- Email: [email protected]
Theoretical modeling of swift-ion-beam amorphization: Application to LiNbO3
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 249 (2006) 118–121 www.elsevier.com/locate/nimb
Theoretical modeling of swift-ion-beam amorphization: Application to LiNbO3 F. Agullo´-Lo´pez b
a,b,*
, G. Garcı´a b, J. Olivares
b,c
a Departamento de Fı´sica de Materiales, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain Centro de Microana´lisis de Materiales (CMAM), Universidad Auto´noma de Madrid, Campus Cantoblanco, 28049 Madrid, Spain c ´ ptica ‘‘Daza de Valde´s’’, CSIC, Serrano 121, 28006 Madrid, Spain Instituto de O
Available online 12 May 2006
Abstract A general theoretical framework to describe crystal damage and amorphization under swift-ion-beam irradiation is discussed. Main physical channels accounting for electron energy losses are considered: (a) phonon generation (heating) and (b) exciton localization and deexcitation. Two alternative schemes are analyzed. In one of them, intrinsic defects are generated through a direct thermal process, i.e. through channel (a). Above a certain threshold rate of electronic energy deposition the temperature reaches the melting point and the crystal becomes amorphous. In the other scheme, defects are generated through non-radiative de-excitation of excitons, formed during the final stage of electron slowing down. This may require to overcome a certain potential energy barrier. In this case the necessary thermal energy is provided by coupling to channel (a). The two schemes are discussed in the light of available experimental information. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.80.x; 61.80.Az; 61.80.Jh Keywords: Swift heavy ion irradiation; Ion amorphization; LiNbO3
1. Introduction Single swift ions incident on a crystalline material can create latent (amorphous) tracks along their trajectory [1,2]. The generation of those tracks is a consequence of a strong electronic excitation followed by efficient energy transfer to the ionic lattice. A definite electronic stopping power threshold is required for such a process. At fluences where overlapping of individual tracks occurs a homogeneous amorphous layer is generated. This has been clearly inferred from recent experiments on silicon [3,4], oxygen [5,6] and fluorine [7] irradiations. Moreover, the boundary of the amorphous layer moves on increasing fluence [3,4,7]. * Corresponding author. Address: Departamento de Fı´sica de Materiales, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain. Tel.: +34 91 4973635; fax: +34 91 4973623. E-mail address: [email protected] (F. Agullo´-Lo´pez).
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.03.094
This effect has been interpreted as a reduction of threshold with fluence as a consequence of pre-amorphization damage. An analytical model has been recently proposed [8] to account for this behavior in silicon irradiated LiNbO3. It is based on thermally induced generation of defects and subsequent rapid quenching. The theoretical analysis of the lattice damage, and eventually amorphization, caused by heavy electronic excitation is a formidable task. The situation appears not only in ionbeam irradiation but also in strong electronic and laser excitation. In this paper, we present a rather general framework for modeling the damage and amorphization induced by swift-ion irradiation. It combines concepts of the exciton [9–11] and thermal spike [12,13] models and it includes our previous approach [8] as a particular case. The proposed framework is described in a schematic way (at a semiquantitative level) and is intented to emphasize the key features of any detailed microscopic model.
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
2. Main phenomenological features Experiments [3–7] suggest that any adequate theoretical framework should take into account the following features: (a) Electronic excitation causes permanent latent damage that is cumulative. Whenever the electronic stopping power Se overcomes a threshold value Sth, each ion generates a latent (amorphous) track along its trajectory. For irradiation fluences where individual latent tracks overlap an homogeneous amorphous layer is produced. (b) The threshold for track formation depends on the damage caused by prior fluence / so that the thickness of the amorphous layer increases with /. (c) The dependence of the induced defect concentration on Se is strongly nonlinear (superlinear). (d) There are indications that the nuclear collision damage may influence the amorphization process.
3. General theoretical framework In the electronic losses regime (i.e. ignoring nuclear collisions), the energy deposited by the bombarding ion, proportional to Se, is initially converted into kinetic energy of the excited electronic system, Fig. 1. Subsequently, this energy is transferred to the ionic lattice and eventually leads to creation of defects. For a given material one expects that the defect concentration c should be a function of Se and /. When a critical concentration, c = cm, is reached, the lattice collapses into an amorphous phase. In order to determine the function c(Se, /) one should take
BOMBARDING IONS
FORMATION OF EXCITED ELECTRON CLOUD
ELECTRON THERMALIZATION (a)
PHONON CREATION
(b)
into account the main processes, Fig. 1, during the slowing down of electronic motion: (a) energy transfer to the crystal lattice (phonon creation), (b) localization of the excitation energy as trapped excitons and finally, (c) recombination of these excitons. This latter step leads to either light emission (radiative recombination), phonon creation (heating) or atomic displacement. The two latter processes arise from non-radiative recombination. Within this general scheme, one may consider two different alternative mechanisms to account for defect formation. 4. Thermally induced generation of defects Disorder is generated through channel (a), i.e. by thermal heating and subsequent quenching (Fig. 1(a)). This corresponds to a thermal spike approach as in our previous model [8]. The key information is the temperature profile in the spike T (t, r, Se) that can be considered gaussian T ðt; r; S e Þ ¼ T 0 er
(HEATING)
(LOCALIZATION)
COOLING
EXCITATION DECAY
PHONONS/DEFECTS
Fig. 1. Flow diagram showing the processes following an ion impact event.
þ T S.
cðr; S e Þ ¼ A expfe=k Tb ðr; S e Þg.
ð1Þ
ð2Þ
e being the thermal formation energy of the defects and A a constant. Using Eqs. (1) and (2) one obtains [8] the total number of defects per unit length Z 1 gðS e Þ ¼ 2p rcf Tb ðr; S e Þg dr ð3Þ 0
generated (at a given depth) by a single ion impact with stopping power Se. In order to avoid the divergence of the integral the defect concentration (at RT) should be substracted from c. Then, the differential equation governing the irradiation-induced growth in the average defect concentration c at depth z is ð4Þ
/ being the impact density (fluence). Once the solution c(Se, /) has been obtained, one can readily determine the conditions for the generation of an amorphous layer. Taking into account the dependence Se(z) obtained from a SRIM program [14], one may calculate the thickness of the amorphous layer as a function of fluence through the equation cðS e ðzÞ; /Þ ¼ cm .
LIGHT
2 =2a2 ðtÞ
a being the width of the distribution and TS the crystal temperature. The variable t stands for the time elapsed (during cooling) from the maximum temperature T0 the spike axis. Local defect concentration is assumed to correspond to thermal equilibrium at the maximum temperature Tb ðr; S e Þ reached at a distance r from the track axis, that can be readily calculated [8,12]. The strong nonlinearity in the dependence c(Se) is, then, provided by the Arrhenius law that yields the local defect concentration c(r, Se),
c ¼ gfS e ðzÞg/.
EXCITON FORMATION
119
ð5Þ
The model has been described in detail in a previous work [8]. It is in reasonable accordance with experimental data on Si-irradiated (7.5 MeV) LiNbO3 at fluences 61014 cm2, as
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
120
by the heating channel. In order to proceed with the analysis one may assume that the total number of excitons formed by one single impact at a depth z is Nexc(z) = Se(z)/I, I being an average ionization energy. Moreover, we will consider that excitons are homogeneously trapped at regular lattice sites (homogeneous nucleation) and all of them are equivalent for defect creation. The model could be extended to cover the case where excitons are trapped at certain defective sites, previously existing or created by the irradiation (heterogeneous nucleation). The growth rate of the (volume) density of defects nD(r, z) in the track generated by non-radiative exciton decay writes, Fig. 2. Depth of the crystal–amorphous boundary as a function of fluence by the thermal model (continuous line) and the exciton model (dashed line). Parameters used in both models are: kTm/e = 0.2, Sth = 5.2 keV/nm, a0 = 5.5 nm (m0 = 1013 s1). Experimental data points are included for comparison.
illustrated in Fig. 2. At higher fluences the agreement with experiment deteriorates and one should, possibly, include in the right side of Eq. (2) an additional term g(SN) / SN associated to nuclear collisions (stopping power SN). 5. Non-radiative exciton decay mechanism In this alternative scheme we assume that defects are generated by non-radiative de-excitation of localized excitons (Fig. 1(b)). This type of process is well known in alkali halides as the mechanism responsible for generation of lattice defects under ionizing radiation (X-rays and UV light) [15] and it has been also proposed [9–11] to describe swiftion damage. To understand the process one may invoke an adiabatic level scheme as illustrated in Fig. 3. The excited state may lead to molecular dissociation and atomic displacement, either directly, or after overcoming a certain potential energy barrier. This latter case provides a superlinear generation rate g(Se), due to coupling between the two energy-loss channels (a) and (b). In other words, the temperature needed to overcome the barrier is provided
dnD ðr; zÞ dnexc ðr; zÞ ¼ ¼ m0 nexc expfe=kT g; dt dt
ð6Þ
where nexc is the corresponding exciton density and m0 a frequency factor. The integration of (6) only becomes tractable under several rough approximations. We assume that the thermal activation factor is determined by the maximum local temperature Tb ðr; S e Þ reached under the excitation spike (as in our previous model [8]). Moreover, we consider that during track cooling, each point remains at that maximum temperature during the same time s (regardless of position) to allow for the non-radiative relaxation of the excitons. After that period the exciton relaxation proceeds via light emission. This assumption may be qualitatively justified by the simulations of the time evolution of the thermal spike [13]. Finally, it is reasonable to consider that the exciton distribution follows the spike profile at the beginning of the cooling stage. Then, one can go a step forward and assume for simplicity a constant radial exciton density nexc ðr; zÞ ¼
N exc ðzÞ
for r < a0 ; 2 pað0Þ nexc ðr; zÞ ¼ 0 for r > a0
ð7Þ
in the track. Then, the total number of defects generated by a single ion impact and unit length is Z að0Þ ^ nD ¼ 2prS e Aexc ee=kT ðr;Se Þ dr ¼ S e gexc ðS e Þ; ð8Þ 0
E
R að0Þ ^ m0 s where Aexc ¼ pað0Þ 2prAexc ee=kT dr is the 2 and g exc ðS e Þ ¼ 0 I generation-rate function appearing in the thermal spike model after substituting Aexc for A. Finally, the equation describing the evolution of the homogeneous (averaged) concentration of defects cðzÞ with fluence / (density of impacts) can be written as
ε
Frenkel pairs
cðzÞ ¼ S e gexc ðS e Þ/;
ð9Þ
Light
Q Fig. 3. Proposed energy level diagram for the localized exciton.
which is very similar to Eq. (2) as used in previous (8), except for the factor Se. It is concluded that, under the assumed strong approximations, the exciton-decay and the thermal spike models, although physically very different, can be mathematically
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
described within a rather similar formal framework. Using the same physical parameters (and m0 = 1013 s1) the predictions of the exciton model for the depth evolution of the amorphous–crystalline boundary with fluence is displayed in Fig. 2, together with the prediction of the thermal model and experimental data. The two curves show a quite similar behavior. A more rigorous analysis of the exciton model is now underway. References [1] R. Spohr, in: K. Bethge (Ed.), Ion Tracks and Microtechnology: Basic Principles and Applications, Vieweg, Braunchsweig, 1990. [2] M. Toulemonde, S. Bouffard, F. Studer, Nucl. Instr. and Meth. B 91 (1994) 108. [3] J. Olivares, G. Garcı´a, F. Agullo´-Lo´pez, F. Agullo´-Rueda, A. Kling, J.C. Soares, Nucl. Instr. and Meth. B 242 (2006) 534. [4] J. Olivares, G. Garcı´a, F. Agullo´-Lo´pez, F. Agullo´-Rueda, A. Kling, J.C. Soares, Appl. Phys. A 81 (2005) 1465.
121
[5] G.G. Bentini, M. Bianconi, M. Chiarini, L. Correra, C. Sada, P. Mazzoldi, N. Argiolas, M. Bazzan, R. Guzzi, J. Appl. Phys. 92 (2002) 6477. [6] G.G. Bentini, M. Bianconi, L. Correra, M. Chiarini, P. Mazzoldi, C. Sada, N. Argiolas, M. Bazzan, R. Guzzi, J. Appl. Phys. 96 (2004) 242. [7] J. Olivares, G. Garcı´a, A. Garcı´a-Navarro, F. Agullo´-Lo´pez, O. Cabllero, A. Garcı´a-Caban˜es, Appl. Phys. Lett. 86 (2005) 183501. [8] F. Agullo´-Lo´pez, G. Garcı´a, J. Olivares, J. Appl. Phys. 97 (2005) 093514. [9] Yu.M. Annenkov, A.M. Pritulov, Sov. Phys. Solid State 23 (1981) 616. [10] N. Itoh, A.M. Stoneham, Nucl. Instr. and Meth. B 146 (1998) 362. [11] N. Itoh, Nucl. Instr. and Meth. B 135 (1998) 175. [12] G. Szenes, Phys. Rev. B 51 (1995) 8026. [13] M. Toulemonde, Ch. Dufour, A. Meftah, E. Paumier, Nucl. Instr. and Meth. B 166–167 (2000) 903. [14] J.F. Ziegler, J.P. Biersack, U. Littmark (Eds.), The Stopping and Ranges of Ions in Solids, Pergamon, New York, 1985. Available from:. [15] F. Agullo´-Lo´pez, R.C. Catlow, P.D. Townsend, Point Defects in Materials, Academic Press, London, 1984.
Nuclear Instruments and Methods in Physics Research B 249 (2006) 118–121 www.elsevier.com/locate/nimb
Theoretical modeling of swift-ion-beam amorphization: Application to LiNbO3 F. Agullo´-Lo´pez b
a,b,*
, G. Garcı´a b, J. Olivares
b,c
a Departamento de Fı´sica de Materiales, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain Centro de Microana´lisis de Materiales (CMAM), Universidad Auto´noma de Madrid, Campus Cantoblanco, 28049 Madrid, Spain c ´ ptica ‘‘Daza de Valde´s’’, CSIC, Serrano 121, 28006 Madrid, Spain Instituto de O
Available online 12 May 2006
Abstract A general theoretical framework to describe crystal damage and amorphization under swift-ion-beam irradiation is discussed. Main physical channels accounting for electron energy losses are considered: (a) phonon generation (heating) and (b) exciton localization and deexcitation. Two alternative schemes are analyzed. In one of them, intrinsic defects are generated through a direct thermal process, i.e. through channel (a). Above a certain threshold rate of electronic energy deposition the temperature reaches the melting point and the crystal becomes amorphous. In the other scheme, defects are generated through non-radiative de-excitation of excitons, formed during the final stage of electron slowing down. This may require to overcome a certain potential energy barrier. In this case the necessary thermal energy is provided by coupling to channel (a). The two schemes are discussed in the light of available experimental information. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.80.x; 61.80.Az; 61.80.Jh Keywords: Swift heavy ion irradiation; Ion amorphization; LiNbO3
1. Introduction Single swift ions incident on a crystalline material can create latent (amorphous) tracks along their trajectory [1,2]. The generation of those tracks is a consequence of a strong electronic excitation followed by efficient energy transfer to the ionic lattice. A definite electronic stopping power threshold is required for such a process. At fluences where overlapping of individual tracks occurs a homogeneous amorphous layer is generated. This has been clearly inferred from recent experiments on silicon [3,4], oxygen [5,6] and fluorine [7] irradiations. Moreover, the boundary of the amorphous layer moves on increasing fluence [3,4,7]. * Corresponding author. Address: Departamento de Fı´sica de Materiales, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain. Tel.: +34 91 4973635; fax: +34 91 4973623. E-mail address: [email protected] (F. Agullo´-Lo´pez).
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.03.094
This effect has been interpreted as a reduction of threshold with fluence as a consequence of pre-amorphization damage. An analytical model has been recently proposed [8] to account for this behavior in silicon irradiated LiNbO3. It is based on thermally induced generation of defects and subsequent rapid quenching. The theoretical analysis of the lattice damage, and eventually amorphization, caused by heavy electronic excitation is a formidable task. The situation appears not only in ionbeam irradiation but also in strong electronic and laser excitation. In this paper, we present a rather general framework for modeling the damage and amorphization induced by swift-ion irradiation. It combines concepts of the exciton [9–11] and thermal spike [12,13] models and it includes our previous approach [8] as a particular case. The proposed framework is described in a schematic way (at a semiquantitative level) and is intented to emphasize the key features of any detailed microscopic model.
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
2. Main phenomenological features Experiments [3–7] suggest that any adequate theoretical framework should take into account the following features: (a) Electronic excitation causes permanent latent damage that is cumulative. Whenever the electronic stopping power Se overcomes a threshold value Sth, each ion generates a latent (amorphous) track along its trajectory. For irradiation fluences where individual latent tracks overlap an homogeneous amorphous layer is produced. (b) The threshold for track formation depends on the damage caused by prior fluence / so that the thickness of the amorphous layer increases with /. (c) The dependence of the induced defect concentration on Se is strongly nonlinear (superlinear). (d) There are indications that the nuclear collision damage may influence the amorphization process.
3. General theoretical framework In the electronic losses regime (i.e. ignoring nuclear collisions), the energy deposited by the bombarding ion, proportional to Se, is initially converted into kinetic energy of the excited electronic system, Fig. 1. Subsequently, this energy is transferred to the ionic lattice and eventually leads to creation of defects. For a given material one expects that the defect concentration c should be a function of Se and /. When a critical concentration, c = cm, is reached, the lattice collapses into an amorphous phase. In order to determine the function c(Se, /) one should take
BOMBARDING IONS
FORMATION OF EXCITED ELECTRON CLOUD
ELECTRON THERMALIZATION (a)
PHONON CREATION
(b)
into account the main processes, Fig. 1, during the slowing down of electronic motion: (a) energy transfer to the crystal lattice (phonon creation), (b) localization of the excitation energy as trapped excitons and finally, (c) recombination of these excitons. This latter step leads to either light emission (radiative recombination), phonon creation (heating) or atomic displacement. The two latter processes arise from non-radiative recombination. Within this general scheme, one may consider two different alternative mechanisms to account for defect formation. 4. Thermally induced generation of defects Disorder is generated through channel (a), i.e. by thermal heating and subsequent quenching (Fig. 1(a)). This corresponds to a thermal spike approach as in our previous model [8]. The key information is the temperature profile in the spike T (t, r, Se) that can be considered gaussian T ðt; r; S e Þ ¼ T 0 er
(HEATING)
(LOCALIZATION)
COOLING
EXCITATION DECAY
PHONONS/DEFECTS
Fig. 1. Flow diagram showing the processes following an ion impact event.
þ T S.
cðr; S e Þ ¼ A expfe=k Tb ðr; S e Þg.
ð1Þ
ð2Þ
e being the thermal formation energy of the defects and A a constant. Using Eqs. (1) and (2) one obtains [8] the total number of defects per unit length Z 1 gðS e Þ ¼ 2p rcf Tb ðr; S e Þg dr ð3Þ 0
generated (at a given depth) by a single ion impact with stopping power Se. In order to avoid the divergence of the integral the defect concentration (at RT) should be substracted from c. Then, the differential equation governing the irradiation-induced growth in the average defect concentration c at depth z is ð4Þ
/ being the impact density (fluence). Once the solution c(Se, /) has been obtained, one can readily determine the conditions for the generation of an amorphous layer. Taking into account the dependence Se(z) obtained from a SRIM program [14], one may calculate the thickness of the amorphous layer as a function of fluence through the equation cðS e ðzÞ; /Þ ¼ cm .
LIGHT
2 =2a2 ðtÞ
a being the width of the distribution and TS the crystal temperature. The variable t stands for the time elapsed (during cooling) from the maximum temperature T0 the spike axis. Local defect concentration is assumed to correspond to thermal equilibrium at the maximum temperature Tb ðr; S e Þ reached at a distance r from the track axis, that can be readily calculated [8,12]. The strong nonlinearity in the dependence c(Se) is, then, provided by the Arrhenius law that yields the local defect concentration c(r, Se),
c ¼ gfS e ðzÞg/.
EXCITON FORMATION
119
ð5Þ
The model has been described in detail in a previous work [8]. It is in reasonable accordance with experimental data on Si-irradiated (7.5 MeV) LiNbO3 at fluences 61014 cm2, as
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
120
by the heating channel. In order to proceed with the analysis one may assume that the total number of excitons formed by one single impact at a depth z is Nexc(z) = Se(z)/I, I being an average ionization energy. Moreover, we will consider that excitons are homogeneously trapped at regular lattice sites (homogeneous nucleation) and all of them are equivalent for defect creation. The model could be extended to cover the case where excitons are trapped at certain defective sites, previously existing or created by the irradiation (heterogeneous nucleation). The growth rate of the (volume) density of defects nD(r, z) in the track generated by non-radiative exciton decay writes, Fig. 2. Depth of the crystal–amorphous boundary as a function of fluence by the thermal model (continuous line) and the exciton model (dashed line). Parameters used in both models are: kTm/e = 0.2, Sth = 5.2 keV/nm, a0 = 5.5 nm (m0 = 1013 s1). Experimental data points are included for comparison.
illustrated in Fig. 2. At higher fluences the agreement with experiment deteriorates and one should, possibly, include in the right side of Eq. (2) an additional term g(SN) / SN associated to nuclear collisions (stopping power SN). 5. Non-radiative exciton decay mechanism In this alternative scheme we assume that defects are generated by non-radiative de-excitation of localized excitons (Fig. 1(b)). This type of process is well known in alkali halides as the mechanism responsible for generation of lattice defects under ionizing radiation (X-rays and UV light) [15] and it has been also proposed [9–11] to describe swiftion damage. To understand the process one may invoke an adiabatic level scheme as illustrated in Fig. 3. The excited state may lead to molecular dissociation and atomic displacement, either directly, or after overcoming a certain potential energy barrier. This latter case provides a superlinear generation rate g(Se), due to coupling between the two energy-loss channels (a) and (b). In other words, the temperature needed to overcome the barrier is provided
dnD ðr; zÞ dnexc ðr; zÞ ¼ ¼ m0 nexc expfe=kT g; dt dt
ð6Þ
where nexc is the corresponding exciton density and m0 a frequency factor. The integration of (6) only becomes tractable under several rough approximations. We assume that the thermal activation factor is determined by the maximum local temperature Tb ðr; S e Þ reached under the excitation spike (as in our previous model [8]). Moreover, we consider that during track cooling, each point remains at that maximum temperature during the same time s (regardless of position) to allow for the non-radiative relaxation of the excitons. After that period the exciton relaxation proceeds via light emission. This assumption may be qualitatively justified by the simulations of the time evolution of the thermal spike [13]. Finally, it is reasonable to consider that the exciton distribution follows the spike profile at the beginning of the cooling stage. Then, one can go a step forward and assume for simplicity a constant radial exciton density nexc ðr; zÞ ¼
N exc ðzÞ
for r < a0 ; 2 pað0Þ nexc ðr; zÞ ¼ 0 for r > a0
ð7Þ
in the track. Then, the total number of defects generated by a single ion impact and unit length is Z að0Þ ^ nD ¼ 2prS e Aexc ee=kT ðr;Se Þ dr ¼ S e gexc ðS e Þ; ð8Þ 0
E
R að0Þ ^ m0 s where Aexc ¼ pað0Þ 2prAexc ee=kT dr is the 2 and g exc ðS e Þ ¼ 0 I generation-rate function appearing in the thermal spike model after substituting Aexc for A. Finally, the equation describing the evolution of the homogeneous (averaged) concentration of defects cðzÞ with fluence / (density of impacts) can be written as
ε
Frenkel pairs
cðzÞ ¼ S e gexc ðS e Þ/;
ð9Þ
Light
Q Fig. 3. Proposed energy level diagram for the localized exciton.
which is very similar to Eq. (2) as used in previous (8), except for the factor Se. It is concluded that, under the assumed strong approximations, the exciton-decay and the thermal spike models, although physically very different, can be mathematically
F. Agullo´-Lo´pez et al. / Nucl. Instr. and Meth. in Phys. Res. B 249 (2006) 118–121
described within a rather similar formal framework. Using the same physical parameters (and m0 = 1013 s1) the predictions of the exciton model for the depth evolution of the amorphous–crystalline boundary with fluence is displayed in Fig. 2, together with the prediction of the thermal model and experimental data. The two curves show a quite similar behavior. A more rigorous analysis of the exciton model is now underway. References [1] R. Spohr, in: K. Bethge (Ed.), Ion Tracks and Microtechnology: Basic Principles and Applications, Vieweg, Braunchsweig, 1990. [2] M. Toulemonde, S. Bouffard, F. Studer, Nucl. Instr. and Meth. B 91 (1994) 108. [3] J. Olivares, G. Garcı´a, F. Agullo´-Lo´pez, F. Agullo´-Rueda, A. Kling, J.C. Soares, Nucl. Instr. and Meth. B 242 (2006) 534. [4] J. Olivares, G. Garcı´a, F. Agullo´-Lo´pez, F. Agullo´-Rueda, A. Kling, J.C. Soares, Appl. Phys. A 81 (2005) 1465.
121
[5] G.G. Bentini, M. Bianconi, M. Chiarini, L. Correra, C. Sada, P. Mazzoldi, N. Argiolas, M. Bazzan, R. Guzzi, J. Appl. Phys. 92 (2002) 6477. [6] G.G. Bentini, M. Bianconi, L. Correra, M. Chiarini, P. Mazzoldi, C. Sada, N. Argiolas, M. Bazzan, R. Guzzi, J. Appl. Phys. 96 (2004) 242. [7] J. Olivares, G. Garcı´a, A. Garcı´a-Navarro, F. Agullo´-Lo´pez, O. Cabllero, A. Garcı´a-Caban˜es, Appl. Phys. Lett. 86 (2005) 183501. [8] F. Agullo´-Lo´pez, G. Garcı´a, J. Olivares, J. Appl. Phys. 97 (2005) 093514. [9] Yu.M. Annenkov, A.M. Pritulov, Sov. Phys. Solid State 23 (1981) 616. [10] N. Itoh, A.M. Stoneham, Nucl. Instr. and Meth. B 146 (1998) 362. [11] N. Itoh, Nucl. Instr. and Meth. B 135 (1998) 175. [12] G. Szenes, Phys. Rev. B 51 (1995) 8026. [13] M. Toulemonde, Ch. Dufour, A. Meftah, E. Paumier, Nucl. Instr. and Meth. B 166–167 (2000) 903. [14] J.F. Ziegler, J.P. Biersack, U. Littmark (Eds.), The Stopping and Ranges of Ions in Solids, Pergamon, New York, 1985. Available from: