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A simple model for the formation of compressive stress in thin films by ion bombardment
30
Thin Solid Films, 226 (1993) 30-34
A simple model for the formation of compressive stress in thin films by ion bombardment C. A. Davis School of Physics, University of Sydney, Sydney NSW 2006 (Australia) (Received July 31, 1992; accepted October 28, 1992)
Abstract A simple model is proposed to explain the formation of compressive Stress in thin films deposited with simultaneous bombardment by energetic ions or atoms. Consideration of a steady state in which stress formation by knock-on implantation of film atoms is balanced by thermal spike excited migration of implanted atoms shows that the stress a is proportional to [Y/(1-v)]EJ/2/(R/j+kES/3), where E is the ion energy, R the net depositing flux, j the bombarding flux, k a material-dependent parameter, Y the film material Young's modulus and v the Poisson ratio. This formula is used to explain seemingly contradictory experimental results in the literature. For small values of the normalized flux j / R the stress is proportional to the square root of the ion energy. Larger values of the normalized flux cause the stress to go through a maximum with increasing ion energy, and a power law decrease in stress with ion energy is predicted for large normalized fluxes.
1. Introduction Compressive stress arises when a growing film is b o m b a r d e d by atoms or ions with energies of tens or hundreds o f electronvolts by a process of "atomic peening" [1]. The energetic ions cause atoms to be incorporated into spaces in the growing film which are smaller than the usual atomic volume [2] and this leads to an expansion o f the film outwards from the substrate. In the plane of the film, however, the film is not free to expand and the entrapped atoms cause macroscopic compressive stress. The conditions of energetic b o m b a r d m e n t which lead to the production of compressive stress also favour the formation o f dense thin films with properties approaching those of the bulk material [3]. Unfortunately, excessive compressive stress can cause adhesion failure and other undesirable effects. Reduction o f compressive stress is therefore an important technological issue. However, McKenzie [4] has pointed out that a certain level of compressive stress can be desirable. For example, the anisotropic nature of the compressive stress in thin films can lead to preferential orientation [5], while the hydrostatic component of the stress can lead to formation of metastable film phases such as tetrahedral a m o r p h o u s carbon [6]. The formation of compressive stress thus needs to be understood so that it m a y be controlled to suit particular applications. Despite its importance, however, the formation of compressive stress is not well understood [7]. This is partly caused by the difficulty of quantifying important
0040-6090/93/$6.00
parameters such as ion energy and flux in m a n y experiments. In addition, qualitatively different behaviours m a y be observed under apparently similar conditions. For example, the stress is often found to increase linearly with the square root of the ion energy [8], but this is not always the case. Indeed, the stress has even been observed to decrease [9] as the ion energy increases. The purpose of this paper is to present a simple mathematical model of the formation of compressive stress in thin films. The predictions of the model are discussed and compared to experimental data from the literature.
2. Theory 2. I. The model Consider the structureless film of density N (equal to the bulk density) which is shown in Fig. 1. The film is deposited at a rate d, is at zero temperature and is being b o m b a r d e d by a normally incident flux j of ions or atoms which all have equal kinetic energy E which is less than about 1 keV. Since ions and atoms are equivalent for the purpose of stress formation, the energetic species will henceforth be denoted as "ions". In the model presented here the compressive stress is assumed to be caused by film atoms which are implanted below the surface of the film by knock-on processes. In addition, thermal spikes are assumed to reduce the stress by causing relaxation of the implanted
© 1993 - - Elsevier Sequoia. All rights reserved
C. A. Davis / Formation of compressive stress by ion bombardment
2.3. Stress relaxation
Ions: energy E flux I
~
F
i
l
m
growth: deposition rate d depositing flux R
Fig. 1. Schematic diagram of a film growing at a rate d, with simultaneous bombardment by a flux j of ions with energy E.
atoms. Although the details of knock-on implantation and thermal spikes may be complex, simple models are presented below which give a clear picture of the overall effects. It will be seen that the stress scaling derived from the steady state condition may be easily modified to take into account modification or extension of these models. In order to formulate a general model which does not require the details of particular cases, implantation and relaxation processes are not considered in great detail. Furthermore, the effects of film temperature, microscopic and macroscopic film structure, and film composition are not considered at all. It should be noted that bombardment by ions with a normalized energy of less than a few electronvolts per atom leads to films containing a significant fraction of voids [10]. In addition, Muller [11] has predicted that for large energies the film density may also be reduced because of sputtering of atoms from near the surface of the growing film. These voids lead to a tensile stress state dependent mainly on the details of the film structure. Therefore only dense films containing no voids are modelled here. This is the most important restriction on the applicability of the model.
The compressive stress caused by implanted atoms is associated with an increase in the strain energy of the film. Therefore a reduction in the stress by movement of the implanted atoms to the film surface is energetically favoured. The implanted atoms, however, are prevented from moving by the replusive force exerted by the surrounding atoms. The implanted atoms, then, are in metastable locations within a few nanometres of the film surface. It is therefore likely that atoms which acquire more than some excitation energy E0 will escape from their metastable position to the surface of the film. A significant fraction of the energy of a bombarding ion is transferred to violent motion of the atoms in the area of the impact. This intense local heating, or "thermal spike" [15], can provide the energy required to release implanted atoms from their metastable positions within the film. Seitz and Koehler [12] have calculated that the number na of atoms which will receive more than the excitation energy when an energy Q is deposited in a thermal spike is given by [ Q'~5/3
na=O.O16p~-~o)
(2)
where p is a material-dependent parameter which is of order 1. For the energies of less than 1 keV considered here, each energetic impact will produce only one thermal spike and therefore Q can be approximated by the ion kinetic energy E. The rate riR per unit area with which relaxation occurs is proportional to the number o f atoms which acquire more than the excitation energy from each energetic impact, the fraction o f these atoms which are in metastable positions, and the rate of ion impact. If it is assumed that every implanted atom which receives more than the excitation energy will migrate to the film surface then hR is given by •
n.
E
5/3
nR = O.O16p~J(~oo )
2.2. Knock-on implantation Knock-on implantation occurs when an energetic ion transfers sufficient energy to film atoms to implant them below the surface of the film. The implanted atom is not necessarily an interstitial [ 1] and is rarely associated with a lattice vacancy. It should be noted that in order to initiate this process the incoming ion must have more than some threshold energy Ecnt [12]. The magnitude of Ecnt will depend on the details of the film structure and composition. Windischmann [13] has used the knock-on linear cascade theory o f forward sputtering developed by Sigmund [ 14] to show that the rate h i per unit area with which atoms are implanted below the film surface is approximately related to the ion flux and energy by tii oc-jE 1/2
31
(1)
(3)
In practice, however, not all atoms which receive more than the excitation energy will migrate to relax the stress and Eo should therefore be considered to be an effective excitation energy.
2.4. The steady state compressive stress The stress may now be calculated by assuming that there is a balance between implantation and relaxation processes so that the density n o f implanted atoms is constant with time. The rate per unit area with which implanted atoms are incorporated into the film is R(n/N), where R = Nd is the total rate per u n i t area with which atoms are added to the growing film. In addition, the net rate of implantation is given by the difference between the rate
C. A. Davis / Formation of compressive stress by ion bombardment
32
of implantation and the rate of activated relaxation. The steady state condition therefore implies that n
hi -- fR = R ~.
(4)
Equations (1), (3) and (4) can now be solved to give the fraction of implanted atoms in the film as
n E a/2 A[ ~: R/j + O.O16p(E[Eo) 5/3
(5)
As a first-order approximation, the volumetric strain • is assumed to be proportional to the fraction n / N of implanted atoms in the film [ 13]. For a thin film the stress is related to the strain by a = [Y I(1 - v ) ] • where Y is the Young's modulus of the film material and v the Poisson ratio [13]. The stress a is thus found to be
y E 1/2 a(E) oc 1 -- v R/j + kE 5/3
(6)
where k = O.O16pEo-5/3. Note that this result only holds for bombardment by monoenergetic ions with energy greater than Ecnt. A range of ion energies may be taken into account by integrating over all energies greater than Ecrit to give of)
a =
-~ a(E)f(E) dE
(7)
Ecrit
where f ( E ) dE is the number of ions with energy between E and E + dE. It should also be noted that the linear relationship assumed between stress and strain will not apply for strains approaching the yield threshold of the material. Strains exceeding the yield threshold will cause plastic flow of the film material which will prevent the stress from increasing further.
3. Discussion
3.1. Comparison with Experiment Figure 2 shows the stress predicted by eqn. (6) for three values of the normalized flux j/R. In general the stress rises to a maximum with increasing energy and then decreases. This behaviour arises as a result of the competition between implantation and stress relaxation. The proportionality constant in eqn. (6) will be denoted by x and is not known since it depends on the relationship between the volumetric strain and the fraction of implanted atoms. Although the value of k is constrained by the fact that the values of p and E0 should be physically reasonable, it will strongly depend on the details of the film composition and structure. Thus eqn. (6) effectively contains two undetermined parameters. In order to make quantitative predictions
s 2
1
4 3
~. 2
8
1 .
.
.
.
i
200
.
.
.
.
i
400
.
.
.
.
i
600
.
.
.
.
i
800
.
.
.
.
i
i000
Ion Energy (eV) Fig. 2. The variation in compressive stress with ion energy predicted by eqn. (6), for R/j = 1, 5, and 20, assuming that E o = 8 eV and
p=l.
based on the model, these parameters must be determined experimentally. One method for determining the parameters x and k is to use them as fitting parameters for a least-squares fit to data for which the energy, ion flux and deposition rate are known. Ideally, the data should be fitted for a variety of (known) values of the normalized flux j / R and energy E. Since such data are currently unavailable, only a qualitative comparison between the theory and experiment will be made here. In future experiments the quantities E and j / R should be determined whenever possible. The interpretation of existing measurements may be clarified in the light of the theory presented above. For example, if the stress increases with the square root of the ion energy (or the substrate bias), then the normalized flux ratio may be presumed to be low. A change in the stress behaviour so that a maximum is observed in the stress as the bias is increased may then be related to an increase in the normalized flux. It should be noted, however, that factors not considered here may cause significant effects. In particular, changes in the film density, structure or composition may cause significant changes in the film stress. Figure 3 shows two sets of experimental data with normalized flux approximately equal to 1. Assuming that p = 1, the data may be fitted by eqn. (6) if the excitation energy Eo is approximately 11 eV for the ion-assisted A1N films and about 3 eV for the filtered arc deposited tetrahedral amorphous carbon films. Figure 4 shows stress data for filtered arc deposited TiC which shows some of the main limitations of the theory presented here. It should be noted that the ion energy is not necessarily equal to the negative substrate bias: in the filtered arc the plasma potential may be negative [17] and this would make the actual energy lower than the substrate potential. Furthermore, the
C. A. Davis / Formation of compressive stress by ion bombartbnent
33
10
8
6
> o~
4
~J
es_ o
2
0 0
200
400
600
800
1000
1200
Ion Energy (eV)
Fig. 3. Experimental measurements of the compressive stress in tetrahedral amorphous carbon deposited by filtered cathodic arc [6] (B), and AIN deposited by ion-assisted deposition [9] (@). The data were fitted by eqn. (6), assuming that E0 = 3 eV (---) for the amorphous carbon and E0 = 11 eV ( ) for the A1N, with p = 1 and R/j = 1 for both curves.
normalized flux and film density are not known. Equation (6) is shown in comparison with the data assuming E 0 = 14 eV and R / j = 10. Although the fit is poor, the rapid rise in stress and the reduction in stress at large energies are correctly modelled. At low bias the stress is lower than predicted; this m a y be due to the fact that the film is not sufficiently dense to support large stresses. At high bias the stress decreases more rapidly than predicted; again, this m a y be due to a reduction in the film density caused by sputtering [18]. Nevertheless, the
0.035
0.02
iE
.=
0.015 0.01
3.2. Scaling considerations: implications for stress control When the normalized flux is sufficiently low that R / j is large compared with k E 5/3 then eqn. (6) can be approximated by y j E 1/2 a(E) oc (8) 1 -v Nd It should be noted that this is similar to the result predicted by Windischmann [13]. In this theory, however, the stress is found to be proportional to j/d, whereas Windischmann predicted a linear dependence on j only. F o r a very large normalized flux the term R]j will be negligible and the stress can be approximated by
0.03
.- 0.025
model is able to reproduce qualitatively previously unexplained features of the data such as the fact that the stress goes through a m a x i m u m as the substrate bias increases.
.S.
Y 1 a(E) oc 1 - v k E 7/6
,..1 0.005 I
I
I
I
I
I
100
200
300
400
500
600
700
Negative Substrate Bias (V)
Fig. 4. Measured compressive stress in TiC deposited by filtered cathodic arc [16]. The prediction of eqn. (6) is shown, assuming Eo = 1 4 e V , R [ j = 1 0 a n d p = l .
(9)
Equations (8) and (9) suggest two simple methods for reducing compressive stress. The most obvious strategy is to reduce the normalized flux. F o r l o w normalized fluxes, the stress m a y be approximated by eqn. (8) and is therefore lowest for low energies. The problem with this strategy is that the benefits of ion-assisted deposition m a y be reduced for low normalized energies
C. A. Davis / Formation of compressive stress by ion bombardment
34
j E / R [ 10]. For a high flux deposition system, the stress is given by eqn. (9) and may therefore be reduced by increasing the ion energy. In some cases, however, ion energies sufficiently large to reduce the stress may also cause significant damage or reduced film density [18]. Yehoda et al. [10] have shown that dense unvoided films are produced when the normalized ion energy is greater than a threshold value of about 5 eV atom -1. Furthermore, it seems reasonable that collision cascade densification [18] will occur at energies which are below the threshold for implantation since atomic rearrangement requires less energy than implantation. It may be, then, that dense, stress-free films can be produced at useful deposition rates and low substrate temperatures by providing a sufficiently large flux of ions which all have energies below Ecnt. A high flux source of low energy ions based on the filtered cathodic arc which has recently been developed [19] may enable this to be achieved in the near future. By differentiating eqn. (6) it can be shown that the maximum stress is given by trmax OC(j/R)
7/10
(10)
and is obtained at an energy
= \TjkJ
(11)
Since no implantation will occur when Em,x is less than Ecrit, then the largest possible value of am.~ occurs when Em,x = E~nt. Thus the magnitude of the maximum stress increases nearly linearly with j / R up to a critical normalized flux which depends on the values of Ecnt and k. At the same time the energy of the stress maximum decreases with the flux ratio until it reaches E~nt. The magnitude of the compressive stress can thus be maximized by bombarding the growing film with a sufficiently large normalized flux of ions with kinetic energy slightly greater than E~i t.
4. Conclusions A simple model for the formation of compressive stress has been presented. The net stress results from a competition between stress formation by knock-on implantation of film atoms below the film surface and stress relaxation by thermal spike excited processes. In order to concentrate on the underlying mechanisms of compressive stress formation, processes which depend on the specific details of particular systems have not been considered, Assumption of a steady state has led to a simple mathematical model which qualitatively predicts previously unexplained features of currently available experimental results.
The model predicts that the magnitude of the compressive stress is strongly dependent on the ion energy, with the form of the energy dependence determined by the normalized flux j/R, where R is the net depositing flux and j is the bombarding flux. This enables the different stress-energy relationships seen in the literature to be explained in terms of variations in the normalized flux. For example, the model predicts that the stress is proportional to the square root of the ion energy for low normalized fluxes. Higher normalized fluxes, however, cause the stress to go through a maximum with increasing ion energy with a power law decrease for large energies and fluxes. In addition, the model provides a basis for interpretation of existing experimental data as well as guidance for the design of future experiments. Scaling considerations lead to a variety of strategies which may be used to minimize or maximize the compressive stress.
Acknowledgments The author wishes to acknowledge the guidance and inspiration of Dr. D. R. McKenzie. Helpful discussions with Mr. N. Marks are also gratefully acknowledged as is the financial support of an Australian Postgraduate Research Award.
References 1 F. M. d'Heurle and J. M. E. Harper, Thin Solid Films, 171 (1989) 81. 2 0 . Knotek, R. Elsing, G. Kramer and F. Jungblut, Surf. Coat. Tedmol., 46 (1991) 265. 3 P. J. Martin, Vacuum, 36 (1986) 585. 4 D. R. McKenzie, .4ppl. Opt., (1992) submitted. 5 T. Nakano, S. Baba, A. Kobayashi, A. Kinbara, T. Kajiwara and K. Watanabe, J. Vac. Sci. Technol. A, 9 (1991) 547. 6 D. R. McKenzie, D. A. Muller and B. A. Pailthorpe, Phys. Rev. Left., 67 (1991) 773. 7 H. Windischmann, J. Vac. Sci. Technol. A, 9 (1991) 2431. 8 D. Nit, J. Vac. Sci. Technol. A, 4 (1986) 2954. 9 P. J. Martin, R. P. Netterfield, T. J. Kinder and A. Bendavid, Appl. Opt., (1992), in the press. 10 J. E. Yehoda, B. Yang, K. Vedam and R. Messier, J. Vac. Sci. Technol. A, 6 (1988) 1631. II K.-H. Muller, J. Appl. Phys., 59 (1986) 2803. 12 F. Seitz and J. S. Koehler, Solid State Phys., 3 (1956) 305. 13 H. Windischmann, J. AppL Phys., 62 (1987) 1800. 14 P. Sigmund, in R. Behrisch (ed.), Sputtering by Particle Bombardment, vol. l, Springer, Berlin, 1981, p. 49. 15 K.-H. Muller, J. Vac. Sci. Technol. A, 4 (1986) 184. 16 P. J. Martin, R. P. Netterfield, T. J. Kinder and L. Descotes, Surf. Coat, TechnoL, 49 (1991) 239. 17 C. A. Davis and I. J. Donnelly, J. Appl. Phys., 72 (1992) 1740. 18 K.-H. Muller, Appl. Phys. A, 40 (1986) 209. 19 M. K. Puchert, C. A. Davis, D. R. McKenzie and B. W. James, J. Vac. Sci. Technol., (1992), in the press.
Thin Solid Films, 226 (1993) 30-34
A simple model for the formation of compressive stress in thin films by ion bombardment C. A. Davis School of Physics, University of Sydney, Sydney NSW 2006 (Australia) (Received July 31, 1992; accepted October 28, 1992)
Abstract A simple model is proposed to explain the formation of compressive Stress in thin films deposited with simultaneous bombardment by energetic ions or atoms. Consideration of a steady state in which stress formation by knock-on implantation of film atoms is balanced by thermal spike excited migration of implanted atoms shows that the stress a is proportional to [Y/(1-v)]EJ/2/(R/j+kES/3), where E is the ion energy, R the net depositing flux, j the bombarding flux, k a material-dependent parameter, Y the film material Young's modulus and v the Poisson ratio. This formula is used to explain seemingly contradictory experimental results in the literature. For small values of the normalized flux j / R the stress is proportional to the square root of the ion energy. Larger values of the normalized flux cause the stress to go through a maximum with increasing ion energy, and a power law decrease in stress with ion energy is predicted for large normalized fluxes.
1. Introduction Compressive stress arises when a growing film is b o m b a r d e d by atoms or ions with energies of tens or hundreds o f electronvolts by a process of "atomic peening" [1]. The energetic ions cause atoms to be incorporated into spaces in the growing film which are smaller than the usual atomic volume [2] and this leads to an expansion o f the film outwards from the substrate. In the plane of the film, however, the film is not free to expand and the entrapped atoms cause macroscopic compressive stress. The conditions of energetic b o m b a r d m e n t which lead to the production of compressive stress also favour the formation o f dense thin films with properties approaching those of the bulk material [3]. Unfortunately, excessive compressive stress can cause adhesion failure and other undesirable effects. Reduction o f compressive stress is therefore an important technological issue. However, McKenzie [4] has pointed out that a certain level of compressive stress can be desirable. For example, the anisotropic nature of the compressive stress in thin films can lead to preferential orientation [5], while the hydrostatic component of the stress can lead to formation of metastable film phases such as tetrahedral a m o r p h o u s carbon [6]. The formation of compressive stress thus needs to be understood so that it m a y be controlled to suit particular applications. Despite its importance, however, the formation of compressive stress is not well understood [7]. This is partly caused by the difficulty of quantifying important
0040-6090/93/$6.00
parameters such as ion energy and flux in m a n y experiments. In addition, qualitatively different behaviours m a y be observed under apparently similar conditions. For example, the stress is often found to increase linearly with the square root of the ion energy [8], but this is not always the case. Indeed, the stress has even been observed to decrease [9] as the ion energy increases. The purpose of this paper is to present a simple mathematical model of the formation of compressive stress in thin films. The predictions of the model are discussed and compared to experimental data from the literature.
2. Theory 2. I. The model Consider the structureless film of density N (equal to the bulk density) which is shown in Fig. 1. The film is deposited at a rate d, is at zero temperature and is being b o m b a r d e d by a normally incident flux j of ions or atoms which all have equal kinetic energy E which is less than about 1 keV. Since ions and atoms are equivalent for the purpose of stress formation, the energetic species will henceforth be denoted as "ions". In the model presented here the compressive stress is assumed to be caused by film atoms which are implanted below the surface of the film by knock-on processes. In addition, thermal spikes are assumed to reduce the stress by causing relaxation of the implanted
© 1993 - - Elsevier Sequoia. All rights reserved
C. A. Davis / Formation of compressive stress by ion bombardment
2.3. Stress relaxation
Ions: energy E flux I
~
F
i
l
m
growth: deposition rate d depositing flux R
Fig. 1. Schematic diagram of a film growing at a rate d, with simultaneous bombardment by a flux j of ions with energy E.
atoms. Although the details of knock-on implantation and thermal spikes may be complex, simple models are presented below which give a clear picture of the overall effects. It will be seen that the stress scaling derived from the steady state condition may be easily modified to take into account modification or extension of these models. In order to formulate a general model which does not require the details of particular cases, implantation and relaxation processes are not considered in great detail. Furthermore, the effects of film temperature, microscopic and macroscopic film structure, and film composition are not considered at all. It should be noted that bombardment by ions with a normalized energy of less than a few electronvolts per atom leads to films containing a significant fraction of voids [10]. In addition, Muller [11] has predicted that for large energies the film density may also be reduced because of sputtering of atoms from near the surface of the growing film. These voids lead to a tensile stress state dependent mainly on the details of the film structure. Therefore only dense films containing no voids are modelled here. This is the most important restriction on the applicability of the model.
The compressive stress caused by implanted atoms is associated with an increase in the strain energy of the film. Therefore a reduction in the stress by movement of the implanted atoms to the film surface is energetically favoured. The implanted atoms, however, are prevented from moving by the replusive force exerted by the surrounding atoms. The implanted atoms, then, are in metastable locations within a few nanometres of the film surface. It is therefore likely that atoms which acquire more than some excitation energy E0 will escape from their metastable position to the surface of the film. A significant fraction of the energy of a bombarding ion is transferred to violent motion of the atoms in the area of the impact. This intense local heating, or "thermal spike" [15], can provide the energy required to release implanted atoms from their metastable positions within the film. Seitz and Koehler [12] have calculated that the number na of atoms which will receive more than the excitation energy when an energy Q is deposited in a thermal spike is given by [ Q'~5/3
na=O.O16p~-~o)
(2)
where p is a material-dependent parameter which is of order 1. For the energies of less than 1 keV considered here, each energetic impact will produce only one thermal spike and therefore Q can be approximated by the ion kinetic energy E. The rate riR per unit area with which relaxation occurs is proportional to the number o f atoms which acquire more than the excitation energy from each energetic impact, the fraction o f these atoms which are in metastable positions, and the rate of ion impact. If it is assumed that every implanted atom which receives more than the excitation energy will migrate to the film surface then hR is given by •
n.
E
5/3
nR = O.O16p~J(~oo )
2.2. Knock-on implantation Knock-on implantation occurs when an energetic ion transfers sufficient energy to film atoms to implant them below the surface of the film. The implanted atom is not necessarily an interstitial [ 1] and is rarely associated with a lattice vacancy. It should be noted that in order to initiate this process the incoming ion must have more than some threshold energy Ecnt [12]. The magnitude of Ecnt will depend on the details of the film structure and composition. Windischmann [13] has used the knock-on linear cascade theory o f forward sputtering developed by Sigmund [ 14] to show that the rate h i per unit area with which atoms are implanted below the film surface is approximately related to the ion flux and energy by tii oc-jE 1/2
31
(1)
(3)
In practice, however, not all atoms which receive more than the excitation energy will migrate to relax the stress and Eo should therefore be considered to be an effective excitation energy.
2.4. The steady state compressive stress The stress may now be calculated by assuming that there is a balance between implantation and relaxation processes so that the density n o f implanted atoms is constant with time. The rate per unit area with which implanted atoms are incorporated into the film is R(n/N), where R = Nd is the total rate per u n i t area with which atoms are added to the growing film. In addition, the net rate of implantation is given by the difference between the rate
C. A. Davis / Formation of compressive stress by ion bombardment
32
of implantation and the rate of activated relaxation. The steady state condition therefore implies that n
hi -- fR = R ~.
(4)
Equations (1), (3) and (4) can now be solved to give the fraction of implanted atoms in the film as
n E a/2 A[ ~: R/j + O.O16p(E[Eo) 5/3
(5)
As a first-order approximation, the volumetric strain • is assumed to be proportional to the fraction n / N of implanted atoms in the film [ 13]. For a thin film the stress is related to the strain by a = [Y I(1 - v ) ] • where Y is the Young's modulus of the film material and v the Poisson ratio [13]. The stress a is thus found to be
y E 1/2 a(E) oc 1 -- v R/j + kE 5/3
(6)
where k = O.O16pEo-5/3. Note that this result only holds for bombardment by monoenergetic ions with energy greater than Ecnt. A range of ion energies may be taken into account by integrating over all energies greater than Ecrit to give of)
a =
-~ a(E)f(E) dE
(7)
Ecrit
where f ( E ) dE is the number of ions with energy between E and E + dE. It should also be noted that the linear relationship assumed between stress and strain will not apply for strains approaching the yield threshold of the material. Strains exceeding the yield threshold will cause plastic flow of the film material which will prevent the stress from increasing further.
3. Discussion
3.1. Comparison with Experiment Figure 2 shows the stress predicted by eqn. (6) for three values of the normalized flux j/R. In general the stress rises to a maximum with increasing energy and then decreases. This behaviour arises as a result of the competition between implantation and stress relaxation. The proportionality constant in eqn. (6) will be denoted by x and is not known since it depends on the relationship between the volumetric strain and the fraction of implanted atoms. Although the value of k is constrained by the fact that the values of p and E0 should be physically reasonable, it will strongly depend on the details of the film composition and structure. Thus eqn. (6) effectively contains two undetermined parameters. In order to make quantitative predictions
s 2
1
4 3
~. 2
8
1 .
.
.
.
i
200
.
.
.
.
i
400
.
.
.
.
i
600
.
.
.
.
i
800
.
.
.
.
i
i000
Ion Energy (eV) Fig. 2. The variation in compressive stress with ion energy predicted by eqn. (6), for R/j = 1, 5, and 20, assuming that E o = 8 eV and
p=l.
based on the model, these parameters must be determined experimentally. One method for determining the parameters x and k is to use them as fitting parameters for a least-squares fit to data for which the energy, ion flux and deposition rate are known. Ideally, the data should be fitted for a variety of (known) values of the normalized flux j / R and energy E. Since such data are currently unavailable, only a qualitative comparison between the theory and experiment will be made here. In future experiments the quantities E and j / R should be determined whenever possible. The interpretation of existing measurements may be clarified in the light of the theory presented above. For example, if the stress increases with the square root of the ion energy (or the substrate bias), then the normalized flux ratio may be presumed to be low. A change in the stress behaviour so that a maximum is observed in the stress as the bias is increased may then be related to an increase in the normalized flux. It should be noted, however, that factors not considered here may cause significant effects. In particular, changes in the film density, structure or composition may cause significant changes in the film stress. Figure 3 shows two sets of experimental data with normalized flux approximately equal to 1. Assuming that p = 1, the data may be fitted by eqn. (6) if the excitation energy Eo is approximately 11 eV for the ion-assisted A1N films and about 3 eV for the filtered arc deposited tetrahedral amorphous carbon films. Figure 4 shows stress data for filtered arc deposited TiC which shows some of the main limitations of the theory presented here. It should be noted that the ion energy is not necessarily equal to the negative substrate bias: in the filtered arc the plasma potential may be negative [17] and this would make the actual energy lower than the substrate potential. Furthermore, the
C. A. Davis / Formation of compressive stress by ion bombartbnent
33
10
8
6
> o~
4
~J
es_ o
2
0 0
200
400
600
800
1000
1200
Ion Energy (eV)
Fig. 3. Experimental measurements of the compressive stress in tetrahedral amorphous carbon deposited by filtered cathodic arc [6] (B), and AIN deposited by ion-assisted deposition [9] (@). The data were fitted by eqn. (6), assuming that E0 = 3 eV (---) for the amorphous carbon and E0 = 11 eV ( ) for the A1N, with p = 1 and R/j = 1 for both curves.
normalized flux and film density are not known. Equation (6) is shown in comparison with the data assuming E 0 = 14 eV and R / j = 10. Although the fit is poor, the rapid rise in stress and the reduction in stress at large energies are correctly modelled. At low bias the stress is lower than predicted; this m a y be due to the fact that the film is not sufficiently dense to support large stresses. At high bias the stress decreases more rapidly than predicted; again, this m a y be due to a reduction in the film density caused by sputtering [18]. Nevertheless, the
0.035
0.02
iE
.=
0.015 0.01
3.2. Scaling considerations: implications for stress control When the normalized flux is sufficiently low that R / j is large compared with k E 5/3 then eqn. (6) can be approximated by y j E 1/2 a(E) oc (8) 1 -v Nd It should be noted that this is similar to the result predicted by Windischmann [13]. In this theory, however, the stress is found to be proportional to j/d, whereas Windischmann predicted a linear dependence on j only. F o r a very large normalized flux the term R]j will be negligible and the stress can be approximated by
0.03
.- 0.025
model is able to reproduce qualitatively previously unexplained features of the data such as the fact that the stress goes through a m a x i m u m as the substrate bias increases.
.S.
Y 1 a(E) oc 1 - v k E 7/6
,..1 0.005 I
I
I
I
I
I
100
200
300
400
500
600
700
Negative Substrate Bias (V)
Fig. 4. Measured compressive stress in TiC deposited by filtered cathodic arc [16]. The prediction of eqn. (6) is shown, assuming Eo = 1 4 e V , R [ j = 1 0 a n d p = l .
(9)
Equations (8) and (9) suggest two simple methods for reducing compressive stress. The most obvious strategy is to reduce the normalized flux. F o r l o w normalized fluxes, the stress m a y be approximated by eqn. (8) and is therefore lowest for low energies. The problem with this strategy is that the benefits of ion-assisted deposition m a y be reduced for low normalized energies
C. A. Davis / Formation of compressive stress by ion bombardment
34
j E / R [ 10]. For a high flux deposition system, the stress is given by eqn. (9) and may therefore be reduced by increasing the ion energy. In some cases, however, ion energies sufficiently large to reduce the stress may also cause significant damage or reduced film density [18]. Yehoda et al. [10] have shown that dense unvoided films are produced when the normalized ion energy is greater than a threshold value of about 5 eV atom -1. Furthermore, it seems reasonable that collision cascade densification [18] will occur at energies which are below the threshold for implantation since atomic rearrangement requires less energy than implantation. It may be, then, that dense, stress-free films can be produced at useful deposition rates and low substrate temperatures by providing a sufficiently large flux of ions which all have energies below Ecnt. A high flux source of low energy ions based on the filtered cathodic arc which has recently been developed [19] may enable this to be achieved in the near future. By differentiating eqn. (6) it can be shown that the maximum stress is given by trmax OC(j/R)
7/10
(10)
and is obtained at an energy
= \TjkJ
(11)
Since no implantation will occur when Em,x is less than Ecrit, then the largest possible value of am.~ occurs when Em,x = E~nt. Thus the magnitude of the maximum stress increases nearly linearly with j / R up to a critical normalized flux which depends on the values of Ecnt and k. At the same time the energy of the stress maximum decreases with the flux ratio until it reaches E~nt. The magnitude of the compressive stress can thus be maximized by bombarding the growing film with a sufficiently large normalized flux of ions with kinetic energy slightly greater than E~i t.
4. Conclusions A simple model for the formation of compressive stress has been presented. The net stress results from a competition between stress formation by knock-on implantation of film atoms below the film surface and stress relaxation by thermal spike excited processes. In order to concentrate on the underlying mechanisms of compressive stress formation, processes which depend on the specific details of particular systems have not been considered, Assumption of a steady state has led to a simple mathematical model which qualitatively predicts previously unexplained features of currently available experimental results.
The model predicts that the magnitude of the compressive stress is strongly dependent on the ion energy, with the form of the energy dependence determined by the normalized flux j/R, where R is the net depositing flux and j is the bombarding flux. This enables the different stress-energy relationships seen in the literature to be explained in terms of variations in the normalized flux. For example, the model predicts that the stress is proportional to the square root of the ion energy for low normalized fluxes. Higher normalized fluxes, however, cause the stress to go through a maximum with increasing ion energy with a power law decrease for large energies and fluxes. In addition, the model provides a basis for interpretation of existing experimental data as well as guidance for the design of future experiments. Scaling considerations lead to a variety of strategies which may be used to minimize or maximize the compressive stress.
Acknowledgments The author wishes to acknowledge the guidance and inspiration of Dr. D. R. McKenzie. Helpful discussions with Mr. N. Marks are also gratefully acknowledged as is the financial support of an Australian Postgraduate Research Award.
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