Ultrafast Laser Processing of Diamond Materials: A Review

Diamond laser engineering is of great importance for designing devices, which find applications in radiation sensing and quantum technologies. A review of the present state of the art of experimental and theoretical studies on ultrashort laser irradiation of diamond is presented. For a wide range of laser parameters, the optimization of laser-induced electronic, optical and structural modifications of diamond requires quantitative understanding of the microscopic processes underlying the high electronic excitation in the material.

1 Introduction

Diamond is an indirect wide band gap semiconductor which has numerous exceptional material parameters like extreme hardness and thermal conductivity not matched by any solid [1, 2], high electric breakdown field, high electron and hole mobilities [3], high carrier saturation rate, possible implementation of several color centers with applications in quantum technologies [4, 5] and high biocompatibility. The speed of sound propagation in diamond is also high. Some basic characteristics of diamond are summarized in Table 1.

TABLE 1

TABLE 1. Basic characteristics of diamonds.

Diamond has extraordinary optical characteristics. It is transparent in the ultraviolet, visible, infrared and microwave spectral regions. The absorption coefficient of diamond depends on the laser wavelength and on the properties of the carbon material as shown in Figure 1 [6]. Some crystals are almost transparent to laser light, while others strongly absorb it. Due to the high refractive index in the visible and ultraviolet ranges, diamond is a useful material for dielectric [7] and hybrid nanophotonic devices [8–10]. When irradiated with intense ultrashort laser pulses diamond can be converted into a short-lived plasmonic state enabling the photoexcitation and the propagation of surface plasmon-polaritons (SPPs). Possible applications are spatial phase modulation, high-speed optical switching and saturable absorption [11, 12]. The availability of advanced technologies for the production of pure single-crystal diamond [13, 14] allows to study the formation of electron–hole liquid (EHL) [15, 16] in the material which can help to reveal the mechanisms of phase transitions occuring in a strongly excited diamond. These technologies have also allowed the formation of atomically flat diamond (111) surfaces that can subsequently undergo graphitization [17].

FIGURE 1

FIGURE 1. Dependence of the optical absorption coefficient on the laser wavelength [6].

Currently, diamond can be obtained not only in the form of natural monocrystals. Technologies have been developed that provide the opportunity to grow synthetic diamonds with parameters and properties similar or even better than those of natural crystals. There are several ways to obtain diamond crystals. The oldest one is based on high-pressure high-temperature (HPHT) compression of a high purity carbon source into a metal solvent [18]. This allows growing a single or polycrystalline diamond, generally with inclusions of impurities, mainly nitrogen that degrades the electrical and optical properties of the material. The method to obtain the purest material [19] uses chemical vapor deposition (CVD) of diamond from different gas mixtures, mainly hydrogen with a percentage of hydrocarbons and possibly oxygen. The size of such crystals is much larger than the size of the crystals obtained from HPHT synthesis and their thickness varies from a few nanometers to several centimeters. Homoepitaxial growth can also be feasible with the CVD method. Synthetic diamond possesses the properties of the purest natural diamond. For instance, homoepitaxial tissue-equivalent CVD diamond has substituted for natural diamond as a base material for dosimeters used in radiotherapy [20]. Homoepitaxially grown monocrystalline diamond is fabricated with a maximum size of about 1 cm² due to the availability of diamond substrates and its application is limited. Although comparatively large area monocrystalline samples are obtained via heteroepitaxial growth, their electronic quality is still a subject of intense investigation [21]. Diamond materials can vary greatly in characteristics according to the inclusions of impurities and structural defects. In comparison to the traditional techniques for laser processing intense ultrashort laser pulses interact with diamond non-linearly, minimizing thermal diffusion and providing low ablation threshold, reduced thickness of graphite layer on the ablated surface and high spatial resolution beyond the diffraction limit [22–24]. Precise femtosecond laser profiling of diamond crystals resulted in fabrication of X-ray refractive lenses [25]. The technology of laser induced periodic surface structures (LIPSS) [26] is also developing successfully. Thanks to it, there has been an attempt to create high-quality anti-reflective surfaces on diamond with Raman lasers [27]. Laser processing is also used to create a variety of microstructures inside the bulk of diamond. Due to the movement of the focus of the laser beam through the crystal, it is possible to make structures of different shapes and sizes [28–30]. 3D architecture has been proposed for radiation hard detectors with timing capabilities and for dosimeters with high spatial resolution applied in radiotherapy. This involves fabricating columnar electrodes in the diamond bulk by laser irradiation [31]. New high-quality methods of processing diamond will open the way for using this material in new industries.

2 Diamond Surface Treatment

Pulsed laser processing of the surface of diamond can result into a complete conversion of diamond to monocrystalline graphite as demonstrated by Raman spectroscopy [32]. Since the absorption coefficient of graphitic materials is higher than that of different forms of diamond in a broad spectral range, the decrease of optical transmittance serves as an experimental indication of surface graphitization of diamond [6, 33]. Figure 2 shows the change in the transmittance of CVD polycrystalline diamond as a function of the number of irradiating picosecond pulses for varying laser fluence. For the highest fluence the decrease of transmittance starts from the first irradiating pulse while the number of pulses required to induce surface graphitization increases with decrease of laser fluence. Raman spectroscopy showed that surface ablation of polycrystalline diamond occurred without significant graphitization when femtosecond laser pulses were used [34]. In [35] LIPSS were formed on CVD mono-crystalline diamond surface perpendicular to the direction of laser polarization with spatial periods 125 and 400 nm corresponding to femtosecond laser irradiation with wavelengths λ = 248 and λ = 800 nm, respectively. Wires of millimeter length composed of 150 nm ripples (high spatial frequency–HSF LIPSS) perpendicular to the direction of laser polarization were obtained after irradiation of synthetic single-crystal diamond with near-infrared femtosecond laser pulses with fluences close to the ablation threshold [36]. Three types of LIPSS were formed after irradiation with femtosecond laser pulses in the visible and near infrared spectral range–first one with subwavelength periodicity, the second one with spacial period ∼λ/3 and the third one with a spatial period close to the laser wavelength (low spatial frequency LSFL LIPSS) [37]. It was also found that the spatial periods of HSF LIPSS increase from 170 to 190 nm when laser fluence increases 1.5 times [38]. The experimental results showed that the spatial periods of LIPSS depend on the wavelength of the laser, the polarization of the laser electric field, the number of pulses and the laser fluence.

FIGURE 2

FIGURE 2. Transmission properties of the CVD diamond sample with varying fluence and number of laser pulses [30].

High quality diamond nano-structures with surface roughness below 1.4 nm were produced by using a Ti: Sapphire laser pulses of 130 fs at a repetition rate of 1 kHz at a central wavelength of 800 nm. Surface structures in the 100–400 nm range [27] were produced for λ = 400 nm. The ablation threshold was 2 J/cm², which corresponds to previous studies in which 100 fs pulses were used with λ = 400 nm [39]. Figure 3A shows the values of laser fluence and scanning speed used for the fabrication of low-roughness LSF LIPSS with period of ∼400 nm. The high quality LSF structures shown in Figure 3B were formed at the irradiated location on the diamond surface for fluences ranging from 2 to 6 J/cm² and scanning speeds from 100 to 300 μm/s. Figure 3C shows a lineout of the structures along the x-axis obtained by AFM measurements. When there are deviations from the obtained set of parameters, the optical quality of structures is reduced. At energies exceeding 6 J/cm², intense ablation occurs. As a result, HSF structures predominate in the treated areas. It was demonstrated that when high number of laser pulses and low laser fluence were used HSF ripples were predominantly formed [40], while LSF structures were obtained when the number of pulses was low and the fluence was high. It was found that the difference in the threshold fluences for the formation of the two types of ripples was 5.4 J/cm². In [41] 200 fs laser pulses with wavelength 515 nm were used to obtain coarse and fine ripples on monocrystalline diamond surface. Figures 4A–C show the ablative craters produced on the surface and the formation of LSF (coarse CR) and HSF (fine FR) LIPSS structures with a period ∼ 100 nm, staring for number of laser pulses N > 3,4. For N > 10 the shallow FR and CR ripples become very pronounced (Figure 4B). Coarse ripples disappear for N > 100 at the laser modified spot center due to ablation as seen in Figure 4D. Both LSFL and HSFL were oriented perpendicularly to laser polarization within the ablation crater. The dependence of HSF and LSF LIPSS on the number of pulses N was also obtained (Figure 5). Studies of the optical characteristics of laser-treated diamonds were performed. The laser setup used is based on Spectra-Physics Tsunami S with wavelength 800 nm, peak power >0.7 W, pulse duration 100 fs and repetition rate 80 MHz. The formation of LIPSS oriented perpendicular to the direction of polarization took place using a laser beam with a fluence of 20 J/cm² [42]. Structures were formed with a ripple period in the range of 170 ± 10 nm with structure length greater than 5 μm. In Figure 6A Raman spectra of CVD diamond samples before and after the laser treatment are shown. In the inset, a clearly defined diamond peak for the untreated sample (black line) is shown at 1,332.2 cm⁻¹, typical of diamond [43]; the treated sample (red line) shows a shift of the peak to 1,333.1 cm⁻¹. This is interpreted as an indication of compressive stress caused by laser irradiation. No peak is exhibited at 1,560 cm⁻¹ indicating absence of graphitic phase. In Figure 6B the transmitance of laser irradiated polycrystalline diamond sample in comparison to an untreated diamond sample is shown. Since laser irradiation induces defect states in the sample, the transmission of diamond decreases by more than 80% in the visible range (from 32 to 4.5% for 500 nm) and by 70% in the IR range (from 50.2 to 11.7 for 1,500 nm).

FIGURE 3

FIGURE 3. (A) Formation of diamond structures depending on laser fluence and scanning speed. (B) Morphology of the nano-ripples for different scanning speeds at a fixed fluence. Scale bar = 1 μm. (C) Line-out of the surface structures [27].

FIGURE 4

FIGURE 4. SEM images of ablation crater edge (ABL), fine (FR) and coarse (CR) ripples formed within the craters on diamond surface for different number of pulses: 10 (A), 30 (B), 100 (C), and 300 (D) [41].

FIGURE 5

FIGURE 5. Pulse number-dependence of the variation of CR and FR periods. The dotted lines represent qualitatively the cumulative trends [41].

FIGURE 6

FIGURE 6. Comparison of Raman spectra (A), transmittance spectra (B), absorbance spectra (C) of CVD diamond before and after fs-laser treatment [42].

After laser treatment, the absorbance of the polycrystalline diamond sample increases in the entire wavelength range under consideration. For the visible and IR wavelengths it increases up to 80% as seen in Figure 6C. In the UV range (close to the value of diamond indirect bandgap of 5.47 eV) the absorbance is close to 95%. Femtosecond laser treatment leads to the appearance of surface defect states in the band gap of diamond, which can act as traps or recombination centers [29, 44–46].

3 Diamond Bulk Treatment

Besides the surface treatment of diamond crystals, it is also possible to process the bulk of the material [29, 47, 48]. Three-dimensional conductive microstructures at any depth inside bulk diamond [30] and of arbitrary shape can be created by translating the sample through the focus of the laser pulse. Raman microspectroscopy showed that the laser modified material is a mixture of amorphous and graphitic sp² bonded carbon with sp³ bonded diamond [30, 49]. A strong correlation between the conductivity of the obtained structures and their graphitic content was also evidenced [32]. The complete laser-induced phase transformation of diamond bulk into sp² carbon is hindered by the fact that it produces both compressive and tensile stresses due to substantial difference in densities of two carbon allotropes. As a result the conductive structures usually consists of a large number of micro-and nano-inclusions of sp²-carbon in sp³ diamond matrix and the conductivity of the laser-treated area is likely to be controlled by a percolation mechanism [32]. A novel technique demonstrated laser induced controlled generation of layered extended defects in diamond. To study the structural morphology of a single-crystal diamond, a titanium-sapphire laser was used, generating 140 fs pulses at a wavelength of 800 nm with a repetition rate of 1 kHz. Figure 7A [50] shows how the sample mounted on the XYZ translation table was scanned at a raster speed of 50 μm/s along the Y axis with a simultaneous slow movement 0.6 μm/s along the Z axis away from the laser. As a result, an opaque “plate” of thickness ∼8 μm was fabricated inside the diamond sample. After processing, SEM microscopy of the polished surface (110) of the sample (Figure 7B) showed numerous transparent spots of irregular shape distributed in the XZ plane of the laser-modified region. It was found that their number and size depend on the processing conditions and their occurrence was due to the predominant graphitization along the cracks formed in diamond in the path of the laser beam. Cracks were most likely to occur along the {111} planes [50, 51]. Scanning Spreading Resistance Microscopy (SSRM) revealed that the nanocracks had higher conductivity with respect to the main modified path. Based on this observation it was concluded that the graphitization processes was mainly localized inside the nanocracks forming thin graphitic-carbon sheets while the gaps between them were filled by diamond. The state-of-the-art experimental research has achieved the fabrication of electrically conductive paths along multiple nano-scale poorly connected sp²-bonded graphitic columns and a network of strain-relieving micro-cracks [52]. The control of the modified material conductivity is of paramount importance for technological applications.

FIGURE 7

FIGURE 7. (A) Microstructuring of diamond bulk with femtosecond laser pulses, (B) transmission optical micrographs of side view of the modified region [50].

The design of optical wave-guides in the bulk of a diamond crystal is also investigated based on the principles of diamond bulk graphitization. The obtained electrically conductive graphitized phase has been successfully used to fabricate radiation detectors [53]. Due to the lower density of the sp² phase as compared to the density of the sp³ phase, a strong localized stress field is created in the surrounding pure diamond, which could act as an optical waveguide. In Type II waveguide, the optical mode confinement was achieved between two closely spaced and parallel laser-inscribed conducting wires. The scheme of fabrication of the waveguide is shown in Figure 8 [54]. The type II waveguides have high polarization dependence permitting strong transmission for vertically polarized light while the transmission for the horizontal polarization state of light is negligible. Waveguides were fabricated using a 790 nm wavelength laser pulse of duration 100 fs and a repetition rate of 1 kHz. By improving and complicating the design of the waveguide, a Y splitter can be obtained from the Type II waveguides as shown in Figure 9A [55]. At the entrance, there is a single type II waveguide and at a distance of 1 mm, it is divided into two. As a result, two waveguides are formed separated by 20 μm. The outputs coupling ratio can be varied by translating the input fiber along the z-axis. A splitting ratio of 50:50 is seen in the near field mode image (Figure 9D). In this case, the overall transmission of the structure is 7 dB, which corresponds to a propagation loss of 18 dB/cm. For the manufacture of Type III waveguides that support both polarization modes, it is possible to form tubular structures, as shown in Figure 10 [55]. The radius of such structures is 10 μm. Laser-written 32 graphite tracks are arranged in a circular manner with a center-to-center spacing 2 μm between each two tracks. The waveguides had an insertion loss of 5.5 dB for the vertical polarization mode and 11.5 dB for the horizontal polarization mode. Type III waveguides were multimode for both polarizations, with minimal losses. Such structures cannot be fabricated without the use of adaptive optics. Due to the symmetrical structure of the Type III waveguide, it is insensitive to polarization. Due to the asymmetry of the graphite tracks as well as the asymmetry of the laser focus, a difference in transmission is formed for vertical and horizontal polarization. The sides of the waveguide are continuous, and the bottom and top contain unmodified sections that separate the graphite tracks. The ability to manufacture waveguides in the bulk of the material and integrate them into optical fiber systems is a promising topic. However, propagation losses are relatively high compared with optical nanocavities obtained by plasma etching [56]. It is nevertheless apparent that the plasma shaping of microstructures in diamond requires a complex process of masking and plasma etching on the surface of a sample. Laser writing of waveguides in the diamond bulk is on the contrary a real time process and the graphitization of the material can follow arbitrary paths inside the diamond bulk. This can result in a new technology of programmable laser shaping of a number of optoelecronic devices, once the properties of the modified material are properly understood and controlled.

FIGURE 8

FIGURE 8. Type II waveguide geometry and transverse view optical microscope image of type II modification in diamond. Creative Commons Attribution License [54].

FIGURE 9

FIGURE 9. (A) Schematic view of the vertical Y-splitter. Transmission images of the input (B) and output (C) facet of the splitter. (D) An image of the output near-field mode with a 50:50 power splitting [55].

FIGURE 10

FIGURE 10. (A) Microscopy image of the side surface of the type III waveguide focused under the surface of the sample.(B) Microscopy image of the side surface of the type III waveguide focused on the surface of the sample. (C) Topdown transmission microscopy image of the graphitic tracks comprising the waveguide. Image of the end of the waveguide with (D) vertical and (E) horizontal polarization [55].

In the last few years, active investigations of 3D diamond detectors have been carried out [57, 58]. Femosecond laser processing in the bulk of the diamond [59] leads to the formation of graphitic electrodes. This 3D placement of the electrodes greatly improves the radiation resistance of the detectors compared to conventional electrodes [60]. When machining and fabricating structures in the bulk, the laser beam must focus in the bulk of the diamond. This is quite difficult to do due to the presence of optical aberrations. Due to the refraction of light on the surface of the diamond, there is a depth-dependent shift of focus and a decrease in its intensity [61]. In order to avoid this effect, adaptive optics aberration correction is used to remove distortions [62].

The effect of laser processing parameters on the properties of graphite wires formed inside synthetic mono- and polycrystalline diamonds was investigated [63]. Wires were fabricated in CVD diamond samples using different light polarizations, processing speeds and laser pulse energies. A comparison of their quality was made by measuring their electrical resistance and radius. For many of the columns a barrier potential was observed. The output current was negligible until the applied voltage reached a threshold value. For values of the applied voltage exceeding the threshold voltage the wires exhibited ohmic behavior [63]. Graphite wires with larger diameter and lower resistance and barrier potential were formed with higher pulse energies. The lowest barrier potential and resistance were obtained in wires produced by a large number of repetitions of the graphitization process and slow scanning. The properties of wires were almost independent of the polarization of the laser beam. The correction of the optical aberrations by adaptive optics, aimed to control the laser spot size during fabrication made a great impact on the quality of the formed wires. Applying aberration correction resulted in a negligible barrier potential and a minimum value of the wire resistance. The obtained results were reproduced by multiple measurements [63]. Determining of the barrier potential could be a very important factor for the characterization of graphite wires fabricated by emerging methods, such as the use of Bessel beams [64] as well as for the design of wire junctions [65, 66] or gaps in the conductive network [67].

4 Effect of the Electronic Kinetics on Graphitization of Diamond

Amorphization of diamond by ion irradiation has been experimentally studied by means of spectroscopic or electron microscopy characterization [68–70]. Solid phase transition of diamond induced by femtosecond free-electron laser irradiation has also been evidenced and simulated [71–73]. However, in spite of the comparable level of excitation of the electron system in the femtosecond laser irradiated spots, no noticeable structure transformations were detected in diamond irradiated with swift heavy ions (SHI) [74–78]. This can be ascribed to the conditions of initial excitation–photons vs. ion impact as well as to the difference in the spatial and temporal scales of the initial electronic excitations (10 fs–1 ns and a cylindrical laser spot 1 × 1 μm on the surface of the sample vs. 0.1–10 fs and a diameter 5–50 nm in SHI track of 20–100 μm length). The latter leads to different channels of energy dissipation to the diamond lattice [79].

It has been proposed [80], that in the case of laser excitation, the electron density can be kept above critical for times sufficient for nonthermal melting due to the micrometer scale of the laser irradiated region.

On the contrary, the SHI tracks have an angstrom to nanometric size and due to the very fast lateral spreading of electrons, their density drops quickly. Hence, relatively little energy is transmitted to the diamond lattice in the nanometric vicinity of the SHI trajectory. Due to the high thermal conductivity of the diamond lattice, this excess lattice energy quickly disappears without starting the structural transformations in the SHI track. The lifetime of the excited electronic system during laser irradiation is quite longer, so a large amount of energy is transferred to the diamond lattice. As a result, there are non-thermal effects that destabilize the lattice itself.

A lower level of laser fluence is needed to initiate graphitization in neutron predamaged diamond samples [81]. This points out to a large influence of the concentration of defects on the kinetics of excitation and relaxation of laser irradiated diamond. The neutron-induced defects enhance the coupling of excited electrons with diamond lattice, while decreasing heat diffusion from the laser spot to the lattice itself. Consequently, energies delivered to the electronic subsystem and the lattice and lifetimes of the excitations increase. Finally, this results in an enhanced graphitization of the pre-damaged diamond material, compared to the undamaged one.

5 Overview of Theoretical Modeling

We present an overview of the advanced theoretical approaches aiming at understanding the fundamental physical processes of femtosecond laser interaction with semiconductors and dielectrics that are relevant and important for ultrafast diamond processing applications. In this regime of material modification, the different stages during the transformation of the material properties can be studied using time-resolved techniques. In compementarity to experimental findings, the theoretical methods describe the evolution of strongly nonlinear and non-equilibrium carrier dynamics in diamond and the change of its optical, electronic and structural properties. The ultimate goal is to identify an accurate parameter free theoretical approach suitable for interpretation and prediction of real–time experiments and for establishing of a connection between complex microscopic response and macroscopic properties of the photoexcited substrate.

5.1 Electronic Band Structure

The extraordinary physical and chemical properties of diamond are determined by its crystal structure. Diamond is a Group IV element with six electrons occupying a ground state configuration: 1s²2s²2p². Diamond crystal comprises a face-centered cubic lattice structure with a lattice constant equal to the cube edge length a₀ = 0.3567 nm. Two carbon atoms which are displaced from each other by a translation of a₀ (1/4, 1/4, 1/4) along a body diagonal are associated with each lattice vertex. Each carbon atom forms a strong sp³-hybridized covalent bond with its four neighboring carbon atoms in this regular stable tetrahedral configuration [82]. The C-C bond length is 0.1545 nm at 298 K. The high strength of the C-C bond and its short length are responsible for the extreme hardness of diamond and its large elastic moduli.

The theoretical calculation of the energy band structure of diamond is of key importance for interpreting the material’s optical properties. The band structure represents the allowed energies of an electron as a function of crystal momentum k. The electronic energy gap Eg is defined as the minimum energy required to excite an electron from a filled valence band to an empty conduction band. Pseudopotentials or their form factors were calculated from crystalline energy levels using empirical pseudopotential [83] or nonlocal pseudopotential [84] methods to obtain band structures for Group IV elements for the accurate interpretation of optical reflectivity and photoemission experimental data [85]. Crystalline diamond band structure and electronic properties were calculated applying tight binding model with Hartree-Fock wave functions [86, 87] and two ab initio methods–Hartree-Fock (HF) and density functional theory (DFT) within the generalized gradient approximation (GGA) [88, 89].

The state-of-the-art accurate calculations of the electronic structure of solids combine density functional theory (DFT) [90] and many-body perturbation theory (MBPT) [91]. DFT describes the structure of materials in their electronic ground states and provides reliable initial wave functions and valence eigenenergies, but an unreliable bandgap. For that reason, the DFT wave functions are post-treated by MBPT in the GW approximation to DFT for incorporating many-electron self-energy effects [92]. The combined approach delivers electronic bandgaps within 0.1–0.4 eV of the experimentally obtained ones [93].

Most electronic structure studies are performed for static lattices that are implicitly assumed to be at 0 K. Experimental investigations of material properties, though, are carried out at finite temperatures. Temperature affects the nuclear motion in the material leading to lattice dynamics, which in turn alters the electronic energies in the solid. The electron-phonon interaction whose major contribution stems from the lattice dynamical behavior is included in density functional total energy calculations with ab initio pseudopotentials via the formalism of Allen [94] to obtain the temperature dependent electronic structure of diamond and silicon [95]. An ab initio method based on the all electron quasiparticle (QP) self-consistent (QPSC) GW approximation and Keldysh time-loop Green’s function approach was applied for calculations of diamond electronic structure at finite temperatures [96]. The ground state electronic properties of diamond were calculated from first principles using local density approximation (LDA), generalized gradient approximations (GGA) functionals [97] and an improved quasiparticle self-consistent GW (QSGW) method [98, 99]. In Figure 11, the band structure of diamond for crystal momentum varying along the ΓX and ΓL directions in the Brillouin zone is shown. The empirical pseudopotential model used in the calculation [100] reproduces the principal energy gaps quantitatively. The absolute energy position of the conduction band minimum corresponds to an indirect gap of 5.42 eV. This is in very good quantitative agreement with the experimental result. Since in long-wavelength approximation a momentum changing transition cannot be induced by the laser, the minimal energy required to cross the energy gap at the Γ point is 7 eV.

FIGURE 11

FIGURE 11. Electronic energy bands of diamond along Δ and Λ symmetry lines [100].

5.2 Laser-Induced Processes

An ultrashort laser pulse (10^–15 s–10⁻¹² s) generates high peak electric fields (>10⁹ V/m), which are orders of magnitude larger than the Coulomb fields binding electrons to atoms and correspond to laser intensities exceeding 10¹² W/cm². The highest frequency at which the electronic system responds to the strong driving field is determined by the material band gap E g., In this excitation regime, nonlinear absorption for laser frequencies below the direct bandgap energy of diamond is induced and the material absorbs energy from the field due to the creation of electron-hole pairs across the material band gap.

The initial processes following the conventional near-infrared (NIR) to near-ultraviolet (NUV) wavelength ultrashort-pulse laser irradiation of diamond are carrier photoionization and photoexcitation, buildup of coherent polarization and carrier thermalisation. Energy deposited onto the electron system on a timescale shorter than the electron-phonon scattering time may induce ultrafast non-thermal structural changes directly driven by electronic excitation and related non-linear processes. For example, when a certain fraction of valence electrons are excited to the conduction band, they are removed from the bonds and move to the interstitial regions. This causes the weakening of the chemical bonds and affects the cohesion of diamond material. If the electrons diffuse away, the positively charged ions left behind repel each other and cause lattice instability.

On ultrashort time scale, shorter than that of momentum scattering, the generated electrons and holes are described by a coherent superposition of states in the valence and conduction bands. At increased field strength a high number of electron-hole pairs is generated leading to optical breakdown of the material. Ultrafast optical breakdown is a complex, strongly nonlinear threshold phenomenon involving a number of microscopic prossesses and leading to structural modifications of the irradiated material that are highly controllable and reproducible. Conventionally the laser fluence, defined as the time integral of the laser intensity over the pulse length, which produces the critical electron density for which the generated electron plasma becomes reflective, is defined as the optical breakdown threshold (OBT) fluence for the fixed pulse duration [101]. Properly calculated material bandgap and dielectric function near the laser frequency are important for obtaining the OBT. An alternative way to determine the OBT is defined by the averaged kinetic energy deposited onto the electrons [102].

Depending on the pulse duration, scattering with phonons may initiate during the irradiation leading to field-assisted collisional effects (e.g., making possible inditect transitions in conduction band), dephasing of coherence and carrier-phonon scattering. Usually permanent structural effects on the surface including carrier removal and ablation occur for interaction times longer than a picosecond, i.e., after the conclusion of the ultrashort pulse and involves energy transfer between the carriers and the lattice.

5.2.1 Photoionization and Photoexcitation

Photoionization and photoexcitation of electrons are the main mechanisms of laser-pulse energy deposition, which subsequently causes ultrafast transformation of the crystal lattice. The conventional theoretical description of strong-field ionization and interband transitions in solids is based on the Keldysh approach [103–105] that provides an analytical expression for the cycle-averaged interband transitions rate induced by a monochromatic electric field of arbitrary strength. A dimensionless Keldysh parameter relating the bandgap energy of the material with the ponderomotive energy of an electron hole pair is used to separate the interband transitions into adiabatic tunneling, diabatic tunneling and multiphoton excitations. In the adiabatic tunneling regime, the electron goes through the partially suppressed potential barrier without changing its total energy, while in the multiphoton regime transitions occur without any penetration into classically forbidden regions. In the regime of diabatic tunneling the electron wave function cannot adjust to the fast changes of the external potential, so it gradually acquires energy that facilitates its transition into the continuum of conduction states. In this method, the two-band approximation was used, assuming that the electric field predominantly excites electrons from the highest valence band to the lowest conduction band. Second order of two-band k × p-perturbation theory, which provides the dispersion relation between energy and momentum of Bloch electrons, predicted that the band gap monotonically increases with increase of the electron crystal momentum. A nonperturbative parameter which is the ratio of the ponderomotive and photon energies, determines the number of additional photons that must be involved in the excitation of an electron-hole pair in order to overcome the increase of the band gap due to intraband motion. Another key approximation of this theory is the neglect of the Coulomb interaction between an electron excited to the conduction band and the positively charged hole left behind in the valence band.

The transition between adiabatic states, induced by ultrashort laser pulses of arbitrary waveform was studied using quantum mechanical formalism which was a generalization of the Keldysh theory of photoionization [106]. It was demonstrated that electron dynamics in strong field resulted from a nontrivial combination of interband and intraband motion and that photionization depended sensitively on details of the substrate band structure [106]. The Time Dependent Schrodinger Equation (TDSE) in single active electron approximation and velocity gauge, combined with empirical pseudopotential method for realistic band structure calculation, was applied to model the high electronic excitation in CVD polycrystalline diamond irradiated by 30 fs infrared laser pulses with varying intensity below and close to the graphitization threshold [107]. The modeling was carried out under the same conditions as the experimental procedure used to produce three-dimensional conductive paths in the bulk of diamond. It was found that in the weak coupling regime (laser intensity I ∼ 1 TW/cm²) the scaling of the conduction electron density with driving laser intensity strongly deviates from the conventional multiphoton ionization due to Stark effect creating a highly polarizable transient state in the bulk of the material that ionizes very efficiently. In the strong coupling regime (I ∼ 50 TW/cm²), nonadiabatic effects of electron-hole pair excitation result in the generatation of high carrier densities which cause the optical breakdown of diamond. The comparison of theoretical prediction with experimental data of laser-generated charge as a function of laser energy density provides a good quantitative agreement over six orders of magnitude.

An analytical model based on TDSE with a parabolic two-band structure was reported to give consistent results with the experimental data on anisotropy of multiphoton carrier generation in monocrystalline diamond [108]. The energy transfer mechanism from a few cycle intense mid-infrared laser pulse to the electronic system in the bulk of diamond and the subsequent optical breakdown in the material is studied via TDSE with model band structure in [109]. Ultrafast modification of the band structure of wide-band-gap solids was studied using TDSE in length gauge [110]. In [107–109] dipole approximation was used and electron-electron and electron-phonon collisions were neglected.

The density matrix method leading to Semiconductor Bloch equations (SBE) for the time evolution of electron and hole occupations and microscopic polarization was applied to describe interband and intraband carrier excitation and ultrafast dephasing in semiconductors and diamond in particular [111–114]. This approach requires input parameters on various levels of sophistication and can incorporate a different number of energy bands. For example quasi-energies are defined for a single electron (hole) in the crystal lattice and they are modified when Coulomb interaction of carriers is included in the model. SBEs are derived within the Hatree Fock approximation and they incorporate many-body Coloumb effects and phase-space filling. Going beyond this approximation requires higher-order correlation functions leading to further renormalization of interactions and energies.

Time-dependent density functional theory (TDDFT) [115] as an extension of DFT provides the band structure of diamond and describes the real time-dependent quantum dynamics of interacting electrons at the first-principles level treating the mechanisms of multiphoton and tunneling ionization and including dynamical screening. Norm-conserving pseudopotentials are used to account for the interactions between the valence electrons and the ionic cores. The static Kohn-Sham equation is first solved to determine energy bands, density of states, and forces acting on the ions in the ground state. The time evolution of the electronic orbitals is obtained following the excitaion of the time-dependent spatially uniform electric field of the ultrashort-pulsed laser. The transmitted field inside the bulk substrate includes the polarization field of the medium. Since semi-local exchange correlation functionals are used in the calculations, i.e. the adiabatic local-density approximation (ALDA) or generalized gradient approximations (GGA), the method is limited for pulse durations shorter than the electron-electron and the electron-phonon relaxation time. The relaxation related to energy transfer from electrons to ions was also ignored. In [115] 16 fs laser pulses with varying wavelengths were used to determine optical breakdown in diamond. It was found that when laser intensity is low diamond exhibits dielectric response with static dielectric function obtained from the ratio of the external to the total electric field. It was shown that in the weak coupling regime the energy deposition in diamond as a function of intensity follows the perturbative scaling wich agrees with Keldysh theory. For peak laser intensities (I > 70 TW/cm²), the deposited energy exhibits abrupt increase which is consistent with resonant energy absorption when optical breakdown occurs. This effect is also exhibited by the phase shift between the total electric field and the applied electric field. The optical breakdown, reached when plasma frequency matches the laser frequency, is found to be limited by emerging plasma oscillations at the conclusion of the laser pulse and screening of the electrons in the conduction band. In [116] electronic excitation in diamond following the irradiation of intense ultrashort laser pulse was investigated by TDDFT for various wavelengths and laser intensities in order to describe the energy transfer mechanisms. The calculated results were compared with the excitation rates from Keldysh theory and with the energy deposition rates in other dielectric materials possessing different bandgap energies and cohesion mechanisms.

The strong non-linear and ultrafast dynamics of electrons results in high-order harmonic generation [100, 117–120] and in optical-field-induced transient and reversible currents below the bulk damage threshold [121–124]. It was experimentally found and theoretically confirmed [113] that high harmonic generation in bulk solids is characteristic of a nonperturbative quantum interference process that involves electrons from multiple valence bands. Photoexcitation, nonlinear optical properties, induced ultrafast currents and generation of high harmonics in diamond irradiated by ultrashort infrared ultrashort laser pulses were described by an extended TDDFT accounting for coupling of the excited electronic system to external (“environmental”) degrees of freedom that could be phonons or lattice defects. The results of the calculation were compared with the results from SBE calculation [120].

5.2.2 Optical Properties

The electric fields of ultrashort intense laser pulses are able to modify the optical properties of a solid substantially and nondestructively. The transient changes of optical properties provide valuable information on the prompt changes of electronic and atomic structure of irradiated solids. The optical response of band gap materials varies with the driving laser intensity. For low peak intensities a dielectric response is exhibited. With increase of intensity, nonlinear response prevails and high-order perturbation theories should be used for calculating the response in the frequency domain [125]. For very strong electric fields of the laser the material’s optical properties become similar to those of a nonequilibrium electron-hole plasma [126]. In the strong coupling regime the polarization response of a solid is no longer described by a set of linear and nonlinear susceptibilities and should be modelled quantum mechanically using Shrodinger equation which is only feasible in one or two active electrons approximation for the treatment of many electron systems hence approximations are required. As described in previous sections, density functional theory (DFT) with an approximated semi-local exchange and correlation functional is used to describe electrons in the ground state and time-dependent DFT (TDDFT) is required to simulate excited electrons. For electron dynamics, the real-time formulation of TDDFT can account for response to strong perturbations. Going beyond TDDFT it is possible to describe more precisely optical properties of solids, for example by using the time dependent Bethe–Salpeter equation. The optical responses of infinitely periodic systems to a spatially uniform external field are characterized by the complex dielectric function.

Non-local empirical-pseudopotential method was used to calculate the optical absorption in type I diamond [84] and a good agreement with the position of the main peak of the imaginary part of the complex dielectric function was found but its magnitude showed a significant discrepancy with experiment. The momentum and energy dependent Lindhard dielectric function of diamond was obtained using full band calculation based on the local empirical pseudopotential method [127] via MC calculation. The frequency-dependent dielectric functions for the crystals in the diamond lattice structure were obtained by a real-space approach to TDDFT within the adiabatic local-density approximation, implementing full-potential linear combination of atomic orbitals (LCAO). The dielectric constants at optical frequencies were also obtained and were in good agreement with experiments. It was found that the calculated dielectric functions reproduced accurately the experimental spectral features [128]. The linear response of diamond was calculated via TDDFT using a real-space representation of the electron wave functions and calculating the carrier dynamics in real time [129]. After obtaining the ground-state electron orbitals and the periodic Coulomb potential in the absence of an electric field, an impulsive external electric field was applied and the polarization electric field was determined by the real-time evolution of the time dependent Kohn Sham (TDKS) equation. The inverse frequency dependent dielectric function was related to the Fourier transform of the total electric field (including the induced polarization field). This approach was further developed and the frequency dependent electric conductivity and the dielectric function of a group IV semiconductor were obtained by solving the TDKS equation for the electron dynamics following the excitation of a pump pulse. It was shown that the real part of the dielectric function exhibits a metallic like response at low frequencies reflecting the generation of dense electron-hole pair excitations [130]. Similar approach was applied for the manipulation of the dielectric function of diamond in nonequilibrium state induced by an intense and single-cycle ultrafast laser field with duration of a several femtoseconds [131]. As in the previous study for silicon [130], for high intensities negative divergence of the real part of the dielectric function at low frequencies was observed. The anisotropy of dielectric tensor in excited diamond when going beyond the linear response regime was also demonstarted [131]. Using real-space, real-time TDDFT calculations it was demonstrated that the permittivity tensor of the diamond can be effectively tuned by an ultrashort laser pulse to exhibit hyperbolic or elliptic response for a particular light frequency [132].

The time-dependent subcycle modulation of the dielectric function close to the energy gap of diamond was calculated by TDDFT in the frame of a numeric pump-probe experiment as a manifestation of the time-resolved Dynamical Franz Keldysh effect (Tr-DFKE). This ultrafast modulation was found to be phase shifted with respect to the applied probe field [133]. The sub-femtosecond optical response of polycrystalline diamond irradiated by intense, 250 attosecond pump pulses was studied by transient absorption spectroscopy. It was shown that the infrared pulse induces oscillations in the imaginary part of the dielectric function and consequently in the absorbance of the material. Ab initio calculations performed by coupling time-dependent density functional theory (TDDFT) in real time with Maxwell’s equations reproduced the experimental results [134]. Experimental and theoretical results showed that the strong field induces tens of orders of magnitude increase of the conductivity that may lead to a state of semi-metallization, which is reversible on a femtosecond time scale [135].

The complex dielectric function and transient optical properties such as reflectivity, transmission and absorption were calculated by a hybrid MC method [136] and it was shown that they could be used for diagnostics of electronic and structural transitions occurring in irradiated semiconductors.

In [38] the dielectric function of diamond irradiated with 515 fs infra red laser pulse was obtained via the time-dependent plasmon pole-approximation for the density-density correlation function of coulombically interacting electron gas assuming that the linear response of photoexcited diamond can be described by a free-carrier Drude response. Most recently, the linear response and the modulation of optical properties of diamond were calculated using a multiscale method combining self-consitenly Time dependent Kohn Sham equation to describe microscopic electron dynamics, Maxwell equations to describe macroscopic propagation of the laser pulse and first-principles Ehrenfest (classical) Molecular dynamics approach to describe the ultrafast lattice dynamics [137, 138].

A first principles method was used (GW approximation to DFT and an effective Hamiltonian including the electron-hole interaction extracted from Bethe-Salpeter equation for the electron-hole pair wave function) to determine the frequency-dependent imaginary part of the dielectric function in diamond by iteratively applying the effective Hamiltonian on an electron-hole pair state [139]. Recently the optical absorption spectra for diamond were obtained through TDDFT based on nonempirical hybrid functionals that were designed to correctly reproduce the dielectric function. The comparison with the combined state-of-the-art GW calculations and the Bethe-Salpeter equation (BSE-GW) shows close agreement for the transition energies and the main features of the spectra [140].

5.2.3 Imapact Ionization and Auger Recombination

After photoelectrons are excited into the conduction band across the direct band gap (with well defined energies above the conduction band minimum) they experience electron-electron collisions, while absorbing additional energy from the laser field. If a substantial fraction of the carriers occupy higher lying conduction bands with energy above the threshold for impact ionization (specified by the indirect gap of diamond equal to 5.4 eV), the highly energetic electrons can collisionally ionize electrons into lower energy states. Following the collision, the impact electrons lose part of their energy, transferring it to promote valence electrons in the conduction band. Thus, once a threshold energy is reached by conduction electrons, impact ionization (II) may occur on a femtosecond time scale resulting in carrier multiplication through the decay of high energy electrons into lower energy states while generating electron-hole pairs [141]. When carrier densities become high Auger recombination (AR), related to impact ionization through the principle of detailed balance as an inverse microscopic prossesses, may become significant. In a direct Auger process, an electron recombines with a hole, but instead of emitting a photon the process results in the excitation of another carrier to a higher-energy state [141].

The impact ionization rate in diamond was calculated by Monte-Carlo (MC) simulation which included a band structure obtained from an empirical pseudopotential method and frequency- and momentum-dependent Lindhard dielectric function [127]. Recently the ultrafast impact ionization in diamond irradiated by mid-infrared femtosecond laser pulses was experimentally observed by transient transmission spectroscopy and photoluminescence measurements [142] revealing that the excited carrier population incerases two times in comparison to the initial one for peak intensity of 2.5 TW/cm². Supporting TDDFT simulations of electron dynamics showed that the photoexcited carriers could reach the energy threshold for impact ionization in diamond under the experimental conditions. Scattering mechanisms were not included to describe theoretically ultrafast impact ionization. Even though the kinetic energy distribution of the photoexcited carriers in CVD diamond extended above the required threshold, the process of impact ionization was strongly suppressed by the high population of the conduction-band low-energy states due to Pauli blocking and electrons could gain sufficiently high energies from the laser field [107]. A first-principles approach based on screened exchange density functional theory (DFT) was applied for the calculation of II and AR rates in semiconductors [143]. MC techniques were used to calculate impact ionization and Auger recombination rates in diamond [71] irradiated by X ray femtosecond pulses.

On ultrashort time scale, photoionization alone or together with ultrafast impact ionization generate high electron-hole plasma density, leading to multiplication of conduction electron number and an efficient energy transfer from the laser to the conduction electrons. When the absorbed laser energy exceeds the melting or cohesive energy of the material, optical breakdown occurs which is also considered as a precursor to laser ablation. In [38] for the 200 fs pulse duration used in the experiment and intensities lower than 50 TW/cm² it was found that the electron density produced by photoionization is below the critical one. The calculated density of conduction states at the end of the pulse showed that substantial fraction of carriers occupy higher lying conduction bands with energy above the threshold for impact ionization and may collisionally promote valence electrons in the conduction band. The intensity-dependent impact ionization rate was obtained as a weighed average of the field-free rate [127] using the calculated density of conduction states in diamond.

5.2.4 Coherent Phonon Generation and Ultrafast Phase Transitions

As shown in the experimental part of the review it is important to know on what time scale the structural transformations occur. Since the typical electron-lattice energy relaxation time is 10^–10–10^–12 s, the incident femtosecond laser pulse deposits energy onto the electron system creating a transient highly non-equilibrium state in which initially the nuclei remain fixed at the lattice positions and electronic excitation and vibrational degrees of freedom are completely decoupled. Consequently, the laser induced redistribution of the electronic occupation modifies the interatomic bonding that results in coherent phonon generation [137, 138, 144–147], solid-to-solid phase transitions [148, 149], thermal phonon squeezing [150], and nonthermal melting [151–153].

Femtosecond time-resolved pump-probe reflection experiments in semiconductors revealed large-amplitude oscillations with periods that were characteristic of lattice vibrations. The frequency of the oscillations in the observed reflection matched one of the optical phonon modes, which indicated that this particular phonon mode was coherently excited. Under the assumption that the electronically excited system rapidly comes to quasiequilibrium in a time shorter than nuclear response times, a theoretical model of displacive excitation of coherent phonons (DECP) describes the evolution of a coherent phonon amplitude in the presence of nonequilibrium photocarriers with driving force, which is a function of the carrier density and the temperature of the carrier distribution. A rapid increase in the carrier density during the optical excitation, initiates the oscillating motion of the coherent amplitude [144]. The excitation of coherent optical phonons was modeled via adiabatic TDDFT [146]. It was theoretically predicted by ab-initio molecular-dynamics simulations [150] that femtosecond-laser-induced collective atomic motions precede complete atomic disordering (nonthermal melting) as a function of laser fluence. The instability of the diamond and zinc-blende structures due to the excitation of a dense electron-hole plasma were described under the assumption that the photo-excited plasma thermalizes rapidly, due to the strong electron-electron interaction. Its effect on the cohesive energy of the diamond structure together with the lattice distortions caused by phonons were analyzed [154]. When a substantial fraction of electron–hole pairs is generated in the material, there is a strong modification of the electron density distribution. Consequently, the ions experience strong repulsive inter-atomic forces due to the dramatic changes of the potential energy surface (PES). Using a tight binding Hamiltonian for the electronic occupations it was shown that the functional form of the potential energy surface (PES) strongly depends on the occupation of the electronic levels. The modified interatomic forces can cause coherent atomic motions. If they are sufficiently strong, they can induce permanent structural modification of crystalline diamond on a very short time scale [73, 144, 155–157] without invoking significant energy transfer between electrons and ions and without significant increase of the ionic temperature. Hence, the ultrafast disordering was regarded as nonthermal phase transition [158, 159]. Density functional theory approaches were used to account for the changes in the PES following electron–hole-pair excitation [160, 161]. Laser excited ab initio PES was obtained under the assumption that the electronic system has a particular temperature following the excitation and the subsequent thermalization. Thus, electronic occupations were considered to obey a Fermi distribution at the resulting electronic temperature and Kohn–Sham equations were solved self-consistently under this constraint. The electronic entropy was determined and PES was obtained by applying a generalized Born–Oppenheimer approximation. Recently a numerical pump-probe multiscale calculation was used to simulate coherent phonon generation in diamond [137]. The calculations demonstrated that the driving force on the ions of the crystal lattice was generated as a consequence of the change in the electron density which was dependent on the strength of the applied field and did not arise from the direct laser field-ion interaction. The ground state electron density in the (011) plane of diamond was compared with the transient electron density induced by the ultrashort laser pulse excitation. It was demonstrated that the applied field induced a shift in the electron bonding direction in respect to the ground state in which the covalent bonds were situated between the ions of the lattice. The shift was dependent on the direction of the applied electric field. It was also found that the forces exerted on the ions were independent of the field direction and all the ions in the chosen plane of the material were weakened. It was found that the generated coherent phonons induced anisotropy in the refractive index of diamond as well as a modulation in the reflection of the probe pulse [138].

5.2.5 Carrier Relaxation

For times greater than 10^–12 s, the energy absorbed by the electronic subsystem dissipates into the ionic degrees of freedom via electron-phonon coupling, causing the ionic temperature to increase. The ionic temperature can rise beyond the melting point, which would result in the disordering of the lattice in few picoseconds. This transition is ultrafast thermal melting since conduction electrons can cause a permanent structural damage of the material by transferring their kinetic energy to the lattice. At these time scales the ultrafast dynamics of photocarriers in irradiated semiconductors was described by theoretical methods varying from model equations using parametrization of the distribution functions [162–165], direct integration of Boltzmann equation [166,167] or Monte Carlo simulations [168–171]. Calculation approaches were chosen based on their ability to include and interpret on microscopic level the carrier-carrier and carrier-phonon scattering processes, which strongly depend on the experimental conditions (e.g., electron-electron effects depend on the conduction electron density). Complementarity of experimental results and theoretical calculations provide relevant data on the required scattering rates and coupling constants [172–174].

On short time scales, both coherent and incoherent processes have to be included. The buildup of polarization and the Coulomb interaction between carriers requires a full quantum mechanical treatment. Quantum kinetic theoretical methods used to describe the coherent phenomena are non-equilibrium Green’s-functions techniques [172–174] and density-matrix theory [175, 176]. For time dependent processes such as ultrashort-pulsed laser irradiation of semiconductor materials the density matrix method provides quantities directly comparable with physical observables.

Quantum kinetic-type approaches, two temperature models and rate equations have been applied to account for all main relaxation processes [177–179]. In quantum-kinetic approach, the starting point is a Hamiltonian that describes the electron and the phonon subsystems and their coupling to first and second order in the atomic displacements using second-quantized formalism. The matrix elements in the Hamiltonian provide the strength of the coupling between the electron and the phonon subsystems. This approach has important shortcomings when used for predictive calculations, because the electronic part Hamiltonian relies on the assumption that the studied system can be described in terms of well-defined quasiparticle excitations and that the phonon term is included within the harmonic and the adiabatic approximations. In addition, the single-particle eigenvalue of an electron with a crystal momentum k in a band n, the frequency of a lattice vibration with crystal momentum q in the branch λ, the phonon dispersion and the strength of the coupling between the electron and the phonon subsystems have to be determined using other methods. Phonon dispersion relations, electron phonon coupling strength and electron-phonon matrix elements in diamond can be calculated using Density Functional Theory, entirely from first principles [180–183]. In addition, nonadiabatic corrections to phonon dispersion relations and temperature dependence of optical spectra in direct and indirect-gap semiconductors can be obtained. Density functional perturbation theory DFPT enables the calculations of vibrational frequencies and eigenmodes at arbitrary wave vectors in the Brillouin zone. Electron-phonon scattering processes and electron-phonon renormalization of band structure can be computed from the lattice dynamical properties obtained from DFT or DFPT within the local density approximation (LDA) [184–191]. Combined with the Boltzmann or the Kadanoff-Baym equations [189–191], the dynamics in materials can be studied. The temperature-dependent renormalization of the direct band gap of diamond arising from electron-phonon interactions and the temperature-dependent broadening of the direct absorption edge in the material were calculated in [186]. The unperturbed band structures were obtained from DFT within the GW approximation, and the lattice dynamics was described within density-functional perturbation theory.

In DFT approach though, the evaluation of the electron-phonon matrix elements uses the assumption that the interaction between electrons and nuclei is governed by the effective Kohn-Sham potential and depends on the exchange and correlation functional. The definition of phonons relies on the Born- Oppenheimer approximation. Since electron-phonon interaction can modify both the electronic structure and the lattice dynamics of a solid, and these modifications will in turn affect the coupling between electrons and phonons, a complete theory of interacting electrons and phonons must be self-consistent.

More elaborate first principle approaches based on many-body perturbation theories discussed in previous sections, provide an ab initio description of both the photo-excitation and of the time dependence of the carrier scattering rates. They allow including on the same footing the change of carrier and phonon populations due to both excitation and scattering which for the case of ultrashort laser pulse excitation may overlap in time. Approaches such as many body Green’s functions and GW plus Bethe Salpeter equation [192,193] may provide accurate parameter-free interpretation of real time pump-probe experiments used to capture electron excitation, electron spatio-temporal distribution on the femtosecond scale, transient optical response, the electron collision time and laser-induced electron-hole plasma lifetime [194, 195]. A first-principles version of the Kadanoff-Baym equations [196,197] employing collision approximation [198] to reduce the non-equilibrium Green’s function dynamics to a single time variable was used to reproduce the carrier dynamics in bulk silicon as observed in a two photons photo-emission experiment. The equation for the evolution of the non-equilibrium electron (hole) occupation contains a coherent generation term, which describes the creation of carriers via the interaction of the polarization and the laser electric field, and a term describing how carriers dissipate energy. This equation is coupled to the polarization equation self-consistently. The generation term is derived from Green’s functions approach based on DFT Kohn-Sham Hamiltonian and includes Hartree and exchange correlation effects. The expressions for carrier-carrier and carrier phonon lifetimes that govern the carrier relaxation include the statically screened Coulomb interaction and the screened ionic potential derivatives, calculated within density-functional perturbation theory [199, 200] and formally correspond to the carrier-carrier and carrier-phonon scattering in the semi classical Boltzmann equation obtained using Fermi Golden Rule. The dependence of the relaxation terms on the calculated occupations was obtained self consistently at each time step.

5.2.6 Ultrafast Laser Ablation

Laser ablation is a complex phenomenon that includes the initial stage of carrier photoexcitation, collisional ionization, thermalization and relaxation and the transition stage of development of structural changes and removal of matter. Ultrafast laser ablation in diamond was studied experimentally and theoretically [22, 24, 201–211]. In [201] femtosecond time resolved reflectivity and transmission pump-probe measurements of the laser excited substrate indicated an ultrafast melting transition from crystalline diamond to a highly reflective dense plasma. The threshold fluence for ablation of diamond with 620-nm, 90-fs laser pulses was found to be 0.6 J/cm². Raman spectra and SEM images of CVD diamond ablated with a number of femtosecond laser pulses near threshold fluence revealed no evidence of graphite or amorphous carbon and clean ablation with sharp interfaces and little thermal damage. In [202] molecular dynamic simulations performed on the basis of time-dependent, many-body potential energy surfaces derived from a microscopic electronic Hamiltonian were used to obtain laser induced femtosecond structural changes in diamond. The results showed smoothly increasing ablation threshold for increasing pulse duration. In [212] the optical damage and surface ablation threshold fluences as a function of pulse width for femtosecond laser irradiated silicon oxide were obtained from a TDDFT calculation. The threshold fluence for optical damage was determined by comparing the transferred energy to the energy needed to melt the quartz substrate. The calculated ablation threshold fluence was compared to experimental data to demonstare the significance of the nonthermal ablation mechanism and the agreement between the measured and theoretically calculated ablation craters.

First-principles electromagnetic theory predicted that when the laser fluence exceeds that of the ablation threshold, laser induced periodic surface structures (LIPSS) can be formed [213] as a universal phenomenon. It was shown that the periodicity of the formed ripples was close to the laser wavelength and orthogonal to the laser polarization. Near-wavelength ripples (low-spatial-frequency LIPSS–LSFLs) can be formed when a small number of intense pulses irradiate semiconductors and dielectrics via inhomogeneous energy deposition resulting from the interference between the femtosecond laser pulse scattered on permanent or laser-induced (phase transition of diamond to carbon phase) surface relief roughness and the incident pulse [214]. For irradiation with femtosecond laser pulses, in the regime of optical breakdown fluences, the photogenerated dense electron-hole plasma (EHP) induces a transient change of the optical properties (the permittivity) of the substrate surface and it exhibits metallic behavior [215]. The interaction of the incident pulse with the photogenerated plasma at the interface between the insulator and the conductor leads to plasma oscillations coupled to electromagnetic fields, which can support a highly confined surface electromagmetic wave–surface plasmon (SP). These short wavelength longitudinal surface plasmons could also interfere with the incident laser pulse or among themselves inducing surface ripples with periods much lower than the laser wavelength [35, 216].

The transition of diamond into plasmonically active material and the subsequent formation of ripples on its surface caused by SPP-laser interference were investigated via first principle approaches [41]. The numerical modeling of periodic laser energy deposition on photo-excited diamond surface showed the formation of fine ripples oriented perpendicularly to the laser polarization in a narrow range of intensities above the optical breakdown threshold. The transient increase of the carrier density above this threshold resulted in the formation of near-wavelength surface ripples. Quantitative agreement with the experimentally observed periods of the periodic structures was obtained. The transient efficacy factor describing the efficacy with which the preexisting surface roughness induces inhomogeneous energy absorption at the LIPSS wave vector κ = (κ_x, κ_y) incorporated the variation of the dielectric constant of the photoexcited diamond. The two dimensional efficacy factor map of LIPSS formation as a function of the normalized LIPSS wave vector components is shown in Figure 12 for two values of the transient bulk plasma frequency in monocrystalline diamond irradiated by a 515 nm, 200 fs laser pulse [41]. LIPSS were formed where the efficacy factor map exhibits sharp features (minima and maxima). In Figure 12A for laser frequency ω_L close to the surface plasma frequency ω_p, the efficacy factor has large contribution due to excitation of the surface plasmon resonance. In this regime, the deposition of laser energy at the surface plasmon wavevector favors the formation of fine ripples with spatial periods around 100 nm. Figure 12B shows that when the transiently increasing electron-hole plasma number density results in large and negative value of the dielectric constant at the laser wavelength, the observed feature in the efficacy map is a circle with κ = 1 and can be associated with the formation of near-wavelength surface ripples oriented perpendicularly to the laser polarization.

FIGURE 12

FIGURE 12. [41] Two dimensional intensity graph of the logarithm of diamond transient efficacy factor as a function of the normalized to the laser wavelength LIPSS wave vector components. In (A) ω_p = 1.5 ω_L and (B) ω_p = 1.5 ω_L.

5.2.7 Summary and Outlook

In the experimental part of our review, it was revealed that ultrashort intense laser pulses provide a unique technique for precise diamond micromachining, mainly due to high electric field strengths capable of inducing nonlinear absorption in transparent wide band gap materials as diamond, due to minimized thermal effects and due to short processing times. It was demonstrated that this technique could be used for effective surface nanostructuring or fabrication of three-dimensional structures inside diamond, which allows rapid prototyping of integrated photonic and electronic devices such as photon-enhanced thermionic emitters, electrochemical electrodes and sensors. The confinement of the laser energy absorption to the focal volume situated on top of or inside the bulk of the material is performed experimentally by tight focusing and by tuning the laser parameters such as wavelength, energy density, pulse repetition rate, angle of incidence and polarization state. Surface nanostructuring results in significant modifications of the optical and electronic properties of diamond. One major experimental result is tailoring the optical transmissivity of diamond surfaces via creating laser induced periodic surface structures (LIPSS) with high degree of accuracy of structure periodicity, morphology and aspect ratio achieved by varying the laser parameters. An important experimental result is the fabrication of three-dimensional buried optical waveguides in diamond by focusing high repetition rate femtosecond laser pulses in the material. Despite this innovative capability, the fabricated waveguides still have high extinction coefficient.

Highly conductive graphitic wires with resistivity approaching that of polycrystalline graphite and capable of following any 3D path were fabricated in diamond. Conductive nanostructures of graphenic carbon were produced inside single crystal diamond by femtosecond laser irradiation and the ratio between the sp³ and sp² carbon in the laser-modified heterophase was estimated. These achievements are directly applicable to the design of three dimensional diamond detectors. The current drawback is that diamond detectors with three-dimensional architecture are strongly limited by the excessive resistivity of the columnar electrodes, which increase the noise of the response and degrade the time resolution. The issue for three-dimensional diamond dosimeters is the uniformity of response of the single pixels, which is impaired by the uncontrolled properties of the columnar electrodes and the surrounding diamond lattice subjected to a high stress condition due to the lower density of the formed sp² phase. As a major improvement of these issues, namely reducing the electrode resistance, the design, the fabrication and the characterization of timing 3D diamond sensors, with electrodes obtained through a pulsed laser graphitization technique combined with the application of adaptive optics was achieved recently.

In the theoretical part of the review, it is revealed that both mature theoretical and computational methods and advanced time resolved experimental techniques contribute to the fundamental understanding of the microscopic processes involved in ultrafast processing of diamond as a wide band gap semiconductor. The mechanisms are essentially governed by femtosecond pulsed laser-electronic system interaction that can trigger ultrafast electronic, optical and material response and affect the pulse propagation in the modified medium.

Some of the advanced theoretical methods can be used to model successfully the conditions for occurance of ultrafast phase transitions, optical breakdown and ablation in diamond starting from the material’s ground state band structure, and accurately describing initial carrier excitation and subsequent carrier dynamics induced by the ultrafst intense irradiation. Typically the calculations provide time- (and space)- dependent electron density and current, polarization, electronic excitation energies, energy deposited in a unit cell of the substrate, number density of excited electron-hole pairs, transient distribution function of conduction photoelectrons, and forces acting on the ions as functions of time. Optical properties are described at a different level of sophistication. Multiscale models are developed which include self-consistent description of carrier and lattice dynamics and pulse propagation.

The importance of treating some of the involved microscopic processes on an equal footing is demonstrated in view of achieving a deeper theoretical insight and guidance to experiments. Part of the advanced approaches incorporate at first principles level scattering and relaxation processes occurring in parallel with or following carrier ionization and excitation. Limitations of the described theoretical approaches are pointed out and the possibility of interfacing some of methods are discussed.

For instance, in order to engrave three dimensional electrodes in diamond bulk for the design of radiation sensors, extremely short laser pulses are used. In this irradiation regime, close to the experimental conditions, theoretical description of the phase transformation from diamond to conductive phase in the bulk of the material might be closely achieved using the first principles multiscale methods. On the other hand, it is critical for the applications to control the connection between conductive domains, which can be realized via laser annealing of the modified material. In this case it is important to use the first principle approaches based on many-body perturbation theories which successfully describe coupled electron and phonon dynamics. Recent developments in these approaches could also help predict and control the number of defects induced in the diamond bandgap during fabrication which is also of paramount importance for the efficiency of the produced devices.

Author Contributions

VK and IG contributed to the review of experimental methods and TA contributed to the review of theoretical models, some of the experimental methods and the overall text of the manuscript.

Funding

The authors VK and IG declare that this study received funding from NoviNano Lab, LLC (Ukraine). The funder had the following involvement with the study: the description of part of the experimental data.

Conflict of Interest

VK and IG are employees of NoviNano Lab, LLC.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors acknowledge Cost Action CA17126. Support is acknowledged from the Bulgarian National Science Fund under Contract No. KP-06-KOST, No. DN-18/7 and No. DN-18/11 (TA). T.A. expresses gratitude to Professor S. Sciortino from the Department of Physics of Florence for useful discussions on the experimental aspects of the physics of laser-irradiated diamond.

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