{'fraction_non_alphanumeric': 0.03599564465509511, 'fraction_numerical': 0.021584576955101516, 'mean_word_length': 4.8677189026681695, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Softening the ultra-stiff: controlled variation of Young's modulus in single crystal diamond

A combined experimental and numerical study on the variation of the elastic properties of defective single-crystal diamond is presented for the first time, by comparing nano-indentation measurements on MeV-ion-implanted samples with multi-scale modeling consisting of both ab initio atomistic calculations and meso-scale Finite Element Method (FEM) simulations. It is found that by locally introducing defects in the 2x10^18 - 5x10^21 cm-3 density range, a significant reduction of, as well as of density, can be induced in the diamond crystal structure without incurring in the graphitization of the material. Ab initio atomistic simulations confirm the experimental findings with a good degree of confidence. FEM simulations are further employed to verify the consistency of measured deformations with a stiffness reduction, and to derive strain and stress levels in the implanted region. Combining these experimental and numerical results, we also provide insight into the mechanism responsible for the depth dependence of the graphitization threshold in diamond. This work prospects the possibility of achieving accurate tunability of the mechanical properties of single-crystal diamond through defect engineering, with significant technological applications, i.e. the fabrication and control of the resonant frequency of diamond-based micromechanical resonators.


Introduction
Diamond is an extremely attractive material for a broad range of technological applications due to its unique physical and chemical properties. In particular with regards to its extreme mechanical and thermal properties, in the past years several works were focused on developing mechanical structures and resonators in diamond either by MeV ion implantation [1,2] or by reactive ion etching [2][3][4][5][6] with the purpose of taking advantage of its high mechanical hardness, stiffness and thermal conductivity. Moreover, diamond hosts a wide variety of luminescent defect centres [7,8] that can act as stable single photon emitters at room temperature or as optically addressable solid-state spin-qubits [9,10]. A challenging goal in this field is to efficiently couple negatively charged nitrogen-vacancy centres to resonant mechanical structures [11][12][13]. For advanced applications in nano-opto-mechanical devices, the prospect of being able to modify and finely tune the mechanical properties of diamond is therefore particularly appealing. In the case of other -carbon-based materials (e.g. carbon nanotubes or graphene), the effect of structural defects on their macroscopic mechanical properties has been studied both experimentally [14,15] and theoretically [16,17], while a significant gap is present in the case of bulk diamond. Here, we report such a systematic study, showing that of diamond can be effectively achieved by defect engineering through MeV ion irradiation.
Ion beam lithography based on MeV [18][19][20] or keV ions [21,22] has emerged in the past years as one of the most promising techniques used in the micro-and nano-machining of diamond for different applications. It is known that ion implantation induces structural modifications (at low damage densities, consisting primarily of vacancies and interstitials [23]) and local mass density variations in the diamond crystal, which in turn result in mechanical deformations, including surface swelling [24] and local stresses [25,26]. A mixed analytical/numerical approach to estimate these stresses has been developed, providing -p. 4 4 good agreement with experimental measurements [27]. One important issue that remains to be adequately addressed, however, is the variation of the elastic properties of damaged diamond as a function of induced damage density. This is particularly relevant for the , which is expected to vary between that of pristine diamond (i.e. 1135 GPa [28,29]) and that of amorphous carbon (i.e. 20 GPa in a fully amorphized phase [30]). Clearly, this large variation in the elastic properties is likely to strongly affect the modeling of implantation-induced stresses. Attempts have been made to experimentally derive the variation of elastic properties in irradiated diamond [26], but only indirect estimations with limited accuracy were obtained. This significant lack of experimental evidence is partly due to the value of the pristine material, which makes it difficult to probe its mechanical properties. A related open question is represented by the relatively high uncertainty found in the literature on the value of the so-called amorphisation threshold , i.e. the damage level (usually parameterized with a vacancy density, as calculated via the SRIM Monte Carlo code [31]) above which the diamond lattice is permanently amorphized, and subsequently graphitizes upon thermal annealing. It is often hypothesized that the large variability of this parameter is related to the depth of implantation [32], as well as to self-annealing effects [33], but so far no unequivocal evidence of these effects has been provided.
In this work we present the first systematic study of the controlled variation of the

Raman characterization
Micro-Raman spectroscopy was employed to assess the degree of amorphization/graphitization in the regions of the diamond sample which were characterized by the highest ion-induced damage density, i.e. in correspondence with the end-of-range Bragg peak. The measurements were performed in the same cross-sectional geometry adopted for the nanoindentation measurements, i.e. the probing laser beam was focused across the upper surface of the sample and scanned across the ion-beam-induced damage profile (see Fig. 1 as a reference). A similar approach has been adopted in previous Raman studies in ion-implanted diamond [32,35]. An inVia Raman micro-spectrometer (Renishaw) was employed, with a = 514 nm excitation laser beam focused onto a micrometric spot with a 80 objective. The laser power focused on the probed region was 2.5 mW. A PC-controlled stage allowed the sample displacement across the three directions, and thus the mapping of the Raman signal with micrometric resolution. The experimental results are reported in Section 4.1.

Nanoindentation measurements
The lateral implantation geometry shown in Fig Pharr [37]. Indentation measurements were performed in the linear elastic range, both in unimplanted and in ion-implanted areas. As an example, Fig. 2

Ab initio simulations
An atomistic approach was employed to simulate the damage-induced modification of from first principles. To this purpose, the quantum-mechanical ab initio code CRYSTAL14 [38,39] was employed, which allows the prediction of the structural and elastic properties of a defected material [40] through a [41]. In this context, a supercell is defined as a multiple of the unit cell that contains a vacancy at its body center. The CRYSTAL software creates a bulk system by the repetition of such a defective supercell, so that (differently to the real physical system) the with respect to the strain are evaluated in CRYSTAL as a numerical derivative of an analytical gradient [42,43]. The s thus derived from the elastic constants tensor. Point symmetry was used at all stages of the calculation to reduce the number of components of the elastic tensor to be considered. For each irreducible strain, a deformation was applied to the system, so that N strained configurations were defined according to a strain step. After a loop of N strained configurations, the energy gradients were fitted with single-value-decomposition routines and the first derivatives were determined numerically.
Hartee-Fock (HF) and B3LYP Hamiltonians were adopted [44,45], both of which have been shown to produce particularly accurate results for this kind of system [44], and the described procedure was adopted for both cases. The neutral vacancy in diamond can occur in three spin states, i.e. S z = 0, 1, 2 [46,47]. The spin state of the defect was however found to have a negligible influence on the elastic properties, so that the reported data only refer to the lowest-energy spin state S z = 0.

Mesoscale model
The dependence of structural/mechanical properties of damaged diamond have previously been estimated using a phenomenological mesoscale model accounting for damage saturation effects at high fluences, based on a rule of mixtures approach [27].
According to this model, when accounting for defect recombination in the crystal, the vacancy density V in the depth direction x can be expressed as: 10 10 where (x) is the linear vacancy density calculated numerically using the SRIM code, F is the implantation fluence and is the saturation vacancy density. The latter parameter can be derived as follows: let us consider an initial mass M and volume V 0 of diamond that after implantation expands to a volume V containing n defects (i.e. vacancies), each assigned the same volume v for sake of simplicity. The vacancy density is therefore V = n/V. By neglecting the mass of the implanted ions, the mass density of the implanted diamond volume By rearranging, we obtain: where f = n·v/V = · V is the volume fraction of vacancies, which correctly implies that = d for f = 0, and = 0 for f = 1. In fact, in our case an upper bound exists for f due to defect recombination effects [24] corresponding to a vacancy density saturation value V = . This -p. 11 11 vacancy density saturation value corresponds to an experimentally determined mass density saturation value of aC = 2.06 g cm -3 [48]. By rearranging Eq. (4) and estimating v as the inverse of the atomic density of diamond ( = 1.77 10 23 cm -3 ), we can derive as: consistently with previous works on 2 MeV H + implantations [49]. This vacancy density is shown in the inset of Fig. 1. The mass density variation (x) can thus be written as: The e from the vacancy density can be derived from Quantized Fracture Mechanics (QFM) [17,50] for the case of single isolated (non-interacting) vacancies: where is an empirical factor related to defect shape and interaction [17].  (8) In this way, in an unconstrained expansion the volume variation would be inversely proportional to the density variation. The diamond sample was considered mechanically were accounted for using Eqs. (6) and (7). The same functional form, based on a rule of

Finite Element simulations
where d = 0.07 and aC = 0.34 are s of pristine diamond and amorphous carbon, respectively [51]. can be observed. On the other hand, at frequencies lower than the 1 st order transition, a broad band between 1000 cm -1 and 1300 cm -1 appears at the highest damage densities (blue plot), which is commonly attributed to sp 3 defects and in heavily stressed diamond [53,54]. As for the features at frequencies above the first order transition, it is worth noting that the Raman peak at 1370 cm -1 closely resembles the D band at 1350 cm -1 commonly associated to tetrahedral amorphous carbon characterized by a high sp 3 content [55], while no G band at 1580 cm -1 is observed in our spectra. More likely, the 1370 cm -1 feature can be related (with a slight frequency shift attributed to the effect of different local stress fields) to a feature observed at 1390 cm -1 in highly stressed diamond regions close to laser-induced graphitized areas, which was attributed to sp 3 carbon allotropes (i.e. Z-carbon and hexagonal diamond) subjected to high mechanical stress [56]. A detailed analysis of the obtained Raman spectra is beyond the scope of this work, but it is nonetheless worth remarking that all of the above-mentioned attributions are compatible with a purely-sp 3 diamond-like phase, thus confirming that even in correspondence with the Bragg peak the damaged diamond structure is not subjected to a graphitization process. The progressive reduction of the first-order Raman transition is accompanied by its broadening and shift to lower frequencies. The latter feature is reported in Fig. 4(b), where the peak frequency is reported. The first-order Raman peak corresponding to the Bragg peak is positioned at 1321 cm -1 . In [32], the redshift of the first-order Raman line in defective diamond has been correlated to vacancy density, estimated -p. 14 14 by the authors in a linear approximation as the product between the linear vacancy density predicted by SRIM and the implantation fluence. If such a correlation is applied to the measured 1321 cm -1 value, an estimated vacancy density at the Bragg peak of 4.5 10 21 cm -3 is obtained in the above-mentioned linear approximation. If we consider that the SRIM-predicted linear vacancy density in such region is 4.5 10 4 cm -1 , we can obtain an estimation of the implantation fluence of (4.5 10 21 cm -3 ) / (4.5 10 4 cm -1 ) = 1 10 17 cm -2 , in striking agreement with the corresponding experimental value (see Section 2.1). Again, these correspondences confirm that the highly damaged regions of the sample can be still considered as a highly defective sp 3 diamond phase rather than a graphitized phase.

Nanoindentation results
Firstly, an unimplanted region of the sample was probed with the nano-indenter, yielding a value of E r = (600 ± 45) which is compatible with values from the literature [28,29]. This measured reduced modulus value GPa, which can be adopted to express the relations between principal stresses ii and strains ii in the generalized 3D [57].  Fig. 1).

Density/stiffness variations and surface swelling results
Experimental and numerical results for the relative variation of mass density and presented in Figs. 6a and 6b, respectively. For the mass density variation in Fig. 6a, the predictions of the model (see Eq. (7), allowing the determination of the empirical value = 4.46, which is reasonably close to the reference value of indicated for carbon nanotubes [17]. The numerical predictions ab initio simulations were then compared to experimental nanoindentation measurements, as reported in Fig. 6b.
Both predictions are compatible with the experimental data within their respective uncertainties, although the theoretical predictions tend to systematically under-estimate the reduction in You . This tendency is confirmed by a previous experimental dataset from [26] (also included in Fig. 6b), which also falls below numerical predictions.
This systematic underestimation by theoretical models can be reasonably attributed to the fact that in all of the above-mentioned approaches only the effect of induced isolated vacancies is modelled, while the possible effects of more complex defect aggregates are disregarded.
Nonetheless, the compatibility between theoretical and experimental results is striking, thus indicating that (at least at the reported damage densities, i.e. below 10 22 vacancies cm -3 ) the isolated vacancy defect indeed plays a prominent role in the modification of the mechanical properties of diamond. strain-driven process [48], i.e. it occurs in regions where a defined strain threshold ( 16%) is exceeded. Therefore, such a strain (or equivalently, stress) value must be estimated with the highest accuracy. Here, as an example we report in Fig. 7c [23,32,33,[58][59][60][61][62][63][64][65]. The general (but so far undemonstrated) understanding is that such a threshold increases at increasing implantation depth due to higher internal pressures that do not allow relaxation to graphitic structures [32]. Here we show quantitatively that in fact the opposite is true, i.e.

Conclusions
A systematic study of the variation of the structural and mechanical properties of