- Email: [email protected]
Damage accumulation in ion-irradiated ceramics
ARTICLE IN PRESS
Vacuum 81 (2007) 1352–1356 www.elsevier.com/locate/vacuum
Damage accumulation in ion-irradiated ceramics Jacek Jagielskia,b,, Lionel Thome´c a
Institute of Electronic Materials Technology, Wolczynska 133, PL-01-919 Warszawa, Poland b The Andrzej Soltan Institute for Nuclear Studies, PL-05-400 Swierk/Otwock, Poland c Centre de Spectrome´trie Nucle´aire et de Spectrome´trie de Masse CNRS-IN2P3, Universite´ Paris-Sud, F-91405 Orsay Campus, France
Abstract The radiation damage induced by energetic ions is one of the basic problems related to ion implantation technology. Each incoming ion creates a huge number of defects, so that high irradiation fluences eventually lead to complete amorphization of the material. Despite the obvious importance of this process, theoretical descriptions of the kinetics of radiation damage accumulation are still incomplete and limited to single-step processes only. In the present paper, we propose a new approach for the description of the radiation damage build-up in crystals, which is based on the concept of subsequent destabilization of the crystalline phases. We compare the presented model to experimental results of damage accumulation kinetics measured for SiC, ZrO2 and MgAl2O4, i.e. cases when one, two or three stages of defects accumulation were observed. r 2007 Elsevier Ltd. All rights reserved. keywords: Amorphization; Crystals; Models; Irradiation
1. Introduction The radiation damage caused by the slowing-down of energetic ions in solids is a particularly important characteristic of ion implantation. It is thus not surprising that this topic was so intensively studied from the very beginning of this technique. Various aspects related to this question were studied: the type of defects formed, their influence on the phase structure of irradiated solids, the role of the process temperature, of the mass and energy of projectiles, etc. Among them, one of the most interesting subjects is the investigation of the kinetics of damage formation, i.e. the dependence of the amount of accumulated damage (fD) versus the ion fluence (F). A measure of fD is often provided using the Rutherford backscattering/ channeling (RBS/C) technique [1], a method able to quantitatively assess the level of damage in irradiated Corresponding author. Institute of Electronic Materials Technology, Wolczynska 133, PL-01-919 Warszawa, Poland. Tel.: +48 502095524; fax: +48 22 864 54 96. E-mail address: [email protected] (J. Jagielski).
0042-207X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2007.01.059
crystals. A large variety of dependencies (going from an exponential rise to a sigmoidal variation) were reported. The challenging and still not fully resolved task is then to reproduce the variations of fD with an appropriate theoretical description. To that purpose, several models were developed in the past, such as the direct impact and the cascade overlap both proposed by Gibbons [2], the direct-impact/cascade overlap [3,4], the direct-impact/ defect stimulated [5] and the nucleation and growth [6,7]. All the above models are based on phenomenological descriptions. Typical problems with their use are: inability to reproduce multi-step damage accumulation kinetics and, in the case of cascade overlap model, necessity to consider a large number of ion impacts to reproduce the observed disorder. The aim of the present paper is to present a new insight into the problem of the disordering kinetics in ionirradiated crystals. A model based on multiple steps of simple accumulation of damage is proposed. This model provides a good agreement with the experimental data in cases when one, two or three steps were observed in damage accumulation kinetics. We will present such a
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
description, discuss the results obtained and, finally, propose some physical mechanisms which may explain observed dependencies. 2. Single-step models of damage accumulation Various phenomenological models were developed in order to reproduce the different types of damage accumulation behaviors observed in ion-irradiated materials [8]. Two main approaches used were: direct impact (DI) and damage accumulation (DA) (both schematically presented in Fig. 1). The DI description assumes that each ion impact creates a given amount of disorder in the target. The damage accumulation process is thus controlled by the probability that an incoming ion will hit an unperturbed volume of the crystal. This type of processes is described by an exponential-rise-to-maximum dependency. The damage measured at the end of irradiation, i.e. generally at saturation, is then the result of the recovery of the various disordered volumes created by incident ions. Thus, the degree of damage increases mostly linearly at the beginning of irradiation and tends to saturate when the major part of the target is already disordered. On the contrary, in the DA description, it is assumed that the measured disorder at a given ion fluence results from the accumulation of many ion impacts in the same volume. The target is thus mostly unaffected by irradiation up to an ion fluence for which a percolation of the various disordered regions leads to a sharp increase of the accumulated damage parameter. A rather general description of the damage build-up was provided by Gibbons [2]. In this description, the disordering kinetics follows the general expression: " # n1 X ðsFÞk f D ¼ f D ð1Þ 1 expðsFÞ , (1) k! k¼0
disordering cross-section and n is the number of ion impacts needed to damage the material. Note that n ¼ 1 in Eq. (1) is characteristic of the DI process when each ion creates a damaged volume in the crystal structure (curve DI in Fig. 1). For n41, i.e. when multiple impacts into the same volume of the crystal are needed to create damage, the Gibbons’ formula describes the cascade overlap model (curve CO in Fig. 1). A combination of DI and cascade overlap descriptions was proposed by Dennis and Hale and later by Webb [3,4]. In the DI/cascade overlap model, the damage accumulation is described by the formula f D ¼ f D ð1Þ½1 ð1 þ st F sa FÞ expðst FÞ,
(2)
where st is the total damage cross-section for disordering and amorphization. In order to reproduce the sigmoidal dependencies of damage accumulation process in Si, SiGe and SiC, a DI/ defect stimulated model [5] (DI/DS in Fig. 1) was proposed, where fD is given by 1 ðs þ ss Þ f D ¼ f D ð1Þ , (3) s þ sa exp½ðs þ ss ÞF
0.6
where ss is the cross-section for stimulated amorphization. All the above models yield sigmoidal or exponential-riseto-maximum shaped dependencies of accumulated damage versus the irradiation fluence and suffer from similar weaknesses. First of all, none of these models take into account cases when more than one step is observed in the damage accumulation kinetics. Such a situation was observed in, e.g., irradiated MgAl2O4 or ZrO2 crystals [9,10]. Moreover, at low irradiation fluences of low-energy (up to several hundreds of keV) ions, one frequently observes the saturation of radiation damage at low (fDo0.1) level [9,10]. In consequence at low fluences, one can expect negative values of dfD/dF (a sublinear dependency), whereas all models yield positive values (with the only exception for the direct-impact model, in which no saturation at low fluences is exhibited). Finally, when a sharp increase of fD is observed at relatively high irradiation fluences, one needs to artificially increase the number of overlapping cascades (n in Gibbons’ formula) up to unrealistic values reaching a few tens in order to reproduce damage accumulation kinetics. All these considerations can be summarized by a strong need to propose a new, alternative description of the damage accumulation in ion-irradiated crystals.
0.4
3. Multi-step damage accumulation
0.2
The main weakness of the models described in the previous part of the paper is related to their inability to reproduce damage accumulation kinetics composed of numerous steps. A typical example of such a dependency is the kinetics measured by RBS/C for magnesium aluminate spinel crystals irradiated with 300 keV Cs or Xe ions (Fig. 2) [9]. Note that in this figure, the fluence axis
where fD is the amount of disorder in the irradiated layer, fD(N) is the value of fD measured at saturation (fD ¼ 1 in the case where irradiation leads to amorphization), s is the Direct Impact (DI) Cascade Overlap (CO) (n=2) Direct Impact/Defect Stimulated DI/DS Defect Accumulation (DA)
1.2 Accumulated damage
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1.0 0.8
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Normalized fluence Fig. 1. Schematic representation of the various models of damage accumulation.
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
0.8
0.6 Cs irradiation Xe irradiation stage 1 stage 2 stage 3 fit
0.4
0.2
0.0 0
2
4
6 8 Dose (dpa)
10
12
Fig. 2. Damage accumulation kinetics of spinel crystal irradiated with 300 keV Cs2+ ions (open circles) and 300 keV Xe2+ ions (open triangles). The solid line represents a fit according to Eq. (6). The dashed line represents stage 1, the dashed-dotted line fits stage 2 and the dashed double dotted holds for stage 3 of the radiation damage accumulation process.
was recalculated to a dose scale (dpa, displacement per atom) using conversion factors calculated by using the TRIM code [11]. The damage accumulation kinetics shown in Fig. 2 exhibits three distinctive steps, which obviously cannot be reproduced by using the single-step models described in the previous section. A new model was thus elaborated to account for most of the damage accumulation kinetics reported in the literature. Let us analyze step by step the processes occurring in irradiated crystals. At low irradiation fluences, the defects, mostly simple Frenkel pairs [12], are formed in an unperturbed crystalline matrix. Taking into account the very high mobility of self-interstitial atoms (SIA) at room temperature [13], it is likely that SIAs migrate to defect sinks (essentially the crystal surface) leaving vacancies within the damaged layer. The first step of the damage accumulation process can thus be regarded as a vacancy concentration build-up. Such a description is consistent with the results of EPR, UV [14], positron annihilation [10] and swelling [15] measurements. When the vacancy concentration increases, the probability of recombination of SIAs with vacancies increases as well. In consequence, one observes a saturation of the concentration of defects at a level determined by the effective recombination volume. At low fluences, the description of damage accumulation can thus be made in the frames of a simple defect accumulation model [16,17] fd ¼
f sat d;1 ½1 f sat d;1
expðs1 FÞ,
(4)
is the saturation value in the first step of the where damage accumulation process. Taking into account that the effective recombination volume for simple Frenkel pairs is rather large (about 1 nm3), one can expect that the value of f sat d;1 should be rather small.
The accumulation of vacancies leads to the formation of free volume within the crystal structure, hence to the generation of a high compressive stress. Since the crystal structure cannot indefinitively support continuously increasing stresses, at a given irradiation fluence, a phase transformation must thus occur to relax the stress. In other words, the stress increases the free energy of the crystalline phase which becomes higher than that of other possible atomic configurations. When the energy of the original crystalline phase is no longer the lowest one, this structure is destabilized and each ion impact leads to its transformation into a configuration which has a lowest free energy. The transformation process is controlled by the probability that a given ion hits an unperturbed part of the material, so that the corresponding kinetics is described by Eq. (5) (exponential rise to maximum), with the only difference for the existence of an irradiation fluence threshold F1. This threshold corresponds to the irradiation fluence at which the destabilization of the original crystalline phase takes place. Thus, the second step can be reproduced with the equation sat sat f d;2 ¼ f sat d;1 þ ðf d;2 f d;1 Þ½1 expðs2 ðF F1 ÞÞ.
(5)
The crucial point of this hypothesis relies in the assumption that the stress caused by the formation of free volume in an irradiated sample triggers the phase transformation. This hypothesis was checked by experiments performed on thin (150 mm thick) spinel crystals where the damage accumulation kinetics was determined by RBS/C and the evolution of the stress in the irradiated layer was measured with the Stoney method [18]. The results are shown in Fig. 3, which shows fast increase of the stress for low irradiation fluences followed by a sharp decrease when the dose exceeds 2 dpa. At the same dose of 2 dpa, an abrupt rise of the level of accumulated damage is noted. Thus, it is clear that the second step in the accumulation of radiation damage in irradiated spinel crystals is caused by the stress increase. The solid line joining experimentally measured fD values is a fit made by using a combination of Eqs. (4) and (5). One can note that, 0.8
3.5 3.0
0.6
2.5 2.0
0.4
1.5 1.0
0.2
0.5
fD fit Stress
0.0
Stress (GPa)
Accumulated damage
1.0
Accumulated damage
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0.0 -0.5
0
10
20 Dose (dpa)
30
40
Fig. 3. Damage accumulation kinetics and stress in the surface layer measured for thin spinel crystals irradiated with 300 keV Ar2+ ions.
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
f d;1 ¼ f sat d;1 ½1 expðs1 FÞ
for 0oFoF1 ;
sat sat f d;2 ¼ f sat d;1 þ ðf d;2 f d;1 Þ½1 expðs2 ðF F1 ÞÞ
for F1 oFoF2 ;
sat sat f d;i ¼ f sat d;i1 þ ðf d;i f d;i1 Þ½1 expðsi ðF Fi1 ÞÞ
for Fi1 oF:
fd ¼
a
0.8 0.6 0.4 0.2 0.0 0
1.0
2
4 6 Fluence (x1013 cm-2)
8
10
0.4
0.5
b
0.8 0.6 0.4 0.2 0.0 0.0
0.1
0.2
0.3 Dose (dpa)
c
1.0 0.8 0.6
The final formula including necessary normalization factors is thus n X
1.0
Accumulated damage
in thin samples, the crystalline structure persists to much higher irradiation fluences and the third step in damage accumulation is not observed. This result is likely due to a partial relaxation of the stress due to sample bending. A similar approach has been used to reproduce the results obtained in thick samples where three steps were observed (Fig. 2). A very good agreement between experimental results and calculations points to the conclusion that the description based on several steps of simple damage accumulation provides a much better agreement with experimental data than the models described in Section 2. In conclusion, the new approach is based on a sequence of several damage accumulation steps, each stage being described by a simple accumulation of disorder. A new stage occurs when the current structure of the matrix is no longer characterized by the lowest free energy; transformations are thus triggered by the destabilization of a given structure. This destabilization may be due to various mechanisms: stress accumulation, formation of defect clusters, increase of the concentration of impurity atoms acting as damage stabilizers, etc. In irradiated spinel crystals, the possible sequence of structural transformations is likely the following: crystalline structure-crystalline structure with point defects-crystalline structure with randomized cationic network [19] or dislocation network-amorphous structure. The equations describing the sequence of damage accumulation can be written as follows:
1355
0.4 0.2
ðf sat d;i
f sat d;i1 ÞG½1
expðsi ðF Fi1 ÞÞ,
(6)
i¼1
f sat d;i
where is the level of damage saturation in the ith step. G is a function which transforms negative values into 0 and leaves the positive values unchanged. The proposed model offers numerous advantages with respect to the previous ones. First of all, this description is able to reproduce multi-step damage accumulations, as the ones occurring in spinel or zirconia crystals. Secondly, the same formalism can be applied to damage accumulation processes composed of one, two or more stages. Some examples are shown in Fig. 4, which presents the damage accumulation kinetics for spinel crystals irradiated with swift, heavy ions (Fig. 4a, single-step amorphization kinetics), SiC (Fig. 4b—two steps) and ZrO2 (Fig. 4c— three steps), the latter two materials being irradiated with low energy (hundreds of keV) ions. The interpretation in terms of microstructural evolution is straightforward: when the crystalline structure is destabilized, each ion impact transforms a given volume of the material into a new,
0.0 0
20
40
60
80
Dose (dpa) Fig. 4. Damage accumulation kinetics for: (a) spinel crystals irradiated with 450 MeV Xe ions (single-step damage accumulation), (b) SiC irradiated with 160 keV Cs ions (two-step damage accumulation) and (c) ZrO2 irradiated with 150 keV Cs ions (three-step damage accumulation). Solid lines are fits to experimental data according to Eq. (6).
stable configuration. Most likely, the displacement cascades created by incoming ions collapse to form a new atomic configuration; the volume transformed by a single ion impact should thus be comparable to the volume of the core of a displacement cascade. The size of the cascade core is directly related to the cross-section for damage production and can be calculated from fit to the experimental results. For instance in spinel crystal irradiated with swift, heavy ions (Fig. 4a), the effective cross-section was calculated from the fit to damage accumulation kinetics
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J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
and was compared to track diameter measured by using transmission electron microscopy technique revealing a very good agreement [20]. Finally, the same description can be applied to phase transformations in ion-implanted materials. We tested successfully this formula for boronor nitrogen-implanted iron. Consequently, the concept based on single-impact description of phase transformations triggered by the destabilization of the host structure can likely be applied not only for damage accumulation processes but, more widely, to the description of phase transformations in implanted or irradiated solids.
samples (as those used in RBS/C experiments) and thin foils used in TEM experiments. Acknowledgments Financial support from French–Polish collaboration program no. 6391 (research contract IN2P3/1/2006) and grant from Polish Ministry of Science and Higher Education (contract 44/IN2/2006/03) are greatly acknowledged. References
4. Summary A new concept of the description of damage accumulation kinetics based on subsequent destabilization of crystalline structures of host material has been presented. This model allowed us to reproduce the accumulation of radiation damage in several cases characterized by one, two or three steps in the damage accumulation process. In contrast with most of theories previously described in the literature, our model uses the simple approach of directimpact mechanism. The novelty relies in the assumption that the damage accumulation process is composed of a series of transformations. In irradiated spinel crystals, a first step can be interpreted as the accumulation of point defects in the crystalline structure. The formation of free volume leads to an increase of the stress, the high stress level triggering the structural transformation (randomizing of the cationic sublattice or formation of a dislocation network) that is accompanied by a partial stress release. The mechanism triggering the third step in the damage accumulation process has not yet been identified. It seems, however, very likely that it can also be attributed to increasing stress level as in thin samples susceptible to bending this last step has not been observed. The partial stress release in thin samples may also explain the large discrepancies observed in damage accumulation in bulk
[1] Feldman LC, Mayer JW, Picraux ST, editors. Materials analysis by ion channeling. New York: Academic Press; 1982. [2] Gibbons JF. IEEE 1972;60:1062. [3] Dennis JR, Hale EB. J Appl Phys 1978;49:1119. [4] Webb RP, Carter G. Radiat Eff 1981;59:69. [5] Hecking N, Heidemann KF, te Kaat E. Nucl Instrum Methods B 1986;15:760. [6] Campisano SU, Coffa S, Rainieri V, Priolo F, Rimini E. Nucl Instrum Methods B 1993;80/81:514. [7] Bolse W. Nucl Instrum Methods 1998;B141:133. [8] Weber WJ. Nucl Instrum Methods B 2000;166–167:98. [9] Thome´ L, Jagielski J, Binet C, Garrido F. Nucl Instrum Methods B 2000;166–167:258. [10] Thome´ L, Fradin J, Jagielski J, Gentils A, Enescu SE, Garrido F. Eur Phys J Appl Phys 2003;24:37. [11] Biersack JP, Haggmark LG. Nucl Instrum Methods 1980;174:257. [12] Dearnaley G, Freeman JH, Nelson RS, Stephen J. Ion implantation. Amsterdam: North-Holland; 1973. [13] Nolfi FV, editor. Phase transformations during irradiation. New York: Applied Science Publishers; 1983. [14] Costantini JM, Beuneu F, Gourier D, Trautmann C, Calas G, Toulemonde M. J Phys: Condens Matter 2004;16:3657. [15] Fukushima Y, Yano T, Maruyama T, Iseki T. J Nucl Mater 1990;175(3):203. [16] Vook FL, Stein HJ. Radiat Eff 1969;2:23. [17] Weber WJ. J Nucl Mater 1981;98:206. [18] Stoney GG. Proc R Soc Lond 1909;82:172. [19] Devanathan R, Yu N, Sickafus KE, Nastasi M. J Nucl Mater 1996;232:59. [20] Thome´ L, Jagielski J, Gentils A, Nowicki L, Garrido F. Nucl Instrum Methods 2006;242:643.
Vacuum 81 (2007) 1352–1356 www.elsevier.com/locate/vacuum
Damage accumulation in ion-irradiated ceramics Jacek Jagielskia,b,, Lionel Thome´c a
Institute of Electronic Materials Technology, Wolczynska 133, PL-01-919 Warszawa, Poland b The Andrzej Soltan Institute for Nuclear Studies, PL-05-400 Swierk/Otwock, Poland c Centre de Spectrome´trie Nucle´aire et de Spectrome´trie de Masse CNRS-IN2P3, Universite´ Paris-Sud, F-91405 Orsay Campus, France
Abstract The radiation damage induced by energetic ions is one of the basic problems related to ion implantation technology. Each incoming ion creates a huge number of defects, so that high irradiation fluences eventually lead to complete amorphization of the material. Despite the obvious importance of this process, theoretical descriptions of the kinetics of radiation damage accumulation are still incomplete and limited to single-step processes only. In the present paper, we propose a new approach for the description of the radiation damage build-up in crystals, which is based on the concept of subsequent destabilization of the crystalline phases. We compare the presented model to experimental results of damage accumulation kinetics measured for SiC, ZrO2 and MgAl2O4, i.e. cases when one, two or three stages of defects accumulation were observed. r 2007 Elsevier Ltd. All rights reserved. keywords: Amorphization; Crystals; Models; Irradiation
1. Introduction The radiation damage caused by the slowing-down of energetic ions in solids is a particularly important characteristic of ion implantation. It is thus not surprising that this topic was so intensively studied from the very beginning of this technique. Various aspects related to this question were studied: the type of defects formed, their influence on the phase structure of irradiated solids, the role of the process temperature, of the mass and energy of projectiles, etc. Among them, one of the most interesting subjects is the investigation of the kinetics of damage formation, i.e. the dependence of the amount of accumulated damage (fD) versus the ion fluence (F). A measure of fD is often provided using the Rutherford backscattering/ channeling (RBS/C) technique [1], a method able to quantitatively assess the level of damage in irradiated Corresponding author. Institute of Electronic Materials Technology, Wolczynska 133, PL-01-919 Warszawa, Poland. Tel.: +48 502095524; fax: +48 22 864 54 96. E-mail address: [email protected] (J. Jagielski).
0042-207X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2007.01.059
crystals. A large variety of dependencies (going from an exponential rise to a sigmoidal variation) were reported. The challenging and still not fully resolved task is then to reproduce the variations of fD with an appropriate theoretical description. To that purpose, several models were developed in the past, such as the direct impact and the cascade overlap both proposed by Gibbons [2], the direct-impact/cascade overlap [3,4], the direct-impact/ defect stimulated [5] and the nucleation and growth [6,7]. All the above models are based on phenomenological descriptions. Typical problems with their use are: inability to reproduce multi-step damage accumulation kinetics and, in the case of cascade overlap model, necessity to consider a large number of ion impacts to reproduce the observed disorder. The aim of the present paper is to present a new insight into the problem of the disordering kinetics in ionirradiated crystals. A model based on multiple steps of simple accumulation of damage is proposed. This model provides a good agreement with the experimental data in cases when one, two or three steps were observed in damage accumulation kinetics. We will present such a
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
description, discuss the results obtained and, finally, propose some physical mechanisms which may explain observed dependencies. 2. Single-step models of damage accumulation Various phenomenological models were developed in order to reproduce the different types of damage accumulation behaviors observed in ion-irradiated materials [8]. Two main approaches used were: direct impact (DI) and damage accumulation (DA) (both schematically presented in Fig. 1). The DI description assumes that each ion impact creates a given amount of disorder in the target. The damage accumulation process is thus controlled by the probability that an incoming ion will hit an unperturbed volume of the crystal. This type of processes is described by an exponential-rise-to-maximum dependency. The damage measured at the end of irradiation, i.e. generally at saturation, is then the result of the recovery of the various disordered volumes created by incident ions. Thus, the degree of damage increases mostly linearly at the beginning of irradiation and tends to saturate when the major part of the target is already disordered. On the contrary, in the DA description, it is assumed that the measured disorder at a given ion fluence results from the accumulation of many ion impacts in the same volume. The target is thus mostly unaffected by irradiation up to an ion fluence for which a percolation of the various disordered regions leads to a sharp increase of the accumulated damage parameter. A rather general description of the damage build-up was provided by Gibbons [2]. In this description, the disordering kinetics follows the general expression: " # n1 X ðsFÞk f D ¼ f D ð1Þ 1 expðsFÞ , (1) k! k¼0
disordering cross-section and n is the number of ion impacts needed to damage the material. Note that n ¼ 1 in Eq. (1) is characteristic of the DI process when each ion creates a damaged volume in the crystal structure (curve DI in Fig. 1). For n41, i.e. when multiple impacts into the same volume of the crystal are needed to create damage, the Gibbons’ formula describes the cascade overlap model (curve CO in Fig. 1). A combination of DI and cascade overlap descriptions was proposed by Dennis and Hale and later by Webb [3,4]. In the DI/cascade overlap model, the damage accumulation is described by the formula f D ¼ f D ð1Þ½1 ð1 þ st F sa FÞ expðst FÞ,
(2)
where st is the total damage cross-section for disordering and amorphization. In order to reproduce the sigmoidal dependencies of damage accumulation process in Si, SiGe and SiC, a DI/ defect stimulated model [5] (DI/DS in Fig. 1) was proposed, where fD is given by 1 ðs þ ss Þ f D ¼ f D ð1Þ , (3) s þ sa exp½ðs þ ss ÞF
0.6
where ss is the cross-section for stimulated amorphization. All the above models yield sigmoidal or exponential-riseto-maximum shaped dependencies of accumulated damage versus the irradiation fluence and suffer from similar weaknesses. First of all, none of these models take into account cases when more than one step is observed in the damage accumulation kinetics. Such a situation was observed in, e.g., irradiated MgAl2O4 or ZrO2 crystals [9,10]. Moreover, at low irradiation fluences of low-energy (up to several hundreds of keV) ions, one frequently observes the saturation of radiation damage at low (fDo0.1) level [9,10]. In consequence at low fluences, one can expect negative values of dfD/dF (a sublinear dependency), whereas all models yield positive values (with the only exception for the direct-impact model, in which no saturation at low fluences is exhibited). Finally, when a sharp increase of fD is observed at relatively high irradiation fluences, one needs to artificially increase the number of overlapping cascades (n in Gibbons’ formula) up to unrealistic values reaching a few tens in order to reproduce damage accumulation kinetics. All these considerations can be summarized by a strong need to propose a new, alternative description of the damage accumulation in ion-irradiated crystals.
0.4
3. Multi-step damage accumulation
0.2
The main weakness of the models described in the previous part of the paper is related to their inability to reproduce damage accumulation kinetics composed of numerous steps. A typical example of such a dependency is the kinetics measured by RBS/C for magnesium aluminate spinel crystals irradiated with 300 keV Cs or Xe ions (Fig. 2) [9]. Note that in this figure, the fluence axis
where fD is the amount of disorder in the irradiated layer, fD(N) is the value of fD measured at saturation (fD ¼ 1 in the case where irradiation leads to amorphization), s is the Direct Impact (DI) Cascade Overlap (CO) (n=2) Direct Impact/Defect Stimulated DI/DS Defect Accumulation (DA)
1.2 Accumulated damage
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1.0 0.8
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Normalized fluence Fig. 1. Schematic representation of the various models of damage accumulation.
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
0.8
0.6 Cs irradiation Xe irradiation stage 1 stage 2 stage 3 fit
0.4
0.2
0.0 0
2
4
6 8 Dose (dpa)
10
12
Fig. 2. Damage accumulation kinetics of spinel crystal irradiated with 300 keV Cs2+ ions (open circles) and 300 keV Xe2+ ions (open triangles). The solid line represents a fit according to Eq. (6). The dashed line represents stage 1, the dashed-dotted line fits stage 2 and the dashed double dotted holds for stage 3 of the radiation damage accumulation process.
was recalculated to a dose scale (dpa, displacement per atom) using conversion factors calculated by using the TRIM code [11]. The damage accumulation kinetics shown in Fig. 2 exhibits three distinctive steps, which obviously cannot be reproduced by using the single-step models described in the previous section. A new model was thus elaborated to account for most of the damage accumulation kinetics reported in the literature. Let us analyze step by step the processes occurring in irradiated crystals. At low irradiation fluences, the defects, mostly simple Frenkel pairs [12], are formed in an unperturbed crystalline matrix. Taking into account the very high mobility of self-interstitial atoms (SIA) at room temperature [13], it is likely that SIAs migrate to defect sinks (essentially the crystal surface) leaving vacancies within the damaged layer. The first step of the damage accumulation process can thus be regarded as a vacancy concentration build-up. Such a description is consistent with the results of EPR, UV [14], positron annihilation [10] and swelling [15] measurements. When the vacancy concentration increases, the probability of recombination of SIAs with vacancies increases as well. In consequence, one observes a saturation of the concentration of defects at a level determined by the effective recombination volume. At low fluences, the description of damage accumulation can thus be made in the frames of a simple defect accumulation model [16,17] fd ¼
f sat d;1 ½1 f sat d;1
expðs1 FÞ,
(4)
is the saturation value in the first step of the where damage accumulation process. Taking into account that the effective recombination volume for simple Frenkel pairs is rather large (about 1 nm3), one can expect that the value of f sat d;1 should be rather small.
The accumulation of vacancies leads to the formation of free volume within the crystal structure, hence to the generation of a high compressive stress. Since the crystal structure cannot indefinitively support continuously increasing stresses, at a given irradiation fluence, a phase transformation must thus occur to relax the stress. In other words, the stress increases the free energy of the crystalline phase which becomes higher than that of other possible atomic configurations. When the energy of the original crystalline phase is no longer the lowest one, this structure is destabilized and each ion impact leads to its transformation into a configuration which has a lowest free energy. The transformation process is controlled by the probability that a given ion hits an unperturbed part of the material, so that the corresponding kinetics is described by Eq. (5) (exponential rise to maximum), with the only difference for the existence of an irradiation fluence threshold F1. This threshold corresponds to the irradiation fluence at which the destabilization of the original crystalline phase takes place. Thus, the second step can be reproduced with the equation sat sat f d;2 ¼ f sat d;1 þ ðf d;2 f d;1 Þ½1 expðs2 ðF F1 ÞÞ.
(5)
The crucial point of this hypothesis relies in the assumption that the stress caused by the formation of free volume in an irradiated sample triggers the phase transformation. This hypothesis was checked by experiments performed on thin (150 mm thick) spinel crystals where the damage accumulation kinetics was determined by RBS/C and the evolution of the stress in the irradiated layer was measured with the Stoney method [18]. The results are shown in Fig. 3, which shows fast increase of the stress for low irradiation fluences followed by a sharp decrease when the dose exceeds 2 dpa. At the same dose of 2 dpa, an abrupt rise of the level of accumulated damage is noted. Thus, it is clear that the second step in the accumulation of radiation damage in irradiated spinel crystals is caused by the stress increase. The solid line joining experimentally measured fD values is a fit made by using a combination of Eqs. (4) and (5). One can note that, 0.8
3.5 3.0
0.6
2.5 2.0
0.4
1.5 1.0
0.2
0.5
fD fit Stress
0.0
Stress (GPa)
Accumulated damage
1.0
Accumulated damage
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0.0 -0.5
0
10
20 Dose (dpa)
30
40
Fig. 3. Damage accumulation kinetics and stress in the surface layer measured for thin spinel crystals irradiated with 300 keV Ar2+ ions.
ARTICLE IN PRESS J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
f d;1 ¼ f sat d;1 ½1 expðs1 FÞ
for 0oFoF1 ;
sat sat f d;2 ¼ f sat d;1 þ ðf d;2 f d;1 Þ½1 expðs2 ðF F1 ÞÞ
for F1 oFoF2 ;
sat sat f d;i ¼ f sat d;i1 þ ðf d;i f d;i1 Þ½1 expðsi ðF Fi1 ÞÞ
for Fi1 oF:
fd ¼
a
0.8 0.6 0.4 0.2 0.0 0
1.0
2
4 6 Fluence (x1013 cm-2)
8
10
0.4
0.5
b
0.8 0.6 0.4 0.2 0.0 0.0
0.1
0.2
0.3 Dose (dpa)
c
1.0 0.8 0.6
The final formula including necessary normalization factors is thus n X
1.0
Accumulated damage
in thin samples, the crystalline structure persists to much higher irradiation fluences and the third step in damage accumulation is not observed. This result is likely due to a partial relaxation of the stress due to sample bending. A similar approach has been used to reproduce the results obtained in thick samples where three steps were observed (Fig. 2). A very good agreement between experimental results and calculations points to the conclusion that the description based on several steps of simple damage accumulation provides a much better agreement with experimental data than the models described in Section 2. In conclusion, the new approach is based on a sequence of several damage accumulation steps, each stage being described by a simple accumulation of disorder. A new stage occurs when the current structure of the matrix is no longer characterized by the lowest free energy; transformations are thus triggered by the destabilization of a given structure. This destabilization may be due to various mechanisms: stress accumulation, formation of defect clusters, increase of the concentration of impurity atoms acting as damage stabilizers, etc. In irradiated spinel crystals, the possible sequence of structural transformations is likely the following: crystalline structure-crystalline structure with point defects-crystalline structure with randomized cationic network [19] or dislocation network-amorphous structure. The equations describing the sequence of damage accumulation can be written as follows:
1355
0.4 0.2
ðf sat d;i
f sat d;i1 ÞG½1
expðsi ðF Fi1 ÞÞ,
(6)
i¼1
f sat d;i
where is the level of damage saturation in the ith step. G is a function which transforms negative values into 0 and leaves the positive values unchanged. The proposed model offers numerous advantages with respect to the previous ones. First of all, this description is able to reproduce multi-step damage accumulations, as the ones occurring in spinel or zirconia crystals. Secondly, the same formalism can be applied to damage accumulation processes composed of one, two or more stages. Some examples are shown in Fig. 4, which presents the damage accumulation kinetics for spinel crystals irradiated with swift, heavy ions (Fig. 4a, single-step amorphization kinetics), SiC (Fig. 4b—two steps) and ZrO2 (Fig. 4c— three steps), the latter two materials being irradiated with low energy (hundreds of keV) ions. The interpretation in terms of microstructural evolution is straightforward: when the crystalline structure is destabilized, each ion impact transforms a given volume of the material into a new,
0.0 0
20
40
60
80
Dose (dpa) Fig. 4. Damage accumulation kinetics for: (a) spinel crystals irradiated with 450 MeV Xe ions (single-step damage accumulation), (b) SiC irradiated with 160 keV Cs ions (two-step damage accumulation) and (c) ZrO2 irradiated with 150 keV Cs ions (three-step damage accumulation). Solid lines are fits to experimental data according to Eq. (6).
stable configuration. Most likely, the displacement cascades created by incoming ions collapse to form a new atomic configuration; the volume transformed by a single ion impact should thus be comparable to the volume of the core of a displacement cascade. The size of the cascade core is directly related to the cross-section for damage production and can be calculated from fit to the experimental results. For instance in spinel crystal irradiated with swift, heavy ions (Fig. 4a), the effective cross-section was calculated from the fit to damage accumulation kinetics
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J. Jagielski, L. Thome´ / Vacuum 81 (2007) 1352–1356
and was compared to track diameter measured by using transmission electron microscopy technique revealing a very good agreement [20]. Finally, the same description can be applied to phase transformations in ion-implanted materials. We tested successfully this formula for boronor nitrogen-implanted iron. Consequently, the concept based on single-impact description of phase transformations triggered by the destabilization of the host structure can likely be applied not only for damage accumulation processes but, more widely, to the description of phase transformations in implanted or irradiated solids.
samples (as those used in RBS/C experiments) and thin foils used in TEM experiments. Acknowledgments Financial support from French–Polish collaboration program no. 6391 (research contract IN2P3/1/2006) and grant from Polish Ministry of Science and Higher Education (contract 44/IN2/2006/03) are greatly acknowledged. References
4. Summary A new concept of the description of damage accumulation kinetics based on subsequent destabilization of crystalline structures of host material has been presented. This model allowed us to reproduce the accumulation of radiation damage in several cases characterized by one, two or three steps in the damage accumulation process. In contrast with most of theories previously described in the literature, our model uses the simple approach of directimpact mechanism. The novelty relies in the assumption that the damage accumulation process is composed of a series of transformations. In irradiated spinel crystals, a first step can be interpreted as the accumulation of point defects in the crystalline structure. The formation of free volume leads to an increase of the stress, the high stress level triggering the structural transformation (randomizing of the cationic sublattice or formation of a dislocation network) that is accompanied by a partial stress release. The mechanism triggering the third step in the damage accumulation process has not yet been identified. It seems, however, very likely that it can also be attributed to increasing stress level as in thin samples susceptible to bending this last step has not been observed. The partial stress release in thin samples may also explain the large discrepancies observed in damage accumulation in bulk
[1] Feldman LC, Mayer JW, Picraux ST, editors. Materials analysis by ion channeling. New York: Academic Press; 1982. [2] Gibbons JF. IEEE 1972;60:1062. [3] Dennis JR, Hale EB. J Appl Phys 1978;49:1119. [4] Webb RP, Carter G. Radiat Eff 1981;59:69. [5] Hecking N, Heidemann KF, te Kaat E. Nucl Instrum Methods B 1986;15:760. [6] Campisano SU, Coffa S, Rainieri V, Priolo F, Rimini E. Nucl Instrum Methods B 1993;80/81:514. [7] Bolse W. Nucl Instrum Methods 1998;B141:133. [8] Weber WJ. Nucl Instrum Methods B 2000;166–167:98. [9] Thome´ L, Jagielski J, Binet C, Garrido F. Nucl Instrum Methods B 2000;166–167:258. [10] Thome´ L, Fradin J, Jagielski J, Gentils A, Enescu SE, Garrido F. Eur Phys J Appl Phys 2003;24:37. [11] Biersack JP, Haggmark LG. Nucl Instrum Methods 1980;174:257. [12] Dearnaley G, Freeman JH, Nelson RS, Stephen J. Ion implantation. Amsterdam: North-Holland; 1973. [13] Nolfi FV, editor. Phase transformations during irradiation. New York: Applied Science Publishers; 1983. [14] Costantini JM, Beuneu F, Gourier D, Trautmann C, Calas G, Toulemonde M. J Phys: Condens Matter 2004;16:3657. [15] Fukushima Y, Yano T, Maruyama T, Iseki T. J Nucl Mater 1990;175(3):203. [16] Vook FL, Stein HJ. Radiat Eff 1969;2:23. [17] Weber WJ. J Nucl Mater 1981;98:206. [18] Stoney GG. Proc R Soc Lond 1909;82:172. [19] Devanathan R, Yu N, Sickafus KE, Nastasi M. J Nucl Mater 1996;232:59. [20] Thome´ L, Jagielski J, Gentils A, Nowicki L, Garrido F. Nucl Instrum Methods 2006;242:643.