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Cumulative approaches to track formation under swift heavy ion (SHI) irradiation: Phenomenological correlation with formation energies of Frenkel pairs
Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
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Cumulative approaches to track formation under swift heavy ion (SHI) irradiation: Phenomenological correlation with formation energies of Frenkel pairs M.L. Crespillo a,b,⇑, F. Agulló-López a,⇑, A. Zucchiatti a a b
Centro de Microanálisis de Materiales, CMAM-UAM, Cantoblanco, Madrid 28049, Spain Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, United States
a r t i c l e
i n f o
Article history: Received 27 September 2016 Received in revised form 9 December 2016 Accepted 11 December 2016
Keywords: Radiation-induced damage mechanisms Ion-solid interaction Radiation-induced point defects Frenkel pair formation energies Order to disorder microstructure phase transformation Swift heavy ion irradiation Ion track formation Electronic stopping power threshold Cumulative defect models Thermal spike model
a b s t r a c t An extensive survey for the formation energies of Frenkel pairs, as representative candidates for radiation-induced point defects, is presented and discussed in relation to the cumulative mechanisms (CM) of track formation in dielectric materials under swift heavy ion (SHI) irradiation. These mechanisms rely on the generation and accumulation of point defects during irradiation followed by collapse of the lattice once a threshold defect concentration is reached. The physical basis of those approaches has been discussed by Fecht as a defect-assisted transition to an amorphous phase. Although a first quantitative analysis of the CM model was previously performed for LiNbO3 crystals, we have, here, adopted a broader phenomenological approach. It explores the correlation between track formation thresholds and the energies for Frenkel pair formation for a broad range of materials. It is concluded that the threshold stopping powers can be roughly scaled with the energies required to generate a critical Frenkel pair concentration in the order of a few percent of the total atomic content. Finally, a comparison with the predictions of the thermal spike model is discussed within the analytical Szenes approximation. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction The most representative consequence of electronic damage under SHI irradiations [1–4] is the production of linear heavily disordered regions (amorphous tracks) around the ion trajectory. The radius R of the track increases with the ion stopping power, in practice the electronic part Se. Extrapolation to R = 0 determines a threshold stopping Sth for track formation. An extensive (but not exhaustive) list of threshold values, Sth(exp), extracted from the experiments, is included in the left side of Table 1. It needs to be remarked that threshold values depend on the precision of the extrapolation process, are known to be velocity dependent and are often attributed quite different values from different authors. Large uncertainties cannot therefore be excluded on the values reported in Table 1. Measured threshold lie in the range 1–50 keV/nm, roughly corresponding to a deposited energy of ⇑ Corresponding author at: Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, United States (M.L. Crespillo). E-mail addresses: [email protected] (M.L. Crespillo), [email protected] (F. AgullóLópez). http://dx.doi.org/10.1016/j.nimb.2016.12.022 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
around 1–50 eV/molecule, comparable with the cohesive energy of solids. Most thresholds for dielectric crystals are below 20 keV/nm. The lowest values (<5 keV/nm) are reported for some oxides, such as SiO2, ZrSiO4, YIG, KGW and some halides, whereas highest values (>25 keV/nm) are found for semiconductors. For some binary ceramic oxides having closed shells cations, MgO, Al2O3, UO2, as well as for diamond, carbides (SiC), and nitrides (Si3N4), the threshold is comparable to those for radiation-hard semiconductors. As far as we know, a definite threshold value for track formation has not been measured for diamond. Most of the models proposed to understand and predict the track generation processes are based on the thermal spike concept, i.e. a hot region around the ion trajectory containing a high concentration of excited electron carriers. In particular, the thermal spike model (TSM) [5–7], which relies on the local melting of the material caused by the abrupt temperature rise in the spike, has been extensively investigated. It qualitatively accounts for many gross features of track formation, such as threshold stopping power, track radius, etc. A different, less explored approach, also rooted in the concept of thermal spike, relies on the generation and accumulation of
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Table 1 Thermal properties, experimental values of amorphization threshold stopping power Sth (Exp), and formation energies for intrinsic Frenkel-pair defects (V stands for vacancy and I for interstitial) in various crystalline materials. Also listed are the amorphization threshold (Sth (Szenes)) predicted from Eq. (5) and the calculated total energy (Edef) stored as Frenkel pairs in the spike volume per unit length of the trajectory. Material
Thermal properties and thresholds a0 [nm]
k [nm]
Defects formation energies Sth (Exp) [keV/nm]
C [J/g K]
Refs.
Frenkel pairs
Formation energy [eV] EF (in bold)
Refs.
[g/cm3]
Sth (Szenes) [keV/nm]
C0 [%]
a1 [nm]
Edef [keV/nm]
(V0Si + I0Si) (V0Ge + I0Ge) (V0Ga + I0Ga), (V0As + I0As) (V0P + I0P), (V0In + I0In) 1 (V1+ Cl + ICl ), 1+ + I (V1 Na Na) 1 (V1+ F + IF ) 1 (V1+ F + IF ) 2+ (V2 Ca + ICa) 2 (V2+ O + IO ), 1 (V1+ O + IO ), (V0O + I0O) 2 (V2+ O + IMg) Schottky pair (V0O) Single vacancy defect. No calculations for Frenkel pairs (V0O) Single vacancy defect. No calculations for Frenkel pairs (V0O + I0O), (V0Al + I0Al) 2 (V2+ O + IO ) 3+ (V3 Al + IAl ) (V0O + I0O) 2 (V2+ O + IO ), (V0Zr + I0Zr), 4 (V4+ Zr + IZr ) (V0O + I0O), 2 (V2+ O + IO ), (V0Hf + I0Hf), 4 (V4+ Hf + IHf )
7.42 6.41 7.0 7.80 5.17 6.88 4.602, 2.881 2.34–2.40 2.6–2.8 6, 8.5, 9.2 13.3, 11.93, 10.28 7.5 4.4
[42] [42] [45]
4.05 2.58 33.86
3.70 4.81 5.77
1.72 1.40 14.47
[47]
9.86
4.68
1.73
[29,50]
1.71
5.76
0.30
[29] [53]
10.06 41.07
2.28 3.18
0.63 2.21
[55,56]
14.61
2.52
3.12
[59]
7.96
2.56
0.61
6.6, 8.5
[60,61]
0.04
14.93, 25.32 11.38, 13.22 8.95, 9.0 7.57, 7.3 4.11, 20.05, 9.45 8.34, 4.69, 23.36, 10.13
[63]
3.36
2.59
0.37
[65,66]
52.28
3.67
6.73
[65,68]
53.72
4.18
10.18
(V0O + I0O), (V0U + I0U) 2 (V2+ O + IO ), 2+ (V2 Sr + ISr ), 4+ (V4 Ti + ITi ) 2 (V2+ O + IO ), 4+ (V4 Ti + ITi ), 2+ (V2 Ba + IBa) 2 (V2+ + I O O ), 5+ (V5 Nb + INb), 1+ (V1 Li + ILi )
4.02, 13.15 5.14, 7.88, 14.20 8.98, 15.12, 11.88 6.84, 12.5, 1.86
[73]
1.78
4.27
0.27
[75]
31.25
3.41
2.61
[77,78]
56.93
2.57
4.38
[77,78]
59.76
3.28
1.74
2 (V2+ O + IO ), 3+ (V3 Y + IY ), 3+ (V3 Al + IAl ) 2 (V2+ O + IO ), 3+ (V3 Y + IY ), 3+ (V3 Fe + IFe ) IO + V2 O + IO 0 0 (VO + IO ), (V0Si + I0Si), (V0Zr + I0Zr)
10.16, 12.57, 12.72 3.55, 8,0.45, 5.93 13 7.3, 22.8, 23.9
[85]
47.50
3.94
2.89
[87]
62.41
4.54
1.59
3+ (V3 Al + IAl )
10
[93]
4.26
3.69
2.32
(VC0 + IC0), (V0Si + I0Si) (V0C + I0C) lowest energy interstitial (V0C + I0C) (V0N + I0N), (V0Ga + I0Ga)
8.64 (9.07), 12.81 23
[97,98]
6.29
3.81
3.19
3.03
3.47
1.10
Si Ge GaAs
7.2 9.85 7.4
2.33 5.32 5.3
2.92 3.12 1.16
28 34 36
0.9 0.34 0.17
[28,41] [43] [28,44]
InP
7.95
4.5
0.98
12
0.17
[28,46]
NaCl
12
2.16
2
10.00
0.43
[17,48,49]
LiF CaF2
3.8 3.8
2.64 3.18
0.61 0.4
4.20 4.80
1 0.32
[18,51] [52]
MgO
4
3.58
1.94
15.8
0.63
[54]
5.8
4.23
0.73
5.1
0.31
[57,58]
c-SiO2
4
2.62
0.60
1.61
0.42
[7]
Al2O3
8
3.7
0.76
23
18
[2,54,62]
ZrO2
4
5.68
0.94
13
0.2
[64]
HfO2
4.5
9.65
1.23
20
0.12
[67]
SnO2 UO2
5.5 9
6.95 10.96
1.1 0.96
6 22
0.17 0.23
[69,70] [71,72]
SrTiO3
4.5
5.11
0.41
7.2
0.14
[74]
BaTiO3
4.5
6.01
0.37
22
0.1
[76]
4.3
0.35
6.0
0.17
[7,20,79]
7.27 5.4 6.14 7.08 4.56
0.09 0.046 0.3 0.09 0.11
4 5.40 16.00 7.2 6.00
0.04 0.022 0.1 0.025 0.042
[80] [70] [81] [82] [83,84]
TiO2
4.5
LiNbO3
4.5
KGd(WO4)2(KGW) BaFe12O19 ZnFe2O4 Gd3Ga5O12(GGG) Y3Al5O12(YAG)
3.7
4.5
8.2 4.5 4.6 4.6
Y3Fe5O12(YIG)
4.5
5
5.12
0.08
3.00
0.034
[86]
4
3.6 4.6
0.4 0.44
13 2.50
0.17 0.14
[88] [23]
2.7
0.01
3.8
0.9
[91,92]
7 7 7
3.26 3.27 3.21
4.8 1 4.8
34.00 15 33.00
0.6 0.18 0.6
[27] [94] [95,96]
Diamond
8
3.5
2
[99]
Graphite GaN
8 7
2.2 6.15
2 0.3
[102] [27,104]
MgAl2O4(spinel) ZrSiO4 (zircon)
MicaKAl2(AlSi3O10) (F,OH)2 AlN Si3N4 SiC
4.5
CM model
q
4.3
2.5 23.6
13.7 8.2, 12.3
[89] [90]
[100,101] [103] [105]
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irradiation-induced point defects around the ion trajectory [8–12]. The physical background of the cumulative model is similar to that for the mostly investigated thermal spike (melting) model. The two models are comparatively illustrated in the sketch of Fig. 1. A first preliminary theoretical analysis of this novel cumulative approach (CM model) was developed and applied to LiNbO3 [8]. In that work the formulation was quite general and no specific defect structures were invoked. Although the model has received so far scarce attention of researchers to account for track formation and SHI damage, some of its ideas have described quite successfully other related phenomena, such as the sputtering yields under SHI irradiation [13–15]. To stress the potential relevance of the CM models in relation to ion-beam damage, one may comment that they can provide a reasonable explanation for quite a number of experimental features, namely: (a) the complex structure of tracks consisting of an amorphous core surrounded by a defective halo [16–18]; (b) the kinetics of amorphization as a function of fluence which does not obey a Poisson, as often assumed, but an Avrami-type law presenting an initial incubation stage [16,19]; (c) the effect of prior irradiation on the measured threshold value [8,20]. Finally, one should remark that molecular dynamics (MD) simulations [21– 24] appear better consistent with the CM model and, indeed, confirm that after an ion impact a large number of point defects are formed. These simulations describe in microscopic detail the modifications in the structure of the material until they lead to a heavily disordered or quasi-liquid phase [23]. All those considerations make a further analysis of the CM model opportune if not necessary. The purpose of this work is to analyse the cumulative approach by considering specific irradiation-induced defects and adopting a more general phenomenological strategy than in Ref. [8]. We first offer a critical survey of formation energies for Frenkel pairs as main irradiation-induced point-defects that are expected to be relevant for the CM model. The data refer to a broad range of crystalline materials, including semiconductors and dielectrics (halides and oxides). Using the data of that survey we explore possible correlations between such defect properties and the measured threshold values. The results of the analysis, although are not fully quantitative, offer reasonable support to consider the cumulative approach as a reasonable alternative to the more extensively investigated melting models. It is expected that this broader phenomenological perspective helps to appreciate the value and potential of the CM model and may stimulate a deeper quantitative analysis for specific materials.
2. The concept of thermal spike The thermal spike concept proposed in the early days of radiation damage physics [25,26] describes the process of energy transfer from the SHI to the material through initial excitation of a dense electron cloud by Coulomb interaction with the electrons of the material. Subsequently, the excited electrons transfer energy to the phonon system and cause a high rise of temperature in the surrounding volume (thermal spike). Regardless of the particular model being investigated, the thermal spike concept provides the basis for any deep analysis of the lattice modification around the ion trajectory, and has been extensively and deeply analysed by Toulemonde and co-workers [5,7,27]. Our comparative phenomenological study uses the Szenes analytical approximation, which ignores the very fast electronic dynamics (<0.1 ps after the ion impact). It assumes [6,28] an initial Gaussian distribution for the transversal lattice temperature profile once the electronlattice thermal equilibrium has been reached:
Tðr; tÞ ¼ T B þ
2 r2 a0 hSe aðtÞ e 2 2 aðtÞ pa0 qC
ð1Þ
TB being the background temperature (generally the room temperature); C and q are the specific heat and density of the material; h is a heating efficiency, i.e. the fraction of the deposited energy Se invested in heating. a0 is an empirical parameter that characterizes the initial (t = 0) extension of the spike and is determined by the dynamics of electron transport from the ion trajectory in the time scale of the electron-lattice interaction 61 ps. It is roughly equivalent to the electron-phonon mean free path k [5,7,15] used in the two-temperatures (2T) formulations of the thermal spike. It is assumed that after t = 0 the lattice temperature profile follows the thermal transport equation during spike cooling. Table 1 (left side) lists the experimental or estimated values of the parameters for the spike appearing in Eq. (1). Where a0 is not available, the k value [5,7,15] will be used in our calculations. For semiconductors (Ge, Si, AsGa, InP) and hard ceramics (AlN, GaN) k values have been taken from a recent analysis of experimental data [27]. They are larger than for oxides, as expected from the higher electron mobilities (1300 cm2 V1 s1 for Si, 8500 cm2 V1 s1 for AsGa, 900 cm2 V1 s1 for SiC and 440 cm2 V1 s1 for GaN). Since the temperatures reached in the spike are much higher than the Debye temperature, the heat capacities have been given the classical Einstein value, C = 25 J/(mol K), for an insulator material.
Fig. 1. An idealized representation of the two track formation models considered in this work. On the left, the track volume is melted; on the right, defects are accumulated with different concentrations in the track volume (core) and halo.
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
3. The defect accumulation model (CM): survey of defect formation energies for Frenkel pairs One of the physical consequences of swift ion-beam irradiation is the thermally-assisted generation of lattice disorder around the ion trajectory, in the form of point defects, as a consequence of the high temperatures reached in the spike. This effect, which is associated to the high electronic energy deposited by the ion, is rooted in thermodynamical arguments and it has been extensively discussed in books and monographs (e.g. Ref. [29]). The subsequent very fast cooling (quenching stage) freezes the high temperature defect concentration or a certain fraction of it, that remain in the crystal on returning to the background temperature. These defect-generation processes and their subsequent migration have been discussed at length in connection with irradiation-induced sputtering [13–15], although not in relation to damage. During swift-heavy-ion (SHI) bombardment, the primary products of damage are point defects similar to those generated by elastic nuclear collisions [29,30], although the mechanisms for defect formation are expected to be of a different (electronic) character, whatever they might be. In the cumulative models it is proposed that the formation of amorphous tracks results from the accumulation and clustering of those point defects around the ion trajectory i.e. at the single impact level, followed by the collapse and amorphization of the lattice structure once a critical concentration is reached [8,31]. They offer a reasonable alternative to melting models, since point defects and tracks are, both, created during ion-beam irradiation, and so one expects that they are intimately coupled. Moreover, it is well known that point defects migrate and may escape from the material (sputtering process) or, alternatively, aggregate and lead to amorphization. Although a universal quantitative formalism for the CM is not yet available, the threshold stopping power Sth to start amorphization should depend on the concentration level of point defects generated by irradiation as well as on the critical value required for lattice collapse (defect-induced phase transition) [31]. As to the first factor, a key parameter in relation to thermally-induced disorder is the formation energy (enthalpy) of the defects. In principle, the simplest radiation-induced defects are expected to be vacancies and interstitials, either isolated or associated in pairs [29]. We will focus on a particular and relevant type of paired defect: the vacancy-interstitial pair (Frenkel pair). This is the primary defect created by elastic collision damage and, therefore, one might establish a certain parallelism between nuclear (collision) and electronic damage. Experimental values are difficult to obtain and often not reliable; so for our comparative analysis we have selected theoretical values listed in Table 1. Despite creation energies strongly depend on the details of the calculations, the Frenkel pairs values (right side of Table 1) are more reliable and robust [32] than those for other simple defects (e.g. Schottky pairs). In fact, the chemical and electronic (Fermi level) potentials, involved in the calculations for the individual vacancies and interstitials of the pair, cancel for Frenkel pairs. The calculated formation energies EF of Frenkel pairs are mostly in the range 2.5–15 eV and are strongly dependent on the charge state of the members of the pair, and calculation method. For our analysis we have considered those pairs presenting the lowest formation energy although this point may be revised in future analyses. On the other hand, the experimental Sth values span from 2 keV/nm to more than 30 keV/nm. Within the cumulative scheme, track formation should occur when the non-equilibrium concentration of generated defects at the spike reaches a critical value and triggers a defect-assisted phase transition or lattice collapse. The theoretical framework to describe such transition has been investigated by Fecht [31] using simplified phase diagrams incorporating the concentration as a thermodynamical coordinate and
23
concentration-dependent Gibbs free energies. For simple intermetallic compounds his model predicted that a critical relative vacancy concentration for melting instability was around 0.08. Analyses for other materials are not available and may strongly differ for different compounds in accordance with their different binding structures and cohesive energies. We are going to adopt, here, a simple phenomenological approach to compare different crystals. In line with the extensive survey of formation energies presented in this work (Table 1) for Frenkel pairs as the main radiation-induced defect, our present objective is to explore at a phenomenological level whether the cumulative model may provide the right trend for the variation of stopping power threshold for a broad range of materials. The main point of the analysis, as in the previous work [8], is to assume a Arrhenius-law behaviour for the local concentration of Frenkel pairs reached in thermal equilibrium at a temperature T,
cF ¼
nF EF ¼ v exp N 2kT
ð2Þ
nF and cF being, respectively, the absolute and relative concentrations corresponding to an atomic concentration N, and EF stands for the formation energy of a single non-bound Frenkel pair. The above relation takes into account the configurational entropy contribution to the free energy [29] through the pre-exponential factor v that is not very different from 1 and depends on the number of interstitial configurations around the displaced host atom. For simplicity, it will be taken as v = 1 in all cases. This kind of Arrhenius analysis is quite general and it has been often used to obtain the defects concentration (e.g. vacancies) reached after a high temperature treatment and has been instrumental for the determination of the formation energies of point defects in a broad range of materials [29]. To account for the threshold stopping power (Sth) under SHI irradiation we have to consider the maximum temperature (T0) reached at the axis of the spike (r = 0) at t = 0 that, according to Eq. (1), when the heating efficiency factor h is assumed to be 1 as in many previous works [28], is given by:
T0 ¼ TB þ
Sth
qC r0
;
r0 ¼ pa20
ð3Þ
In the cumulative model the threshold stopping power Sth is obtained when the concentration cF at the spike axis (r = 0) reaches a critical value c0 that, according to the analysis by Fecht [31], triggers a local defect-assisted phase transition into an amorphous state. This initial amorphization is enhanced and extends up to a finite radius for stopping powers higher than the threshold value. The theoretical prediction of such a threshold concentration is a very hard problem and it may strongly depend on the particular material. It is out of the scope of the present work. Anyhow, the Frenkel pair concentration reached at threshold Sth, induced by a peak temperature rise T0 in the spike, writes:
! EF EF ¼ exp c0 ¼ exp 2kT 0 2kT B þ 2k qSCthr0 EF qC r0 ffi exp 2kSth
ð4Þ
In order to test this analysis we have plotted in Fig. 2 the experimentally measured Sth versus the smallest values for EF listed in Table 1. In spite of the high dispersion of data there is a rough correlation between the two magnitudes, allowing to conclude that the threshold electronic stopping roughly scales with the formation energy of the Frenkel pair. This is a main conclusion of the present work. Materials with low EF, such as halides (LiF, CaF2), present low thresholds, whereas those with EF P10 eV, as AlN,
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M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
a1 = 3.24) the a0/a1 ratio is always between 1 and 2. In other words the energy stored in the defective spike is between 1 and 4 times more concentrated than that involved in heat and implies stored energies associated to the lattice distortion of 1–4 eV nm3. A further quantitative analysis cannot be performed at this stage and would, indeed, require consideration of specific crystals. Anyhow, the relation between the defect and temperature profiles is a relevant question, which has to do with the heating efficiency parameter h introduced by Szenes, when comparing experimental threshold values with those determined from melting (see next Section). In our opinion, the rough correlation shown in Fig. 2, as well as the above numerical arguments, provide significant stimulus to consider the CM model as a possible alternative to melting models. Fig. 2. Experimentally measured Sth(exp) for materials versus the formation energy of a single non-bound Frenkel pair EF. The 10% error bar is only indicative of the possible variation of values.
SiC, diamond and some hard oxides as MgO and Al2O3, show high or very high thresholds (no threshold has been found for diamond). The graph shows a few data point that markedly depart from the general observed trend. They correspond to complex oxides such as zircon, quartz, and YAG, and should deserve particular analysis. Aside from experimental uncertainties one may here consider that other mechanisms (e.g. exciton recombination) are significantly contributing to the damage [2,33–35]. In accordance with Eq. (4) one can evaluate the threshold Frenkel pair concentration, obtaining values in the order of a few percent (see Table 1) and other in the order of tens percent, showing large differences from material to material. It must be considered that these values are strongly dependent on the defect energy calculations that may cause a large variability of results, making therefore difficult the quantitative predictions. However, although the comparison may be unjustified these values are in range with those obtained in Fecht’s work [31] for some intermetallic alloys (102). They correspond to absolute atomic concentrations at the spike axis of 1020–1021 cm3 0.1– 1 nm3. It is important to realize that, in accordance with Eq. (4), the radial profile of defect concentration, having a half-width a1, is generally sharper than that corresponding to the spike temperature a0. The a0 and a1 parameters have been extracted from fitting Eq. (1) and (2) with a function
y ¼ y0 expðr 2 =a2 Þ . The example of Si is given in Fig. 3. The a1 fitted values are reported in Table 1, showing that, with the only exception of Al2O3 (a0/
Fig. 3. Radial distributions of temperature T(r) and Frenkel-pair concentration c(r) for crystalline Si, calculated from Eqs. (1) and (2), respectively, and fitted with the exponential function y = y0 exp(r2/a2).
4. Comparison with the thermal spike (melting) approach We may now, summarily, compare the CM approach with the melting model, using the same analytical Szenes approximation [6,28] as for the spike temperature profile. In the standard thermal spike model, crystal amorphization starts when the maximum temperature reached in the spike at the axis of the trajectory (T0) is equal to the melting point (TM). Then, a threshold stopping power Sth is derived from Eq. (5)
Sth ¼ r0 qCðT M T B Þ=h
ð5Þ
The values for Sth are listed, as Sth(Szenes) in Table 1 (left side) together with those for the experimental threshold, listed as Sth(exp). The relation between the two magnitudes is plotted in Fig. 4. One sees that, in spite of the dispersion of data, the theoretical values predicted by use of Eq. (5), roughly follow the experimental trend although they are around a factor 10 smaller than experimental ones. It may be illustrative in relation to a quantitative comparison between the CM and thermal spike (melting) models, to compare the energy spent in producing Frenkel-pair defects over the spike volume when the irradiation-induced Frenkel pair concentration at the peak of temperature (T0) reaches the critical value c0 (Eq. (4)). The CM calculation has been performed integrating the Gaussian distributed defect concentration between 3a1 and 3a1 so to account for 96.6% of the total defects. Their results are plotted in Fig. 5 and show that the energy stored in the material as Frenkel pairs is higher than that spent for melting (Szenes model, compare with Fig. 4) and closer (but still somewhat lower) to the experimental threshold. In spite of the uncertainties in, both calculations and experiments, the results suggest that the predictions of the CM model reasonably compete with those from the melting model as to track formation thresholds.
Fig. 4. Relation between the experimental threshold stopping power in various materials (Sth (Exp)) and the values calculated from the Szenes model (Sth (Szenes), from Eq. (5)). The 10% error bar is only indicative of the possible variation of values.
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
Fig. 5. Total energy Edef stored as Frenkel pairs in the spike volume per unit length of the trajectory when the concentration on the spike axis is c0 (see text) plotted against the amorphization threshold stopping power Sth (exp). The 10% error bar is only indicative of the possible variation of values.
4.1. Summary and outlook In summary, the defect accumulation models yield a roughly correct qualitative trend with regard to the Se dependence of the threshold, but they need a sound theoretical formulation in the line of the work by Fecht. Possibly this is only achievable by considering specific materials where the structure and properties are well understood. Moreover, no remarkable incompatibility is observed between the CM and thermal spike predictions. Such observation is likely associated to the fundamental relation between defects and melting [36]. An advantage of the cumulative approach over melting models is that the former can deal more conveniently with the synergy between atomic collision and electronic mechanisms [37], which is one of the hot topics in present research. Moreover, it is more suited to deal with thermally or irradiation-induced back-reactions and recrystallization processes [38–40] that may lead to marked enhancements in the measured threshold values. As a final remark one should consider the very low threshold measured for some oxides, such as zircon. This is likely associated to the operation of different damage and track formation mechanisms not considered here. Particularly, the excitonic mechanism has been definitely established for SiO2, halides and some chalcogenides [33,34].
Acknowledgements M.L. Crespillo gratefully acknowledges support from the University of Tennessee Governor’s Chair program.
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Cumulative approaches to track formation under swift heavy ion (SHI) irradiation: Phenomenological correlation with formation energies of Frenkel pairs M.L. Crespillo a,b,⇑, F. Agulló-López a,⇑, A. Zucchiatti a a b
Centro de Microanálisis de Materiales, CMAM-UAM, Cantoblanco, Madrid 28049, Spain Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, United States
a r t i c l e
i n f o
Article history: Received 27 September 2016 Received in revised form 9 December 2016 Accepted 11 December 2016
Keywords: Radiation-induced damage mechanisms Ion-solid interaction Radiation-induced point defects Frenkel pair formation energies Order to disorder microstructure phase transformation Swift heavy ion irradiation Ion track formation Electronic stopping power threshold Cumulative defect models Thermal spike model
a b s t r a c t An extensive survey for the formation energies of Frenkel pairs, as representative candidates for radiation-induced point defects, is presented and discussed in relation to the cumulative mechanisms (CM) of track formation in dielectric materials under swift heavy ion (SHI) irradiation. These mechanisms rely on the generation and accumulation of point defects during irradiation followed by collapse of the lattice once a threshold defect concentration is reached. The physical basis of those approaches has been discussed by Fecht as a defect-assisted transition to an amorphous phase. Although a first quantitative analysis of the CM model was previously performed for LiNbO3 crystals, we have, here, adopted a broader phenomenological approach. It explores the correlation between track formation thresholds and the energies for Frenkel pair formation for a broad range of materials. It is concluded that the threshold stopping powers can be roughly scaled with the energies required to generate a critical Frenkel pair concentration in the order of a few percent of the total atomic content. Finally, a comparison with the predictions of the thermal spike model is discussed within the analytical Szenes approximation. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction The most representative consequence of electronic damage under SHI irradiations [1–4] is the production of linear heavily disordered regions (amorphous tracks) around the ion trajectory. The radius R of the track increases with the ion stopping power, in practice the electronic part Se. Extrapolation to R = 0 determines a threshold stopping Sth for track formation. An extensive (but not exhaustive) list of threshold values, Sth(exp), extracted from the experiments, is included in the left side of Table 1. It needs to be remarked that threshold values depend on the precision of the extrapolation process, are known to be velocity dependent and are often attributed quite different values from different authors. Large uncertainties cannot therefore be excluded on the values reported in Table 1. Measured threshold lie in the range 1–50 keV/nm, roughly corresponding to a deposited energy of ⇑ Corresponding author at: Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, United States (M.L. Crespillo). E-mail addresses: [email protected] (M.L. Crespillo), [email protected] (F. AgullóLópez). http://dx.doi.org/10.1016/j.nimb.2016.12.022 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
around 1–50 eV/molecule, comparable with the cohesive energy of solids. Most thresholds for dielectric crystals are below 20 keV/nm. The lowest values (<5 keV/nm) are reported for some oxides, such as SiO2, ZrSiO4, YIG, KGW and some halides, whereas highest values (>25 keV/nm) are found for semiconductors. For some binary ceramic oxides having closed shells cations, MgO, Al2O3, UO2, as well as for diamond, carbides (SiC), and nitrides (Si3N4), the threshold is comparable to those for radiation-hard semiconductors. As far as we know, a definite threshold value for track formation has not been measured for diamond. Most of the models proposed to understand and predict the track generation processes are based on the thermal spike concept, i.e. a hot region around the ion trajectory containing a high concentration of excited electron carriers. In particular, the thermal spike model (TSM) [5–7], which relies on the local melting of the material caused by the abrupt temperature rise in the spike, has been extensively investigated. It qualitatively accounts for many gross features of track formation, such as threshold stopping power, track radius, etc. A different, less explored approach, also rooted in the concept of thermal spike, relies on the generation and accumulation of
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Table 1 Thermal properties, experimental values of amorphization threshold stopping power Sth (Exp), and formation energies for intrinsic Frenkel-pair defects (V stands for vacancy and I for interstitial) in various crystalline materials. Also listed are the amorphization threshold (Sth (Szenes)) predicted from Eq. (5) and the calculated total energy (Edef) stored as Frenkel pairs in the spike volume per unit length of the trajectory. Material
Thermal properties and thresholds a0 [nm]
k [nm]
Defects formation energies Sth (Exp) [keV/nm]
C [J/g K]
Refs.
Frenkel pairs
Formation energy [eV] EF (in bold)
Refs.
[g/cm3]
Sth (Szenes) [keV/nm]
C0 [%]
a1 [nm]
Edef [keV/nm]
(V0Si + I0Si) (V0Ge + I0Ge) (V0Ga + I0Ga), (V0As + I0As) (V0P + I0P), (V0In + I0In) 1 (V1+ Cl + ICl ), 1+ + I (V1 Na Na) 1 (V1+ F + IF ) 1 (V1+ F + IF ) 2+ (V2 Ca + ICa) 2 (V2+ O + IO ), 1 (V1+ O + IO ), (V0O + I0O) 2 (V2+ O + IMg) Schottky pair (V0O) Single vacancy defect. No calculations for Frenkel pairs (V0O) Single vacancy defect. No calculations for Frenkel pairs (V0O + I0O), (V0Al + I0Al) 2 (V2+ O + IO ) 3+ (V3 Al + IAl ) (V0O + I0O) 2 (V2+ O + IO ), (V0Zr + I0Zr), 4 (V4+ Zr + IZr ) (V0O + I0O), 2 (V2+ O + IO ), (V0Hf + I0Hf), 4 (V4+ Hf + IHf )
7.42 6.41 7.0 7.80 5.17 6.88 4.602, 2.881 2.34–2.40 2.6–2.8 6, 8.5, 9.2 13.3, 11.93, 10.28 7.5 4.4
[42] [42] [45]
4.05 2.58 33.86
3.70 4.81 5.77
1.72 1.40 14.47
[47]
9.86
4.68
1.73
[29,50]
1.71
5.76
0.30
[29] [53]
10.06 41.07
2.28 3.18
0.63 2.21
[55,56]
14.61
2.52
3.12
[59]
7.96
2.56
0.61
6.6, 8.5
[60,61]
0.04
14.93, 25.32 11.38, 13.22 8.95, 9.0 7.57, 7.3 4.11, 20.05, 9.45 8.34, 4.69, 23.36, 10.13
[63]
3.36
2.59
0.37
[65,66]
52.28
3.67
6.73
[65,68]
53.72
4.18
10.18
(V0O + I0O), (V0U + I0U) 2 (V2+ O + IO ), 2+ (V2 Sr + ISr ), 4+ (V4 Ti + ITi ) 2 (V2+ O + IO ), 4+ (V4 Ti + ITi ), 2+ (V2 Ba + IBa) 2 (V2+ + I O O ), 5+ (V5 Nb + INb), 1+ (V1 Li + ILi )
4.02, 13.15 5.14, 7.88, 14.20 8.98, 15.12, 11.88 6.84, 12.5, 1.86
[73]
1.78
4.27
0.27
[75]
31.25
3.41
2.61
[77,78]
56.93
2.57
4.38
[77,78]
59.76
3.28
1.74
2 (V2+ O + IO ), 3+ (V3 Y + IY ), 3+ (V3 Al + IAl ) 2 (V2+ O + IO ), 3+ (V3 Y + IY ), 3+ (V3 Fe + IFe ) IO + V2 O + IO 0 0 (VO + IO ), (V0Si + I0Si), (V0Zr + I0Zr)
10.16, 12.57, 12.72 3.55, 8,0.45, 5.93 13 7.3, 22.8, 23.9
[85]
47.50
3.94
2.89
[87]
62.41
4.54
1.59
3+ (V3 Al + IAl )
10
[93]
4.26
3.69
2.32
(VC0 + IC0), (V0Si + I0Si) (V0C + I0C) lowest energy interstitial (V0C + I0C) (V0N + I0N), (V0Ga + I0Ga)
8.64 (9.07), 12.81 23
[97,98]
6.29
3.81
3.19
3.03
3.47
1.10
Si Ge GaAs
7.2 9.85 7.4
2.33 5.32 5.3
2.92 3.12 1.16
28 34 36
0.9 0.34 0.17
[28,41] [43] [28,44]
InP
7.95
4.5
0.98
12
0.17
[28,46]
NaCl
12
2.16
2
10.00
0.43
[17,48,49]
LiF CaF2
3.8 3.8
2.64 3.18
0.61 0.4
4.20 4.80
1 0.32
[18,51] [52]
MgO
4
3.58
1.94
15.8
0.63
[54]
5.8
4.23
0.73
5.1
0.31
[57,58]
c-SiO2
4
2.62
0.60
1.61
0.42
[7]
Al2O3
8
3.7
0.76
23
18
[2,54,62]
ZrO2
4
5.68
0.94
13
0.2
[64]
HfO2
4.5
9.65
1.23
20
0.12
[67]
SnO2 UO2
5.5 9
6.95 10.96
1.1 0.96
6 22
0.17 0.23
[69,70] [71,72]
SrTiO3
4.5
5.11
0.41
7.2
0.14
[74]
BaTiO3
4.5
6.01
0.37
22
0.1
[76]
4.3
0.35
6.0
0.17
[7,20,79]
7.27 5.4 6.14 7.08 4.56
0.09 0.046 0.3 0.09 0.11
4 5.40 16.00 7.2 6.00
0.04 0.022 0.1 0.025 0.042
[80] [70] [81] [82] [83,84]
TiO2
4.5
LiNbO3
4.5
KGd(WO4)2(KGW) BaFe12O19 ZnFe2O4 Gd3Ga5O12(GGG) Y3Al5O12(YAG)
3.7
4.5
8.2 4.5 4.6 4.6
Y3Fe5O12(YIG)
4.5
5
5.12
0.08
3.00
0.034
[86]
4
3.6 4.6
0.4 0.44
13 2.50
0.17 0.14
[88] [23]
2.7
0.01
3.8
0.9
[91,92]
7 7 7
3.26 3.27 3.21
4.8 1 4.8
34.00 15 33.00
0.6 0.18 0.6
[27] [94] [95,96]
Diamond
8
3.5
2
[99]
Graphite GaN
8 7
2.2 6.15
2 0.3
[102] [27,104]
MgAl2O4(spinel) ZrSiO4 (zircon)
MicaKAl2(AlSi3O10) (F,OH)2 AlN Si3N4 SiC
4.5
CM model
q
4.3
2.5 23.6
13.7 8.2, 12.3
[89] [90]
[100,101] [103] [105]
22
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
irradiation-induced point defects around the ion trajectory [8–12]. The physical background of the cumulative model is similar to that for the mostly investigated thermal spike (melting) model. The two models are comparatively illustrated in the sketch of Fig. 1. A first preliminary theoretical analysis of this novel cumulative approach (CM model) was developed and applied to LiNbO3 [8]. In that work the formulation was quite general and no specific defect structures were invoked. Although the model has received so far scarce attention of researchers to account for track formation and SHI damage, some of its ideas have described quite successfully other related phenomena, such as the sputtering yields under SHI irradiation [13–15]. To stress the potential relevance of the CM models in relation to ion-beam damage, one may comment that they can provide a reasonable explanation for quite a number of experimental features, namely: (a) the complex structure of tracks consisting of an amorphous core surrounded by a defective halo [16–18]; (b) the kinetics of amorphization as a function of fluence which does not obey a Poisson, as often assumed, but an Avrami-type law presenting an initial incubation stage [16,19]; (c) the effect of prior irradiation on the measured threshold value [8,20]. Finally, one should remark that molecular dynamics (MD) simulations [21– 24] appear better consistent with the CM model and, indeed, confirm that after an ion impact a large number of point defects are formed. These simulations describe in microscopic detail the modifications in the structure of the material until they lead to a heavily disordered or quasi-liquid phase [23]. All those considerations make a further analysis of the CM model opportune if not necessary. The purpose of this work is to analyse the cumulative approach by considering specific irradiation-induced defects and adopting a more general phenomenological strategy than in Ref. [8]. We first offer a critical survey of formation energies for Frenkel pairs as main irradiation-induced point-defects that are expected to be relevant for the CM model. The data refer to a broad range of crystalline materials, including semiconductors and dielectrics (halides and oxides). Using the data of that survey we explore possible correlations between such defect properties and the measured threshold values. The results of the analysis, although are not fully quantitative, offer reasonable support to consider the cumulative approach as a reasonable alternative to the more extensively investigated melting models. It is expected that this broader phenomenological perspective helps to appreciate the value and potential of the CM model and may stimulate a deeper quantitative analysis for specific materials.
2. The concept of thermal spike The thermal spike concept proposed in the early days of radiation damage physics [25,26] describes the process of energy transfer from the SHI to the material through initial excitation of a dense electron cloud by Coulomb interaction with the electrons of the material. Subsequently, the excited electrons transfer energy to the phonon system and cause a high rise of temperature in the surrounding volume (thermal spike). Regardless of the particular model being investigated, the thermal spike concept provides the basis for any deep analysis of the lattice modification around the ion trajectory, and has been extensively and deeply analysed by Toulemonde and co-workers [5,7,27]. Our comparative phenomenological study uses the Szenes analytical approximation, which ignores the very fast electronic dynamics (<0.1 ps after the ion impact). It assumes [6,28] an initial Gaussian distribution for the transversal lattice temperature profile once the electronlattice thermal equilibrium has been reached:
Tðr; tÞ ¼ T B þ
2 r2 a0 hSe aðtÞ e 2 2 aðtÞ pa0 qC
ð1Þ
TB being the background temperature (generally the room temperature); C and q are the specific heat and density of the material; h is a heating efficiency, i.e. the fraction of the deposited energy Se invested in heating. a0 is an empirical parameter that characterizes the initial (t = 0) extension of the spike and is determined by the dynamics of electron transport from the ion trajectory in the time scale of the electron-lattice interaction 61 ps. It is roughly equivalent to the electron-phonon mean free path k [5,7,15] used in the two-temperatures (2T) formulations of the thermal spike. It is assumed that after t = 0 the lattice temperature profile follows the thermal transport equation during spike cooling. Table 1 (left side) lists the experimental or estimated values of the parameters for the spike appearing in Eq. (1). Where a0 is not available, the k value [5,7,15] will be used in our calculations. For semiconductors (Ge, Si, AsGa, InP) and hard ceramics (AlN, GaN) k values have been taken from a recent analysis of experimental data [27]. They are larger than for oxides, as expected from the higher electron mobilities (1300 cm2 V1 s1 for Si, 8500 cm2 V1 s1 for AsGa, 900 cm2 V1 s1 for SiC and 440 cm2 V1 s1 for GaN). Since the temperatures reached in the spike are much higher than the Debye temperature, the heat capacities have been given the classical Einstein value, C = 25 J/(mol K), for an insulator material.
Fig. 1. An idealized representation of the two track formation models considered in this work. On the left, the track volume is melted; on the right, defects are accumulated with different concentrations in the track volume (core) and halo.
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
3. The defect accumulation model (CM): survey of defect formation energies for Frenkel pairs One of the physical consequences of swift ion-beam irradiation is the thermally-assisted generation of lattice disorder around the ion trajectory, in the form of point defects, as a consequence of the high temperatures reached in the spike. This effect, which is associated to the high electronic energy deposited by the ion, is rooted in thermodynamical arguments and it has been extensively discussed in books and monographs (e.g. Ref. [29]). The subsequent very fast cooling (quenching stage) freezes the high temperature defect concentration or a certain fraction of it, that remain in the crystal on returning to the background temperature. These defect-generation processes and their subsequent migration have been discussed at length in connection with irradiation-induced sputtering [13–15], although not in relation to damage. During swift-heavy-ion (SHI) bombardment, the primary products of damage are point defects similar to those generated by elastic nuclear collisions [29,30], although the mechanisms for defect formation are expected to be of a different (electronic) character, whatever they might be. In the cumulative models it is proposed that the formation of amorphous tracks results from the accumulation and clustering of those point defects around the ion trajectory i.e. at the single impact level, followed by the collapse and amorphization of the lattice structure once a critical concentration is reached [8,31]. They offer a reasonable alternative to melting models, since point defects and tracks are, both, created during ion-beam irradiation, and so one expects that they are intimately coupled. Moreover, it is well known that point defects migrate and may escape from the material (sputtering process) or, alternatively, aggregate and lead to amorphization. Although a universal quantitative formalism for the CM is not yet available, the threshold stopping power Sth to start amorphization should depend on the concentration level of point defects generated by irradiation as well as on the critical value required for lattice collapse (defect-induced phase transition) [31]. As to the first factor, a key parameter in relation to thermally-induced disorder is the formation energy (enthalpy) of the defects. In principle, the simplest radiation-induced defects are expected to be vacancies and interstitials, either isolated or associated in pairs [29]. We will focus on a particular and relevant type of paired defect: the vacancy-interstitial pair (Frenkel pair). This is the primary defect created by elastic collision damage and, therefore, one might establish a certain parallelism between nuclear (collision) and electronic damage. Experimental values are difficult to obtain and often not reliable; so for our comparative analysis we have selected theoretical values listed in Table 1. Despite creation energies strongly depend on the details of the calculations, the Frenkel pairs values (right side of Table 1) are more reliable and robust [32] than those for other simple defects (e.g. Schottky pairs). In fact, the chemical and electronic (Fermi level) potentials, involved in the calculations for the individual vacancies and interstitials of the pair, cancel for Frenkel pairs. The calculated formation energies EF of Frenkel pairs are mostly in the range 2.5–15 eV and are strongly dependent on the charge state of the members of the pair, and calculation method. For our analysis we have considered those pairs presenting the lowest formation energy although this point may be revised in future analyses. On the other hand, the experimental Sth values span from 2 keV/nm to more than 30 keV/nm. Within the cumulative scheme, track formation should occur when the non-equilibrium concentration of generated defects at the spike reaches a critical value and triggers a defect-assisted phase transition or lattice collapse. The theoretical framework to describe such transition has been investigated by Fecht [31] using simplified phase diagrams incorporating the concentration as a thermodynamical coordinate and
23
concentration-dependent Gibbs free energies. For simple intermetallic compounds his model predicted that a critical relative vacancy concentration for melting instability was around 0.08. Analyses for other materials are not available and may strongly differ for different compounds in accordance with their different binding structures and cohesive energies. We are going to adopt, here, a simple phenomenological approach to compare different crystals. In line with the extensive survey of formation energies presented in this work (Table 1) for Frenkel pairs as the main radiation-induced defect, our present objective is to explore at a phenomenological level whether the cumulative model may provide the right trend for the variation of stopping power threshold for a broad range of materials. The main point of the analysis, as in the previous work [8], is to assume a Arrhenius-law behaviour for the local concentration of Frenkel pairs reached in thermal equilibrium at a temperature T,
cF ¼
nF EF ¼ v exp N 2kT
ð2Þ
nF and cF being, respectively, the absolute and relative concentrations corresponding to an atomic concentration N, and EF stands for the formation energy of a single non-bound Frenkel pair. The above relation takes into account the configurational entropy contribution to the free energy [29] through the pre-exponential factor v that is not very different from 1 and depends on the number of interstitial configurations around the displaced host atom. For simplicity, it will be taken as v = 1 in all cases. This kind of Arrhenius analysis is quite general and it has been often used to obtain the defects concentration (e.g. vacancies) reached after a high temperature treatment and has been instrumental for the determination of the formation energies of point defects in a broad range of materials [29]. To account for the threshold stopping power (Sth) under SHI irradiation we have to consider the maximum temperature (T0) reached at the axis of the spike (r = 0) at t = 0 that, according to Eq. (1), when the heating efficiency factor h is assumed to be 1 as in many previous works [28], is given by:
T0 ¼ TB þ
Sth
qC r0
;
r0 ¼ pa20
ð3Þ
In the cumulative model the threshold stopping power Sth is obtained when the concentration cF at the spike axis (r = 0) reaches a critical value c0 that, according to the analysis by Fecht [31], triggers a local defect-assisted phase transition into an amorphous state. This initial amorphization is enhanced and extends up to a finite radius for stopping powers higher than the threshold value. The theoretical prediction of such a threshold concentration is a very hard problem and it may strongly depend on the particular material. It is out of the scope of the present work. Anyhow, the Frenkel pair concentration reached at threshold Sth, induced by a peak temperature rise T0 in the spike, writes:
! EF EF ¼ exp c0 ¼ exp 2kT 0 2kT B þ 2k qSCthr0 EF qC r0 ffi exp 2kSth
ð4Þ
In order to test this analysis we have plotted in Fig. 2 the experimentally measured Sth versus the smallest values for EF listed in Table 1. In spite of the high dispersion of data there is a rough correlation between the two magnitudes, allowing to conclude that the threshold electronic stopping roughly scales with the formation energy of the Frenkel pair. This is a main conclusion of the present work. Materials with low EF, such as halides (LiF, CaF2), present low thresholds, whereas those with EF P10 eV, as AlN,
24
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
a1 = 3.24) the a0/a1 ratio is always between 1 and 2. In other words the energy stored in the defective spike is between 1 and 4 times more concentrated than that involved in heat and implies stored energies associated to the lattice distortion of 1–4 eV nm3. A further quantitative analysis cannot be performed at this stage and would, indeed, require consideration of specific crystals. Anyhow, the relation between the defect and temperature profiles is a relevant question, which has to do with the heating efficiency parameter h introduced by Szenes, when comparing experimental threshold values with those determined from melting (see next Section). In our opinion, the rough correlation shown in Fig. 2, as well as the above numerical arguments, provide significant stimulus to consider the CM model as a possible alternative to melting models. Fig. 2. Experimentally measured Sth(exp) for materials versus the formation energy of a single non-bound Frenkel pair EF. The 10% error bar is only indicative of the possible variation of values.
SiC, diamond and some hard oxides as MgO and Al2O3, show high or very high thresholds (no threshold has been found for diamond). The graph shows a few data point that markedly depart from the general observed trend. They correspond to complex oxides such as zircon, quartz, and YAG, and should deserve particular analysis. Aside from experimental uncertainties one may here consider that other mechanisms (e.g. exciton recombination) are significantly contributing to the damage [2,33–35]. In accordance with Eq. (4) one can evaluate the threshold Frenkel pair concentration, obtaining values in the order of a few percent (see Table 1) and other in the order of tens percent, showing large differences from material to material. It must be considered that these values are strongly dependent on the defect energy calculations that may cause a large variability of results, making therefore difficult the quantitative predictions. However, although the comparison may be unjustified these values are in range with those obtained in Fecht’s work [31] for some intermetallic alloys (102). They correspond to absolute atomic concentrations at the spike axis of 1020–1021 cm3 0.1– 1 nm3. It is important to realize that, in accordance with Eq. (4), the radial profile of defect concentration, having a half-width a1, is generally sharper than that corresponding to the spike temperature a0. The a0 and a1 parameters have been extracted from fitting Eq. (1) and (2) with a function
y ¼ y0 expðr 2 =a2 Þ . The example of Si is given in Fig. 3. The a1 fitted values are reported in Table 1, showing that, with the only exception of Al2O3 (a0/
Fig. 3. Radial distributions of temperature T(r) and Frenkel-pair concentration c(r) for crystalline Si, calculated from Eqs. (1) and (2), respectively, and fitted with the exponential function y = y0 exp(r2/a2).
4. Comparison with the thermal spike (melting) approach We may now, summarily, compare the CM approach with the melting model, using the same analytical Szenes approximation [6,28] as for the spike temperature profile. In the standard thermal spike model, crystal amorphization starts when the maximum temperature reached in the spike at the axis of the trajectory (T0) is equal to the melting point (TM). Then, a threshold stopping power Sth is derived from Eq. (5)
Sth ¼ r0 qCðT M T B Þ=h
ð5Þ
The values for Sth are listed, as Sth(Szenes) in Table 1 (left side) together with those for the experimental threshold, listed as Sth(exp). The relation between the two magnitudes is plotted in Fig. 4. One sees that, in spite of the dispersion of data, the theoretical values predicted by use of Eq. (5), roughly follow the experimental trend although they are around a factor 10 smaller than experimental ones. It may be illustrative in relation to a quantitative comparison between the CM and thermal spike (melting) models, to compare the energy spent in producing Frenkel-pair defects over the spike volume when the irradiation-induced Frenkel pair concentration at the peak of temperature (T0) reaches the critical value c0 (Eq. (4)). The CM calculation has been performed integrating the Gaussian distributed defect concentration between 3a1 and 3a1 so to account for 96.6% of the total defects. Their results are plotted in Fig. 5 and show that the energy stored in the material as Frenkel pairs is higher than that spent for melting (Szenes model, compare with Fig. 4) and closer (but still somewhat lower) to the experimental threshold. In spite of the uncertainties in, both calculations and experiments, the results suggest that the predictions of the CM model reasonably compete with those from the melting model as to track formation thresholds.
Fig. 4. Relation between the experimental threshold stopping power in various materials (Sth (Exp)) and the values calculated from the Szenes model (Sth (Szenes), from Eq. (5)). The 10% error bar is only indicative of the possible variation of values.
M.L. Crespillo et al. / Nuclear Instruments and Methods in Physics Research B 394 (2017) 20–27
Fig. 5. Total energy Edef stored as Frenkel pairs in the spike volume per unit length of the trajectory when the concentration on the spike axis is c0 (see text) plotted against the amorphization threshold stopping power Sth (exp). The 10% error bar is only indicative of the possible variation of values.
4.1. Summary and outlook In summary, the defect accumulation models yield a roughly correct qualitative trend with regard to the Se dependence of the threshold, but they need a sound theoretical formulation in the line of the work by Fecht. Possibly this is only achievable by considering specific materials where the structure and properties are well understood. Moreover, no remarkable incompatibility is observed between the CM and thermal spike predictions. Such observation is likely associated to the fundamental relation between defects and melting [36]. An advantage of the cumulative approach over melting models is that the former can deal more conveniently with the synergy between atomic collision and electronic mechanisms [37], which is one of the hot topics in present research. Moreover, it is more suited to deal with thermally or irradiation-induced back-reactions and recrystallization processes [38–40] that may lead to marked enhancements in the measured threshold values. As a final remark one should consider the very low threshold measured for some oxides, such as zircon. This is likely associated to the operation of different damage and track formation mechanisms not considered here. Particularly, the excitonic mechanism has been definitely established for SiO2, halides and some chalcogenides [33,34].
Acknowledgements M.L. Crespillo gratefully acknowledges support from the University of Tennessee Governor’s Chair program.
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