1. Introduction

With the development of plasma diagnostics, data links, distributed fiber-based temperature and strain sensors, the integration of optical fibers (OF) has been further encouraged in harsh environments such as nuclear power plants, fusion and high energy physics facilities [1]. Because of the radiation in these envinronments, OF experience damaging phenomena that result in an attenuation of the transmission of optical signals. The development of OF technology for the above mentioned applications could tremendously benefit of multiscale modeling tools allowing to estimate the vulnerability of OF-based devices, especially for those environments where a direct testing is not possible as e.g in fusion facilities [2]. An indispensable condition for a successfull modeling is to deeply understand the nature of the radiation induced damaging at the atomic level. To this aim, nowadays first-principles techniques, by allowing direct comparison with optical absorption or electron paramagnetic resonance (EPR) spectroscopy data, are very insightful [3–6].

A convenient and common way for fabricating the light-guiding core of OF consists in doping vitreous silica (v-SiO₂) with vitreous germania (v-GeO₂) so to raise the index of refraction with respect to the one of the cladding [7]. The procedures (e.g. drawing) involved in the OF fabrication are known to generate germanium defects such as the germanium lone pair center (GLPC) and the Ge neutral oxygen vacancy [3, 8, 9]. The presence of these precursors defects is responsible for the generation under irradiation of other Ge defects, some still debated, that cause the degradation of the OF [8, 10–15]. Moreover little is known about the mechanisms by which the so-called Ge-E′, Ge(1), and Ge(2) paramagnetic centers are generated from the ionization of the GLPC, although several experimental data indicate that the latter is very relevant for the radiation response of Ge-doped silica [16–19]. The Ge(1) and Ge-E′ centers are universally attributed to an electron localized at a four-fold Ge atom and to a Ge paramagnetic defect analogous of the $E_{\gamma }^{\prime }$ in SiO₂, respectively [6, 10, 20, 21]. By contrast the origin of the Ge(2) defect, detected in both Ge-doped SiO₂ [22–29], and high purity v-GeO₂ [30, 31], is very controversial and first principles calculations have recently been invoked to solve the debate [10]. In analogy to the Ge(I) and Ge(II) centers in α-quartz [32], Griscom has recently suggested that the Ge(2) and Ge(1) centers are two energetically inequivalent configurations of a single trapped-electron at a Ge site [10, 11, 33]. In particular, the thermal behavior of the relative Ge(1)/Ge(2) concentration and the comparison of g-values provide support in favor of the analogy between Ge(1) and Ge(II) and between Ge(2) and Ge(I) [10, 33]. An alternative explanation of the Ge(2) center assumed the latter to originate from the ionisation of a two-fold Ge atom which is known as the GLPC⁺ hypothesis [23–25]. More recently, Agnello et al. [28] have given support to the GLPC⁺ hypothesis by showing that a correlation exists between the concentration of radiation induced Ge(2) paramagnetic centers and the content of the neutral GLPC.

In this work, we present a first-principles investigation of the EPR parameters of Ge paramagnetic centers in v-SiO₂ aiming at identifying the origin of the highly debated Ge(2) center. We deduce, by analysing our g-values distributions obtained for a large set of defect configurations, that the Ge(2) center, observed in both Ge-doped silica and pure v-GeO₂, may arise from the Ge forward-oriented [34, 35] configuration (Ge-FO). In such a configuration, the unpaired spin is localized at a three-fold coordinated Ge atom that forms a weak bond with a three-fold O atom, about 0.2 Å longer than its other two Ge-O bonds. We thus establish that not only are Ge-E′ centers the analogue of $E_{\gamma }^{\prime }$ centers in silica, but also the Ge(2) center could be considered as the Ge counterpart of the $E_{\alpha }^{\prime }$ center in v-SiO₂ [36]. On the other hand our calculations show that a four-fold coordinated Ge atom that traps an electron does not provide g-values compatible with those of the Ge(2) center, and should be regarded only as responsible for the Ge(1) center. Moreover we obtain GLPC configurations by relaxing the neutralized Ge-FO configurations. Viceversa, GLPC⁺ configurations are shown to relax back, with no energy barrier, to the departing Ge-FO configurations thus supporting the precursory GLPC structural origin of the Ge(2) center.

2. Theoretical methodology and modeling details

The calculations carried out in this work are based on density functional theory. The Becke-Lee-Yang-Parr exchange-correlation functional has been adopted [37, 38]. We employ norm-conserving Trouiller-Martins pseudopotentials designed for calculations within the gauge including projector augmented wave (GIPAW) method [39, 40]. Kohn-Sham wavefunctions are expanded in a basis of plane waves up to a cutoff energy of 80 Ry [40]. The wavefunctions were expanded at the Γ point of the Brillouin zone, as justified by the large size and the large band gap of our systems [3, 41]. Geometry optimizations and EPR parameters have been obtained by means of spin-polarized calculations as implemented in the Quantum-Espresso (QE) package [42]. The EPR parameters are calculated by exploiting the GIPAW method as available in the QE package. Germanium isotropic hyperfine couplings (Fermi contacts) are obtained by including a scalar relativistic correction while core-relaxation effects are not considered [42]. The periodic Ge-doped silica supercells (107 atoms and 570 electrons) used in this work are derived from those we previously generated to investigate silicon oxygen deficient centers (SiODC) in v-SiO₂ [43–45]. We generated germanium oxygen deficient center (GeODC) configurations by replacing silicon atom with germanium for the silicon site carrying the unpaired spin. In this way we obtained 48 Ge puckered, 11 Ge unpuckered and 5 Ge-FO configurations. We subsequently relaxed the structure of each configuration using the Broyden-Fletcher-Goldfarb-Shanno algorithm [42]. A force threshold of 0.00075 Ryd/bohr has been adopted. We obtained 7 single-trapped electron (STE) at a four-fold Ge atom configurations by following the methodology outlined in [46]. The density of the Ge-doped supercells (2.25 g/cm³) is only slighlty larger (by 2%) with respect to that of the undoped system. We also obtained five Ge-FO in pure v-GeO₂ by replacing every silicon atom with germanium in each Ge-FO configuration of Ge-doped silica, and then by performing a variable-cell structural relaxation [42] necessary to achieve the correct GeO₂ density. In fact the relaxed GeO₂ cell shows a volume expansion of about 5% with respect to the original SiO₂ cell. The so-generated GeO₂ supercells have a density of about 3.64 g/cm³ in good agreement with the experimental one [47]. Finally, an evaluation of size effects has been done by taking advantage of a larger size silica model with 144 atoms [48]. In this model we could generate a Si-FO configuration [45] successively converted in to a Ge-FO configuration as explained here above. The results for the g-tensor (g₁ = 2.0018, g₂ = 1.9982, g₃ = 1.9877) and Fermi contact reasonably agree with those we obtain on average with the smaller 108 atoms model (see next Sect.). Furthermore g₁₂ = 0.0036 and g₁₃ = 0.0141 both agree within less than one standard deviation (std) from the average we found for the Ge-FO in the adopted 108 atoms model indicating that g differences are weakly dependent on size effects [49]. Still the latter might affect g-values e.g. the g₁ here above differs by three std with respect to the 108 atoms model. The use of a larger supercell can improve the agreement with the experiment (up to ~ 1000 ppm) but at a considerably higher computational cost, meanwhile is not expected to change the conclusion of the present analysis that is mainly based on g-values differences.

3. Results

3.1. Structural properties of the Ge-E′ -like, STE and Ge-FO configurations

In Ge-doped v-SiO₂, the Ge-O bond length in the Ge-E′ -like configurations (i.e. Ge puckered and unpuckered [21,50]) is found to be ∼ 1.70 Å sligthly shorter than the experimental estimate of 1.73 Å in v-GeO₂ [51]. Similarly, the two normal Ge-O bonds of the Ge-FO have a bond length of 1.69 Å, while the bond between the three-fold Ge and the three-fold O is about 1.89 Å [Fig. 1(a)] [34]. The average dangling bond angle O-Ge-O angle in Ge-FO configurations is about 106°, slightly smaller than the expected value for sp³ hybridization i.e. 109.4°. More specifically, in these configurations, the lowest O-Ge-O angle is about 98.4° with a standard deviation (std) of 2.7°. The largest O-Ge-O angle (between the two normal Ge-O bonds) and the last one are on average ∼112.4° and 107.2° with standard deviations of ~4°. In Fig. 1(d) we show the the spin density of the Ge-FO configuration that closely resembles the one previously calculated for the Ge-E′ center [20, 21]. The structure of Ge-FO configurations in pure v-GeO₂ is almost unchanged with respect to Ge-FO in Ge-doped silica. Only minor changes are observed, the most relevant one concerning the longest Ge-O bond (1.88 Å) that is shortened by ∼0.01 Å. In the STE configurations [Fig. 1(c)], the geometry of the Ge tetrahedron is distorpted as consequence of the localization of an extra electron [6, 20, 52]. An illustration of the spin density of the STE configurations is shown in Fig. 1(f). In particular, in our configurations one of the Ge-O-Ge angle opens up to an average value of ∼154.6° with a std of 5.6°. Correspondingly the two Ge-O bonds forming the latter angle are stretched up to 1.85 Å, while the remaining Ge-O bonds are considerably shorter i.e. 1.74 Å. The corresponding Ge-O-Ge angle is also quite narrow ∼ 98° with a std of 2°.

 

Fig. 1 Schemes of the average atomic structure of (a) the Ge-FO configuration, (b) the ionized (non-relaxed) two-fold Ge configuration and (c) the single-trapped electron (STE) configuration at a fourfold Ge atom. O (black discs), Si (circles) and Ge (gray disc) atoms and the unpaired electron (dot) are shown together with Ge-O bond lengths. Ball and stick models and spin densities (shadowed) of a (d) Ge forward-oriented (FO) configuration, (e) a two-fold Ge configuration, and (f) a STE configuration. O atoms (red), Si atoms (light brown) and Ge atom (violet) are shown.

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3.2. EPR parameters of the Ge-E′ -like and Ge-FO configurations

In Table 1, we show the results of the ⁷³Ge Fermi contact [A_iso(⁷³Ge)] and g-tensor calculations of Ge-E′ -like and Ge-FO configurations in Ge-doped v-SiO₂. The Ge-E′ -like configurations give rise to a distribution of g-values whose average is in reasonably good agreement with the experimental data reported for the Ge-E′ center in Ge-doped silica [28]. The average g₁ and g₂ values suffer of a small (about one std) overestimation of 700 ppm with respect to the experimental estimates. The latter overestimation, while still reflected in the g₁₃ value, is absent in the g₁₂ value that is found in good agreement with the experiment. The calculated A_iso(⁷³Ge) of Ge-E′ -like configurations on average underestimates by ~10% (1.9 mT, Table 1) the experimental value [53]. The average g principal values of Ge-FO are in fair agreement with available experimental values for the Ge(2) center [29, 54] showing overestimations at most by ~1000 ppm. This is comparable to the overestimation we observe between the calculated g-values of the Ge-E′ -like configurations and the experimental ones of the Ge-E′ center (Table 1). It is worth noting that for the Ge(2) center the measure of the g₃ principal value is straightforward, as this corresponds to the position of a well resolved peak in the EPR spectrum. Thus, the accuracy for g₃ is higher than the one obtained for g₂, the estimate of which requires a more involved analysis. In fact, the Ge(2) EPR spectrum is usually obtained after subtraction of an appropriate fraction of the Ge(1) and Ge-E′ signals and g₂ is then obtained by finding the first zero crossing value on the high field side of the main peak [29, 55]. This likely explains the rather large difference (1100 ppm) in g₂ values seen in experiments [29, 54]. The g₁₃ value is known with a very good accuracy as it corresponds to the whole width of the Ge(2) signal and is not biased by offset errors that could affect the experimental estimate of the g principal values. Quite remarkably, the calculated g₁₃ of Ge-FO configurations coincides with the experimental g₁₃ values of the Ge(2) center (Table 1). Moreover, although the calculated g₁₂ suffers by a 1000 ppm overestimation with respect to [29], it is in excellent agreement with the value obtained by [54].

Table 1. Configuration type, calculated average g-values and Fermi contacts [A_iso(⁷³Ge)] of Ge-FO and Ge-E′ -like and STE configurations in Ge-doped v-SiO₂ compared to available experimental data for the Ge(2), Ge-E′ and Ge(1) centers. Theoretical results are also given for the non-relaxed GLPC⁺ configuration [Fig. 1(b)] and Ge-FO configurations in pure v-GeO₂. Standard deviations and experimental errors (when available) are given in parenthesis. (g₁₂ = g₁ − g₂ and g₁₃ = g₁ − g₃).

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In Figs. 2(a) and 2(b) for each GeODC configuration, we show the A_iso(⁷³Ge) plotted versus g₂ and g₃ principal values. The plot reveals the existence of two distributions of EPR parameters, the first one pertaining to the Ge dangling bond as in puckered and unpuckered configurations and corresponding to the Ge-E′ center (g₂ ∼ 1.995, g₃ ~ 1.994). A second distribution with markedly different average g₂ and g₃ values (Table 1) arises from Ge-FO configurations. Hence Figs. 2(a) and 2(b) strongly suggest that puckered and Ge-FO configurations give rise to two well distinct paramagnetic centers. Because of the systematic error of the Fermi contact mentioned above, both distributions should be shifted down by ∼10% [56]. By doing so, it becomes even more evident that the EPR parameters of the Ge-FO configurations are close enough to the corresponding ones of the Ge(2) center (less than three std) to allow an assignment of the Ge(2) to the Ge-FO configurations. The reliability of this assignment is further supported by Fig. 2(c) where we plot the relative differences, Δg_i, between the experimental g-values of the Ge(2) and the Ge-E′ centers versus the corresponding differences calculated for the Ge-FO and Ge-E′ -like configurations. The agreement between theory and experiment is fair for the g₃ value and very good for the g₁ and g₂ principal values. In Fig. 3, we compare g₂ and g₃ values obtained for each GeODC configuration with those previously obtained for the corresponding SiODC configuration [45]. In this way, for each SiODC defect configuration we can easily determine the effect of the substitution of Si with Ge and we can easily establish which Si/Ge center will arise from e.g. Si/Ge puckered and Si/Ge forward-oriented configurations. Figure 3 not only supports the structural analogy between Si-E′ (i.e. $E_{\gamma }^{\prime }$) and Ge-E′ that arise from E′ -like configurations (i.e. Si/Ge puckered and unpuckered [45]), but also suggests that the Ge(2) originates from a Ge defect analogous to the one that gives rise to the $E_{\alpha }^{\prime }$ center (i.e. the Si-FO configuration).

 

Fig. 2 A_iso(⁷³Ge) Fermi contacts of Ge-E′ -like (squares) and FO (circles) configurations plotted vs (a) g₂ and (b) g₃ principal values. Experimental data (discs) for Ge-E′ and Ge(2) centers are taken from [54,55]. (c) Experimental relative difference between (Δg_i) g-values of Ge(2) and Ge-E′ centers plotted versus calculated average Δg_i differences between Ge-FO and Ge-E′-like configurations. The size of the symbols represents the experimental error as in [28], while horizontal error bars shows the theoretical spread of the distributions of g₂ and g₃ values. The dotted line indicates identity between theory and experiments.

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Fig. 3 Calculated (a) g₂ and (b) g₃ principal values obtained [45] for SiODC configurations [E′ -like (squares) and FO (circles)] plotted vs g₂ and g₃ values obtained for the corresponding GeODC configuration. Experimental data (discs) for Si-E′, Ge-E′, $E_{\alpha }^{\prime }$ and Ge(2) centers are taken from [10, 36, 54].

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3.3. STE configurations at a fourfold coordinated Ge site

In Tab. 1 we give the g-values and ⁷³Ge Fermi contact calculated for our STE configurations [Fig. 1(c)]. As seen for the Ge-E′ and Ge-FO, the calculated ⁷³Ge Fermi contact slighty underestimates (∼8%) the experimental value of the Ge(1) center, and g-values overestimate their experimental counterpart on average by ∼1000 ppm. In Figs. 4(a) and 4(b) we show the existence of a dependence of the g₁₃ value and of the wide O-Ge-O angle of the STE configuration on the distorption δ_V of the Ge tetrahedron (defined as the ratio between the volume of the Ge tetrahedron and the volume of the ideal Ge tetrahedron with same average Ge-O bond length [57]). The larger is the O-Ge-O angle the smaller is δ_V i.e. the Ge tetrahedron becomes more distorpted for small δ_V values [57].

 

Fig. 4 Calculated (a) g₁₃ value vs distorption parameter δ_V of the Ge tetrahedron [57] and (b) O-Ge-O angle vs δ_V in our STE configurations. (c) Comparison of the g-values of the STE (filled squares) and Ge-FO (discs) configurations with respectively the g-values of the Ge(II) and Ge(I) centers detected in irradiated Ge-doped quartz [32]. Superimposed (empty symbols) comparison of the g-values of the Ge(1) and Ge(2) centers with respectively the g-values of the Ge(II) and Ge(I) [10,55]. The dotted line represents 100% agreement. A rigid shift of −900 ppm has been applied to calculated g-values of STE and GeFO configurations.

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Fig. 4(a) suggests that one would need a very low δ_V around 0.5 to have g₁₃ = 0.0140 as observed for the Ge(2). However a δ_V ∼ 0.5 corresponds to an unphysical tetrahedron where the largest Ge-O-Ge angle should be about 180° and the angle between the other bonds should be lower than 70° so that oxygen atoms would become rather close (only ~1.7 Å). Moreover the small differences observed between g-values of Ge(I) and Ge(II) in Ge-quartz on the contrary do not need such big distortions. In fact from [32] we derive for Ge(II): g₁₃ = 0.0078 while for Ge(I): g₁₃ = 0.0083 which indicates a rather small change in the distortion degree of the Ge tetrahedron or equivalently a small change in the O-Ge-O angle. From Fig. 4(b) we estimate it to be about 5° to 10°.

In Fig. 4(c) we show the comparison of the g-values of the STE and Ge-FO configurations with respectively the g-values of the Ge(II) and Ge(I) centers detected in irradiated Ge-doped quartz [32]. A rigid shift of −900 ppm has been applied to the calculated g-values of STE and Ge-FO configurations in order to avoid residual size effects. The comparison shows a good agreement between our STE results and the Ge(II) experimental data. Concerning the comparison of the Ge-FO results vs Ge(I) data we do not find a similar agreement. By superimposing the comparison of the g-values of the Ge(1) and Ge(2) centers with respectively the g-values of the Ge(II) and Ge(I) [10, 55], we instead find that the agreement of our Ge-FO configurations with Ge(2) experimental data is much more evident than with Ge(I) data (e.g. the difference between g₃ of Ge(I) and Ge-FO is one order of magnitude larger than the difference of g₃ calculated between Ge(II) and STE configurations).

3.4. Ge-FO configurations in pure v-GeO₂

We hereafter extend the discussion on the Ge(2) center by considering Ge-FO configurations in v-GeO₂ and by comparing our results (Table 1) to the few available experimental data [30, 31]. In particular in [30], g₁ and g₂ values are not explicitly given and thus considerably limiting our discussion. In pure v-GeO₂ we note an increase of all the g-values with respect to the Ge-FO values reported for Ge-doped silica. By contrast the ⁷³Ge Fermi contact (−24.6 mT) is not affected by the change of the surrounding matrix as its value only shows a negligible variation. For the g₁ value, part of the increase (∼500 ppm) can be explained as an effect of the GeO₂ environment vs the SiO₂ environment as suggested by experimental results for the Ge-E′ center in quartz GeO₂ [31, 58] with respect to Ge-doped quartz SiO₂ [59]. An overestimation (~1000 ppm) as previously seen for Ge-FO in Ge-doped silica should also affect the calculated g₁ value of Ge-FO in GeO₂. We note that, for Ge-FO in GeO₂, the g₁₃ is slightly smaller than the corresponding value calculated in Ge-doped silica, consistently with the tendency observed by [29] and with the g₁₃ value of 0.0137 observed for the largest Ge content [29]. The calculated average g₃ value is 1.9889 with a std of 0.0011. Moreover, in v-GeO₂, the g₃ is shifted by 0.0011 with respect to the corresponding value for Ge-FO in Ge-doped silica (Table 1) [60]. The latter shift, though considerably smaller, is still compatible with the experimental shift of 0.0017 observed between the g₃ of the Ge(2) center in GeO₂ (1.9885) and in SiO₂ (1.9868) glasses [10, 30].

3.5. EPR parameters of the GLPC⁺

The ionized two-fold Ge atom (GLPC⁺) is one of the most popular structural models that have been proposed for the Ge(2) center in the past two decades [23, 24]. To investigate such an explanation of the Ge(2) center, we generated GLPC⁺ configurations by first neutralizing and relaxing the atomic structure of our Ge-FO configurations [Fig. 1(a)]. The two normal Ge-O bonds of the Ge-FO configuration are stretched on average by 0.05 Å with a std of 0.02 Å, while the longest Ge-O bond is further stretched on average by 0.57 Å with a std of 0.28 Å [Fig. 1(b)], up to ~ 2.92 Å for the most stretched configuration. In this latter configuration the two closest non-bonded oxygen atoms are both separated from the Ge atom by about 2.9 Å. Once we obtained these neutral two-fold Ge atom configurations, we proceeded by positively charging them. The spin density in these configurations [Fig. 1(e)] is mainly localized at the Ge atom and its two O nearest neighbors. With no further relaxation of the atomic structure, we calculated the EPR parameters of our GLPC⁺ configurations. The average g-values and A_iso(⁷³Ge) do not agree with the experimental data available for the Ge(2) center [54, 55]. In fact, the g-values indicate a too large rhombicity with respect to that of the Ge(2) center (we obtained g₁₂ = 0.011 and g₁₃ = 0.032). Moreover, the absolute value of the ⁷³Ge Fermi contact (19.3 mT) is smaller by about 5 mT with respect to the average obtained in Ge-FO configurations (Table 1).

We also checked that as we allow the structure of the positively charged two-fold Ge configurations [Fig. 1(b)] to relax, we obtain again the Ge-FO configurations and the EPR results given in the first row of Table 1. The energy gain of this relaxation, from the GLPC⁺ configuration to the Ge-FO, is on average 1.5 eV. These findings suggest that as soon as the two-fold Ge atom is ionized by the incoming radiation it can quickly relax, with no energy barrier, to a Ge-FO configuration. On the other hand, if a free electron is trapped by a Ge-FO configuration then the structure will relax into a two-fold Ge atom configuration, i.e., a GLPC center.

4. Conclusion

In conclusion, our study, by analyzing the EPR parameters distributions of a large amount of defect configurations, provides evidence for an assignment of the Ge(2) center to Ge-FO configurations, while no evidence is found in favor of a Ge(2) model as given by a single-trapped electron at four-fold Ge atom. At variance, we show that the Ge(2) center may arise from an unpaired spin localized at a three-fold Ge atom which is bonded to a three-fold oxygen atom through a weak Ge-O bond, 0.2 Å longer than usual Ge-O bonds in GeO₂. The assignment is further supported by the fact that the GLPC center easily converts into a Ge-FO configuration and viceversa. Such an interconversion mechanism is consistent with the experimental observation that the radiation induced generation of Ge(2) centers requires the precursory presence of two-fold Ge atoms [22, 28]. Yet, the scenario of Ge defects in silica is still far from being fully clarified e.g. a challenging task for future investigations being the understanding of the thermal behavior of defects reported in [33].