- Email: [email protected]
Different methods for the determination of damage profiles in Si from RBS-channeling spectra: a comparison
s
Nuclear Instruments and Methods in Physics Research B I 18 ( 1996) I28- 132 __ __
NOM
B
Beam Interactions with Materials A Atoms
@ ELSEVIER
Different methods for the determination of damage profiles in Si from RBS-channeling spectra: a comparison E. Albertazzi
a, M. Bianconi
a**, G. Lulli a, R. Nipoti a, M. Cantiano
b
a CNR-lsfituto LAMEL, Viu Gobetti IOI. I-40129 Bologna, Ituly b Consorzio OPTEL-InP. Cittadella della Ricercu. I-72100 Brindisi. Italy Abstract RBS-channeling spectra of deep ion implants in silicon were analyzed to extract the displaced atoms depth profiles by different methods. In all cases it was assumed that defects in as-implanted samples can be described as atoms randomly displaced from the lattice sites. In the first method, based on the two beam model, the dechanneling induced by defects was calculated either linearly or following a recently developed semi-empirical formula. In the second method the analyzing beam was divided into a greater number of components to follow the transverse energy distribution of the ions. Finally a three-dimensional Monte Carlo code containing a detailed description of each ion path was used to identify the limits of the previous approaches. It is shown that when high amounts of damage are considered all the methods produce essentially the same profiles. On the contrary they considerably disagree in other cases. Moreover, Monte Carlo calculations indicate that to obtain reliable results it is necessary to take into account a correct description of the channeling energy loss process during ion penetration into a disordered crystal.
1. Introduction
energy ion implantation in Si. The basic assumption that the damage produced by ion implantation in semiconductors consists in randomly distributed displaced atoms was used in the above models; this is reasonable in the range of damage concentration here considered. In the case of deep implants, the key topic is the description of the energy loss of the channeled beam. In the Monte Carlo code the use of an impact-parameter dependent stopping power allowed us to follow the energetic history of each ion. In the other models the assignment of a proper stopping value to each beam appeared quite complicated, so we used an average channeling stopping reduction cr dependent on the total amount of damage. As usually, the normalized yield x (the ratio of the aligned spectrum to the random one) as a function of the RBS energy (channel) rather than the actual spectrum was considered in the calculations. Due to the above considerations on the stopping power we avoided using any correction to the experimental normalized yield to account for the different cross section [4] and channel width [5] between channeled and random trajectories.
Many important applications of high energy ion implantation are concerned with the production of insulating or buried doped layers in the technology of silicon electronic devices [l]. The study of these implantation processes offers the possibility to extend the investigation of basic phenomena such as ion penetration and implantation damage in crystalline semiconductors to the MeV energy regime [2]. Moreover, the strong interest in compound materials such as SiGe and Sic for electronic application, requires the transfer of the know-how accumulated on Si. RBS-channeling is one of the most important and widely used techniques for the analysis of damage in as-implanted semiconductors [3]. However, in spite of its consolidated use, many aspects concerning the extraction of a reliable defect depth profile from experimental spectra deserve further investigation. The first question is the choice among the great number of methods, each working in a limited range of experimental conditions, for the evaluation of the dechanneling induced by defects [3]. In this work three different approaches based on the two beam (TB) model, a multi beam (MB) model and a Monte Carlo (MC) simulation were considered and applied to a typical study of high
2. Experimental
Corresponding author. Tel. + 39 51 6399140, fax + 39 51 6399216, e-mail: [email protected].
The experimental data concern a set of (100) Si samples implanted with 800 keV As+ ions. The implantation doses (a) 5 X lOI cm-‘, (b) 1 X 1014 cm-*,(c) 2.5 X lOI cm-*, (d) 5 X 1014 cm-* were chosen to produce an
l
0168-583X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)01489-6
E. Albertazzi et al./Nucl. Instr. and Meth. in Phys. Res. B 118 (1996) 128-132 integral damage Nd > lOI cm-’ (multiple/plural scattering regime) and relative defect concentrations ranging from 0 to 1 (amorphization). RBS analyses were performed by a 2 MeV He+ beam in both random and (100) channeling geometry. To our knowledge there are no experimental data of channeling energy loss for He in (100) Si in the MeV energy range. Some preliminary results, obtained by measuring the energy shift of the Si/SiO, interface signal between random and (100) aligned spectrum in a SOI/SIMOX (silicon-on-insulator formed by separation-by-implantation-of-oxygen) sample, gave an average channeling to random stopping reduction u = 0.7 at 2 MeV. Random stopping power values were taken from Ref. [6].
3. Models 3.1. Two beams T%e first method considered is based upon the assumption that the beam can be divided into two dynamical components: the dechanneled fraction xd( Z) which follows random trajectories into the crystal and the channeled fraction I - ,y,(z). This simple model has been used since the first applications of RBS-channeling for the determination of implantation damage [7]. The normalized yield of a damaged sample (randomly displaced atoms) is given by x(z)
=xd(z)
+ [l
-xdwl~dw~
(‘1
where n,( ;) is the relative defect concentration at the depth z. Let the integral disorder be so high that multiple or plural scattering can be considered as the main source of dechanneling. In this case the dechanneled fraction xd is a function of depth z, ion energy E and of the integral relative defect concentration (integral defect density)
NJ 2) = i n,(r)dr. 0
(2)
This follows by the consideration that the dechanneling basically depends on the number of small angle collisions only. So, for a given depth and energy, xd is independent of the shape of the relative defect concentration profile n,(z) and, in particular, it is equal to the dechanneling generated by an amorphous surface layer of thickness Nd. There are several ways to determine the functional dependence of xd. Here two approaches were considered. The first one [4] simply assumes a linear dependence of the dechanneling on the integral disorder, that is x~(z> = X,(Z) + pNd(z), where xv is the virgin (undamaged) spectrum normalized yield and /3( E, Nd) is a fitting parameter to be determined experimentally. This method, in spite of its crude approximations, often represents the first practical approach to analyze new materials, while the applica-
129
tion of more sophisticated methods would require a good knowledge of the physical parameters determining their dechanneling behavior. A second approach, proposed in Ref. [8], states that, for a given depth, Nd and E can be scaled in some way to give a single dechanneling formula. This formula can be constructed either theoretically by any reliable method able to estimate the dechanneling produced by different amorphous Si surface layers [9,10], or experimentally [8]. Once an approach for the dechanneling has been chosen, standard calculations [5,7] make use of an iterative procedure to extract the defect depth profile nd from the experimental spectra by reversing Eq. ( 1). Self-consistency is attained when the calculated profile drops to (and remains at) zero in deep perfect layers. Alternatively, Bq. ( 1) can be directly used to simulate the whole spectrum for a given input profile. In this case a trial-and-error procedure has to be applied until the best fit of the experimental spectrum is obtained. This allows one to take into account the effects of energy resolution and straggling, isotopes and the surface peak; such a procedure is of the utmost importance if surface defective layers are considered.
3.2. Multi beams This approach is based on the original work of Girtner et al. [I I]. The whole available range in transverse energy E, is divided into a finite number of intervals. Consequently, the relevant quantity of this model is g,(z), the relative number of ions at depth z with transverse energy between El,,_, and E,,i. The normalized RBS yield is given by
where n, and iFipd are the relative probabilities to hit a lattice atom or a point defect (displaced atom) respectively. The transverse energy discretization allows for the numerical solution of the dechanneling master equation [12] describing the g,( z) behavior with depth Z, as
dg,(z)
~
dz
= CQ;,(Z)gj(Z). j
The key step of this numerical solution is the calculation of the dechanneling matrix Qij containing the contribution of electronic scattering, nuclear scattering with thermally vibrating string atoms and nuclear scattering with point defects. The Qij components due to nuclear scattering were calculated with a Monte Carlo procedure, while the Qjj electronic component was obtained computing integrals representing ensemble averages. This approach gives a better description of the channeling process when compared with the hvo beams model.
II. RBS. ETC.
130
E. Albertaui et al./Nucl. Instr. and Meth. in Phys. Res. B 118 (1996) 128-132
3.3. Monte Carlo model The Monte Carlo code used for the simulation of RBS-channeling spectra was developed at LAMEL as an extension of a previous program [13] originally designed for the simulation of ion implantation in crystalline materials. The full 3D simulation of ion penetration in the lattice, under the binary collision approximation, is similar to the one described by Robinson and Torrens 1141. The main advantage of a three-dimensional approach is that it can naturally describe analyzing beam directions far from a major crystal axis. For the calculation of elastic scattering the Ziegler, Biersack and Littmark interatomic Universal potential is used [15]. The 3D thermal vibrations of atoms are simulated by Gaussian distributions of uncorrelated displacements from the lattice equilibrium position, with mean square value calculated by Debye theory or taken from experiments. An approximate impact-parameter dependent stopping power is used to describe the electronic energy loss of the ions, based on the empirical approach of Gen and Robinson [ 161, as modified in Refs. [ 17,181, which guarantees that the stopping is normalized to the random velocity dependent stopping of Ref. [6] when averaged on the whole range of impact parameters. The parameter which determines the shape of the energy loss curve as a function of the impact parameter is chosen in order to reproduce the experimentally measured o value. Small deflections due to multiple electron scattering are calculated with the formula used in Ref. [ 191 where A E (the energy transferred to the electrons) is substituted by fAE (typically f = 0.5) to account that only single electron collisions produce an angular deviation of the ion. The program gives as output the normalized close encounter probability [20,21] as a function of depth and RBS energy. The close encounter probability is accumulated in the energy channels, corrected for the cross section effects and the obtained spectrum is normalized, channel by channel, to the one calculated in the same way for an amorphous target. Profiles of damage nd( z) or fully amorphous layers can be inserted as an input in the crystalline target. Amorphous regions are treated as a random distribution of target atoms; the simulation scheme in this case is similar to the one used in TRIM code [ 151. Partially damaged regions are treated as a mixture of crystalline and amorphous regions; at each collision step the occurrence of an amorphous region is statistically determined according to a probability equal to n,(z).
4. Results and discussion For the determination of the physical parameters to be used in the calculations both MB and MC models were applied to the simulation of the spectrum of a virgin (undamaged) Si sample. The same values of the one-dimensional thermal vibration amplitude u, = 0.08 .& and of
L
0.6
0.8 I.0 Energy (MeV)
1.;
Fig. 1. RBS-channeling spectra performed by 2 MeV He+ beam in (100) alignment for (a) 5X lOI As/cm-*, (b) 1 X IO“’ As/cm-*, (c) 2.5 X lOI As/cm- *, (d) 5X lOI As/cm-* asimplanted Si samples. The random and aligned spectra for a virgin sample are shown for comparison.
the surface disordered layer, equal to 10” cm-‘, were determined for both models. The parameters of multiple electron scattering were independently adjusted in each model to reproduce the dechanneling behavior of the virgin sample at large depths. As a check, the above programs were applied to the simulation of the a-Si/c-Si spectra, used in Ref. [8] to derive the semi-empirical formula for the dechanneling. Results agree within kO.01 normalized yield, even if MB seems to have some problem rising by the attainment of statistical equilibrium when the beam penetrates the crystalline region. The programs based on two and multi beams models were compared by analyzing the 2 MeV (100) channeling spectra of the arsenic implanted Si samples displayed in Fig. 1. In order to reduce the number of variables, a standard (automatic) extraction of the damage profiles was performed for both programs neglecting isotopes, surface peak, energy resolution and straggling, thus limiting the comparison to the dechanneling behavior. The defect depth profiles for the implanted samples are shown in Fig. 2. A channeling stopping power reduction (Y= 0.7 was used for samples (a) and (b), whereas (Y= 1 was employed in the other cases. The parameter /3 for the linear TB calculation was chosen either to approach zero concentration at the end-of-range region (about 1 urn> or at least to give no negative defect concentration values before this depth. The depth dependence of the dechanneling formula of the TB method was obtained in Ref. [8] by using the random stopping power and this is probably the reason of the inconsistent trend in Fig. 2a. In this case, where Nd is just above the lower limit of 1 X 10” cm-‘, the three calculations agree within 27% integral damage (20% peak concentration) but this difference may not be acceptable for many applications. In Fig. 2b the calculations still result in rather different profiles (23% integral damage, 14% peak
E. Albertazzi
0.0
0.5
1.0
0.0
et al./Nucl.
0.5
Instr. and Meth. in Phys. Rex B 118 (1996) 128-132
131
1.0
Depth(~4 Fig. 2. Smoothed defect depth profiles extracted from the spectra of Fig. I by the multi beam model (dotted line) and by the linear (heavy continuous line) and semi-empirical (light continuous line)
approaches based on the two beams model.
concentration). Here the best self-consistency is attained by the TE%dechanneling formula. The defect depth profiles for the (c) and (d) implanted samples (Figs. 2c and 2d respectively) are very similar in both cases. This indicates that, when high amount of damage (Nd > 1 X IO’* cm-2) together with high peak defect concentration (n, -+ 11 is considered, the direct scattering contribution to the normalized yield is so strong as to make negligible the eventual discrepancy in the dechanneling estimated by the different calculations. The defect depth profiles obtained by the TB dechanneling formula, once recalculated to take into account energy resolution and straggling as well as surface peak, were used in the MC code as a starting point for a trial-and-error simulation procedure of the experimental spectra. The outputs of the first step of this procedure (Fig. 3) allow for a critical analysis of the data obtained with TB and MB calculations. The spectra of the two highest doses (Figs. 3c and 3dl are already well simulated by the first step of MC calculation. In these cases, as the stopping power rapidly increases toward the random value (so that the average (Y value is close to 11, no important correction to the relative yield is needed. Some problems arise in the case of less damaged samples as reported in Figs. 3a and 3b. Here the input profiles lead to a quite wrong estimation of the normalized yield. Actually, in the former case the main contribution to the spectrum comes from the random fraction of the beam and the error lies probably in the calculation of the dechanneling as already underlined in the discussion of Fig. 2a. In the latter case, where the direct scattering contribution to the spectrum is important, the lack of the correction to the normalized yield induces an overestimation of the TB profile in the peak region. The defect depth profile that gives the best fit of the experimental spectra (a) following trial-and-error MC cal-
Energy (MeVf 3. Normalized yield of the spectra of Fig. 1 (light line) compared with the smoothed Monte Carlo simulations (heavy line) based on the defect depth profiles obtained by the two beams dechanneling formula (Fig. 2).
culation is shown in Fig. 4a, together with the starting profile obtained by the TE! dechanneling formula. The integrals Nd differ about 8%. The same comparison for the sample (b) is displayed in Fig. 4b. Here, the two defect depth profiles have the same integral value but they differ in the shape (9% at the peak position). There is another important result that must be stressed. The TE3 dechanneling formula profile for sample (b) satisfies the self-consistence criterion (i.e. the model is able, in this case, to simulate almost perfectly the experimental spectrum). On the other hand the best fit for MC code was obtained with a quite different profile. This means that the self-consistence criterion is not a test for the validity of the model.
J
Fig. 4. Defect depth profiles for spectra (a> and (b) obtained by Monte Carlo trial and error procedure (heavy line) and two beams dechanneling formula (light line).
11.RBS. ETC.
E. Albertaui
132
et al./Nucl.
lnstr. and Meth. in Phys. Res. B 118 (1996) 128-132
5. Conclusions
In this work different methods for the evaluation of the defect depth profile in implanted silicon from RBS-channeling spectra, from the simple linear calculation to a sophisticated Monte Carlo code, were applied to a set of Si samples where high energy arsenic impIantation of different fluences was performed in order to create different levels of damage over a wide depth range. It has been demonstrated that, when large amounts of damage are considered, the simplest models, without any particular correction, give about the same results as the more complicated ones. On the other hand, for the less damaged spectra the calculation based on the two and multi beams models lead to rather different defect depth profiles. Each of the above methods is affected by some problems that only partially are due to the rough description of the energy loss process. In this case, the Monte Carlo method seems to be undoubtedly the most complete procedure presently available to produce a detailed simulation of RBS-channeling spectra even though its reliability should be confirmed by a comparison with alternative analytical techniques.
Acknowledgements
The authors would like to thank Prof. A. Camera, Dr. C. Cellini and Mr. A. Sambo for their collaboration on the RBS measurements performed at the AN2000 facility of Laboratori Nazionali di Legnaro and Prof. A. Uguzzoni for his help on the MB model.
References [ll J.O. Borland (1993) 28.
and R. Koelsch,
Solid State Techn.
36 (12)
[2] O.W. Holland and C.W. White, Nucl. Instr. and Meth. B 59/60 (1991) 353. [3] L.C. Feldman, J.W. Mayer and ST. Picraux, Material Analysis by Ion Channeling (Academic Press, New York, 1982). [4] J.F. Ziegler, J. Appl. Phys. 43 (1972) 2973. [51 W-K. Chu, J.W. Mayer and M-A. Nicolet, Backscattering Spectrometry (Academic Press, New York, 1978). [6] J.F. Ziegler, Helium: Stopping Powers and Ranges in all Elements (Pergamon, New York, 1977). [71 E. Bogh, Can. J. Phys. 46 (1968) 653. [8] M. Bianconi, R. Nipoti. M. Cat&no, A. Gasparotto and A. Sambo, Nucl. Instr. and Meth. B 84 (1994) 507. [9] P. Sigmund and K.B. Winterbon, Nucl. Instr. and Meth. 119 (1974) 541. [IO] D. Schmaus, F. Abel, M. Bruneaux, C. Cohen, A. L’Hoir, G. Della Mea, A.V. Drigo, S. Lo Russo and G.G. Bentini, Phys. Rev. B 19 (1979) 5581. 1111 K. G’tier, K. Hehl and G. Schlotzhauer, Nucl. Instr. and Meth. B 4 (1984) 55. [12] E. Bonderup, H. Esbensen, J.U. Andersen and H.E. Schiott, Radiat. Eff. 12 (1972) 261. [13] A. Mazzone, Nucl. Instr. and Meth. B 34 (1988) 22. [I41 M.T. Robinson and 1.M. Torrens, Phys. Rev. B 9 (1974) 5008. [15] J.F. Ziegler and J.P. Biersack, The Stopping and Range of Ions in Solids (Pergamon, New York, 1982). 1161 O.S. Oen and M.T. Robinson, Nucl. Instr. and Meth. 132 ( 1976) 647. [I71 E.H.A. Dekempener, P.C. Zalm, G. Van Hoften and J. Politiek, Nucl. Instr. and Meth. B 48 (1990) 224. [18] M. Posselt, Nucl. Instr. and Meth. B 80/81 (1993) 28. [19] P.J.M. Smulders and D.O. Boerma, Nucl. Instr. and Meth. B 29 (1987) 471. [201 J.H. Barret, Phys. Rev. B 3 (1971) 1527. [211 A. Dygo, W.N. Lemrard and I.V. Mitchell, Nucl. Instr. and Meth. B 90 ( 1994) 142.
Nuclear Instruments and Methods in Physics Research B I 18 ( 1996) I28- 132 __ __
NOM
B
Beam Interactions with Materials A Atoms
@ ELSEVIER
Different methods for the determination of damage profiles in Si from RBS-channeling spectra: a comparison E. Albertazzi
a, M. Bianconi
a**, G. Lulli a, R. Nipoti a, M. Cantiano
b
a CNR-lsfituto LAMEL, Viu Gobetti IOI. I-40129 Bologna, Ituly b Consorzio OPTEL-InP. Cittadella della Ricercu. I-72100 Brindisi. Italy Abstract RBS-channeling spectra of deep ion implants in silicon were analyzed to extract the displaced atoms depth profiles by different methods. In all cases it was assumed that defects in as-implanted samples can be described as atoms randomly displaced from the lattice sites. In the first method, based on the two beam model, the dechanneling induced by defects was calculated either linearly or following a recently developed semi-empirical formula. In the second method the analyzing beam was divided into a greater number of components to follow the transverse energy distribution of the ions. Finally a three-dimensional Monte Carlo code containing a detailed description of each ion path was used to identify the limits of the previous approaches. It is shown that when high amounts of damage are considered all the methods produce essentially the same profiles. On the contrary they considerably disagree in other cases. Moreover, Monte Carlo calculations indicate that to obtain reliable results it is necessary to take into account a correct description of the channeling energy loss process during ion penetration into a disordered crystal.
1. Introduction
energy ion implantation in Si. The basic assumption that the damage produced by ion implantation in semiconductors consists in randomly distributed displaced atoms was used in the above models; this is reasonable in the range of damage concentration here considered. In the case of deep implants, the key topic is the description of the energy loss of the channeled beam. In the Monte Carlo code the use of an impact-parameter dependent stopping power allowed us to follow the energetic history of each ion. In the other models the assignment of a proper stopping value to each beam appeared quite complicated, so we used an average channeling stopping reduction cr dependent on the total amount of damage. As usually, the normalized yield x (the ratio of the aligned spectrum to the random one) as a function of the RBS energy (channel) rather than the actual spectrum was considered in the calculations. Due to the above considerations on the stopping power we avoided using any correction to the experimental normalized yield to account for the different cross section [4] and channel width [5] between channeled and random trajectories.
Many important applications of high energy ion implantation are concerned with the production of insulating or buried doped layers in the technology of silicon electronic devices [l]. The study of these implantation processes offers the possibility to extend the investigation of basic phenomena such as ion penetration and implantation damage in crystalline semiconductors to the MeV energy regime [2]. Moreover, the strong interest in compound materials such as SiGe and Sic for electronic application, requires the transfer of the know-how accumulated on Si. RBS-channeling is one of the most important and widely used techniques for the analysis of damage in as-implanted semiconductors [3]. However, in spite of its consolidated use, many aspects concerning the extraction of a reliable defect depth profile from experimental spectra deserve further investigation. The first question is the choice among the great number of methods, each working in a limited range of experimental conditions, for the evaluation of the dechanneling induced by defects [3]. In this work three different approaches based on the two beam (TB) model, a multi beam (MB) model and a Monte Carlo (MC) simulation were considered and applied to a typical study of high
2. Experimental
Corresponding author. Tel. + 39 51 6399140, fax + 39 51 6399216, e-mail: [email protected].
The experimental data concern a set of (100) Si samples implanted with 800 keV As+ ions. The implantation doses (a) 5 X lOI cm-‘, (b) 1 X 1014 cm-*,(c) 2.5 X lOI cm-*, (d) 5 X 1014 cm-* were chosen to produce an
l
0168-583X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)01489-6
E. Albertazzi et al./Nucl. Instr. and Meth. in Phys. Res. B 118 (1996) 128-132 integral damage Nd > lOI cm-’ (multiple/plural scattering regime) and relative defect concentrations ranging from 0 to 1 (amorphization). RBS analyses were performed by a 2 MeV He+ beam in both random and (100) channeling geometry. To our knowledge there are no experimental data of channeling energy loss for He in (100) Si in the MeV energy range. Some preliminary results, obtained by measuring the energy shift of the Si/SiO, interface signal between random and (100) aligned spectrum in a SOI/SIMOX (silicon-on-insulator formed by separation-by-implantation-of-oxygen) sample, gave an average channeling to random stopping reduction u = 0.7 at 2 MeV. Random stopping power values were taken from Ref. [6].
3. Models 3.1. Two beams T%e first method considered is based upon the assumption that the beam can be divided into two dynamical components: the dechanneled fraction xd( Z) which follows random trajectories into the crystal and the channeled fraction I - ,y,(z). This simple model has been used since the first applications of RBS-channeling for the determination of implantation damage [7]. The normalized yield of a damaged sample (randomly displaced atoms) is given by x(z)
=xd(z)
+ [l
-xdwl~dw~
(‘1
where n,( ;) is the relative defect concentration at the depth z. Let the integral disorder be so high that multiple or plural scattering can be considered as the main source of dechanneling. In this case the dechanneled fraction xd is a function of depth z, ion energy E and of the integral relative defect concentration (integral defect density)
NJ 2) = i n,(r)dr. 0
(2)
This follows by the consideration that the dechanneling basically depends on the number of small angle collisions only. So, for a given depth and energy, xd is independent of the shape of the relative defect concentration profile n,(z) and, in particular, it is equal to the dechanneling generated by an amorphous surface layer of thickness Nd. There are several ways to determine the functional dependence of xd. Here two approaches were considered. The first one [4] simply assumes a linear dependence of the dechanneling on the integral disorder, that is x~(z> = X,(Z) + pNd(z), where xv is the virgin (undamaged) spectrum normalized yield and /3( E, Nd) is a fitting parameter to be determined experimentally. This method, in spite of its crude approximations, often represents the first practical approach to analyze new materials, while the applica-
129
tion of more sophisticated methods would require a good knowledge of the physical parameters determining their dechanneling behavior. A second approach, proposed in Ref. [8], states that, for a given depth, Nd and E can be scaled in some way to give a single dechanneling formula. This formula can be constructed either theoretically by any reliable method able to estimate the dechanneling produced by different amorphous Si surface layers [9,10], or experimentally [8]. Once an approach for the dechanneling has been chosen, standard calculations [5,7] make use of an iterative procedure to extract the defect depth profile nd from the experimental spectra by reversing Eq. ( 1). Self-consistency is attained when the calculated profile drops to (and remains at) zero in deep perfect layers. Alternatively, Bq. ( 1) can be directly used to simulate the whole spectrum for a given input profile. In this case a trial-and-error procedure has to be applied until the best fit of the experimental spectrum is obtained. This allows one to take into account the effects of energy resolution and straggling, isotopes and the surface peak; such a procedure is of the utmost importance if surface defective layers are considered.
3.2. Multi beams This approach is based on the original work of Girtner et al. [I I]. The whole available range in transverse energy E, is divided into a finite number of intervals. Consequently, the relevant quantity of this model is g,(z), the relative number of ions at depth z with transverse energy between El,,_, and E,,i. The normalized RBS yield is given by
where n, and iFipd are the relative probabilities to hit a lattice atom or a point defect (displaced atom) respectively. The transverse energy discretization allows for the numerical solution of the dechanneling master equation [12] describing the g,( z) behavior with depth Z, as
dg,(z)
~
dz
= CQ;,(Z)gj(Z). j
The key step of this numerical solution is the calculation of the dechanneling matrix Qij containing the contribution of electronic scattering, nuclear scattering with thermally vibrating string atoms and nuclear scattering with point defects. The Qij components due to nuclear scattering were calculated with a Monte Carlo procedure, while the Qjj electronic component was obtained computing integrals representing ensemble averages. This approach gives a better description of the channeling process when compared with the hvo beams model.
II. RBS. ETC.
130
E. Albertaui et al./Nucl. Instr. and Meth. in Phys. Res. B 118 (1996) 128-132
3.3. Monte Carlo model The Monte Carlo code used for the simulation of RBS-channeling spectra was developed at LAMEL as an extension of a previous program [13] originally designed for the simulation of ion implantation in crystalline materials. The full 3D simulation of ion penetration in the lattice, under the binary collision approximation, is similar to the one described by Robinson and Torrens 1141. The main advantage of a three-dimensional approach is that it can naturally describe analyzing beam directions far from a major crystal axis. For the calculation of elastic scattering the Ziegler, Biersack and Littmark interatomic Universal potential is used [15]. The 3D thermal vibrations of atoms are simulated by Gaussian distributions of uncorrelated displacements from the lattice equilibrium position, with mean square value calculated by Debye theory or taken from experiments. An approximate impact-parameter dependent stopping power is used to describe the electronic energy loss of the ions, based on the empirical approach of Gen and Robinson [ 161, as modified in Refs. [ 17,181, which guarantees that the stopping is normalized to the random velocity dependent stopping of Ref. [6] when averaged on the whole range of impact parameters. The parameter which determines the shape of the energy loss curve as a function of the impact parameter is chosen in order to reproduce the experimentally measured o value. Small deflections due to multiple electron scattering are calculated with the formula used in Ref. [ 191 where A E (the energy transferred to the electrons) is substituted by fAE (typically f = 0.5) to account that only single electron collisions produce an angular deviation of the ion. The program gives as output the normalized close encounter probability [20,21] as a function of depth and RBS energy. The close encounter probability is accumulated in the energy channels, corrected for the cross section effects and the obtained spectrum is normalized, channel by channel, to the one calculated in the same way for an amorphous target. Profiles of damage nd( z) or fully amorphous layers can be inserted as an input in the crystalline target. Amorphous regions are treated as a random distribution of target atoms; the simulation scheme in this case is similar to the one used in TRIM code [ 151. Partially damaged regions are treated as a mixture of crystalline and amorphous regions; at each collision step the occurrence of an amorphous region is statistically determined according to a probability equal to n,(z).
4. Results and discussion For the determination of the physical parameters to be used in the calculations both MB and MC models were applied to the simulation of the spectrum of a virgin (undamaged) Si sample. The same values of the one-dimensional thermal vibration amplitude u, = 0.08 .& and of
L
0.6
0.8 I.0 Energy (MeV)
1.;
Fig. 1. RBS-channeling spectra performed by 2 MeV He+ beam in (100) alignment for (a) 5X lOI As/cm-*, (b) 1 X IO“’ As/cm-*, (c) 2.5 X lOI As/cm- *, (d) 5X lOI As/cm-* asimplanted Si samples. The random and aligned spectra for a virgin sample are shown for comparison.
the surface disordered layer, equal to 10” cm-‘, were determined for both models. The parameters of multiple electron scattering were independently adjusted in each model to reproduce the dechanneling behavior of the virgin sample at large depths. As a check, the above programs were applied to the simulation of the a-Si/c-Si spectra, used in Ref. [8] to derive the semi-empirical formula for the dechanneling. Results agree within kO.01 normalized yield, even if MB seems to have some problem rising by the attainment of statistical equilibrium when the beam penetrates the crystalline region. The programs based on two and multi beams models were compared by analyzing the 2 MeV (100) channeling spectra of the arsenic implanted Si samples displayed in Fig. 1. In order to reduce the number of variables, a standard (automatic) extraction of the damage profiles was performed for both programs neglecting isotopes, surface peak, energy resolution and straggling, thus limiting the comparison to the dechanneling behavior. The defect depth profiles for the implanted samples are shown in Fig. 2. A channeling stopping power reduction (Y= 0.7 was used for samples (a) and (b), whereas (Y= 1 was employed in the other cases. The parameter /3 for the linear TB calculation was chosen either to approach zero concentration at the end-of-range region (about 1 urn> or at least to give no negative defect concentration values before this depth. The depth dependence of the dechanneling formula of the TB method was obtained in Ref. [8] by using the random stopping power and this is probably the reason of the inconsistent trend in Fig. 2a. In this case, where Nd is just above the lower limit of 1 X 10” cm-‘, the three calculations agree within 27% integral damage (20% peak concentration) but this difference may not be acceptable for many applications. In Fig. 2b the calculations still result in rather different profiles (23% integral damage, 14% peak
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Depth(~4 Fig. 2. Smoothed defect depth profiles extracted from the spectra of Fig. I by the multi beam model (dotted line) and by the linear (heavy continuous line) and semi-empirical (light continuous line)
approaches based on the two beams model.
concentration). Here the best self-consistency is attained by the TE%dechanneling formula. The defect depth profiles for the (c) and (d) implanted samples (Figs. 2c and 2d respectively) are very similar in both cases. This indicates that, when high amount of damage (Nd > 1 X IO’* cm-2) together with high peak defect concentration (n, -+ 11 is considered, the direct scattering contribution to the normalized yield is so strong as to make negligible the eventual discrepancy in the dechanneling estimated by the different calculations. The defect depth profiles obtained by the TB dechanneling formula, once recalculated to take into account energy resolution and straggling as well as surface peak, were used in the MC code as a starting point for a trial-and-error simulation procedure of the experimental spectra. The outputs of the first step of this procedure (Fig. 3) allow for a critical analysis of the data obtained with TB and MB calculations. The spectra of the two highest doses (Figs. 3c and 3dl are already well simulated by the first step of MC calculation. In these cases, as the stopping power rapidly increases toward the random value (so that the average (Y value is close to 11, no important correction to the relative yield is needed. Some problems arise in the case of less damaged samples as reported in Figs. 3a and 3b. Here the input profiles lead to a quite wrong estimation of the normalized yield. Actually, in the former case the main contribution to the spectrum comes from the random fraction of the beam and the error lies probably in the calculation of the dechanneling as already underlined in the discussion of Fig. 2a. In the latter case, where the direct scattering contribution to the spectrum is important, the lack of the correction to the normalized yield induces an overestimation of the TB profile in the peak region. The defect depth profile that gives the best fit of the experimental spectra (a) following trial-and-error MC cal-
Energy (MeVf 3. Normalized yield of the spectra of Fig. 1 (light line) compared with the smoothed Monte Carlo simulations (heavy line) based on the defect depth profiles obtained by the two beams dechanneling formula (Fig. 2).
culation is shown in Fig. 4a, together with the starting profile obtained by the TE! dechanneling formula. The integrals Nd differ about 8%. The same comparison for the sample (b) is displayed in Fig. 4b. Here, the two defect depth profiles have the same integral value but they differ in the shape (9% at the peak position). There is another important result that must be stressed. The TE3 dechanneling formula profile for sample (b) satisfies the self-consistence criterion (i.e. the model is able, in this case, to simulate almost perfectly the experimental spectrum). On the other hand the best fit for MC code was obtained with a quite different profile. This means that the self-consistence criterion is not a test for the validity of the model.
J
Fig. 4. Defect depth profiles for spectra (a> and (b) obtained by Monte Carlo trial and error procedure (heavy line) and two beams dechanneling formula (light line).
11.RBS. ETC.
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5. Conclusions
In this work different methods for the evaluation of the defect depth profile in implanted silicon from RBS-channeling spectra, from the simple linear calculation to a sophisticated Monte Carlo code, were applied to a set of Si samples where high energy arsenic impIantation of different fluences was performed in order to create different levels of damage over a wide depth range. It has been demonstrated that, when large amounts of damage are considered, the simplest models, without any particular correction, give about the same results as the more complicated ones. On the other hand, for the less damaged spectra the calculation based on the two and multi beams models lead to rather different defect depth profiles. Each of the above methods is affected by some problems that only partially are due to the rough description of the energy loss process. In this case, the Monte Carlo method seems to be undoubtedly the most complete procedure presently available to produce a detailed simulation of RBS-channeling spectra even though its reliability should be confirmed by a comparison with alternative analytical techniques.
Acknowledgements
The authors would like to thank Prof. A. Camera, Dr. C. Cellini and Mr. A. Sambo for their collaboration on the RBS measurements performed at the AN2000 facility of Laboratori Nazionali di Legnaro and Prof. A. Uguzzoni for his help on the MB model.
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