Si (100)/<110> (\({\rm{resistivity}}\,{ > 10}^{3}\)Ω-cm with intrinsic carrier concentration ∼1014 cm−3); electric field polarization was primarily in the (1 1 1) direction. Ex-situ SEM imaging was performed on each damage site. AFM imaging was performed on a subset of damage sites. Cross sectional TEM and spatially-resolved μ-Raman spectroscopy were both performed on a single damage site.

The two traditional simulation approaches

Keldysh simulations were performed using Python simulation code72. This code uses the Keldysh equations described in the supplemental materials. The minimum peak incident laser intensity required for ultrafast melting, i.e. for photoionization of 7% of the valence band electrons, was determined to be the LIDT. Once the conduction band electron density just barely reaches the threshold at the surface, ultrafast melting of the surface layer occurs. In this work, our simulations consider direct transitions from the valence band \(\Gamma \) valley to the conduction band with the positive effective mass. This is indicated through transition number 2 in Fig. 6a. Specific material parameter inputs for silicon are summarized in Supplemental Materials. Additional details are provided in the supplemental materials.

Estimations of the probabilities of direct and indirect transitions

For derivation of the estimation of Eq. (4) of the main text, we assume that the virtual states V1 and V2 are reached via a direct transition by absorption of several laser photons. Correspondingly, the energy gap between the Γ point of the valence bands and the virtual states is evaluated as follows:

$$\begin{array}{c}{E}_{{\rm{V}}1}=\langle \frac{{\Delta }_{{\rm{INDGV}}1}}{\hslash \omega }+1\rangle ,\,{E}_{{\rm{V}}2}=\langle \frac{{\Delta }_{{\rm{INDGV}}2}}{\hslash \omega }+1\rangle ,\end{array}$$
(6)

where ΔINDGv2 = 1.12 eV for the Γ-to-X transition (5 in Fig. 6a), and ΔINDGv1 = 2.00 eV for the Γ-to-L transition (4 in Fig. 6a)34. Probability of the electron transitions from the top Γ-states of the valence bands to the virtual states per time interval δt estimated by the uncertainty relations of Eq. (3) can be evaluated via the multiphoton-ionization rate WV1 and WV2:

$$\begin{array}{c}{P}_{{\rm{V}}1}=\frac{{W}_{{\rm{V}}1}\delta {t}_{{\rm{V}}1}}{{N}_{{\rm{VB}}}},\,{P}_{{\rm{V}}2}=\frac{{W}_{{\rm{V}}2}\delta {t}_{{\rm{V}}2}}{{N}_{{\rm{VB}}}}\end{array}$$
(7)

where NVB is the total population of the involved valence bands. The other process involved into the indirect transitions 4 and 5 (Fig. 6) considers electron-phonon collisions12,19. Probability Pelph of this collision process per time interval δt can be evaluated as follows12,19:

$$\begin{array}{c}{P}_{{\rm{eph}}}={\Gamma }_{{\rm{eph}}}\delta t=\frac{\delta t}{{\tau }_{{\rm{eph}}}}\end{array}$$
(8)

where Γeph = 1/τeph is the rate of electron collisions with polar phonons12,19,34. The total probability of the transitions 4 and 5 can be evaluated in the first approximation as a product of the probabilities of Eqs. (7) and (8) by assuming the electron excitation to the virtual states and the electron-phonon collisions are independent:

$$\begin{array}{c}{P}_{4}={P}_{{\rm{V}}1}{P}_{{\rm{eph}}}=\frac{{W}_{{\rm{V}}1}\delta {t}_{{\rm{V1}}}^{2}}{{N}_{{\rm{VB}}}{\tau }_{{\rm{eph}}}},\,{P}_{5}={P}_{{\rm{V}}2}{P}_{{\rm{eph}}}=\frac{{W}_{{\rm{V}}2}\delta {t}_{{\rm{V2}}}^{2}}{{N}_{{\rm{VB}}}{\tau }_{{\rm{eph}}}}.\end{array}$$
(9)

In Eq. (8), the rate of the electron-phonon collisions considers all the collisions that result in variations of electron momentum, i.e., all contributions to the momentum transfer from phonons to the excited electrons of the virtual states. Characteristic time τeph of the electron-phonon momentum transfer (also referred to as momentum dephasing time) varies from few tens to few hundreds of femtoseconds in typical semiconductors50,51,52 depending on laser and material parameters.

To obtain the estimation of Eq. (4), the probability of the direct Γ-point transitions is evaluated via the inter-band photoionization rate WMP2 as follows:

$$\begin{array}{c}{P}_{2}=\frac{{W}_{MP2}\delta {t}_{i}}{{N}_{VB}},\,i=V1,\,V2.\end{array}$$
(10)

The photoionization rates are estimated using the Keldysh formula16 with corresponding corrections.